Dynamics and Stability of Tethered Satellites at Lagrangian Points

Transcription

1 Dynamics and Stability of Tethered Satellites at Lagrangian Points Final Report Authors: Jesús Peláez, Manuel Sanjurjo, Fernando R. Lucas, Martín Lara Enrico C. Lorenzini, Davide Curreli Daniel J. Scheeres Affiliation: ( ) Universidad Politécnica de Madrid (UPM) Grupo de Dinámica de Tethers ETSI Aeronáuticos Pz Cardenal Cisneros 3 Madrid 84, SPAIN ( ) University of Colorado Aerospace Engineering Sciences 49 UCB Boulder, Colorado, , USA ( ) University of Padova CISAS Centro Interdipartimentale di Studi e Attività Spaziali G. Colombo Via Venezia Padova, ITALY ESA researchers: Claudio Bombardelli, Dario Izzo Date: 4th November 8 Contacts: Name: Jesús Peláez Phone: Fax: j.pelaez@upm.es Name: Dario Izzo Phone: +3() Fax: +3() act@esa.int Ariadna ID: 7/4 Study Duration: 6months Contract Number: 59

2

3 Table of Contents Extended Executive Summary Prologue xi xv I Collinear Lagrangian Points Equations of motion for librating tethers 3. Introduction Scenario. Circular Restricted Three Body Problem Basic references frames Basic forces acting on a mass particle Lagrangian formulation Inert tether Equations for constant tether length Body frame Derivatives Equations for varying tether length Governing equations for a constant length inert librating tether Governing equations for a varying length inert librating tether Hill approach Equations of motion for constant-length rotating tethers 5. Dumbbell Model Basic equations Euler angles as generalized coordinates Bryant angles as generalized coordinates Non-dimensional form of the equations Remarks on the equations Inertial motion Motion relative to the synodic frame Rotating tethers Averaging on rotating tethers Equations for the center of mass motion Equations for the attitude motion Full set of equations for a constant length rotating tether Averaged equations for a constant length rotating tether Hill approach Time evolution of φ and φ iii

4 iv 3 Equations of motion for variable-length rotating tethers Basic equations Euler angles as generalized coordinates Bryant angles as generalized coordinates Non-dimensional form of the equations Averaging of the equations Full set of equations for a varying length rotating tether Averaged equations for a varying length rotating tether Hill approach Solar System Fact Sheets 4 4. Solar System Planets Jovian World Other Planet s Satellites Asteroids Values of λ in different binary systems Jovian Models Introduction to Jupiter environment Reference frames Jovian magnetic field Electronic plasma density model Inner plasmasphere. Inner moonlets Additional details Model for the electrodynamic forces Simplified model Bare Electrodynamic Tethers 6 6. Introduction Non-dimensional equations Simplifications Asymptotic analysis Average tether current Useful power Lorentz torque. Balance condition Reformulation of the asymptotic solution Average tether current Useful power Lorentz torque. Balance condition Rotating tethers. Averaging Electrodynamic drag Lorentz torque Using the Bryant angles for the attitude Appendix A. Solution of the cubic equation (6.5) Halo orbits fundamentals Introduction Background: The Hill problem Equations of motion and equilibria Particular solutions: periodic orbits

5 v 7.4. Linear stability definitions Lyapunov orbits Halo orbits Eight-shaped orbits Family linking Lyapunov and eight-shaped orbits Relevant bibliography in chronological order Halo orbits and fast rotating tethers Equations of motion Constant direction of the angular momentum Lyapunov orbits Eight-shaped orbits Eight-shaped to Lyapunov orbits Halo orbits Summary of results Initial conditions of some stable Halo-derived orbits Varying direction of the angular momentum Periodic Halo orbits for C = Periodic Halo orbits for C = Periodic Halo orbits for C = Periodic, eight-shaped orbits for λ = Periodic orbits of the migrating branch family for λ = Appendix Variational equations Basics on differential corrections computation Stability at the Collinear Lagrangian Points 4 9. Introduction Previous Analysis Equations of Motion Equilibrium equations Equilibrium positions on the plane Oξη Stability analysis for constant length Stabilization of the Collinear Lagrangian Points with variable length tethers 55. Introduction One-dimensional analysis Tether control Simple control strategies Linear Approximation for small values of λ «Ad hoc» control Proporcional control General analysis. Full Problem. Proportional control Control Drawbacks Length Variations Tension Rotating Tethers Rotating tethers. Equilibrium positions Linear stability analysis of the collinear equilibrium positions Varying Length Tether. Linear control strategy

6 vi.6.4 Searching oscillatory stability Asymptotic stability Summary and future research lines Io exploration with electrodynamic tethers 87. Introduction Non rotating electrodynamic tether at equilibrium Rotating electrodynamic tether orbiting Io Optimum Power Generation of an EDT with no ohmic effects a Power Generation in Ionian Orbit Orbit Stability Orbit Control Summary and future research lines II Triangular Lagrangian Points Circular Restricted Three Body Problem: a Commented Review 3. Equations of CRTBP in Cartesian Coordinates Equations of CRTBP in Spherical Coordinates Lagrangian Points Jacobi Integral Equations of motion of a Tether at the Triangular Lagrangian Points 9 3. Reference Frames Roto-translations between Reference Frames Extended Dumbbell Model Gravitational Actions Gravitational Potential Resultant of Gravitational Forces Torque around the center of mass Bare Electrodynamic Tethers Motional Electric Field Analysis Tether Circuit Equation Current and Collection Model Power Extraction Electrodynamic Actions Resultant on the center of mass Torque on the center of mass Equations of motion with Lagrange approach Motion when L/R is Infinitesimally Small Equations of motion in non-dimensional coordinates Hill approximation Dynamics at the Triangular Lagrangian Points: Linear Analysis 3 4. Classical Linear Analysis Procedure of Linear Analysis Application to Triangular Points Tether at Triangulars Inert tether

7 vii 4.3. Orbital motion: classical CRTBP Attitude: Robinson solution Active Electrodynamic Tether New equilibrium positions Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects Inert Tether Orbital Motion: Zero-Velocity Orbits Orbital Motion: Sensitivity to Velocity Errors Active Electrodynamic Tether Non-linear Stability Non-linear Stability at Triangular Points Io environment and Plasma Torus 6 6. Environment Gravitational field and shape parameters Magnetic environment Plasma environment Orbits around Io Electrodynamic tether around Io: numerical simulations Jovian-synchronous orbits around Io Exploration of the Jupiter Plasma Torus with a self-powered Electrodynamic Tether 8 7. Survey on Jupiter Exploration using EDT Introductory Remarks Equilibrium positions in the perturbed CRTBP Equilibrium using a bare EDT Electrodynamic force vs. Gravitational Force Constraints on system parameters Power generation Influence of Tether Size on Equilibrium Locations Power extraction at equilibrium Orbital motion around equilibrium position Center of mass coincident or very closely to the equilibrium point Small oscillations Large oscillations Out-of-plane orbital motion Attitude motion of a librating EDT Attitude with center of mass at equilibrium position Coupling effects between orbital and attitude motion Rotating tethers at triangular points Rotating EDT around equilibrium position Plasma Torus exploration and power extraction Scanning the Plasma Torus Power extraction Drifting orbits Conclusions and promising applications

8 viii 8 Conclusions and Future Works Inert rotating tethers and Halo orbits Inert tethers and equilibrium positions Electrodynamic tethers at Io Electrodynamic tethers and the Plasma Torus References 365 Appendices 37 A Relative motion and inertia forces 373 A. Inertia forces on a mass particle A. Inertia forces on a rigid body A.. Generalized potential A.. Resultant R A..3 Torque M A.3 On the notation used in these pages B Tethered system s mass geometry 383 B. Tether model B. Mass geometry B.. Constant length operation of the tether B.. Varying length operation of the tether B.3 Constant tether length case: Dumbbell Model B.3. Inertia forces in the Dumbbell Model B.3. Generalized potential B.3.3 Resultant B.3.4 Torque at the center of mass G B.4 Variable tether length case: Extended Dumbbell Model B.4. Inertia forces in the Extended Dumbbell Model B.4. Generalized potential B.4.3 Resultant B.4.4 Torque at the center of mass C Gravitational forces 395 C. Mass particle C. Rigid body C.3 Gravitational actions on a rigid tethered system C.3. Gravitational potential V g C.3. Gravitational resultant R g C.3.3 Gravitational torque M g C.3.4 Summary C.4 On the accuracy of two basic assumptions D Kinetic analysis 45 D. Angular momentum of the Dumbbell Model D. Kinetic energy of the Dumbbell Model D.3 Kinetic energy of the Extended Dumbbell Model D.4 Non-dimensional variables D.4. Angular momentum and kinetic energy

9 ix D.4. Inertia forces D.4.3 Gravitational forces of the main primary D.4.4 Gravitational forces of the small primary E Tension 49 E. Introduction E. Forces on the tether E.3 Inert massless tether E.3. Tension in terms of the libration angles (θ, ϕ) E.3. Tension in terms of the Bryant angles (φ,φ,φ 3 ) F EDM Analysis 47 F. Tether Models and Extended Dumbbell Model F. Mass Geometry F.3 Angular Velocity of EDM F.4 Angular Momentum F.5 Kinetic Energy of EDM

10 x

11 Extended Executive Summary This document collects the works carried out under the research project Dynamics and Stability of Tethered Satellites at Lagrangian Points funded by the ESA Ariadna program. The research work was conducted by the team formed by the Technical University of Madrid, the University of Padova and their research collaborators at ESA/ESTEC and the University of Colorado. Due to the size of the team, the variety of issues and the extension of the final report, it is advisable to summarize the main results obtained in the different research lines involved. L 4 L 3 L L E E Equilibrium position with an inert tether L 5 - Figure : Geometry of Lagrangian points The topic of the research stems in part from the fact, pointed out in the pioneering works of Colombo (96) [8] and Farquhar (97) [9, ], that an extended body like a tethered satellite has a specific behavior at a Lagrangian point (see Fig. ). For non-negligible values of the ratio L/R (where L is the tether length and R the distance of the spacecraft from the center of mass of the two primaries) the effect of the higher-order terms of the mutual gravity field play a role in the dynamics and stability of the satellite near a Lagrangian point. These terms can be exploited (by changing the tether length) to stabilize the orbit of a tethered satellite in the proximity of a collinear Lagrangian point that would be otherwise unstable for a compact-body satellite. In Misra [] explored another feature of a tethered satellite, that is, the non-negligible angular momentum of a spinning tethered satellite to stabilize the system at Lagrangian collinear points. In 7 Peláez and Scheeres [8, 9] proposed to use an electrodynamic tether (EDT) to control the position of a tethered satellite near one of the collinear Lagrangian point of the inner moonlets of Jupiter while generating power. The research carried out extends substantially the reach of previous works by analyzing in a systematic way inert and electrodynamic tethers (EDT) and focusing on both the collinear and the xi

12 xii triangular Lagrangian points of the classic restricted three body problem (CRTBP). A new formulation for the analysis of the dynamics of a tethered system has been developed. Based on the Hill approach, it can be used with any kind of tethers (rotating or non-rotating, inert or electrodynamic). This formulation provides interesting results. For instance, it shows that only one parameter a non-dimensional tether length λ, defined in (.49) captures the influence of the tether in the dynamics. An inert rotating tether in the neighborhood of a collinear Lagrangian point, when the rotation plane is parallel to the orbital plane of primaries, has a quite simple dynamics. In effect, that problem is formally equal to the Hill-oblateness problem a Hill problem perturbed by the oblateness of the central body but the J value associated to this virtual oblateness is equal to λ/. Halo orbits around the collinear point.4 L provide excellent opportunities for scientific and exploration missions, most of. the times taking advantage of their unstable character. However, this unstable char-.. acter can be changed when instead of a.4 point mass satellite a rotating tethered system is considered. Specifically, our work establishes the existence and stability of. periodic orbits associated with rotating inert tethers (see Fig ). From a qualitative point of view, these orbits are the natural continuation of periodic motions (Lyapunov, and halo orbits) characteristic of the CRTBP..5 The analysis of the equilibrium positions in the neighborhood of the collinear Lagrangian points has been carried out The unstable character of the collinear libration points does not change with. the presence of the tether. However, following the ideas of Farquhar, the Lagrangian points becomes stable if an appropriate variation of the tether length is accomplished. Reliable control laws have been deduced in this sense; their actual effectivity depends on the particular values of tether length and masses of the binary system considered.. Figure : Halo orbit close to Io, stabilized with a tether length of 7 km (inert tether). The minimum distance to Io is about one half of it radius. Electrodynamic tethers (EDT) are a promising alternative for producing the energy required in any scientific exploration mission to Io and its plasma torus, which are generally handcuffed by a scarcity of power. We propose two alternatives using EDT working in the generator regime: ) a bare self-balanced electrodynamic tether in equilibrium position in the synodic frame Jupiter-Io, and ) a rotating EDT orbiting around Io to generate permanent power and to provide propellantless orbital maneuvering capability. For the first alternative we derive the necessary orbital and tether design conditions for equilibrium and the system performances in terms of power generation. For the second alternative, we investigate two mission scenarios both involving a 5-km-long 5-cm-wide tape tether placed on a stable retrograde equatorial orbit around Io capable to provide kw-level continuous power extracted from the fast rotating Jupiter plasmasphere. In the first scenario the tether current is controlled to provide maximum power generation. The amount of power produced and the impact on the orbit stability is investigated numerically. In the second scenario the current.4

13 xiii z v Io Toward Jupiter ϕ O y E π ωt B x Figure 3: Rotating EDT in retrograde equatorial orbit around Io is controlled in order to reduce or increase the orbital energy of the system with the possibility of reaching escape velocity. Results show that EDT can be used as a permanent power production system in exploration missions to Io and the surrounding plasma torus without compromising the orbital stability. Figure 4: Exploration of the Io plasma torus with an electrodynamic tether. Exploration of Jovian Plasma Torus using an electrodynamic tether has been investigated with a dedicated analysis. Two applications have been investigated: () the first concerning the characteristics and use of new equilibrium positions that emerge when the EDT is activated, and () the second devoted to the exploration of the Io-Torus together with power generation. In the first case, power generation appears to be one of the most attractive features provided by

14 xiv an EDT system considering the weakness of the solar flux at Jupiter. For example, a useful power of a couple of kilowatts can be generated with an EDT of km in these orbital conditions, extracting power from the plasma-sphere. When placed in the new equilibrium positions, the tethered system maintains its orbital energy (or equivalently, it does not loose orbital altitude) as the energy in these orbital conditions is extracted from the plasma sphere co-rotating with Jupiter. In the second case, the electrodynamic model derived for the EDT system was used for the exploration of the Io plasma torus. We proved that this task can be accomplished while at the same time extracting power from the super-rotating Jupiter s plasma sphere. The electrodynamic tether in question is a bare ED tether that has a higher efficiency in collecting electrons from the ionosphere than the large anodic terminations used in past tether missions. Results show (see Fig. 4) that an EDT of moderate length (i.e., of order km) can scan the entire plasma torus in a time frame of a few months, while producing a steady electrical power of several hundred watts for on-board use.

15 Prologue This «Prologue» tries to facilitate the consultation of this final report, by describing its main characteristics; it also illustrates the basic assumptions and the distribution of matters. The report is divided in two main parts Part I. Collinear Lagrangian Points Part II. Triangular Lagrangian Points which address the dynamical behavior of tethered systems in the neighborhood of the Lagrangian points of a given binary system. In the first part, the scenario is the neighborhood of the «Collinear Lagrangian Points»; in the second one, the «Triangular Lagrangian Points». Basically, both parts are self-contained and they use complementary approaches, with some differences in notation, in order to optimize the working time of our team. The last chapter 8, entitled «Conclusions and Future Works», summarizes the main research lines which would be reasonable continuations of the works carried out in this research project. Part I. Collinear Lagrangian Points Chapter, entitled «Equations of motion», contains a short introduction of the Circular Restricted Three Body Problem (CRTBP), the departure point of the analysis. Some basic references of interest are defined and presented in that Section. More details on the CRTBP can be found in chapter at the second part of this report. The constant length tether is modeled using the classical Dumbbell Model in which the tether mass is neglected. Tether mass can be incorporated easily, however, as it is shown in Appendix B. Thus, our governing equations permit simulate the influence of the tether mass if necessary. For tethers of varying length we use an Extended Dumbbell Model in which both ends satellites are modeled as point masses. The equations of motions are deduced in the framework of the Lagrangian formulation. Five generalized coordinates are involved: the three coordinates of the system center of mass and the libration angles (the in-plane and the out-of-plane angles). The equations describe the dynamics of inert librating tethers and they can be completed in order to include other perturbation like, for example, the electrodynamic forces in EDT s. The treatment of the inertia forces is original and different from the classical one (see [7]). The Lagrangian function is simpler although the deduction is more complex. The details can be found in Appendix A for two main cases: the point mass and the rigid body. Obviously, the gauge-invariant property of the Lagrangian functions assures that the final governing equations are the same in both cases. Chapter ends by introducing the Hill approach. Usually ν the reduced mass of the primary around which the tether is moving is small. When working in the neighborhood of the collinear Lagrangian points, the Hill formulation allows a much simpler description of the dynamics and provides an excellent approximated solution. It permits to gain an insight into the evolution of the tethered system and gives significant clues for the analysis of the general case in which ν is of order unity. xv

16 xvi Chapters and 3 are devoted to a deeper analysis of the «Attitude Dynamics» of the Dumbbell Model and the Extended Dumbbell Model. The libration angles are very appropriate for librating tethers but for rotating tethers there are better descriptions of the attitude dynamics. Instead of the libration angles we introduce the Euler angles in sequence 3 (Cardan or Bryant or Krylov angles) which are associated with a body frame constructed on the system angular momentum. An original deduction of the governing equations of the attitude of a rotating tethered system is carried out. It is a step forward when compared with similar approaches like, for example, the one developed in [4] by E. M. Levin. For fast rotating tethers (FRT s) an averaging process is mandatory. To do that, a stroboscopic frame is introduced; such a frame facilitates and organizes in a better way the procedure in the general case. The governing equations are obtained taking the limit in which the angular rate of the tether goes to infinity. Averaged equations for fast rotating tethers are presented using the Hill approach. We show the existence of only one non-dimensional tether length, λ, which captures the influence of the tether size in the dynamics. Chapter 4, «Solar System Fact-Sheets», summarizes some interesting data for different binary systems in the Solar System. The values of the non-dimensional tether length λ as a function of the tether length is shown for several binary systems. An important objective of this study is related with the power generation at Jupiter using bare EDT s. For this reason, in chapter 5, «Jovian Models», we focus on the models that can be used to predict the performances of the bare tethers designed to be used as power supply of a given system. Chapter 6, entitled «Bare Electrodynamic Tethers», collects in detail the theory associated with the electron collection, in the Orbital Motion Limited (OML) regime. Since the Jovian world is the most appropriated place for the operation of an electrodynamic tether a special attention has been paid to ) the asymptotic solution which appears when the ohmic effects becomes negligible and ) the electrodynamic forces (resultant and Lorentz torque) which act on FRT s. Chapters 7 and 8 are devoted to analyze the influence of a FRT on the stability properties of Lyapunov orbits, eight-shaped periodic orbits and Halo orbits. All of them are characteristic periodic motions which take place in the neighborhood of the collinear Lagrangian points. New sets of periodic orbits has been obtained taking the non-dimensional tether length λ as the parameter of the families. Some of the are stable. Chapters 9 and carry out a new formulation to attack the basic problem set out by Colombo, Farquhar and Misra. We reproduce some of the results obtained for these authors in the frame of a new formulation which facilitates to deepen in the analysis. The presence of the tether does not improve the stability properties of the collinear points. However, stability can be provided by changing the tether length following some strategies developed in those chapters. Frequently the non-dimensional tether length λ is small (it depends on the binary system and the actual tether length) for tethers of reasonable lengths. The smallness of λ makes more difficult to implement the strategies developed to control the system. Chapters, entitled «Io exploration with electrodynamic tethers», presents a special analysis of the Io moon. We propose two alternatives using EDT working in the generator regime: ) a bare self-balanced electrodynamic tether in equilibrium position in the synodic frame Jupiter-Io, and ) a rotating EDT orbiting around Io to generate permanent power and to provide propellantless orbital maneuvering capability. For the first alternative we derive the necessary orbital and tether design conditions for equilibrium and the system performances in terms of power generation. For the second alternative, we investigate two mission scenarios both involving a 5-km-long 5-cm-wide tape tether placed on a stable retrograde equatorial orbit around Io capable to provide kw-level useful power extracted from the fast rotating Jupiter plasmasphere. In the first scenario the tether current is controlled to provide maximum power generation. The amount of power produced and the impact on the orbit stability is investigated numerically. In the second scenario the current is controlled in order to reduce or increase the orbital energy of the system with the possibility of reaching escape velocity. Results show that EDT can be used as a permanent power production system in exploration missions to Io and the surrounding plasma torus without compromising the orbital stability.

17 xvii This final Chapter represents the link with the II part of this final report, since the analysis of the plasma torus in the Io orbit can be improved in a significan way by changing the scenario from the collinear points to the triangular points. Part II. Triangular Lagrangian Points The work reported in Part II is focused on the analysis of the dynamics, stability and power extraction capabilities of a bare electrodynamic tether located in proximity of the triangular Lagrangian points. Chapter, «Circular Restricted Three Body Problem: a Commented Review», makes an brief introduction of the circular restricted three body problem. The classical equations of motion governing orbital motion in the 3-body gravitational environment are explicitly derived both in Cartesian and spherical coordinates. The equation to derive Lagrangian point position and the Jacobi integral are also recalled. The principal aim of this chapter is to show notation and conventions adopted in the Part II of this study. In chapter 3, «Equations of motion of Tether at Triangular Points», the basic equations necessary to describe the motion of a variable-length electrodynamic tether in the proximity of triangular Lagrangian points have been explicitly derived. Tether is treated as a rigid dumbbell with two different masses, and it is capable to vary its length. When placed in the proximity of triangular points, the ratio L/R (tether length over the distance of center of mass from the primaries) is neglectable, and equations becomes simpler than in the collinear case. Orbital dynamics is described by the equations of the circular restricted three body problem plus the perturbation due to the electrodynamic force. Attitude dynamics is given by the evolution of the two angles describing the in-plane and out-of-plane orientation of the dumbbell. Chapters 4 and 5, «Dynamics at Triangular Points: Linear Analysis» and «Dynamics at Triangular Points: Investigations of Non-Linear Effects», are both devoted to make a description of the dynamics at triangular points. The D (in-plane) orbital motion of a body placed in the proximity of a triangular point is characterized by a two-eigenfrequncies motion, and its orbit remains stably in a neighborhood of the equilibrium point. The two eigenfrequencies can be easily derived using classical linear analysis techniques. Attitude motion is qualitatively similar to the orbital motion: a dumbbell placed with center of mass coincident with the triangular point oscillates with a twoeigenfrequencies motion for the out-of-plane and in-plane angles. The value of the frequency is a function of the planetary parameter ν, and it reduces to the two-body case when ν =. Non linear effects are also analyzed. Large amplitude oscillations around the equilibrium point exhibit the full non-linear nature of the motion. The orbit around the stable triangular attractor is not more symmetrical when a great angular deviation from the triangular point is imposed as an initial condition. Great horseshoes orbits becomes possible, oscillating between the two points L 4 L 5 of the planetary system. Chapter 6, «Io environment and Plasma Torus», gives a brief survey on environment encountered by the electrodynamic tether at Lagrangian points of Jupiter-Io system. Jupiter and its moons are of particular interest for electrodynamic tethers not only because of the large numbers of Lagrangian points present but also for specific features of one of its moons. Jupiter rotates fast and carries along a large and powerful magnetosphere. The stationary orbit is low and there is relatively dense plasma above the stationary altitude. In particular, Io orbits above the stationary orbit and is inside the Io ionospheric torus in which the plasma density is as high as close to Jupiter itself. The magnetosphere of Jupiter reaches a speed of 74 km/s at Io s orbit while the orbital speed of Io is about 7 km/s. Consequently, the ionosphere is passing ahead of a spacecraft placed at (or co-orbiting with) one of the triangular points at a relative speed close to 57 km/s. The two (stable) triangular Lagrangian points L 4 and L 5 of the Jupiter-Io system lie inside the plasma torus, respectively, ahead (+6ř) and behind (-6ř) Io and are ideal attractors for an EDT to exploit this favorable environment. Chapter 7, «Exploration of the Jupiter Plasma Torus with a self-powered Electrodynamic Tether», is devoted to the analysis of exploration of Jovian Plasma Torus using an electrodynamic tether.

18 xviii Two applications have been investigated, the first concerning the characteristics and use of new equilibrium positions that emerge when the EDT is activated, the second devoted to the exploration of the Io torus together with power generation. In the first case, power generation appears to be one of the most attractive features provided by an EDT system considering the weakness of the solar flux at Jupiter. Note that the tethered system maintains its orbital energy (or equivalently it does not loose orbital altitude) as the energy in these orbital conditions is extracted from the plasma sphere co-rotating with Jupiter. Useful power of a couple of kilowatts can be generated with an EDT of km in these orbital conditions. In the second case, the electrodynamic model derived for the EDT system was used for the exploration of the Io plasma torus. We proved that this task can be accomplished while at the same time extracting power from the super-rotating Jupiter s plasma sphere. The electrodynamic tether in question is a bare ED tether that has a higher efficiency in collecting electrons from the ionosphere than the large anodic terminations used in past tether missions. Results show that an EDT of moderate length (i.e., of order km) can scan the entire plasma torus in a time frame of a few months while producing a steady electrical power of several hundred watts for on-board use.

19 Part I Collinear Lagrangian Points

20

21 Chapter Equations of motion for librating tethers. Introduction The title of this study, Dynamics and Stability of Tethered Satellites at Lagrangian Points, makes reference to Tethered Satellites and Lagrangian Points. Thesetwo essential elements of this work should be appropriately modeled. Taking into account that we shall consider the dynamics of rotating and non-rotating tethers, inert and electrodynamic tethers and also tethers of constant or varying length, we will use an extended dumbbell model, in general, which allows the inclusion of the tether mass or the variation of the tether length, when necessary. In such a model, and for tethers of constant length, the spacecraft turns out to be a rigid body with five degree of freedom. The existence of Lagrangian point requires the existence of a previous binary system. A binary system gives place to a Restricted Three Body Problem, since the mass of the spacecraft is negligible compared with the mass of both primaries. In the most simple case both primaries have circular orbits around their common center of mass. This way, we find the classical Circular Restricted Three Body Problem which constitutes the aim of the next section.. Scenario. Circular Restricted Three Body Problem The analysis will start by describing the dynamics of a mechanical system which is moving in the presence of a system of two primaries. There are three bodies involved in the problem: both primaries and the spacecraft which is modeled as a rigid body in the simplest case. For the sake of simplicity we assume that both primaries have a circular relative motion which is not perturbed by the S/C. This is a realistic approximation based in the small eccentricity of the relative orbits of most binary systems in the Solar System. Moreover, we will assume that the plane where takes place the relative motion of both primaries coincides with the plane of the ecliptic. By way of illustration, Table. shows the values of the eccentricity and inclination for the primaries system formed for Jupiter 3

22 4 Section.: Scenario. Circular Restricted Three Body Problem and its inner moonlets. In general, both primaries can be considered as spherical objects with high accuracy; therefore it is very common to assume that each one of them creates a gravitational field given by the point mass potential. In this work we accept this assumption, but we should not forget, however, that it is a first approximation and it provides an approximated solution and not an exact one. If necessary, it would be possible to improve the analysis adopting more realistic values for the gravitational field (for example, approximating the form of the primaries by a triaxial ellipsoid or just using its lowest degree and order gravity field coefficients). This is the case of the inner Jovian moons whose form is extremely irregular. The main dimensions of Amalthea, for example, are approximately, km. Their gravitational field are also irregular. However, when more than 6 or 7 diameters from the satellite the gravitational field will be very close to the point mass potential. This subject is considered with more detail in Section C.4 taking advantage of the fact that for a tethered system the gravitational potential can be described in some cases by closed forms. Metis Adrastea Amalthea Tebe Eccentricity e Inclination i (degrees) Table.: Eccentricity and inclination of the inner Jupiter moonlets We remember here the Circular Restricted Three Body Problem (CRTBP). This very well known classical problem deals with the dynamics of three point masses under the following assumptions: ) only gravitational interaction acts on the system, ) the mass of one point is negligible compared with the masses of the other two (the primaries) and 3) the relative motion of the primaries, which responds to a classical two-body problem, turns out to be circular. The problem that we face in this work separates from the CRTBP in several aspects. For instance, we have to consider the attitude dynamics of the tethered system, a new aspect which does not appear in the CRTBP. Moreover, the interaction of the spacecraft with the surrounding environment is more complex giving place to additional forces which plays an essential role in the dynamics. This is specially true for the electrodynamic bare tethers in the Jovian world. However, the dynamics is closely related with the CRTBP and it can be considered as an extension of it. We will use an extended Dumbbell Model which permits an additional degree of freedom to take into account a variable length for the tether; the constant length tether will be obtained as a particular case. A clear description of the model provides a better understanding of the validity of the latter analysis; moreover, it makes easy to incorporate new effects to the analysis or to establish, in a more appropriate form, its more interesting continuations.

23 Chapter : Equations of motion for librating tethers 5 In the formulation of the equations of motion the inertia forces play a quite important role. In a classical formulation which can be found in the famous book of Landau (see [7]), for example, the inertia forces can be derived from a generalized potential, and this way a Lagrangian function can be obtained for the basic dynamics of the particle or the rigid body. However, it is posible to obtain a different and simpler Lagrangian function, managing the inertia forces in a different way. To clarify this new procedure the inertia forces have been considered in detail in Appendix A for two main cases: the point mass and the rigid body. Obviously, and taking into account the gauge-invariant property of the Lagrangian functions, the final governing equations are the same in both cases... Basic references frames We summarize here the characteristics of some frames involved in the analysis and we introduce some notation. Let m P and m P be the masses of the main and secondary primaries respectively (m P >m P ); these masses will be clearly dominant. Let G P be the center of mass of both primaries. Any non rotating frame with origin at G P, will be considered as an inertial frame. The relative motion of primaries is circular and it takes place in a plane with constant direction. Let G P x y an inertial frame embedded in this plane. The synodic frame G P xy is rotating around the axis Gz with the constant angular y m P y G P m P ωt x x Figure.: Relative motion of primaries, the inertial frame and the synodic frame velocity: G(mP + m P ) ω = l 3 (.) where l is the distance between both primaries (see Fig..); the primaries are at rest in the synodic frame G P xyz. Since we are interested in the motion of a space system in the neighborhood of the secondary primary (m P ) we will take a new frame Oxyz with origin at the center of mass of this primary and axes parallel to the corresponding axis of the frame G P xyz (see Fig..). In fact, both frames correspond to the same rigid body. This new frame coincides with the orbital frame of the secondary primary in its trajectory around the main primary. Primaries are also at rest in this frame at positions m P (,, ),m P ( l,, )... Basic forces acting on a mass particle Themotionofaparticleofmassm relative to the frame Oxyz is governed by the following forces:

24 6 Section.: Scenario. Circular Restricted Three Body Problem ω z m G P ϱ k ϱ y m P O j m P i νl l l =( ν)l x Figure.: Reference frames gravitational attraction of primaries, givenby F g = Gm P m ϱ 3 ϱ Gm P m ϱ 3 where ϱ =(l + x, y, z), and ϱ =(x, y, z) are the position vectors of the mass m with origin at the primaries m P and m P respectively (ϱ = ϱ, ϱ = ϱ ). This force derive from the potential function Ṽ g = Gm{ m P + m P ϱ ϱ } the inertia forces, since the reference Oxyz is not a Galilean frame. They can be calculated using the expressions (A.-A.) taking into account that in this case: m P ω = ω k, α =, γ O = ( ν)lω i where ν = is the reduced mass of the secondary primary. This way we have: m P + m P centrifugal force, a repulsion of the G P z axis proportional to the mass and the distance: F ( IA = mω [x +( ν)l ] i + y ) j This force derives from the ordinary potential: ϱ Ṽ = mω ( (l + x) + y ), l =( ν)l inertial Coriolis force, which is gyroscopic it does not produce work in the motion of the particle and is given by F c =mω(ẏ, ẋ, )

25 Chapter : Equations of motion for librating tethers 7 This force derives from the generalized potential Ṽ = mω(xẏ yẋ) The whole set of forces derives from the generalized potential G given by: G = Gm{ m P ϱ + m P ϱ } mω ( [x +( ν)l ] + y ) mω(xẏ yẋ) This generalized potential can be decomposed as follows G = Ṽ + Ṽ where Ṽ is the following ordinary and steady potential: Ṽ = Gm{ m P ϱ + m P ϱ } mω ( [x +( ν)l ] + y ) (.) A more detailed analysis of the CRTBP can be found in the second part of this report (Chapter )..3 Lagrangian formulation We classify the whole set of forces acting on the tethered system as follows: the basic forces (gravitational and inertia) the electrodynamic forces the remainder of the forces not included in the previous items It is possible to define a Lagrangian function for the basic forces: L = T G (the details of the derivation of the kinetic energy T and the generalized potential G can be found in the Appendices A-D). The equations governing the motion will be d dt ( L ) L =(Q E j + Q R j ) q j q j mω l where L is the non-dimensional form of the Lagrangian function, q j,j =,...,5 is a generalized coordinate, Q E j is the corresponding generalized force due to the electrodynamic actions and Q R j is the corresponding generalized force due to the remainder of the forces. The Lagrangian function has the following structure: L = L + ɛ L + ɛ 3 L The non-dimensional variables are introduced in Section D.4 of the Appendix D

26 8 Section.4: Inert tether where ɛ = L T /l as defined in (D.6) is usually a small number and the different L i are given by L = (ẋ +ẏ +ż )+ (( ν + x) + y ( ν) )+(xẏ yẋ )+ + ν ρ ρ { L = a ( ϕ +(+ θ) cos ( ν) ϕ)+ ρ 3 P (cos α )+ ν } ρ 3 P (cos α ) { ( ν) L 3 = a 3 ρ 4 P 3 (cos α )+ ν } ρ 4 P 3(cos α ) (.3) (.4) (.5).4 Inert tether For the inert tether the generalized forces Q E j and Q R j vanish, i.e.: Q E j = QR j =, j =,...,5 The system is only acted upon the basic forces. Equations for the main order L : for the inert tether, in the limit ɛ the Lagrangian function reduces to L and the size of the system disappears from the problem. The governing equations become: d dt ( L ẋ ) L x = ẍ ẏ = ν + x ν x ( ν)+x ρ3 ρ 3 d dt ( L ẏ ) L y = ÿ +ẋ = y ν y ρ 3 ( ν) y ρ 3 d dt ( L ż ) L z = z = ν z ρ 3 ( ν) z ρ 3 (.6) (.7) (.8) and they reduce to the classical CRTBP for the motion of the system center of mass G. Equations for the two first orders L + ɛ L : when considering the two first terms in the Lagrangian function we obtain the following governing equations: because L does not depend on (ẋ, ẏ,ż) explicitly. d dt ( L ẋ ) L x = L ɛ x (.9) d dt ( L ẏ ) L y = L ɛ y (.) d dt ( L ż ) L z = L ɛ z (.) d dt ( L θ ) L = θ (.) d dt ( L ϕ ) L = ϕ (.3)

27 Chapter : Equations of motion for librating tethers 9 Equations for the three first orders L + ɛ L + ɛ 3 L 3: when considering the three first terms in the Lagrangian function we obtain the following governing equations: d dt ( L ẋ ) L x = L ɛ x + L 3 ɛ3 x d dt ( L ẏ ) L y = L ɛ y + L 3 ɛ3 y d dt ( L ż ) L z = L ɛ z + L 3 ɛ3 z d dt ( L θ ) L θ = ɛ L 3 θ d dt ( L ϕ ) L ϕ = ɛ L 3 ϕ (.4) (.5) (.6) (.7) (.8) because L 3 does not depend on (ẋ, ẏ,ż, θ, ϕ) explicitly. z u y m G u θ y ϕ z m m G ϕ θ x R x θ R m u 3 G u ϕ R Figure.3: Frame Gu u u 3 attached to the tether when attitude is described with the libration angles θ (in-plane) and ϕ (out-of-plane).4. Equations for constant tether length The governing equations for an inert tether of constant length are collected in Section.5 in page. In those equations we retain the terms of order ɛ and we neglect the terms of order ɛ 3 and smaller..4. Body frame When the attitude dynamics of the tether is described using the libration angles (θ, ϕ) as generalized coordinates, we take as body frame attached to the tether the reference frame defined by the three vectors ( u, u, u 3 ) given by: u = u =cosϕcos θ i +cosϕsin θ j +sinϕ k (.9) u = sin θ i +cosθ j (.) u 3 = sin ϕ cos θ i sin ϕ sin θ j +cosϕ k (.)

28 Section.4: Inert tether Figure.3 shows the three vectors ( u, u, u 3 ) projected onto the more appropriate planes in order to clarify their definitions..4.3 Derivatives The derivatives involved in the equations take the values: ρ x = x ρ, x ( ρ )= x ρ, 3 x ( 3x )= ρ3 ρ, 5 ρ y = y ρ, y ( ρ )= y ρ, 3 y ( 3y )= ρ3 ρ, 5 ρ z = z ρ (.) z ( ρ )= z ρ 3 (.3) z ( 3z )= ρ3 ρ 5 (.4) ρ x = +x, ρ x ( )= +x, ρ ρ 3 x ( 3( + x) )=, ρ 3 ρ 5 ρ y = y, ρ y ( )= y, ρ ρ 3 y ( )= 3y, ρ 3 ρ 5 ρ z = z ρ (.5) z ( )= z ρ ρ 3 (.6) z ( )= 3z ρ 3 ρ 5 (.7) cos α = +x x ρ cos α = y y ρ cos α z = z ρ cos α = ρ cos α ρ + u ρ cos α + ρ i u cos α x cos α + ρ j u cos α y cos α + ρ k u cos α z = x ρ cos α + ρ i u = y ρ cos α + ρ j u = z ρ cos α + ρ k u cos α = ρ ρ cos α + ρ u P (cos α ) θ P (cos α ) ϕ = 3 cos ϕ( ρ ρ u)( ρ u ) = 3 ρ ( ρ u)( ρ u 3) P (cos α ) θ P (cos α ) ϕ = 3 cos ϕ( ρ u)( ρ u) ρ = 3 ( ρ u)( ρ u3) ρ.4.4 Equations for varying tether length When we use the Extended Dumbbell Model we are considering both tether ends as point masses. The governing equations can be obtained using Lagrangian formulation taking into account that: ) the Lagrangian function of the system is the same as in the constant tether length case, and ) now, the parameter ɛ a which captures the influence of the tether length is a function of time. Taking into account the smallness of ɛ we will work with the equations for the two first orders using as Lagrangian Function L = L + ɛ L. From this Lagrangian function

29 Chapter : Equations of motion for librating tethers we obtain the following governing equations: d dt ( L ẋ ) L x = L ɛ x (.8) d dt ( L ẏ ) L y = L ɛ y (.9) d dt ( L ż ) L z = L ɛ z (.3) d L dt (ɛ θ ) L ɛ = θ (.3) d L dt (ɛ ϕ ) L ɛ = ϕ (.3) because L does not depend on (θ, ϕ, θ, ϕ) and L does not depend on (ẋ, ẏ,ż) explicitly. The three first equations, (.8-.3), govern the motion of the center of mass G and they are equal to the corresponding equations for the constant length tether case. There is an important difference since now the parameter ɛ depends on time. The attitude equation, however, experience modification when compared with the corresponding equations for the constant tether length case. Let us consider, for example, the equation (.3); the term ɛ L θ = ɛ a ( + θ)cos ϕ gives place to the following term, in the equation for θ: d L dt (ɛ θ )=ɛ a [ θ cos ϕ ( + θ) ϕ sin ϕ cos ϕ] Taking into account the value of the quotient + d(ɛ a ) ( + dτ θ)cos ϕ ɛ a d(ɛ a ) dτ = I s di s dτ we obtain the following equation for this coordinate: θ cos ϕ ( + θ) I ϕ sin ϕ + s ( + I θ)cosϕ ( ν) =3 s ρ 5 ( ρ u)( ρ u )+3 ν ρ 5 ( ρ u)( ρ u ) A similar procedure provides the governing equation for the coordinate ϕ. Thefullsetof equations for an inert tether with varying length are summarized in Section.6 in page.

30 Section.6: Governing equations for a constant length inert librating tether.5 Governing equations for a constant length inert librating tether ẍ ẏ ( ν)( +x ρ 3 ) x( ν { ( ν) ρ 3 )=ɛ a ρ 5 [3( ρ u)( i u) ( + x)s(cos α)] + ν } ρ 5 [3( ρ u)( i u) xs(cos α)] (.33) ( ν) ÿ +ẋ + y ρ 3 y( ν { ( ν) ρ 3 )=ɛ a ρ 5 [3( ρ u)( j u) ys(cos α)] + ν } ρ 5 [3( ρ u)( j u) ys(cos α)] (.34) z + z ν { ( ν) ( ν) + z ρ 3 ρ 3 = ɛ a ρ 5 [3( ρ u)( k u) zs(cos α)] + ν } ρ 5 [3( ρ u)( k u) zs(cos α)] (.35) ( ν) θ cos ϕ ( + θ) ϕ sin ϕ =3 ρ 5 ( ρ u)( ρ u)+3 ν ρ 5 ( ρ u)( ρ u ) (.36) ϕ +(+ θ) ( ν) sin ϕ cos ϕ =3 ( ρ u)( ρ u3)+3 ν ρ 5 ( ρ u)( ρ u 3) (.37) ρ 5.6 Governing equations for a varying length inert librating tether ) x( ν { ( ν) ρ 3 )=ɛ a ρ 5 [3( ρ u)( i u) ( + x)s(cos α)] + ν } ρ 5 [3( ρ u)( i u) xs(cos α)] (.38) ( ν) ÿ +ẋ + y ρ 3 y( ν { ( ν) ρ 3 )=ɛ a ρ 5 [3( ρ u)( j u) ys(cos α)] + ν } ρ 5 [3( ρ u)( j u) ys(cos α)] (.39) z + z ν { ( ν) ( ν) + z ρ 3 ρ 3 = ɛ a ρ 5 [3( ρ u)( k u) zs(cos α)] + ν } ρ 5 [3( ρ u)( k u) zs(cos α)] (.4) Is θ cos ϕ ( + θ) ϕ sin ϕ = ( + θ)cosϕ +3 ( ρ u)( ρ u)+3 ν ρ 5 ( ρ u)( ρ u ) (.4) ẍ ẏ ( ν)( +x ρ 3 Is ϕ + Is ϕ +(+ θ) sin ϕ cos ϕ =3 Is ( ν) ρ 5 ( ν) ρ 5 ( ρ u)( ρ u3)+3 ν ρ 5 ( ρ u)( ρ u 3) (.4) Note that for a varying length librating tether the parameter ɛ = L d l time. Moreover, some terms of these equations involves the ratio: is a function of the time since the tether length Ld(t) is changing with Is Is = Ld Ld { Λ d ( + 3 cos φ) ( 3sin φ Λ d)} Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λd = in the above expression.

31 Chapter : Equations of motion for librating tethers 3.7 Hill approach Most of the times the parameter ν is small. The Hill approximation leads to more tractable equations and makes easy to understand the main features of the dynamics. We will introduce this approximation by means of the following change of variables: x = ν /3 ξ, y = ν /3 η, z = ν /3 ζ, ρ = ν /3 ˆρ We introduce this change of variables in equations (.38-.4) and then we take the limit ν. The following set of equations is obtained: { } ξ η (3 ˆρ 3 )ξ = λˆρ 5 3Ñ cos ϕ cos θ ξs (Ñ ˆρ ) (.43) θ +(+ θ) { η + ξ + ηˆρ 3 = λˆρ 5 3Ñ cos ϕ sin θ ηs (Ñ ˆρ ) { } ζ + ζ( + ˆρ 3 )= λˆρ 5 3Ñ sin ϕ ζs (Ñ ˆρ ) [ ] Is ϕ tan ϕ +3cosθsin θ = 3Ñ ( ξ sin θ + η cos θ) I s ˆρ 5 cos ϕ I ϕ + [ s ϕ +sinϕcos ϕ ( + I θ) ] +3cos θ s } (.44) (.45) (.46) = 3Ñ ( sin ϕ[ξ cos θ + η sin θ]+ζ cos ϕ) (.47) ˆρ 5 where ˆρ = ξ + η + ζ and the quantity Ñ and the function S (x) are given by Ñ = ξ cos ϕ cos θ + η cos ϕ sin θ + ζ sin ϕ, S (x) = 3 (5x ) (.48) In these equations the new parameter that captures the influence of the tether length is λ = ɛ a, and we assume that λ O() (.49) ν/3 Note that for a varying length tether the parameter ɛ = L d is a function of the time since l the tether length L d (t) is changing with time. Moreover, some terms of these governing equations involve the ratio: I s I s = L d L d { Λ d ( + 3 cos φ) ( 3sin φ Λ d )} Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λ d =in the above expression. For a constant length tether the parameter ɛ is constant and the quotient I s /I s vanishes, that is, I s /I s =.

32 4 Section.7: Hill approach

33 Chapter Equations of motion for constant-length rotating tethers The Attitude Dynamics of the tether can be analyzed more intuitively using the Newton- Euler formulation. The core of the analysis is the angular momentum equation: d H G dt = M G (.) where M G is the resultant of all the external torques applied to the center of mass G of the tethered system and H G is the angular momentum of the system, at G, inthemotion relative to the center of mass. In what follows we will go into details starting by the constant length tether case.. Dumbbell Model.. Basic equations In the Dumbbell Model the system is a rigid body. The angular momentum H G turn out to be H G u 3 m H G = Ī G Ω = I s ( u u) (.) where Ω, the angular velocity of the tether, is given by: Ω = u u + u ( u Ω ) (.3) G u u u Here the unit vector u lies along the tether and u is its time derivative. The angular momentum is normal to the tether. As a consequence, we can take m Figure.: Frame attached to the tether 5

34 6 Section.: Dumbbell Model a reference frame Gu u u 3 attached to the tether where the unit vectors are given by: In this body frame the angular momentum is: u = u, u = u u, u 3 = u u (.4) H G = I s Ω u 3, where Ω = u u = u and the angular momentum equation (.) can be written as follows: d Ω dt u 3 + Ω d u 3 dt = I s M G (.5) Taking the dot product of (.5) with the unit vector u 3 we obtain: d Ω dt = I s ( M G u 3 ) Taking the cross product of (.5) with the unit vector u 3 two times we obtain: d u 3 dt = Ω I s u 3 ( u 3 M G ) The evolution of the unit vector u is given by: Summing up: equations: d u dt = Ω u =( u u) u = Ω u 3 u = Ω u the time evolution of the system is given by the following system of d u dt d u 3 dt d Ω dt = Ω u (.6) = Ω I s u 3 ( u 3 M G ) (.7) = I s ( M G u 3 ) (.8) Note that the vectors u and u 3 given by integrating equations (.6-.8) keeps constant module since u d u = dt d dt ( u u )= u = (by initial conditions) u 3 d u 3 = dt d dt ( u 3 u 3 )= u 3 = (by initial conditions) Moreover, both vectors u and u 3 keep normal each other because: d dt ( u u 3 )= d u u 3 + d u 3 u = u u 3 = (by initial conditions) dt dt

35 Chapter : Equations of motion for constant-length rotating tethers 7 To obtain this last result we have taken into account that the torques acting on the tethered system are normal to the tether, that is, it fulfils the condition M G u = (.9) in all the cases to be considered in the Dumbbell Model. This is true for the gravitational torque, for the inertia torques and for the Lorentz torque (we will give the expression for this last torque later on); these three kinds of torque can appear in the vector M G involved in the right hand side of equations (.6-.8) (later on we will comment this point again). Obviously, it is not necessary to set any equation for the time evolution of the unit vector u since it can be obtained by the orthogonality condition u = u 3 u M G can be written in terms of its com- Taking into account equation (.9) the torque ponents in the body frame as follows: M G = M u + M 3 u 3 This way equations (.6-.8) take the simpler form: d u dt d u 3 dt d Ω dt = Ω u (.) = M Ω I s u (.) = M 3 I s (.) These equations are valid for librating or rotating tethers. However, when the torque M G vanishes equations (.-.) can be easily integrated and the tether is rotating with a constant angular velocity around a fixed direction of the space pointed by u 3 which remains constant. In order to have a librating tether we need, on the right hand side of equations (.-.) the gravitational torque; due to this torque the condition M G = never fulfils... Euler angles as generalized coordinates Let (θ e,ψ e,ϕ e ) be the Euler angles, in sequence, that determine the position of the body frame Gu u u 3 relative to a given reference frame Gx y z. We use here the classical definition of the Euler angles which is based in the line of nodes (see Fig..). In order to pass to the body frame Gu u u 3, starting from the frame Gx y z,we have to give three consecutive rotations: θ e u 3 G z u u u y This section is not necessary and it can be skipped; however we keep it here for convenience x ψ e ϕ e Figure.: Euler angles

36 8 Section.: Dumbbell Model a rotation ψ e, precession angle, around the Gz. This rotation is governed by the matrix cos ψ e sin ψ e R 3 (ψ e )= sin ψ e cos ψ e a rotation θ e, nutation angle, around the line of nodes. This rotation is governed by the matrix R (θ e )= cosθ e sin θ e sinθ e cos θ e a rotation ϕ e, intrinsic rotation angle, around the body axis pointed by u 3. This rotation is governed by the matrix cos ϕ e sin ϕ e R 3 (ϕ e )= sin ϕ e cos ϕ e These three rotations should be given in the order (ψ e,θ e,ϕ e ). The final rotation has associated the matrix Q(ψ e,θ e,ϕ e ) Q = R 3 (ψ e )R (θ e )R 3 (ϕ e ), [ u, u, u 3 ]=[ i, j, k ]Q and it turns out to be Q = cos ψ e cos ϕ e sin ψ e sin ϕ e cos θ e cos ψ e sin ϕ e sin ψ e cos ϕ e cos θ e +sinψ e sin θ e sin ψ e cos ϕ e +cosψ e sin ϕ e cos θ e sin ψ e sin ϕ e +cosψ e cos ϕ e cos θ e cos ψ e sin θ e sin θ e sin ϕ e sin θ e cos ϕ e cos θ e The unit vectors u and u 3 are given by u =(cosψ e cos ϕ e sin ψ e sin ϕ e cos θ e, sin ψ e cos ϕ e +cosψ e sin ϕ e cos θ e, sin θ e sin ϕ e ) u 3 =(sinψ e sin θ e, cos ψ e sin θ e, cos θ e ) Introducing these vectors in equations (.-.) we can deduce, after some algebra with Maple, the equations which provide the time evolution of the Euler angles: dθ e dt = M cos ϕ e (.3) Ω I s dψ e dt dϕ e dt d Ω dt = M Ω I s sin ϕ e sin θ e (.4) = Ω + M Ω I s sin ϕ e cot θ e (.5) = M 3 I s (.6) Obviously, equations (.3-.6) should be integrated from the appropriated initial conditions: at t =: θ e = θ e,ψ e = ψ e, ϕ e = ϕ e, Ω = Ω (.7)

37 Chapter : Equations of motion for constant-length rotating tethers 9 These equations show that the component M of the external torques changes the direction of the angular momentum; when M =, the direction of the angular momentum is constant. However, the component M 3 change the module of the angular momentum. If (p, q, r) are the components of the angular velocity of the tether Ω in the body frame, they can be expressed in terms of the time derivatives of the Euler angles as follows: p = θ cos ϕ + ψ sin θ sin ϕ q = θ sin ϕ + ψ sin θ cos ϕ r = ϕ + ψ cos θ Taking into account the equations (.3-.4) the component q vanishes. An important drawback of this formulation is found in the singularity associated with the Euler angles in θ e =; due to this singularity these equations are not appropriated to describe the dynamics in the neighborhood of θ e =. In fact, this situation appears in this problem when the tether lies into the orbital plane of both primaries, or into a parallel plane. Since this is a quite likely choice in this analysis, in the following section we carry out a new formulation with Euler angles in --3 sequence...3 Bryant angles as generalized coordinates The position of the body axes in the reference frame Gx y z is given by three angles (φ,φ,φ 3 ) which are associated with the following three rotations: a rotation φ around the axis Gx which is governed by the matrix R (φ ): R (φ )= cosφ sin φ sinφ cos φ After this rotation the Gy axis coincides with the straight line s (see.3); this line is the intersection of the plane x =and the plane Gu u. a rotation φ around the straight line s whichisgovernedbythematrixr (φ ): cos φ sinφ R (φ )= sin φ cosφ x u 3 z φ φ G u u u Figure.3: Bryant angles φ 3 φ y s a rotation φ 3 around the axis Gu 3 of the body frame, which is governed by the matrix R 3 (φ 3 ) cos φ 3 sin φ 3 R 3 (φ 3 )= sin φ 3 cos φ 3

38 Section.: Dumbbell Model These three rotations should be given in the order (φ,φ,φ 3 ). The final rotation has associated the matrix Q(φ,φ,φ 3 ) Q = R (φ )R (φ )R 3 (φ 3 ), [ u, u, u 3 ]=[ i, j, k ]Q and it turns out to be cos φ cos φ 3 cos φ sin φ 3 sin φ Q = cos φ sin φ 3 +sinφ sin φ cos φ 3 cos φ cos φ 3 sin φ sin φ sin φ 3 sin φ cos φ sin φ sin φ 3 cos φ sin φ cos φ 3 sin φ cos φ 3 +cosφ sin φ sin φ 3 cos φ cos φ Note that when the three Bryant angles vanish, φ = φ = φ 3 =, the body axes coincide with the axes of the frame Gx y z and the matrix Q turns out to be the unity matrix. In terms of the Bryant angles the unit vectors u and u 3 are given by u =(cosφ cos φ 3, cos φ sin φ 3 +sinφ sin φ cos φ 3, sin φ sin φ 3 cos φ sin φ cos φ 3 ) u 3 =(sinφ, sin φ cos φ, cos φ cos φ ) Introducing these vectors in equations (.-.) we can deduce, after some algebra with Maple, the equations which provide the time evolution of the Euler angles: dφ dt dφ dt dφ 3 dt = M Ω I s cos φ 3 cos φ (.8) = M Ω I s sin φ 3 (.9) = Ω + M Ω I s cos φ 3 tan φ (.) d Ω dt = M 3 I s (.) Obviously, equations (.8-.) should be integrated from the appropriated initial conditions: at t =: φ = φ, φ = φ,φ 3 = φ 3, Ω = Ω (.) These equations also have a singularity when φ = π, that is, when the angular momentum is aligned with the axis Gx ( u e i ). In this case, two angles (φ,φ 3 ) turn out to be associated to the same degree of freedom: the rotation around the axis Gx which is governed by the sum φ + φ 3. However, they are appropriated to study the motion of the tethered system when the tether is in the neighborhood of the orbital plane of both primaries. Note that if (p, q, r) are the components of the angular velocity Ω of the tether in the body frame, they can be expressed in termos of the time derivatives of the Bryant angles as follows: p = φ sin φ 3 + φ cos φ cos φ 3 q = φ cos φ 3 φ cos φ sin φ 3 r = φ 3 + φ sin φ.

39 Chapter : Equations of motion for constant-length rotating tethers Taking into account the equations (.8-.9) the component q vanishes...4 Non-dimensional form of the equations To set the non-dimensional form of the equations we introduce the non-dimensional time τ = ωt. Considering, for example, equations (.8-.); they take the following nondimensional form: dφ dτ = M ω cos φ 3 I s Ω cos φ (.3) dφ dτ = M ω sin φ 3 I s Ω (.4) dφ 3 dτ =Ω + M ω cos φ 3 tan φ I s Ω (.5) dω = M 3 dτ ω I s (.6) where Ω = Ω /ω is the non-dimensional form of Ω.. Remarks on the equations Equations (.6-.8) can be used in two different scenes: ) describing the motion of the system relative to the inertial frame G P x y z, or ) describing the motion of the system relative to the synodic frame Oxyz. Both frames are defined in Section..... Inertial motion In the inertial motion the frame Gx y z used in previous Sections of this Chapter coincides with the frame Gx y z : this frame has origin at the center of mass G and its axes are parallel to the corresponding axes of the inertial frame G P x y z. In equations (.6-.8) the time derivatives should be performed keeping constant the unit vectors ( i, j, k ) of the frame G P x y z. The angular velocity Ω = u u + u ( u Ω ) is the angular rate relative to the inertial frame. Equally, the angular momentum H G is the angular momentum relative to the inertial frame. In equations (.3-.6) (or (.8-.)) the Euler angles fix the position of the body frame Gu u u 3 relative to the frame Gx y z, that is, the attitude of the tether in the inertial frame. On the right hand side of equations (.6-.8), (.3-.6) and (.8-.) the torque M G includes: the gravitational torques of both primaries, the Lorentz torque the torque produced by any other perturbation which could be included in the analysis

40 Section.3: Rotating tethers.. Motion relative to the synodic frame In the motion relative to the synodic frame Oxyz, the frame Gx y z used in previous Sections of this Chapter coincides with the frame Gxyz: this frame has origin at the center of mass G and its axes are parallel to the corresponding axes of the synodic frame Oxyz. In equations (.6-.8) the time derivatives should be performed keeping constant the unit vectors ( i, j, k) of the synodic frame Oxyz. The angular velocity Ω = u u + u ( u Ω ) is the angular rate relative to the synodic frame. Equally, the angular momentum H G is the angular momentum relative to the synodic frame. In equations (.3-.6) (or (.8-.)) the Euler angles fix the position of the body frame Gu u u 3 relative to the frame Gxyz, that is, the attitude of the tether in the synodic frame. On the right hand side of equations (.6-.8), (.3-.6) and (.8-.) the torque M G includes: the torques produced by the inertial forces, the gravitational torques of both primaries, the Lorentz torque the torque produced by any other perturbation which could be included in the analysis.3 Rotating tethers For a rotating tether there are two characteristic times involved in the problem: ) the period of the orbital dynamics of both primaries and ) the period of the intrinsic rotation of the tether. This is because a rotating tether completes a turn around the angular momentum H G in a time which is usually small compared with the orbital period of both primaries. Thus, the value of the non-dimensional variable Ω is usually large, that is, Ω. In suchacasewetalkabouta fast rotating tether. When the variable τ is of order unity, (τ O()), the fast rotating tether has given several turns around the angular momentum H G. Therefore, we can introduce the nondimensional time τ =Ω τ which is associated with the intrinsic rotation of the tether. In equations (.3-.6) we have two different times scale: the variables (φ,φ,φ 3, Ω ) are functions of τ and τ.fortimesτ O() the evolution of the variables with time τ is very small. In the limit Ω the equations simplify and provides the following asymptotic solution: (φ φ, φ φ, Ω =Ω, ) φ 3 τ + φ 3 that is, the angle φ 3 is a fast variable which depends, basically, on the fast time τ.

41 Chapter : Equations of motion for constant-length rotating tethers 3 To find the time evolution of the system in the slow time scale, τ O(), we can use the averaging method. For example, let us assume that u 3 v 3 dφ dτ = f(φ,φ,φ 3, Ω ) is one of equations (.3-.6). The averaged equations takes the form: G φ 3 u < dφ dτ >= π f(φ,φ,φ 3, Ω )dτ π To integrate the function f(φ,φ,φ 3, Ω ) which appears on the right hand side of this relation, the slow variables (φ,φ, Ω ) take constant values and the fast variable φ 3 is approximated by φ 3 τ + φ 3. v v φ 3 u u Figure.4: Stroboscopic frame.4 Averaging on rotating tethers In order to average the equations of motion when we manage rotating tethers, we introduce a stroboscopic frame Gv v v 3 (see Fig..4). For this frame, the unit vector v 3 coincides with the unit vector u 3 and both point into the angular momentum direction. When using the Bryant angles the unit vectors of this new frame have the following coordinates in the frame Gx y z : v =(cosφ, sin φ sin φ, sin φ cos φ ) (.7) v =(, cos φ, sin φ ) (.8) v 3 =(sinφ, cos φ sin φ, cos φ cos φ ) (.9) For a fast rotating tether, the frame Gv v v 3 is attached to the real plane where the tether is rotating; the three unit vectors ( v, v, v 3 ) evolve in the slow time scale τ O() and keep almost-constant values in the fast time scale τ O(). Obviously the unit vector u u along the tether is given by: u =+cosφ 3 v +sinφ 3 v (.3) u = sin φ 3 v +cosφ 3 v (.3) The equations governing the time evolution of a rotating tether relative to the synodic frame are (.3-.38) (they can be found in the Section.5 in page 7). The first three equations govern the motion of the system center of mass G. We will start taking the average of these three equations (.3-.34).

42 4 Section.4: Averaging on rotating tethers.4. Equations for the center of mass motion On the right hand side of equation (.3) we find the term ( ρ u)( i u). Taking into account (.3-.3) we get: that term can be averaged as follows: ρ u =cosφ 3 ( ρ v )+sinφ 3 ( ρ v ) i u =cosφ3 ( i v )+sinφ 3 ( i v ) < ( ρ u)( i u) >= ( ρ v )( i v )+ ( ρ v )( i v ) However, taking into account the values of ( v, v ) given by (.7-.8) and the value of the position vector ρ =(+x) i + y j + z k we obtain the following final value: where the quantity ñ is given by < ( ρ u)( i u) >= x + cos φ ñ sin φ ñ = x sin φ (y sin φ z cos φ )cosφ Following these steps we have obtained the values: < ( ρ u)( i u) > = x + cos φ ñ sin φ < ( ρ u)( i u) > = x ñ sin φ < ( ρ u)( j u) > = y + (sin φ +ñ)cosφ sin φ < ( ρ u)( j u) > = y + ñ cos φ sin φ < ( ρ u)( k u) > = z (sin φ +ñ)cosφ cos φ < ( ρ u)( k u) > = z ñ cos φ cos φ In a similar way the term S (cos α ) can be averaged taking into account the definitions of S (x) =3(5x )/ and cos α =( ρ u)/ρ : <S (cos α ) >= 3 { } 5 ρ [( ρ v ) +( ρ v ) ] This procedure lead to: <S (cos α ) > = <S (cos α ) > = Equations for the attitude motion (ñ ) ρ ) Let us take the equation (.3). In order to average the right hand side of the equation we have to evaluate the value of M = M G u. Since we are studying the motion relative to the synodic frame Oxyz, we have to consider the following torques, all of them included in the vector M G : (ñ ρ

43 Chapter : Equations of motion for constant-length rotating tethers 5 gravitational torque due to the main primary (is given by expression (D.3)) gravitational torque due to the secondary primary (is given by expression (D.7)) torque produced by the inertia forces (is given by expression (B.6)) Main primary: let us start with the gravitational torque due to the main primary as given by expression (D.3). For this torque the value of M is given by: M ml ω = ml ω ( ν) ρ ( ν) ρ { } ( u ρ ) u ρ ɛ a S (cos α )+... { } 3 ( ρ u 3 ) ( ρ ɛ ρ a u ρ ) +... Note that the vector u 3 = v 3 does not depend on φ 3 (see (.9)). As a consequence, taking into account the expressions (.3-.3) for u and u we obtain M 3ml ω Therefore, the averaged value <M cos φ 3 > is ( ν) ρ 5 ɛ a ( ρ u 3 ) {cos φ 3 ( ρ v )+sinφ 3 ( ρ v )} <M cos φ 3 > 3 ml ω ( ν) ρ 5 ɛ a ( ρ u 3 )( ρ v ) The averaged value of equation (.3), for this gravitational torque provides: < M ω cos φ 3 > 3 ( ν) ( ρ u 3 )( ρ v ) I s Ω cos φ Ω ρ 5 cos φ A similar analysis for the equations (.4-.5) provides the values: < M ω sin φ 3 > 3 ( ν) ( ρ u 3 )( ρ v ) I s Ω Ω ρ 5 < M 3 ω I s > and taking into account the values of ρ,and( v, v, v 3 ) we obtain: < M ω cos φ 3 > 3 ( ν) (sin φ +ñ)(cos φ + b) I s Ω cos φ Ω ρ 5 cos φ < M ω sin φ 3 > 3 ( ν) (sin φ +ñ)(y cos φ + z sin φ ) I s Ω Ω ρ 5 < M ω cos φ 3 tan φ > 3 ( ν) (sin φ +ñ)(cos φ + b) I s Ω Ω ρ 5 tan φ < M 3 ω I s > =

44 6 Section.4: Averaging on rotating tethers where the values of (ñ, b) are given by: ñ = x sin φ (y sin φ z cos φ )cosφ b = x cos φ +(y sin φ z cos φ )sinφ Secondary primary: the averaged process in this case leads to the following results: < M ω cos φ 3 > 3 I s Ω cos φ < M ω I s Ω sin φ 3 > 3 < M 3 ω I s > ν ( ρ u 3 )( ρ v ) Ω ρ 5 cos φ ν ( ρ u 3 )( ρ v ) Ω ρ 5 and taking into account the values of ρ, and( v, v, v 3 ) we obtain: < M ω cos φ 3 > 3 I s Ω cos φ < M ω I s < M ω I s Ω sin φ 3 > 3 Ω cos φ 3 tan φ > 3 < M 3 ω I s > ν ñ b Ω ρ 5 cos φ ν ñ(y cos φ + z sin φ ) Ω ρ 5 ν Ω ñ b ρ 5 tan φ Inertia torque: the averaged process in this case leads to the following results: < M ω cos φ ( 3 > + cos φ ) cos φ sin φ cos φ I s Ω cos φ Ω cos φ < M ( ω sin φ 3 > + cos φ ) cos φ sin φ I s Ω Ω < M ( ω cos φ 3 tan φ > + cos φ ) cos φ sin φ cos φ tan φ I s Ω Ω < M 3 ω I s > sin φ sin φ cos φ The equations for an inert, rotating tether, after the average process are collected in ( ) (see Section.6).

45 Chapter : Equations of motion for constant-length rotating tethers 7.5 Full set of equations for a constant length rotating tether ẍ ẏ ( ν)( +x ρ 3 ÿ +ẋ + y ( ν) ρ 3 z + z ν ρ ) x( ν ρ 3 )=ɛ a y( ν ρ 3 )=ɛ a ( ν) + z 3 ρ 3 = ɛ a dφ dτ = M ω Is dφ dτ = M ω Is dφ3 dτ =Ω + M ω Is dω dτ = M 3 ω Is { ( ν) ρ 5 { ( ν) ρ 5 { ( ν) ρ 5 Ω Ω [3( ρ u)( i u) ( + x)s(cos α)] + ν ρ 5 [3( ρ u)( i u) xs(cos α)] [3( ρ u)( j u) ys(cos α)] + ν } ρ 5 [3( ρ u)( j u) ys(cos α)] [3( ρ u)( k u) zs(cos α)] + ν } ρ 5 [3( ρ u)( k u) zs(cos α)] cos φ 3 cos φ sin φ3 Ω } (.3) (.33) (.34) (.35) (.36) cos φ3 tan φ (.37) (.38)

46 8 Section.6: Averaged equations for a constant length rotating tether.6 Averaged equations for a constant length rotating tether ẍ ẏ ( ν)( +x ρ 3 ÿ +ẋ + y ( ν) ρ 3 z + z ν ρ ) x( ν ρ 3 )= ɛ a y( ν ρ 3 )=+ ɛ a ( ν) + z 3 ρ 3 dφ dτ = = ɛ a ( dφ dτ = dω dτ { ν ρ 5 { ν ρ 5 { ν ρ 5 + cos φ cos φ Ω ( [ [ [ + cos φ cos φ Ω =sinφ sin φ cos φ 3(sin φ +ñ)sinφ ( + x)s(ñ ρ 3(sin φ +ñ)cosφ sin φ + ys(ñ ρ 3(sin φ +ñ)cosφ cos φ zs(ñ ρ ) sin φ cos φ ( ν) ) cos φ + 3 Ω sin φ +(y cos φ + z sin φ) ) ) ] + νρ 5 [ ) ] + νρ 5 [ ] + νρ 5 [ (sin φ +ñ)(cos φ + b) ρ 5 cos φ { 3 ( ν) Ω 3ñ sin φ xs(ñ ρ ) ]} 3ñ cos φ sin φ + ys(ñ ρ ) 3ñ cos φ cos φ zs(ñ ρ ) + 3 (sin φ +ñ) ρ 5 ν Ω + 3 ñ b ρ 5 cos φ ν ñ ρ 5 Ω } ]} ]} (.39) (.4) (.4) (.4) (.43) (.44) where the quantities (ñ, b) and the fast variable φ3 are given by ñ = x sin φ (y sin φ z cos φ)cosφ b = x cos φ +(y sin φ z cos φ)sinφ ( dφ3 dτ =Ω + cos φ ) cos φ sin φ cos φ Ω cos φ 3 ( ν) Ω (sin φ +ñ)(cos φ + b) ρ 5 tan φ 3 ν Ω (.45) (.46) ñ b ρ 5 tan φ (.47) Remember: for a fast rotating tether the non-dimensional variable Ω is a large number, that is, Ω.

47 Chapter : Equations of motion for constant-length rotating tethers 9.7 Hill approach Most of the times the parameter ν is small. The Hill approximation leads to more tractable equations and makes easy to understand the main features of the dynamics. We will introduce this approximation by means of the following change of variables: x = ν /3 ξ, y = ν /3 η, z = ν /3 ζ, ρ = ν /3 ˆρ We introduce this change of variables in equations ( ) and then we take the limit ν. The following set of equations is obtained: { } ξ η =(3 ˆρ 3 )ξ λ ˆρ 5 3Ñ sin φ ξs (Ñ ˆρ ) (.48) { } η + ξ = ηˆρ 3 + λ ˆρ 5 3Ñ cos φ sin φ + ηs (Ñ ˆρ ) (.49) { } ζ = ζ( + ˆρ 3 ) λ ˆρ 5 3Ñ cos φ cos φ ζs (Ñ ˆρ ) (.5) ( dφ dτ =cosφ tan φ + Ω cos φ sin φ + 3 sin φ + 3 Ñ B ) ˆρ 5 (.5) cos φ ( ) dφ dτ = sin φ + Ω sin φ cos φ cos φ + 3 (η cos φ + ζ sin φ )(sin φ + Ñ ˆρ 5 ) (.5) dω =sinφ sin φ cos φ (.53) dτ ( dφ 3 dτ =Ω sin φ tan φ cos φ Ω sin φ cos φ + 3 sin φ + 3 Ñ B ) ˆρ 5 tan φ (.54) where ˆρ = ξ + η + ζ and the quantities (Ñ, B) are given by Ñ = ξ sin φ (η sin φ ζ cos φ )cosφ (.55) B = ξ cos φ +(ηsin φ ζ cos φ )sinφ (.56) In these equations the new parameter that captures the influence of the tether length is λ = ɛ a, and we assume that λ O() ν/3 For a fast rotating tether the parameter Ω is large compared with unity, that is, Ω. In many situations, such a condition fulfil very easily; for example, taking Jupiter and Io as primaries, the period of the synodic frame is.769 days. If the rotating tether gives a turn each 5 minutes the value of Ω is greater than. Taking Jupiter and Metis as primaries, the period of the synodic frame is.95 days. If the rotating tether gives a turn each 8.5 minutes the value of Ω is greater than 5. These angular rates are quite acceptable for rotating tethers. As a consequence, it seems reasonable to take the limit Ω in the above equations. However, it should be underlined that the simplified model provided by this limit (Ω ) loses accuracy

48 3 Section.7: Hill approach quickly when the center of mass of the spacecraft approaches to the small primary. In effect, the validity of the model requires that the distance ˆρ roughly fulfil the condition ( ˆρ Ω ) 4 This way we obtain the following simplified model: { ξ η =(3 ˆρ 3 )ξ λ ˆρ 5 3Ñ sin φ ξs (Ñ ˆρ ) { η + ξ = ηˆρ 3 + λ ˆρ 5 3Ñ cos φ sin φ + ηs (Ñ ˆρ ) ζ = ζ( + ˆρ 3 ) } } { λ ˆρ 5 3Ñ cos φ cos φ ζs (Ñ ˆρ ) } (.57) (.58) (.59) dφ dτ =cosφ tan φ (.6) dφ dτ = sin φ (.6) which only have a free parameter, λ, and where the quantity Ñ is given by (.55) that we reproduce here for convenience: Ñ = ξ sin φ (η sin φ ζ cos φ )cosφ (.6) The equations (.57-.6) should be integrated from the initial conditions: at τ =: ξ = ξ,η= η,ζ= ζ, ξ = ξ, η = η, ζ = ζ, φ = φ, φ = φ (.63) When the initial conditions are φ = φ =the solution for the angles φ and φ is φ (τ) =φ (τ), that is, if initially the tether rotates in a plane parallel to the orbital plane of both primaries, the direction of the angular momentum keeps a constant value..7. Time evolution of φ and φ The equations (.6-.6) are decoupled from the remainder and they provide the time evolution of the direction of the angular momentum of the tethered system. The first equation divided by the second provides the following equation of first order: sin φ dφ = sin φ dφ cos φ cos φ that can be integrated once. The solution is cos φ cos φ =cosβ where β is an integration constant with geometric interpretation (see figure.5): is the angle between the unit vector u 3 and the normal to the orbital plane of both primaries, that is: cos β = k u 3

49 Chapter : Equations of motion for constant-length rotating tethers 3 Let u n a unit vector along the line of nodes (intersection of the plane spanned by the vectors ( u, u ) and the plane Gxy which is parallel to the orbital plane of both primaries). This unit vector u n canbeexpressedintwoways: u n = k u 3 k u 3 = sin β (sin φ cos φ i +sinφ j) u n =cosα i +sinα j cos α =cotβtan φ sin α = sin φ sin β Taking the time derivative in the last equation we obtain: cos α dα dτ = cos φ dφ sin β dτ dα dτ = To obtain this result we used the equation (.6) β and the relation cos α =cotβtan φ. The integration provides the solution G α α(τ) =α τ where α is the initial condition for the angle α. α u n This result has a simple dynamic interpretation: i in this model the unit vector u 3 keeps a constant value in any inertial frame; however, in the synodic frame it must rotate with an Figure.5: Line of nodes angular velocity ω k, since the synodic frame is rotating in the inertial space with the angular rate +ω k. This way we reach the following solution for the angles (φ,φ ): u 3 k j sin φ = cos φ = cos α sin β cos β +cos α sin β, sin φ =sinβ sin α (.64) cos β cos β +cos α sin β, cos φ = cos β +cos α sin β (.65) Obviously the constant values (β,α ) are given by the initial conditions: cos β =cosφ cos φ, β [, π ] (.66) sin α = sin φ sin β, cos α = sin φ cos φ sin β (.67)

50 3 Section.7: Hill approach

51 Chapter 3 Equations of motion for variable-length rotating tethers 3. Basic equations In the Extended Dumbbell Model the system is not a rigid body. Considering both end satellites like point masses, the main novelty of this model is the ability of the system to change the tether length in a way fixed in advance. The angular velocity of the tether and its angular momentum H G turn out to be: Ω = u u + u ( u Ω ), H G = Ī G Ω = I s (t)( u u) where u lies along the tether and u is its time derivative. These expressions are equal to the relations (.3) obtained for the Dumbbell Model but now the inertia moment I s is a function of time (different from a constant). The angular momentum of the tether is normal to it. As a consequence, attached to the tether we can take a reference frame Gu u u 3 in the same way that in the Dumbbell Model: thus, the three unit vectors ( u, u, u 3 ) are given by relations (.4). In this body frame the angular momentum is: H G = I s Ω u 3, where Ω = u u = u and the equation (.) can be written as follows: d Ω dt u 3 + Ω d u 3 dt = I s M G I s I s Ω u 3 (3.) Comparing (3.) with the equation (.5) obtained previously for the Dumbbell Model we can conclude that: from the attitude dynamics point of view, to introduce a variable tether length is equivalent to add the external torque I s Ω u 3 which lies along the direction of the angular momentum. 33

52 34 Section 3.: Basic equations As a consequence, the time evolution of the system is given now by the following system of equations: d u = dt Ω u (3.) d u 3 = u 3 ( u 3 M G ) (3.3) dt Ω I s d Ω = I ( M G u 3 ) s Ω (3.4) dt I s I s Obviously, these equations must be integrated from the appropriated initial conditions. It is not necessary to set any equation for the time evolution of the unit vector u since it can be obtained by the orthogonality condition u = u 3 u Note that the body frame ( u (t), u (t), u 3 (t)) obtained from the integration of equations (3.-3.4) keep its orthonormal character since the whole external torque also fulfil the condition (.9). Taking into account equation (.9) the torque M G can be written in terms of its components in the body frame: M G = M u + M 3 u 3 This way equations (3.-3.4) take the simpler form: d u = dt u (3.5) d u 3 = M u dt Ω I s (3.6) d Ω = M 3 dt I s I s Ω I s (3.7) These equations are valid for librating or rotating tethers. However, when the torque M G vanishes equations ( ) can be easily integrated and the tether is rotating with a variable angular velocity around a fixed direction of the space pointed by u 3 which remains constant. Note that the main effect of the varying tether length is to change the angular velocity of the rotating tether; the direction of the angular momentum, however, does not change. In order to have a librating tether we need, on the right hand side of equations ( ) the gravitational torque. In equations (3.4) and (3.7) the ratio I s /I s appears. For a varying length tether, this quotient is given by expression (B.) that we reproduce here for convenience: I s I s = L d L d { Λ d ( + 3 cos φ) ( 3sin φ Λ d )} Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λ d =in the above expression.

53 Chapter 3: Equations of motion for variable-length rotating tethers Euler angles as generalized coordinates Introducing the Euler angles in sequence 3--3 as in Section.., the equations which provide the time evolution of these angles are: dθ e dt = M cos ϕ e (3.8) Ω I s dψ e dt dϕ e dt d Ω dt = M Ω I s sin ϕ e sin θ e (3.9) = Ω + M Ω I s sin ϕ e cot θ e (3.) = M 3 I s Ω (3.) I s I s Obviously, equations (3.8-3.) should be integrated from the appropriated initial conditions (.7). 3.. Bryant angles as generalized coordinates Introducing the Bryant angles as in section..3, the equations which provide the time evolution of these angles are:: dφ dt dφ dt dφ 3 dt d Ω dt = M Ω I s cos φ 3 cos φ (3.) = M Ω I s sin φ 3 (3.3) = Ω + M Ω I s cos φ 3 tan φ (3.4) = M 3 I s I s Ω (3.5) I s Obviously, equations (3.-3.5) should be integrated from the appropriated initial conditions (.) Non-dimensional form of the equations To set the non-dimensional form of the equations we introduce the non-dimensional time τ = ωt. Considering, for example, equations (3.-3.5); they take the following nondimensional form: dφ dτ = M ω cos φ 3 I s Ω cos φ (3.6) dφ dτ = M ω sin φ 3 I s Ω (3.7) dφ 3 dτ =Ω + M ω cos φ 3 tan φ I s Ω (3.8) dω = M 3 dτ ω di s Ω I s dτ I s (3.9)

54 36 Section 3.: Averaging of the equations where Ω = Ω /ω is the non-dimensional form of Ω. From the point of view of the attitude dynamics, it should be noticed that only the rotation rate Ω of the tether is directly influenced by the variation of the tether length, because the term I s /I s only appears in equation (3.9). However, the other variables are also indirectly influenced through the variation of the rotating rate Ω, because all the equations are clearly coupled. Moreover, for a fast rotating tether (Ω ) the sensitivity of the rotation rate with the variation of the tether length seems to be important because in the equation (3.9) the ratio I s /I s is multiplied by the angular rate Ω which is a large number. However, there is another important difference when comparing with the constant tether length case, namely: for the Extended Dumbbell Model, the components of the Coriolis inertia torque included in the terms (M,M 3 ) on the right hand side of the equations are different. There is a new term in (B.34) that not appears in (B.6). The full set of equations governing the dynamics of an inert varying length tether are collected in Section (3.3). 3. Averaging of the equations The averaging procedure follows exactly the same steps as in Section.4. However, there are differences that we should underline. The first important difference appears in the last term of equation (3.9); thus, we have to solve the integral < di s dτ Ω >= π di s I s π dτ I s Ω dφ 3 However, there are two different time scales in the problem and the value of this integral closely depends on the way in which the tether length is changing. It seems reasonable to assume that the tether length is only a function of the slow time scale (τ = O()), that is, in the fast time scale (τ = O()) the tether length keeps a constant value. Under this assumption the above integral turns out to be < di s dτ Ω >= di s I s dτ I s Ω The second difference is due to the torque produced by the inertia Coriolis force that now, in the Extended Dumbbell Model has an additional term; this additional terms must be averaged following the same strategy that in previous cases. The additional term, in non-dimensional variable, is: and their component in the body frame are: M = I s ω u ( k u ) M = M u = I s ω ( u k ), M 3 = M u 3 = I s ω ( u 3 k )

55 Chapter 3: Equations of motion for variable-length rotating tethers Full set of equations for a varying length rotating tether ẍ ẏ ( ν)( +x ρ 3 ÿ +ẋ + y ( ν) ρ 3 z + z ν ρ ) x( ν ρ 3 )=ɛ a y( ν ρ 3 )=ɛ a ( ν) + z 3 ρ 3 = ɛ a dφ dτ = M ω Is dφ dτ = M ω Is dφ3 dτ =Ω + M ω Is dω dτ = M 3 ω Is { ( ν) ρ 5 { ( ν) ρ 5 { ( ν) ρ 5 Ω Ω di s dτ [3( ρ u)( i u) ( + x)s(cos α)] + ν ρ 5 [3( ρ u)( i u) xs(cos α)] [3( ρ u)( j u) ys(cos α)] + ν } ρ 5 [3( ρ u)( j u) ys(cos α)] [3( ρ u)( k u) zs(cos α)] + ν } ρ 5 [3( ρ u)( k u) zs(cos α)] cos φ 3 cos φ sin φ3 Is Ω Ω } (3.) (3.) (3.) (3.3) (3.4) cos φ3 tan φ (3.5) (3.6) Note that for a varying length rotating tether the parameter ɛ = L d l is a function of the time since the tether length L d(t) is changing with time. Moreover, the las term of equation (3.6) involves the ratio: Is Is = Ld Ld { Λ d ( + 3 cos φ) ( 3sin φ Λ d)} Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λd = in the above expression.

56 38 Section 3.4: Averaged equations for a varying length rotating tether 3.4 Averaged equations for a varying length rotating tether ẍ ẏ ( ν)( +x ρ 3 ÿ +ẋ + y ( ν) ρ 3 z + z ν ρ ) x( ν ρ 3 )= ɛ a y( ν ρ 3 )=+ ɛ a 3 + z ( ν) ρ 3 = ɛ a { ν ρ 5 { ν ρ 5 { ν ρ 5 [ [ [ 3(sin φ +ñ)sinφ ( + x)s(ñ ρ 3(sin φ +ñ)cosφ sin φ + ys(ñ ρ 3(sin φ +ñ)cosφ cos φ zs(ñ ρ ) ) ] + νρ 5 [ ) ] + νρ 5 [ ] + νρ 5 [ 3ñ sin φ xs(ñ ρ ) ]} 3ñ cos φ sin φ + ys(ñ ρ ) 3ñ cos φ cos φ zs(ñ ρ ) ]} ]} (3.7) (3.8) (3.9) ( dφ dτ = dφ dτ = dω dτ + cos φ cos φ Ω ( + cos φ cos φ Ω ) sin φ cos φ ) =sinφ sin φ cos φ Ω cos φ + 3 ( ν) Ω sin φ +(y cos φ + z sin φ) Is Is Is Is cos φ cos φ (sin φ +ñ)(cos φ + b) ρ 5 cos φ { 3 ( ν) Ω + 3 (sin φ +ñ) ρ 5 ν Ω + 3 ñ b ρ 5 cos φ ν Ω ñ ρ 5 } + + Is Is Is Is Ω Ω sin φ cos φ (3.3) sin φ cos φ (3.3) (3.3) where the quantities (ñ, b), thequotient Is/Is and the fast variable φ3 are given by ñ = x sin φ (y sin φ z cos φ)cosφ b = x cos φ +(y sin φ z cos φ)sinφ ( + cos φ ) cos φ sin φ cos φ Ω cos φ ( ) dφ3 dτ =Ω Is Is = Ld Ld Λ d ( + 3 cos φ) ( 3sin φ Λ d) 3 ( ν) Ω (sin φ +ñ)(cos φ + b) ρ 5 tan φ 3 ν Ω ñ b ρ 5 tan φ Is Is Ω (3.33) (3.34) sin φ tan φ (3.35) (3.36) Remember: for a fast rotating tether the non-dimensional variable Ω is a large number, that is, Ω. Moreover, for a varying length rotating tether the parameter ɛ = L d l is a function of the time since the tether length L d(t) is changing with time. Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λd =in equation (3.3).

57 Chapter 3: Equations of motion for variable-length rotating tethers 39 The averaged value of equations ( ), for this new torque are: < M ω cos φ 3 > I s sin φ I s Ω cos φ I s Ω cos φ < M ω sin φ 3 > I s sin φ cos φ I s Ω I s Ω < M ω cos φ 3 tan φ > I s sin φ tan φ I s Ω I s Ω < M 3 I s ω > cos φ cos φ I s I s and the averaged equations governing the dynamics of an inert varying length tether are collected in Section 3.4. Comparing the equations ( ) for the varying length case with the equations ( ) for the constant length case we observe that the differences are due to the terms associated with the quotient I s /I s. The third important difference is found in the parameter λ defined by λ = ɛ a ν /3, ɛ = L d l which is constant in the constant tether length case and now it is a function of the time, because ɛ is time depending in the varying tether length case. Let us consider the equation (3.3) that takes the form: dω Ω dτ = Ω [ sin φ sin φ cos φ I s I s cos φ cos φ ] I s di s dτ For a fast rotating tether Ω and this equation adopts the simplified form dω Ω dτ I s di s dτ that can be trivially integrated. This way we obtain: ( ) ( Is Ld Ω (τ) Ω =Ω I s (τ) L d (τ) ) (3.37) This result has a simple dynamic interpretation: the modulus of the angular momentum keeps a constant value, in a first approximation. Thus, the rotation rate of the tether is directly governed by the increase, or decrease, of the tether length. Therefore, the variation of the tether length should not be too large, because otherwise the tether stops being a fast rotating tether. A first sight, it is reasonable to restrict the variation of the tether length requiring that the ratio I s /I s (τ) O() takes values of order unity. However, the equation (3.37) is indicating that the quotient I s /I s is really small for a fast rotating tether. Thus, on the right hand side of equations ( ) the terms associated with the quotient I s /I s are really small, except the one multiplied by Ω in equation (3.3).

58 4 Section 3.5: Hill approach 3.5 Hill approach Taking into account the above comments on the smallness of the quotient I s /I s and neglect all these terms, the Hill approximation provides exactly the same equations ( ) that we obtained for the constant tether length case except the equation (.53); a new equation, namely dω Is =sinφ sin φ cos φ Ω (3.38) dτ I s substitutes for the equation (.53). Remember that now the parameter λ is a function of the time, because ɛ is time depending in the varying tether length case. For a fast rotating tether the evolution of the center of mass and tether attitude is the same as in the case of constant tether length; the differences between both cases are: ) now λ = λ(τ) and ) the time evolution of Ω (τ) is different in both cases. As a consequence, providing that the rotation rate of the tether keeps a high value, the governing equations are: { } λ ˆρ 5 3Ñ sin φ ξs (Ñ ˆρ ) ξ η =(3 ˆρ 3 )ξ { η + ξ = ηˆρ 3 + λ ˆρ 5 3Ñ cos φ sin φ + ηs (Ñ ˆρ ) ζ = ζ( + ˆρ 3 ) } { λ ˆρ 5 3Ñ cos φ cos φ ζs (Ñ ˆρ ) } (3.39) (3.4) (3.4) dφ dτ =cosφ tan φ (3.4) dφ dτ = sin φ (3.43) which only have a free parameter, λ, and where the quantity Ñ is given by: Ñ = ξ sin φ (η sin φ ζ cos φ )cosφ (3.44) The equations ( ) should be integrated from the initial conditions: at τ =: ξ = ξ,η= η,ζ= ζ, ξ = ξ, η = η, ζ = ζ, φ = φ, φ = φ (3.45) When the initial conditions are φ = φ =the solution for the angles φ and φ is φ (τ) =φ (τ), that is, if initially the tether rotates in a plane parallel to the orbital plane of both primaries, the direction of the angular momentum keeps a constant value. In these equations the time evolution of angles (φ,φ ) is decoupled from the other variables and it can be integrated separately. The solution for them is summarized in equations ( ).

59 Chapter 4 Solar System Fact Sheets In previous Sections we highlighted a new parameter λ λ = ɛ a ν /3 which captures the influence of the tether length in the motion of the center of mass for rotating and non-rotating tethers. In an inert tether the coupling between the attitude dynamics and the motion of the system center of mass G takes place through this parameter; for an electrodynamic tether, such a coupling is also influenced by the electrodynamic forces. It is important to detect the solar bodies for which values of λ of order unity lead to reasonable tether lengths. We remember here the definition of the parameter ν, the reduced mass of the small primary: m P ν = m P + m P whichisinvolvedintheparameterλ. We introduce the following values: n = l ν /3 (λ = L d a l n), L λ= = ν /3. =, L λ=. = a na na and we remember the definition of the variables and the value that we are selecting for a : L d : tether length l: distance between primaries a : for a massless tether, and with m = m, Λ d =and φ = π 4 ;thena takes the maximum value: a = 4 L λ= an estimation of the tether length for which λ = L λ=. an estimation of the tether length for which λ =. These values (n, L λ=,l λ=. ) will helps us to detect the bodies of the solar system where a tether would be used, in a reasonable way, to control the dynamics of a spacecraft. For 4

60 4 Section 4.: Solar System Fact Sheets the estimated L λ values, note that lν /3 is the order of magnitud of distances between the collinear Lagrangian points L or L and the small primary m P. Next values have been used in estimation of mass and distances: m3 G = kgs AU = m, for Universal Gravitation Constant for Astronomical Unit Distance

61 Chapter 4: Solar System Fact Sheets Solar System Planets Sun mass: mp = kg Mass mp (kg) l (km) ω (rad/s) ν lc (km) Ld λ=. (km) Ld λ=. (km) L (km) L (km) Sun mp = Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Table 4.: Solar System Planets Dats. Source:

62 44 Section 4.: Jovian World 4. Jovian World Jupiter mass: mp = kg G =6.6759(±.3) m3 kgs Mass mp (kg) l (km) ω (rad/s) ν lc (km) Ld λ=. (km) Ld λ=. (km) L (km) L (km) Jupiter mp = Inner Moonlets Metis Adrastea Amaltea Tebe Galilean Moons Io Europa Ganimede Callisto Table 4.: Main Jovian Satellites Dats. Source:

63 Chapter 4: Solar System Fact Sheets Other Planet s Satellites Mass mp (kg) l (km) ω (rad/s) ν lc (km) Ld λ=. (km) Ld λ=. (km) L (km) L (km) Earth mp = Moon Mars mp = Phobos Deimos Pluto mp = Charon Table 4.3: Some planetary Satellites Source. Source:

64 46 Section 4.3: Other Planet s Satellites Mass mp (kg) l (km) ω (rad/s) ν lc (km) L d λ=. (km) Ld λ=. (km) L (km) L (km) Saturn mp Ring Sepherds = Pan Daphnis Atlas Prometheus Inner Large Moons Mimas Enceladus Tethys Dione Alkyonides Methone Anthe Pallene Outer Large Moons Rhea Titan Hyperion Lapetus Table 4.4: Some planetary Satellites Dats. Fuente:

65 Chapter 4: Solar System Fact Sheets 47 Mass mp (kg) l (km) ω (rad/s) ν lc (km) L d λ=. (km) Ld λ=. (km) L (km) L (km) Uranus mp Irregular Moons = Cordelia Ophelia Bianca Cressida Desdemona Juliet Portia Rosalind Cupid Belinda Perdita Puck Mab Regular Moons Miranda Ariel Umbriel Titania Oberon Table 4.5: Some planetary Satellites Dats. Source:

66 48 Section 4.3: Other Planet s Satellites Mass mp (kg) l (km) ω (rad/s) ν lc (km) L d λ=. (km) Ld λ=. (km) L (km) L (km) Neptune mp Irregular Moons =.78 4 Naiad Thalassa Despina Galatea Larissa Proteus Nereid Regular Moons Triton Table 4.6: Some planetary Satellites Dats. Source:

67 Chapter 4: Solar System Fact Sheets Asteroids Taking as main primary the Sun with mass: mp = kg Mass mp (kg) l (km) ω (rad/s) e ν lc (km) L d λ=. (km) Ld λ=. (km) L (km) L (km) Sun Ceres Pallas Juno Vesta Eugenia Siwa Ida Mathilde Eros Gaspra Icarus Geographos Apollo Chiron Toutatis Itokawa Table 4.7: Some Asteroids Dats. Source:

68 5 Section 4.5: Values of λ in different binary systems 4.5 Values of λ in different binary systems. e-6 e-8 λ e- e- e-4 Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune e Tether length (km) Values of λ for Sun and different planets λ.. e-6 e-8 e- e- Mars-Phobos Mars-Deimos Saturn-Mimas Saturn-Enceladus Saturn-Rhea Saturn-Titan Earth-Moon Neptun-Triton e Tether length (km) Values of λ for different binary systems in the Solar System

69 Chapter 4: Solar System Fact Sheets 5 λ.. e-6 e-8 e- e- Metis Adrastea Amalthea Thebe Io Europa Ganymede Callisto e Values of λ for different binary systems in the Jovian world

70 5 Section 4.5: Values of λ in different binary systems

71 Chapter 5 Jovian Models 5. Introduction to Jupiter environment An important objective of this study is related with the power generation at Jupiter using bare electrodynamic tethers. For this reason, in this chapter we focus on the models that can be used to predict the performances of the bare tethers designed to be used as power supply of a given system. There are two essential elements in the operation of an electrodynamic tether: magnetic field and electronic plasma density. We shall describe here the models that can be used to take into account this two elements in the neighborhood of Jupiter. Another important issue is the harvest of the electrons that surround the conductive tether. The interaction between the tether and the surrounding plasma is a complex matter and the collection of electrons plays a central role in this interaction. As we will show later, if the surrounding plasma and the tether fulfil some conditions it is possible to elaborate simpler models in order to take into account such an interaction in a suitable way. Finally, in any system of two primaries any non-rotating frame with origin at their center of mass can be considered, in a first approximation, as an inertial frame. However, the appropriate selection of the inertial frame makes easy some calculations involved in the theory which is underneath our simulations. We devote some lines to describe more precisely the selection of the inertial frame that we will use in our analysis. We start this Chapter clarifying this point. 5. Reference frames We take two frames linked with Jupiter (see figure 5.): i) Jx 3 y 3 z 3 is the Jupiter System III (965) reference frame and ii) Jx y z is a reference frame with origin at Jupiter and axis parallels to the inertial frame. The Jupiter System III frame was first defined by radio astronomers and it is rotating with the angular rate of the planet s magnetosphere. This rotation time of 9h 55.5m probably reflects the rate at which the solid core of Jupiter rotates, far below the cloud layers 53

72 54 Section 5.3: Jovian magnetic field When Jupiter is the main primary, the small primary will be a given Jovian moon (for example, Metis or Io). Once this Jovian moon has been determined, we select the axis Jx as follows: is the line defined by the radius vector of the Jovian moon when its center of mass crosses the axis Jx 3. Note that this selection determines z the inertial frame G P x y completely. Figure 5. also shows z 3 β the synodic frame Oxyz with origin at the center of mass of the Jovian u m moon. Moreover, the ori- gin of time, t =, is selected in the moment in which the J moonlet just crosses the axis z Jx 3. This way the angles α (between the axes Jx 3 and Jx )and α α α (between the axes Ox and Jx ), O y described in figure 5., are given by x α y 3 y α = ω J t, α = ωt = τ (5.) where ω J rad/s is the sidereal angular velocity of Jupiter (the angular rate of Jupiter System III). x 3 Figure 5.: Reference frames and axis of the magnetic dipole of Jupiter x 5.3 Jovian magnetic field In the neighborhoods of Jupiter the magnetic field is clearly dipolar and its polarity is just the opposite to the polarity of the geomagnetic field. As a consequence, it can be modeled as R B( 3 ( ) J r) =μ m um r 3 3( u m u r ) u r (5.) where μ m =4.7 4 Teslas is the intensity of the dipole, R J = 749 km is the equatorial radius of Jupiter and u m is a unit vector in the direction of the dipole. Let (α =.7,β =9.6 ) the spherical coordinates of the unit vector u m in the frame Jx 3 y 3 z 3. The coordinates of the unit vector u m in the inertial frame Jx y z are u m = sin β cos(ω J t + α ) i sin β sin(ω J t + α ) j cos β k The relation between the frames Oxyz and Jx y z is given by cos ωt, sin ωt,, [ i, j, k]=[ i, j, k ]Q, Q = sin ωt, cos ωt,,,

73 Chapter 5: Jovian Models 55 Therefore, in the frame Oxyz attached to the moonlet the coordinates of u m are u m = sin β cos[ω r t α ] i +sinβ sin[ω r t α ] j cos β k where ω r = ω ω J is the angular velocity of the frame Oxyz relative to Jx 3 y 3 z 3. In accordance with the Hill approximation, the magnetic field in the moonlet orbit is given by B = B () + ν 3 B () +... where each one of the terms B () and B () are given by B () { = B sinβ cos[ωr t α ] i +sinβsin[ω r t α ] j cos β k } RJ 3, B = μ m l 3 (5.3) B () ( = 3B {sin β(ξ cos[ω r t α ]+η sin[ω r t α ]) ζ cos β} i+ +sinβ{ξsin[ω r t α ] η cos[ω r t α ]} j+ ) + { ξ cos β ζ sin β cos[ω r t α ]} k (5.4) Note that the angle β is small (β =9.6 ); moreover, in the expressions ( ) all the terms associated with sin β areperiodic(theperiodist m =π/ω r ). We should expect that the effect of such terms will be balanced, on a long time scale. As a consequence, it is reasonable to simplify the model considering a non-tilted dipole for the magnetic field, i.e., taking the limit β in the above expressions. This way we obtain: B = B k + ν 3 3B (ζ i + ξ k) However, considering the smallness of the parameter ν for the Jovian moons we will neglect the term B () and we will use the expression B = B k (5.5) for the Jovian magnetic field. In this simplified model, the magnetic field in the orbit of a given Jovian moon is constant and only exhibits a k component, normal to the orbital plane of the primaries. 5.4 Electronic plasma density model We will follow the Divine & Garret model provided in their paper Inner plasmasphere. Inner moonlets According with such a model, in the inner plasmasphere of Jupiter ( < electronic plasma density is given by n = exp { r r ( r ) ( ) } λ λ c m 3 H The larger value of ν is ν and it corresponds to Ganymedes r R J < 3.8) the

74 56 Section 5.4: Electronic plasma density model Figure 5.: Divine & Garret model The parameters involved are r =7.68R J, H =.R J, λ c =.3 cos(l l ) where l and λ are the longitude and latitude, in Jupiter System III (965), respectively, and l = n (cm 3 ) 5 5 Metis Adrastea Amalthea Thebe Longitude l in Jx 3 y 3 z 3 Figure 5.3: Electronic plasma density at the Jupiter moonlet orbits This model provides a quasi-constant value for the electronic plasma density at the orbits of the Jupiter moonlets, all of them lying in the equatorial plane of Jupiter. Figure 5.3 shows the variations of n with the longitude of the moonlet in the frame Jx 3 y 3 z 3, for the four

75 Chapter 5: Jovian Models 57 satellites. The maxima and minima values of n are collected in table 5.. As you can see there, the variation between these extreme values is lower than a 7% in all cases. This is an important fact that permit to predict a small variation of the current collected in a bare tether without control. Metis Adrastea Amalthea Thebe Mass (kg) ν Orbital Radius (m) ω (rad/s) B (Teslas) v o (m/s) v p (m/s) v r = v o v p (m/s) E m = v r B (V/m) n cm 3 (Min.) n cm 3 (Max.) n cm 3 (Averg.) Table 5.: Data about the Jovian moonlet We will use another feature of the model of Divine & Garret: the most habitual ions in the region are: sulfur S + (about 7 %) and oxygen O ++ (about %) Additional details Jupiter is rotating and its sideral period is about 9.95 hours. The Jupiter magnetosphere is corotating with the same angular velocity. There is an orbital radius r es for which the inertial orbital velocity and the velocity of the corotating plasma are equals: r es =( T μ π ) 3 69 km =.38RJ An electrodynamic tether working in the generator regime and orbiting, in a circular trajectory, around Jupiter has a velocity relative to the ambient plasma given by v r = v o v p, where v o is the orbital velocity and v p the velocity of the ambient plasma. If the orbital radius r o is lower than r es the orbital and relative velocities have the same sense and the electrodynamic drag is actually electrodynamic drag. This is the case of Metis and Adrastea because they orbits are inside the equatorial circle of radius r es. If the orbital radius r o is greater than r es the orbital and relative velocities have opposite sense and the electrodynamic drag is actually electrodynamic thrust. This is the case of Amalthea and Thebe. The orbital radius of Amalthea, for example, is r 83 km =

76 58 Section 5.5: Model for the electrodynamic forces.536r J >r es. In the neighborhoods of Amalthea we have } v o j m/s v p v r j m/s j m/s As a consequence, if this electrodynamic thrust must be compensated with the gravitational attraction from Amalthea the tether should be placed in front of the moonlet and not behind of it. 5.5 Model for the electrodynamic forces m The electrodynamic forces are distributed along the tether. In the dumbbell model that we are using here they produce a resultant and a torque the Lorentz torque in the system center of mass G. The resultant acting on the system center of mass is given by L f E = I e (ĥ) dĥ( u B) (5.6) where I e (ĥ) is the tether current profile and B is the Jupiter magnetic field, which will be considered constant along the tether (and equal to its value in G). ThetorqueaboutGin- troduced by the electrodynamic forces is M E = u ( u B)J, where J = L (ĥg ĥ)i e(ĥ)dĥ (5.7) When we use the libration angles (θ, ϕ) to describe the tether attitude in the synodic frame (see fig..3) and considering the magnetic field given by (5.3) the electrodynamic drag is { } f E = I m LB cos β cos ϕ w +sinβ a with ĥ G m G ĥ I e (ĥ) f dĥ e Figure 5.4: Electrodynamic forces on the tether I m = I e (ĥ)dĥ L where I m is the averaged tether current and the vectors w and a are given by L w = sin θ i +cosθ j a = sin ϕ sin[ω r t α ] i +sinϕcos[ω r t α ] j+ { +cosϕ sinθcos[ω r t α ]+cosθsin[ω r t α ]} k and the torque about G introduced by the electrodynamic forces takes the form M E = J B { cos β cos ϕ( u w )+sinβ( u a) } When we use the Bryant angles (φ,φ,φ 3 ) to describe the tether attitude, the electrodynamic drag is given by { f E = I m LB cos β w +sinβ } b

77 Chapter 5: Jovian Models 59 where the vectors w and b are given by w = (cos φ sin φ 3 +sinφ sin φ cos φ 3 ) i +cosφ cos φ 3 j b = (sin φ sin φ 3 cos φ sin φ cos φ 3 )sin(ω r t α ) i+ +(sinφ sin φ 3 cos φ sin φ cos φ 3 )cos(ω r t α ) j+ + {cos φ cos φ 3 sin(ω r t α ) (cos φ sin φ 3 +sinφ sin φ cos φ 3 )cos(ω r t α )} k Equally, the Lorentz torque about G takes the form { M E = J B cos β( u w )+sinβ( u b) } 5.5. Simplified model For the theoretical analysis that follows we will assume an aligned dipole model for the Jupiter magnetic field (β = ): B = B k. This way we have: f E = I m LB cos ϕ w or f E = I m LB w Under these assumptions, the resultant of the electrodynamic forces acting onto the S/C take the following nondimensional form ( k =/(mlω )): k f E = χ cos ϕ w, or k f E = χ w (5.8) where the non-dimensional parameter χ is given by χ = I mb L d mlω ν 3 (5.9) In (5.9), L d is the tether length, B the magnetic field in the center of mass of the S/C and I m is the average tether current. The nondimensional parameter χ provides a measure of the magnitude of the electrodynamic forces. This parameter will be assumed of order unity: χ = O(). Lorentz torque: the torque about G introduced by the electrodynamic forces takes the form M E = J B cos ϕ( u w ) or M E = J B ( u w ) (5.) Obviously, the electrodynamic actions depend on the tether current profile I e (h). In order to obtain the final values of these action we should to determine the current profile.

78 6 Section 5.5: Model for the electrodynamic forces

79 Chapter 6 Bare Electrodynamic Tethers 6. Introduction At present, it seems that the largest tethers current appear with the bare tether concept because is more effective collecting electrons than the classical configuration with a conductive balloon at the anodic end. Indeed, the analysis of [3] for the generator mode, found only a discrete gain by adding an electron-collecting device at the end of a bare tether. If the transversal dimensions of the tether are smaller than both the Debye length and the electron gyroradius, I e (ĥ) the electron collection takes place in the Orbital Motion Limited regime (OML) (see []). To describe the electrodynamic performancesofthebaretethers,wewill I B I C follow closely the analysis of [,, A B C ĥ,, 3, 4] for the generator L B L d L B regime of bare electrodynamic tethers. However, to increase the read- e (ĥ) IZ T V ability of this report we include a V p short summary of the bare tether E m L d performances on such a regime. There are two tether segments: the anodic and the cathodic one. In V t E m L B the anodic segment the tether collects electrons; its length is L B and A B C ĥ it is positively biased relative to the Figure 6.: Tether current and potential profiles surrounding plasma. In the cathodic segment the tether collects ions; its length is L d L B and it is negatively biased relative to the surrounding plasma. 6

80 6 Section 6.: Introduction In the deorbiting regime of the tether, the governing equations for the electron collection are: e ĥ<l B : ĥ>l B : di e dĥ = en p π dφ dĥ = e m e Φ (6.) I e σa t E m (6.) di e dĥ = en p π dφ dĥ = e m e Φ μ (6.3) I e σa t E m (6.4) LB Ld LB Here, ĥ is the distance from the anodic end of the tether; Z T the dependent variables are the tether current profile, I e (ĥ), and the potential drop between tether and plasma, Φ(ĥ) = V e t V p. The other parameters are: σ electrical conductivity, e electron charge, p perimeter of the tether cross-section, A t tether cross-section of the conductive part, m e mass of the electron, m i mass of the ions, Figure 6.: Interposed load μ = m e /m i, E m the component of the induced electric field along the tether and n the ambient plasma density. Note that, the external fields provide the free parameters E m and n which are the most important drivers of the electron collection process. Equations (6.-6.4) should be integrated with the following boundary and initial conditions ( ): at ĥ =: I e = (6.5) at ĥ = L d : I e = I C (6.6) at ĥ = L B :Φ=, I e = I B (6.7) In these equations, L B, I B and I C are unknowns and they should be obtained as a part of the solution. Assuming that I C is known, the boundary value problem (6.-6.7) turns out to be mathematically closed and it provides the profiles I e (ĥ), Φ(ĥ) and the parameters L B and I B. However, since I C is unknown we need one additional relation in order to determine this value: the tether circuit equation which is considered in what follows. The plasma contactor (usually a hollow cathode) at the cathodic end of the tether adjust the tether potential to the plasma potential. Let V cc be the potential drop in the cathodic contactor. In addition, let Z T be the impedance of an interposed load at the cathodic end of the tether; this load plays an essential role because it will permit the basic control of the system (see Fig. 6.). Figure 6. is an sketch of the tether profiles in the ideal case in which V cc =.

81 Chapter 6: Bare Electrodynamic Tethers 63 Let ΔV BC be the potential drop due to ohmics effects along the cathodic segment (BC) of the tether: ΔV BC = Ld I e (ĥ)dĥ σa (6.8) t L B The tether circuit equation, see figure 6., turns out to be: V cc + I C Z T +ΔV BC = E m (L d L B ) (6.9) It is the additional relation that closes mathematically the boundary value problem (6.-6.7). 6. Non-dimensional equations To integrate the equations the following non-dimensional variable will be introduced: ς = ĥ [,l t ], l t = L d, L = (m ee m ) /3 L L e 7/3 (3π σh t ) /3 (6.) n I e = I sc i e (ς), ϕ e =Φ/(E m L ), I sc = σe m A t (6.) where L is a characteristic length typical of the bare tethers, I sc is the short circuit current and h t =A t /p is a characteristic transversal length (for a cylindrical tether coincides with the radius of the wire and for a tape can be approximated by its thickness). In nondimensional variables equations (6.-6.4) take the form (ς ς B ) di e dς = 3 ϕe 4 (6.) (ς ς B ) di e dς = 3 4 μ ϕ e (6.3) dϕ e = i e dς (6.4) where ς B = L B /L. In non-dimensional variables, the boundary conditions take the form The potential drop ΔV BC can be written as ΔV BC E m L = lt ς =: i e = (6.5) ς = ς B : ϕ e =, i e = i B (6.6) ς = l t : i e = i C (6.7) ς B i e (ς)dς = lt ς B ( + dϕ e dς )dς = l t ς B + ϕ C and the tether circuit equation (6.9) takes the following non-dimensional form (Ω i C + Ṽcc)l t + ϕ C = (6.8) where Ṽcc = V cc /(E m L d ) is the non-dimensional form of V cc and Ω=Z T /R T is the nondimensional form of the interposed load Z T (here, R T = L d /(σa t ) is the electrical resistance of the tether).

82 64 Section 6.3: Asymptotic analysis Thus, the determination of the tether current profile requires to solve the boundary value problem given by (6.-6.8). The problem must be tackle numerically to obtain: The current profile: i e (ς) =i e (ς; l t, Ω,μ,Ṽcc) The potential drop profile: ϕ e (ς) =ϕ e (ς; l t, Ω,μ,Ṽcc) The length of the anodic segment: ς B = ς B (l t, Ω,μ,Ṽcc) The maximum tether current: i B = i B (l t, Ω,μ,Ṽcc) The current at the cathodic end: i C = i C (l t, Ω,μ,Ṽcc) The details on the integration of the equations (6.-6.8), which do not pose significant problems, can be found in Ref. []. 6.. Simplifications At present, the more promising cathodic contactor are hollow-cathode devices. For them the potential drop V cc takes values in the range 5-3 V, that is, very small compared with the voltage E m L which can be expected in a tether several kilometers long. The value of the parameter μ = m e /m i is small and it depends on the presence of the ions of the different species. For example, in the neighborhoods of the Jovian moonlets, the most abundant ions found in the Jupiter magnetosphere are the atomic oxygen O + (about %) and the atomic sulfur S + (about 7 %); for them the parameter μ takes the values μ /7, /4, respectively. As consequence, we will take μ =/7 and Ṽcc =in the analysis what follows. This way, all the electrodynamic characteristics of the tether become functions of only two non-dimensional parameters: the tether length l t and the interposed load Ω. 6.3 Asymptotic analysis In Jupiter s neighborhood the electronic plasma density n is scarce. As a consequence the current collected in the tether will be small. We can obtain an asymptotic solution neglecting the ohmic losses along the tether. In such a case, eq. (6.4) becomes: dϕ e ϕ e (ς) =ϕ A ς (6.9) dς The zero bias point is given by ς B = ϕ A. By introducing the approximated solution (6.9) in equations (6.-6.3) we obtain: ς<ς B : i e (ς) = { } ϕ 3 A (ϕ A ς) 3 (6.) ς>ς B : i e (ς) = { } ϕ 3 A μ(ς ϕ A ) 3 (6.) The maximum tether current turns out to be i B = ϕ 3 A (6.)

83 Chapter 6: Bare Electrodynamic Tethers 65 and the values at the cathodic end of the tether are: { ϕ 3 A μ(l t ϕ A ) 3 i C = The tether circuit equation takes the form: Ωl t i C + ϕ C = Ωl t }, ϕ C = ϕ A l t (6.3) { ϕ 3 A μ(l t ϕ A ) 3 } + ϕ A l t = and it provides the value of the potential bias ϕ A at the anodic end of the tether, that is: ϕ A = ϕ A (Ω,l t,μ). The solution of this equation can be facilitated considering the small value of the parameter μ. Thuswayweget: ϕ A ϕ A + μ Ωl t (l t ϕ A ) 3 4+3Ωl t ϕ A (6.4) where ϕ A is the only positive root ofthecubicequation x 3 + x = (6.5) Ωl t Ω The roots of this equation depends on the value of the parameter: 4 α = Ω(3l t ) 3 and the positive root, which is obtained in the Appendix A (see page 83), leads to the following solution: ϕ A = l t f(α) (6.6) where the function f(α) is given by f(α) Figure 6.3: Function f(α) vs. α α Exact Asymptotic α<: f(α) = 3 [tan 4 α 3 ( β )+cot 3 ( β ] ) where sin β = α (6.7) α>: f(α) = 3 [ 4 α cos( β 3 )+ 3sin( β ] 3 ) where sin β = α (6.8) For large values of α there is an asymptotic solution which can be obtained from (6.8): f(α) 3 9α = Ω l 3 t, for α Figure shows the aspect of the function f(α) and also this asymptotic solution. Curiously, the asymptotic solution describe very well the function f(α) in a wider range of values. Notice that when the interposed load vanishes (Ω =,α= ), the function f(α) takes the value f(α) =.

84 66 Section 6.3: Asymptotic analysis For small values of α, (i.e., large values of Ω), the expression (6.7) provide the asymptotic expansion: ( α ) ( 3 α ) 4 ( 3 α ) ( α ) 8 ( 3 α ) 3 f(α) Unfortunately, this series has a very slow convergence if α is not very small. Taking into account the expression (6.6) the solution (6.4) becomes: { } ϕ A l t f(α) +μωl 3 ( f(α)) 3 t f(α)(4 + 3Ωl 3 t f (α)) and this expression permit us to asses accurately the small error that we introduce in the analysis when we adopt as solution of the circuit equation the value (6.6): 6.3. Average tether current ϕ A l t f(α) The electrodynamic drag depends on the averaged tether current I m which is given by: I m = Ld lt I e (h)dh = I sc i(ς)dς L d l t Taking into account the asymptotic solution (6.-6.) we obtain for this averaged value: I m = I sc 3 [l tf(α)] 3 (+ 3 ( f(α)) ) Going back to the dimensional variable we obtain: I m = 4 5π n e 3 L 3 d (d w + h) E m G(α), G(α) =[f(α)] 3 (+ 3 ) m ( f(α)) e 6.3. Useful power The useful power that can be obtained from the tether is given by W u = I CZ T = I sc R T Ω i C Taking into account the definition (6.) of the short circuit current this power can be expressed as W u = W u E u (l t, Ω), W u = L σe ma t, E u (l t, Ω) = l t Ωi C(l t, Ω) (6.9) It is more interesting to express the useful energy W u as follows: W u m T = σe m ρ v Ω i C(l t, Ω) (6.3)

85 Chapter 6: Bare Electrodynamic Tethers Ω=.5 Ω=. Ω=.5 Ω=. Ω=.5 Ω= Least square fit Ω i C l t. Ω i C l t Ω Figure 6.4: Function Ω i C (l t, Ω). The left picture shows the dependence on l t for different values of Ω; the right picture shows the dependence on Ω for different values of l t =.5,.,...,5 Here m T is the tether mass; obviously the parameter W u /m T must be maximized in order to obtain the maximum useful energy with the minimum mass; thisis equivalent to maximize the function Ω i C (l t, Ω). This is an involved function of the two variables (l t, Ω) that should be obtained as a part of the solution of the boundary value problem (6.-6.8). Figure 6.4 (left picture) shows the dependence on l t for constant values of Ω; we should underline that Ω i C (l t, Ω) is a monotonic increasing function of l t (actually it reaches a maximum outside of the plot, i.e., for large values of l t which are not interesting). Figure 6.4 (right picture) shows the dependence on Ω for constant values of l t ;itisobviousthat the function reaches a clearly defined maximum for each value of l t. It should be noted that for small values of l t, that curve is very flat andwhenthevalueofω separates from the one corresponding to the maximum the ratio W u /m T remains almost unchanged. This fact makes easy to design the system because the values of l t which appear in the moon orbits are small. However a simplification can be obtained if we realize that the potential drop across the interposed load is given by Ωi C = ϕ C l t = ϕ A l t expression that follows trivially from the tether circuit equation (6.8) and the second of relations (6.3). In such a case, the useful power can be expressed as follows: W u = σe m i C (ϕ A ) ( ϕ A ) (6.3) m T ρ v l t where i C (ϕ A ) is given by the first of relations (6.3). Thus, in this asymptotic analysis and for constant l t, the useful power is a function of only Ω through ϕ A, and its maximum value

86 68 Section 6.3: Asymptotic analysis can be obtained from the equation: d dϕ A ( i C (ϕ A ) ( ϕ ) A ) = l t Taking into account the value of i C (ϕ A ) as given by the first of the relations (6.3), the maximum specific power occurs for the following optimum value of the anodic potential bias: and it turns out to be ( W u ) σe m m T ρ v 5 max ϕ A 3 5 l t{+μ 6 9 } (6.3) ( ) 3 { 3lt ( } 5 3 ) 3 μ (6.33) Whereas the specific power is a function of the two variables (l t, Ω), the maximum value (6.33) is a function of only one variable since on the line of maxima both variables l t and Ω are not independent. In effect, in this asymptotic solution, the line of maxima can be easily obtained if we neglect terms of the order of μ. In such a case we have: [ 5 ϕ A = ϕ A l t f(α) 3 ] 5 l t Ω Figure 6.5: Line of maxima ( W u ) 8 3 m T 5 π max f(α) 3 α where this last value α has been obtained numerically. The line of maxima can be approximated then by [ l 4 t = α 7 Ω ] 3 Figure 6.5 shows the line of maxima given by this expression. Going back to dimensional variables we get for the maximum specific power EmL 3 3 d e3 m e h n ( ( t ρ v 3 ) 3 μ) (6.34) This expression shows clearly that for a given tether mass m T, the maximum power is obtained with a tape (h t h) andwiththe thinner and longer possible configuration. If we only consider configurations which situated on the line of maxima we can write ln(w u )=ln(k )+ln(m T ), where k = EmL 3 3 d e3 m e h n ( ( t ρ v 3 ) 3 μ)

87 Chapter 6: Bare Electrodynamic Tethers 69 If we fix the Jovian moonlet, the length and the thickness h of the tape (h t h) the parameter k above is constant and the logarithm ln(wu ) depends linearly on the logarithm ln(m T ). The expression (6.34) has been used to plot the lower picture of the figure 6.6 which has been elaborated for the orbit of Metis, by taking the following values for the plasma density n = m 3 and the motionless electric field Em =.666 V/m; an Aluminum tape of h =. mm has been considered. In the abscissa-axis figure shows the tether mass m T (in kg); in the ordinate-axis figure shows the useful energy provided by the tether (in kilowatts); in both axes we use a logarithmic scale. On the figure two families of curves have been drawn: the green lines show the variation of the useful power W u with the tether mass m T when the tether length L d is fixed; along these lines the only parameter which changes is the tether width d w. The red lines show the variation of the useful power W u with the tether mass m T when the tether width d w is fixed; along these lines the only parameter which changes is the tether length L d. For the red lines, the slope is larger than for the green lines, this explains why the useful energy is more sensitive to the variations of the tether length L d than to the variations of the tether width d w. Obviously, changing the tether length and width simultaneously is possible to increase W u keeping the tether mass m T constant. In the upper picture of this figure 6.6 the ohmic effects have been considered, that is, instead of using the asymptotic expression (6.34) we obtained the maximum useful power by solving numerically the two boundary value problem given by (6.-6.8). The differences are quite small and this figure confirm that the asymptotic solution obtained by neglecting the ohmic losses in the tether turns out to be an excellent approximation in the case of Metis. L d (km) d w (mm) m T (kg) d w (mm) m T (kg) Table 6.: Some optimized configurations shown in Fig. 6.6 for W u = kw Table 6. shows some configuration providing kw of useful power with the Aluminum tape of h =. mm. The two last columns (in red color) correspond to the asymptotic analysis and the two first column (in blue color) correspond to the exact numerical analysis. The asymptotic solution provides slightly optimistic values with errors lower than 4% for most critical cases, that is, when the tether lengths are the largest and the ohmic effects have an incipient influence.

88 7 Section 6.3: Asymptotic analysis d w = cte L d = cte d w =5mm W u (kw) L d =5km 4 km 3 km km d w =5mm L d =5km km m T (kg) d w = cte L d = cte d w =5mm W u (kw) L d =5km 4 km 3 km km d w =5mm L d =5km (kg) km Figure 6.6: Different optimized configurations in METIS (h =. mm). The lower picture has been obtained using the asymptotic analysis in the limit of negligible ohmic effects. In the upper picture the ohmic effects have been considered. m T

89 Chapter 6: Bare Electrodynamic Tethers Lorentz torque. Balance condition ThetorqueaboutG introduced by the electrodynamic forces is M E = u ( u B)J, where J = Ld Introducing non-dimensional variables, the integral J takes the form (ĥg ĥ)i e(ĥ)dĥ (6.35) J σe m A t L = lt (l t cos φ ς) i e (ς) dς = l t cos φu (l t, Ω) U (l t, Ω) (6.36) where U (l t, Ω) and U (l t, Ω) are defined by the following integrals U (l t, Ω) = lt i e (ς)dς, U (l t, Ω) = lt ςi e (ς)dς (6.37) For a self-balanced tether the Lorentz torque vanishes (J =). The mass angle φ that leads to a balanced tether is given by: cos φ U (l t, Ω) = (6.38) l t U (l t, Ω) In this expression we are assuming, implicitly, that the mass m is at the anodic end of the tether (m at the cathodic end): we are calculating the distance ĥg = L d cos φ from the anodic end to the center of mass G. Taking into account the asymptotic solution (6.-6.) and the approximated expression (6.6) for ϕ A we obtain the following values for U and U : { U = l 5 3 t f 3 (α)[ + 3 ( f(α))] μ } 5 ( f(α)) 5 (6.39) U = l 7 t { 7 4 f 3 (α)[ ( f (α)] μ 7 ( + 5 f(α)) ( f(α)) 5 Neglecting the terms of order μ we obtain for the mass angle φ cos φ ( f (α)) 4 ( f(α)) + 3 [ ] ( f(α)) 5 +3μ f 3 (α)( 5 f(α))(35 8f (α)) When we work on the line of maxima (f(α) 3 5 ) the critical mass angle takes the value cos φ max μ φ max 39 } (6.4) Thus, a self-balanced electrodynamic tether working in the neighborhood of the maximum specific power has a fixed configuration in which the upper segment is about the 6% of the total length (L = L d cos φ.6l d ). It is the upper segment relative to Jupiter outside from the stationary sphere, but it is the lower segment inside such a sphere

90 7 Section 6.4: Reformulation of the asymptotic solution V V t V p χc EmLd EmLd A B C ĥ Figure 6.7: Potential profiles for the tether and the plasma when the ohmic effects are negligible. Definition of the parameter χ c 6.4 Reformulation of the asymptotic solution It is possible to reformulate the problem in a slightly different form which seems more appropriate from the control point of view. The idea is to obtain a new parametric representation of the solution in terms of a new parameter χ c. We have to obtain the following unknowns The current profile: i e (ς) =i e (ς; l t,χ c ) The potential drop profile: ϕ e (ς) =ϕ e (ς; l t,χ c ) The length of the anodic segment: ς B = ς B (l t,χ c ) The maximum tether current: i B = i B (l t,χ c ) The current at the cathodic end: i C = i C (l t,χ c ) The interposed load: Ω=Ω(l t,χ c ) The new formulation attack directly the problem starting from two basic assumptions: The ohmic effects along the tether are negligible Losses at both plasma contactors are negligible The parameter χ c is the potential drop along the interposed load Z T expressed as a fraction of the potential drop E m L d that takes place in the surrounding plasma between both tethers ends (see Fig. 6.7). With the notation that we are using here, the parameter χ c coincides with the product Ωi C, which is, in non-dimensional variables, the potential drop across the interposed load: χ c =Ωi C.

91 Chapter 6: Bare Electrodynamic Tethers 73 It is not difficult to obtain the following solution: The length of the anodic segment: ς B = l t ( χ c ) } The current profile (anodic seg.): i e (ς) = l 3 t {( χ c ) 3 ( χc ς l t ) 3 } (cathodic seg.): i e (ς) = l 3 t {( χ c ) 3 μ ( ς l t +χ c ) 3 The maximum tether current: i B = l 3 t ( χ c ) 3 The current at the cathodic end: i C = { l 3 t ( χ c ) 3 3 } μχ The potential profile: ϕ e (ς) =l t ( χ c ) ς The potential bias (anodic end): ϕ A = l t ( χ c ) The potential bias (cathodic end): ϕ C = χ c l t The interposed load: Ω= χ c i C c which is valid for all values of χ c in the interval [, ] except in the neighborhood of the upper end, when χ c = O(μ) Average tether current The electrodynamic drag depends on the averaged tether current I m which is given by: I m = Ld I e (ĥ)dĥ L = I sc U (l t,χ c ) d l t where the function U (l t,χ c ) is given by G(χ c ) μ = μ =/ U (l t,χ c )= lt i(ς)dς With the new solution the value of U (l t,χ c ) can be calculated and it turns out to be: U = 3 l 5 t {( + 3 χ c) ( χ c ) 3 μ5 } χ 5c χ c Figure 6.8: The tether current gain G(χ c ) due to the interposed load vs. χ c (for μ =and μ =/). Differences are imperceptible This way we obtain the following value for the averaged current I m : { 3 I m = I sc l 3 t ( + } 3 χ μ c) ( χ c ) 3 5 χ 5 c Note that the correction due to the parameter μ is lower than a.% (μ /, χ c < ); as a consequence, we can neglect such a correction.

92 74 Section 6.4: Reformulation of the asymptotic solution Going back to the dimensional variable we obtain: I m = 4 5π n e 3 L 3 d (d w + h) E m G(χ c ), G(χ c )=(+ m e 3 χ c) ( χ c ) 3 (6.4) The function G(χ c ) is shown in figure 6.8. Note that G() = and therefore the function G(χ c ) can be considered as the gain obtained in the tether current due to the existence of the interposed load Useful power The useful power that can be obtained from the tether is given by W u = I CZ T = I sc R T Ω i C Taking into account the definition (6.) of the short circuit current this power can be expressed as W u = W u E u (l t,χ c ), W u = L σe ma t, E u (l t,χ c )=l t χ c i C (l t,χ c ) (6.4) It is more interesting to express the useful energy W u as follows: W u m T = σe m χ c i C (l t,χ c )= ρ v l 3 t σe m ρ v F (χ c ) (6.43)..859 μ =. μ =/ where F (χ c )=χ c { ( χ c ) 3 μχ 3 c }.5 F (χ c) χ c Thus, in this asymptotic analysis and for constant l t, the specific useful power is a function of only Ω through χ c, and its maximum value can be obtained from the equation: d dχ c {F (χ c )} = and it turns out to be Figure 6.9: Function F (χ c ) vs χ c for μ =. and μ =/. Differences are imperceptible F (χ c ) turns out to be: F (χ cmax ) χ cmax 5 ( μ 3 ) The corresponding value of the function ( ( ) 3 ) 3 μ.859

93 Chapter 6: Bare Electrodynamic Tethers 75 Going back to dimensional variables we get for the the maximum specific power ( W u ) 8 3 m T 5 π max Em 3 L3 d e3 m e h n ( ( t ρ v 3 ) 3 μ) expression that coincides with the previous one given in (6.34). In terms of the interposed load Ω the line of maxima is given by the relation Ω max = χ c max i C (χ cmax ) l t = [ 4 5 9Ω ( μ) which exhibits a very good agreement with the relation previously calculated and summarizes in figure 6.5. ] Lorentz torque. Balance condition For the Lorentz torque we should calculate the following integral U (l t,χ c )= lt ςi(ς)dς Taking into account the current profile obtained in this solution this integral adopt the form: { 7 7 U = lt 4 ( ( χ c) χ c ) ( χ c ) 3 μ 5 χ 5 c ( } 7 χ c) Similarly, the balance condition adopt the form: cos φ = 8 35 ( χ c) 5 ( χ c) μ χc ( χ c ) 3 ( 5 ( χ c)) ( 8 35 ( χ c) ) Considering the optimum value for χ cmax 5 ( μ 3 ) the mass angle that balance the tether turns out to be: cos φ = μ = μ Rotating tethers. Averaging In this section we describe the way in which the expressions obtained above for the electrodynamic performances of a bare tether can be used for a fast rotating tether. In such a case, we have to perform the average of the equations and the procedure must be described in detail.

94 76 Section 6.5: Rotating tethers. Averaging v u 3 v 3 u E φ 3 σ = φ 3 γ π E π Tether rotating plane E π γ π v Figure 6.: Motionless electric field 6.5. Electrodynamic drag The electrodynamic drag is given by: f E = Ld I e (ĥ) dĥ = I ml d ( u B ) (6.44) where I m is the spatial averaged value of the tether current which is given by (6.4). When using this expression for the electrodynamic drag, there is an implicit agreement which must be underlined to avoid misundertstanding: the unit vector u along the tether has equal sense than the electric current; therefore it is directed from the cathodic end to the anodic end of the tether. Assuming that we are using the Bryant angles (φ,φ,φ 3 ) to describe the attitude of the tether, the unit vector u along the tether can be expressed as u =cosφ 3 v +sinφ 3 v where φ 3 is a fast variable for a fast rotating tether. Thus, taking into account (6.4) the electrodynamic drag can be expressed as follows: f E =ΥL d G(χ c ) E m (cos φ 3 v B +sinφ 3 v B ) where the quantity Υ, which does not depend on φ 3,isgivenby Υ= 4 5π n e 3 L 3 d (d w + h) m e Let E be the motionless electric field induced by the magnetic field B. It is given by E = v rp B where v rp is the velocity of the spacecraft relative to the surrounding plasma. The parameter which determines the tether current is the component along the tether of the electric field,

95 Chapter 6: Bare Electrodynamic Tethers 77 E m = E u = u ( v rp B) =[ u, v rp, B]. To obtain this value, we project the electric field E on the tether rotating plane (see figure 6.); such a projection turns out to be E π = v 3 ( v 3 E) =A v + A v where the coefficients (A,A ), cartesian coordinates of E π,aregivenby: A = E v =[ v rp, B, v ], A = E v =[ v rp, B, v ] This way, the component E m can be expressed as: E m = E u = E π u =cosφ 3 A +sinφ 3 A Instead of (A,A ), it is better to introduce the polar coordinates (E π,γ π ) defined by: E π = A + A, cos γ π = A E π, sin γ π = A E π They permit to express the electric field component E m in the following form: E m = E π (cos φ 3 cos γ π +sinφ 3 sin γ π )=E π cos(φ 3 γ π )=E π cos σ (6.45) Note that (E π,γ π ) do not depend on the fast angle φ 3. The electrodynamic drag takes the form: f E =ΥL d Eπ G(χ c ) cos(φ 3 γ π ) [cos φ 3 ( v B)+sin φ 3 ( v B)] where the dependence with φ 3 has been highlighted. In order to average this expression for the fast variable φ 3 some comments should be taken into account. For an electrodynamic tether working in the generator regime the parameter E m must be positive (negative values of E m produce no tether current); to fulfil this requirement the angle σ = φ 3 γ π hastobelongtotheinterval: σ [ π, π ] (see equation (6.45)). There are two possible different configurations:. in one end of the tether there is one (only one) cathodic plasma contactor and in the other tether end there is one (only one) anodic plasma contactor (the bare tether);. in this other configuration the tether has two plasma contactors (cathodic and anodic) at both ends of the tether. Configuration.: let us consider one turn of the tether around the unit vector u 3. In such a turn the angle σ describe the interval [ π, π ] only once and we have two possibilities:. the control parameter χ c keeps a constant value. In this case the averaged value of the electrodynamic drag is < f ( ) E >=ΥL d Eπ G(χ c ) F ( v B)+F ( v B)

96 78 Section 6.5: Rotating tethers. Averaging where the factors (F, F ) are given by: F = γπ+ 3π π γ π π F = γπ+ 3π π γ π π π cos σ cos(σ + γπ)dσ = 3π K( )cosγπ cos(φ3 γ π)cosφ 3 dφ 3 = π π cos(φ3 γ π)sinφ 3 dφ 3 = π cos σ sin(σ + γπ)dσ = π π 3π K( where K(x) is the complete elliptic integrals of the first kind defined by K(x) = )sinγπ ( u )( x u ) du, K( ) As a consequence, the averaged electrodynamic drag turns out to by: G(χ c ) < f E >=ΥL d Eπ 3π K( )( E π B ) Taking into account the value of Υ we get: < f E >= 8 5π K( ) n e (d w + h) 3 L 5 m e E d G(χ c)( E π B ) (6.46) π. the control parameter χ c is a function of φ 3. In this case, the electrodynamic drag depends essentially from the following integrals F = γπ+ 3π π F = π γ π π γπ+ 3π γ π π G(χ c ) cos(φ 3 γ π )cosφ 3 dφ 3 = π G(χ c ) cos(φ 3 γ π )sinφ 3 dφ 3 = π π π π and it takes the value < f ) E >=ΥL d Eπ (F ( v B)+F ( v B) π G(χ c ) cos σ cos(σ + γ π )dσ G(χ c ) cos σ sin(σ + γ π )dσ Obviously, the values of the factors (F, F ) depend on the way in which the control parameter χ c evolves in the spin of the tether. Configuration.: let us consider again one turn of the tether around the unit vector u 3. If the tether has duplicated plasma contactors at both ends, in such a turn the angle σ describe the interval [ π, π ] twice: if we start with σ = π when we reach the value σ = π both tether ends swap their roles, the unit vectors ( u, u ) undergo a discontinuity changing suddenly their senses and the angle σ pass abruptly from π to π. Thus, the angle σ describes the interval [ π, π ] again. As a consequence, the averaged electrodynamic drag reaches, in this case, twice as much as the value obtained in the Configuration.

97 Chapter 6: Bare Electrodynamic Tethers Lorentz torque The torque about G introduced by the electrodynamic forces is M E = u ( u B) J, where J = Ld The integral J can be expressed as follows: J = I sc L ( lt cos ) φu U (ĥg ĥ)i e(ĥ)dĥ (6.47) The functions (U,U ) take the values U = l 5 t H (χ c ), H (χ c )= 3 { ( + } 3 χ μ c) ( χ c ) 3 5 χ 5 c U = l 7 t H (χ c ), H (χ c )= 7 4 ( ( χ c) χ c ) ( χ c ) 3 μ 5 χ 5 c ( 7 χ c) This way we obtain for J the following value: J = 8 3π (d w + h)n L 7 e 3 ( d Em cos φh (χ c ) H (χ c ) ) m e (6.48) On the other hand, the vector u ( u B) can be expressed as: u ( u B) = u ( u B) B = (6.49) [ ] cos φ 3 v ( v B)+sinφ 3 cos φ 3 [ v ( v B)+ v ( v B)] + sin φ 3 v ( v B) B We should average the Lorentz torque M E given by (6.47) for the fast variable φ 3 taking into account the values ( ) and the expression E m = E π cos(φ 3 γ π ) for the electric field. We consider again the two possible configuration described for the averaging of the electrodynamic drag: Configuration.: let us consider one turn of the tether around the unit vector u 3. In such a turn the angle σ describe the interval [ π, π ] only once and we have two possibilities:. the control parameter χ c keeps a constant value. Inthiscasewehavetosolve the following integrals: M = γπ + 3π π γ π π M = γπ + 3π π γ π π M 3 = γπ + 3π π γ π π M 4 = γπ + 3π π γ π π cos(φ3 γ π)cos φ 3 dφ 3 = π cos σ cos (σ + γ π)dσ π π cos(φ3 γ π)cosφ 3 sin φ 3 dφ 3 = π π π π cos σ sin(σ + γπ)cos(σ + γ π)dσ cos(φ3 γ π)sin φ 3 dφ 3 = cos σ sin (σ + γ π)dσ π π cos(φ3 γ π) dφ 3 = π cos σdσ π π

98 8 Section 6.5: Rotating tethers. Averaging which take the following values M = Γ π ( + cos γ π ) M = Γ π sin γ π cos γ π M 3 = Γ π ( + sin γ π ) M 4 = Γ π 5 where the constant Γ takes the value ( ) Γ= E( 5 ) K( ), Γ (6.5) and E(x) is the complete elliptic integrals of the second kind defined by x E(x) = u du, E( u ) This way we obtain for the averaged Lorentz torque: < M E > = 8 3π (d w + h)n L 7 e 3 ( d Eπ cos φh (χ c ) H (χ c ) ) m e Γ { } u π ( u π B) v 3 ( v 3 B) 4 B π where the unit vector u π, which lies on the tether rotating plane, is given by: u π = E π =cosγ π v +sinγ π v E π In this case, since the value of χ c is constant, if we select the mass angle φ as follows: cos φ = H (χ c ) H (χ c ) the averaged Lorentz torque is zero and the rotating tether is self-balanced.. the control parameter χ c is a function of φ 3. Inthiscasewehavetosolvethe following integrals: M = π π M = π M 3 = π M 4 = π π π π π π π π ( cos φh (χ c ) H (χ c ) ) cos σ cos (σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σ sin(σ + γ π )cos(σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σ sin (σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σdσ which depend on the way in which the control parameter χ c evolves in the spin of the tether.

99 Chapter 6: Bare Electrodynamic Tethers 8 Configuration.: let us consider again one turn of the tether around the unit vector u 3. If the tether has duplicated plasma contactors at both ends, in such a turn the angle σ describe the interval [ π, π ] twice: if we start with σ = π when we reach the value σ = π both tether ends swap their roles, the unit vectors ( u, u ) undergo a discontinuity changing suddenly their senses and the angle σ pass abruptly from π to π. Thus, the angle σ describes the interval [ π, π ] again. As a consequence, the averaged Lorentz torque reaches, in this case, twice as much as the value obtained in the Configuration Using the Bryant angles for the attitude When using the Bryant angles to describe the attitude of the rotating tethers, the attitude motion is governed by the equations (.3-.6). In this case, instead of average the Lorentz torque directly is better to average the corresponding components which appear on the right hand side of equations (.3-.6). For example, let us consider the averaging of equation (.6); taking into account equation (6.47) the component M 3 of the Lorentz torque turns out to be: M 3 = M E u 3 = J ( B v 3 ) Thus we obtain: < M 3 ω >= <J > ( B v 3 ) I s Similarly, the component M of the Lorentz torque turns out to be: M = M E u = J ( B u )= J [ sin φ 3 ( B v )+cosφ 3 ( B v )] In order to obtain the averaged versions of equations (.3-.5) we have to obtain the following averaged values: which lead to the averaging of the quantities: <M cos φ 3 >, < M sin φ 3 > <J cos φ 3 >, < J sin φ 3 >, < J sin φ 3 cos φ 3 > All the averaged values can be expressed in terms of the integrals M i,i=...4 previously calculated. In what follows, we will describe the equations (.3-.6), assuming that on the right hand sides only the Lorentz torque is involved. Obviously, the right hand sides obtained in this way must be added to the right hand sides of equations: ( ) for a constant length rotating tether (see Section.6) ( ) for a varying length rotating tether (see Section 3.4)

100 8 Section 6.6: Rotating tethers. Averaging Configuration.: let us consider one turn of the tether around the unit vector u 3. In such a turn the angle σ describe the interval [ π, π ] only once and we have two possibilities:. the control parameter χ c keeps a constant value. The averaged attitude equations turn out to be: dφ dτ =ˆɛ b { v +cosγ π ( v 3 u π )} Ω cos φ (6.5) dφ dτ =ˆɛ b { v +sinγ π ( v 3 u π )} Ω (6.5) dφ 3 dτ =Ω ˆɛ tan φ b { v +cosγ π ( v 3 u π )} Ω (6.53) dω = ˆɛ 5( dτ b v 3 ) (6.54) In these equations the parameter ˆɛ is defined as: ˆɛ = 8 3π n (d w + h)l 3 d ΓB mω a e 3 E π m e { cos φh (χ c ) H (χ c ) } (6.55) where Γ is defined in (6.5) and B and the unit vector b are defined by: B = B, b = B B (6.56) The condition cos φh (χ c ) H (χ c )=holds for a self-balanced electrodynamic tether; as a consequence the parameter ˆɛ vanishes and the Lorentz torque does not influence the attitude dynamics.. the control parameter χ c is a function of φ 3. Inthiscasewehavetosolvethe following integrals: M = π π M = π M 3 = π M 4 = π π π π π π π π ( cos φh (χ c ) H (χ c ) ) cos σ cos (σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σ sin(σ + γ π )cos(σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σ sin (σ + γ π )dσ ( cos φh (χ c ) H (χ c ) ) cos σdσ which depend on the way in which the control parameter χ c evolves in the spin of the tether.

101 Chapter 6: Bare Electrodynamic Tethers Appendix A. Solution of the cubic equation (6.5) After the change x = y, the cubic equation (6.5) becomes: l t Ω y 3 + y l3 t Ω 4 = (6.57) and after the change y = z /3 the equation adopt the normal form (without quadratic term) z 3 3 z Ω l 3 t = z3 + pz + q = p = 3, q = 7 4 Ω l 3 t We form the auxiliary quadratic equation S + qs p3 7 = Let (u, v) be the roots of this auxiliary equation. The solution of the equation (6.57) can be expressed as: y = u 3 + v 3 3 If α< then the roots (u, v) turn out to be u = 7 tan ( β ), v = 7 cot ( β ), where sin(β) =α This way we arrive to the solution (6.7). If α> then the roots (u, v) turn out to be u = 7 (cos β i sin β), v = 7 (cos β + i sin β), where sin(β) = α and i stands for the imaginary unit. As a consequence u 3 = {cos( π3 3 β3 )+isin(π3 β3 } ), v 3 = {cos( π3 3 β3 ) i sin(π3 β3 } ) and the solution for y turns out to be y = {cos( β3 3 )+ 3sin( β3 } ) This way we arrive to the solution (6.8).

102 84 Section 6.6: Appendix A. Solution of the cubic equation (6.5)

103 Chapter 7 Halo orbits fundamentals 7. Introduction The two next chapters deal with periodic solutions of the tethered-satellite problem in the vicinity of the collinear libration points. Specifically, an inert tether is considered in fast rotation when compared to the rotation rate of the system, and the applicability of the results is constrained to systems in which the Hill problem assumptions apply. The basics of the Hill problem is recalled in chapter 7, where the well known families of periodic solutions originated from the collinear points are briefly discussed: Lyapunov and eight-shaped orbits, and the Halo orbits bifurcated from the Lyapunov family. A final section lists several references pertinent to the topic. Chapter 8 presents numerical explorations of the tethered-satellite problem. More precisely, we discuss how the known periodic solutions of the Hill problem are modified in the case of tethered satellites. The most favorable configuration is found for tethers rotating parallel to the plane of the primaries, a case in which the attitude of the tethered-satellite remains constant on average. In this case, the effect produced by a non-negligible tether s length is equivalent to introducing a J perturbation on the primary at the origin, or intensifying it if the primary at the origin is an oblate body, and it is shown that either lengthening or shortening the tether may lead to orbit stability. Promising results are found for eight-shaped orbits, but regions of stability are also found for Halo orbits. When the rotation of the tether is not parallel to the plane of the primaries the attitude of the tethered-satellite oscillates (on average) breaking, in general, the periodicity of the orbital motion. Only a selected set of periodic solutions survive, and just a few among them remain as stable orbits when varying the initial attitude. All the stable solutions found for the non-constant attitude motion problem originate from eight-shaped orbits. Finally, the variational equations of the Hill tethered-satellite problem are provided in an appendix in which the essentials of differential corrections algorithms are also included for the benefit of programmers. «He had not been alone to believe in the stability [...]» Joseph Conrad (The End of the Tether) 85

104 86 Section 7.3: Background: The Hill problem Planet z y ζ O ω t x η r ξ Satellite Orbiter Figure 7.: Geometry of the Hill model showing inertial (x, y, z) and rotating (ξ,η,ζ) frames. 7. Background: The Hill problem The Hill problem is a simplification of the three-body problem, which in turn is a simplification of real models. The three-body problem considers the motion of three point masses under their mutual gravitational attraction. In the «restricted» approximation, it is assumed that the mass of one body, the «secondary», is so small that it does not influence the motion of the other two bodies called the «primaries». The«circular» restricted three-body problem (CRTBP) assumes that the primaries evolve in circular orbits around their mutual center of mass. Finally, the Hill problem further simplifies the CRTBP by assuming that: the mass of the primary at the origin (the central body) is small when compared to the mass of the other primary the distance between the primaries is large when compared to the distance of the secondary to the origin Besides its original application to the motion of the Moon, the Hill problem provides a good approximation to the real dynamics of a variety of systems, encompassing the motion of comets, natural and artificial satellites, distant moons of asteroids, or other applications in the realm of dynamical astronomy. Specifically, the Hill model and its variations are useful for describing motion about planetary satellites. Moreover, the Hill problem is an invariant model that does not depend on any parameter, thus, giving broad generality to the results, whose application to different systems becomes a simple matter of scaling. 7.3 Equations of motion and equilibria We consider a rotating frame centered at the smaller primary, with the orbital plane of the primaries materializing the x-y plane and the bigger primary defining the negative x axis direction, the z direction in coincidence with the rotation rate ω of the primaries, and the y

105 Chapter 7: Halo orbits fundamentals 87 direction completing a direct frame. Then, using Hamiltonian formulation the Hill problem is written H = (X X) ω (x X) W (x) (7.) where x =(x, y, z) is the position vector of the secondary, X =(X, Y, Z) is the vector of conjugate momenta velocity in inertial frame. The potential function W is W (x) = ω (3x r )+Gm/r (7.) where r is the distance of the secondary to the origin, and Gm is the gravitational parameter of the smaller primary. The equations of motion in the rotating frame are or, in scalar form d x dx +ω dt dt = ω (ω x)+ x W (7.3) d x dy ω dt dt =3ω x xgm d y dx r 3, +ω dt dt = ygm d z r 3, dt = ω z zgm r 3 (7.4) As Hamiltonian (7.) does not depend explicitly on time, the Hill problem is conservative and accepts the integral ( dx W (x, y, z) + dy + dz ) = dt dt dt C (7.5) where C is the so-called Jacobi constant. Besides, the positions (±r H,, ), wherer H = 3 Gm/(3 ω ) is the Hill radius, are equilibria of the differential system. They are known to be unstable and are called the «collinear points». Note that one may use units of length and time such that ω and Gm are set to one in the new units. Thus, calling (ξ,η,ζ) to the coordinates in the new unit of length and ρ to the radius, the equations of motion are then ξ η =3ξ ξ/ρ 3, η + ξ = η/ρ 3, ζ = ζ ζ/ρ 3, (7.6) where the dot means derivation in the new time scale, and the Hill problem does not depend on any parameter. Now, the Hill radius is simply ρ H =3 /3 and the Jacobi constant in the normalized units is ( C =W (ξ,η,ζ) ξ + η + ζ ). (7.7) Three symmetries are apparent from the simple inspection of the terms occurring in the differential equations 7.6. Thus, if (ξ(t),η(t),ζ(t)) is a particular solution to Eqs. 7.6, then (ξ( t), η( t),ζ( t)), ( ξ( t),η( t),ζ( t)), (ξ(t),η(t), ζ(t)) are also solutions to Eqs That is, we scale time so that t = τ/ω, and length so that x = ξ (Gm/ω ) /3. Therefore, dx/dt = ξ (Gm ω) /3 and C = C (Gm ω) /3.

106 88 Section 7.4: Particular solutions: periodic orbits 7.4 Particular solutions: periodic orbits Among the particular solutions of a dynamical system, periodic orbits are of specific interest because, besides the equilibria solutions, they are the only solutions which evolution is known for all time. Furthermore, the stability character of a periodic orbit can be easily ascertained by integrating its variational equations. For conservative problems periodic solutions are not isolated and, on the contrary, are grouped in families generated for variations of the system parameters. In the case of the Hill problem there are no parameters and, therefore, there exist only «natural» families of periodic orbits, which are generated by variations of the Jacobi constant. Basic families of periodic orbits start with small retrograde and direct rotations on the plane of the primaries around the origin small retrograde rotations on the plane of the primaries around the collinear points small vertical oscillations through the collinear points From the basic families of periodic orbits new families appear as branches that bifurcate at different «critical» orbits (orbits with indifferent stability character) either in the plane of the primaries or out of it Linear stability definitions The stability of a periodic orbit is derived from the eigenvalues of the state transition matrix at the end of one period, which appear in reciprocal pairs (λ, /λ) in Hamiltonian systems. If a given eigenvalue verifies λ > the periodic orbit is unstable. As a consequence, for Hamiltonian systems λ =turns out to be a necessary condition for stability. Since periodic orbits enjoy one trivial eigenvalue λ =, periodic orbits of Hamiltonian systems with three degrees of freedom have four non-trivial eigenvalues. Then, two stability indices k i = λ i +/λ i (i =, ) are normally used, and the condition k, real and k, < applies for linear stability. For planar motions, one index measures the «horizontal» or in-plane stability (that we note k h ), whereas the other (noted k v )showsthe«vertical» stability character of the periodic orbit. At critical values of the stability indices (some non-trivial eigenvalues taking the value λ = ±) new families of periodic orbits may bifurcate from the original one, either in the plane (k h = ±) or orthogonal to it (k v = ±). The representation of the stability indices versus the parameter generator of a family of periodic orbits produce «stability curves» in which the changes in the stability of a family can be followed. The stability curves are usually represented in the real plane, but unstable orbits with complex eigenvalues out of the unit circle have complex stability indices. Bifurcations of families of periodic orbits are not limited to the critical cases k = ±. For <k< new families of periodic orbits may bifurcate from the original one with multiple period. Thus, for eigenvalues λ that are n-th roots of the unity the stability index is k =cos(πd/n), and a bifurcation orbit with k =+results after n-periods.

107 Chapter 7: Halo orbits fundamentals Lyapunov orbits The family of Lyapunov orbits starts from highly unstable small ellipses with retrograde motion around the collinear points. Decreasing values of the Jacobi constant result in orbits of increasing size; several Lyapunov orbits are shown in the left plot of Fig. 7.. The whole family is made of highly unstable orbits with a minimum in the horizontal stability index k h = 4.99 that occurs at C =.375 (see the right plot of Fig. 7.). For a better understanding of the stability diagram in the right plot of Fig. 7. we represented the ordinates in the usual arcsinh scale. 3 y Stability indices Horizontal stability curve Vertical stability curve Orbit period curve Period x 5 5 Jacobi constant Figure 7.: Left: sample orbits of the family of Lyapunov orbits around L (dashed) and L (full line) for, from larger to smaller, C =,,,, 3, 4. Right: stability-period diagram of the family of Lyapunov orbits of the Hill problem. Note the arcsinh scale used for the stability curves. Despite the high instability of the Lyapunov orbits, the vertical stability index k v starts with moderate values although it soon grows reaching high negative values. However, at the beginning of the family this index crosses three times the lines of critical values and a variety of vertically bifurcated families are expected. More specifically, the family of Halo orbits bifurcates out of the plane from a critical Lyapunov orbit Halo orbits The family of Halo orbits starts from a critical orbit of the family of Lyapunov orbits at C =4.53 where k v =, k h = 79.4, andtheperiodist = Approximate initial conditions of the bifurcation orbit are ξ = , η = , η = ζ =

108 9 Section 7.4: Particular solutions: periodic orbits ξ = ζ =. Since this orbit is a Henon s type A v bifurcation orbit, a small variation in the z direction is enough to find a Halo orbit. Figure 7.3 shows the stability-period diagram of the family of Halo orbits (left plot) and one sample stable orbit (right plot). Stability indices Orbit period curve Reflection k stability curve k stability curve Period.4 Η Ζ Jacobi constant Ξ.6 Figure 7.3: Left: stability-period diagram of the family of Halo orbits of the Hill problem. Right: sample stable orbit for C =.8 (the blue and red dots linked by a gray line are the origin and L point, respectively). The family starts with a highly unstable Lyapunov bifurcation orbit and for decreasing values of the Jacobi constant the orbits move from the L point towards the origin with increasing inclination with respect to the plane of the primaries. The instability of the Halo orbits becomes smaller and smaller until finding a reflection orbit at C =.693. After the reflection, Halo orbits enjoy linearly stable character in a short region until changing again to instability at C =.955. It is known that this family continues with orbits that get closer and closer to the origin until its termination in a collision orbit. Finally, one should note that changing ζ by ζ ( ζ by ζ), the equations of motion (7.6) remain unaltered. Therefore, for any three-dimensional solution of the Hill problem there exists also its symmetric orbit with respect to the (ξ,η) plane Eight-shaped orbits The family of eight-shaped periodic orbits starts with small vertical oscillations through the collinear points. Decreasing values of the Jacobi constant produce eight-shaped orbits of increasing size. The orbits of this family are symmetric with respect to the planes ζ =and η =, and cross the plane ζ =only at η =and ξ =. The left plot of Fig. 7.4 shows three sample orbits of this family for, from larger to smaller, C =,, and. This family is made of highly unstable orbits with one of the stability indices, say k, always having very high values. As shown in the right plot of Fig. 7.4, the k index first

109 Chapter 7: Halo orbits fundamentals 9 Η.... Ξ Orbit period curve 6. Ζ Stability indices k stability curve k stability curve Critical point Period Jacobi constant 3. Figure 7.4: Left: sample eight-shaped orbits for C = (red), (magenta), and (blue). Right: stability-period diagram of the family of eight-shaped orbits of the Hill problem. decreases until k = 36.5 (C =.78757) and then continuously grows. The other stability index starts with small values (the minimum k =.6383 occurs at C =.7643), but soon it crosses the critical value k =+(k = 434.3) atc =.543, andthen grows continuously yet with moderate values Family linking Lyapunov and eight-shaped orbits A new family bifurcates from the critical orbit of the family of eight-shaped, periodic orbits. This family exists for increasing values of the Jacobi constant and is made of highly unstable orbits that unfold and get closer and closer to the plane ζ =until ending onto a planar Lyapunov orbit at C =.86, thus linking the family of eight-shaped orbits and the family of Lyapunov orbits. The stability diagram in Fig. 7.5 shows the evolution of the k stability index of this family jointly with the relevant parts of the families of Lyapunov and eight-shaped orbits (left plot); the k stability index always has very high values and it is not shown. The right plot of Fig. 7.5 shows three sample orbits: the eight-shaped and Lyapunov bifurcation orbits, and an intermediate orbit with C =.9. Another family linking Lyapunov and eight-shaped orbits exists, which is made of orbits symmetric to these with respect to the plane η =.

110 9 Section 7.5: Relevant bibliography in chronological order.4 Link Lyapunov vertical stability curve...5 Η..5. Ξ.4.8. Stability indices Ζ shaped k stability curve Jacobi Constant Figure 7.5: Left: stability diagram of the family linking eight-shaped and Lyapunov orbits of the Hill problem. Right: intermediate orbit for C =.9 (magenta), and end orbits. 7.5 Relevant bibliography in chronological order [] G.W. Hill, «Researches in the Lunar Theory», American Journal of Mathematics, Vol. I, 878, pp. 5-6, 9-47, [] W.H. Clohessy, R.S. Whiltshire, «Terminal Guidance System for Satellite Rendezvous», Journal of the Aerospace Sciences, Vol. 7, No. 9, 96, pp [3] M.L. Lidov, «The Evolution of Orbits of Artificial Satellites of Planets under the Action of Gravitational Perturbations of External Bodies», Planetary and Space Science, Vol. 9, No., 96, pp Translated from Iskusstvennye Sputniki Zemli, No. 8, 96, p. 5 ff. [4] Y. Kozai, «Secular Perturbations of Asteroids with High Inclination and Eccentricity», The Astronomical Journal, Vol. 67, No. 9, 96, pp [5] Y. Kozai, «Motion of a Lunar Orbiter», Publications of the Astronomical Society of Japan, Vol. 5, No. 3, 963, pp [6] M. Hénon, «Exploration Numérique du Problème Restreint. II. Masses égales, stabilité des orbites périodiques», Annales d Astrophysique, Vol. 8, No., 965, pp [7] V. Szebehely, Theory of Orbits The Restricted Problem of Three Bodies, Academic Press, New York, 967. [8] M. Hénon, «Numerical Exploration of the Restricted Problem. V. Hill s Case: Periodic Orbits and Their Stability», Astronomy and Astrophysics, Vol., 969, pp

111 Chapter 7: Halo orbits fundamentals 93 [9] Broucke, R.A., «Stability of Periodic Orbits in the Elliptic Restricted Three-Body Problem», AIAA Journal, Vol. 7, No. 6, 969, pp. 3-9 [] Farquhar, R.W., «The utilization of halo orbits in advanced lunar operations», NASA technical note; NASA TN D-6365, 97. [] Siegel, C.L., Moser, J.K., Lectures on Celestial Mechanics, Springer-Verlag, 97. [] Hénon, M., «Vertical Stability of Periodic Orbits in the Restricted Problem. I. Equal Masses», Astronomy and Astrophysics, Vol. 8, 973, p. 45-ff. [3] Hénon, M., «Vertical Stability of Periodic Orbits in the Restricted Problem. II. Hill s Case», Astronomy and Astrophysics, Vol. 3, 974, p. 37-ff. [4] Breakwell, J.V., and Brown, J., «The halo family of three-dimensional periodic orbits in the Earth-Moon restricted three-body problem», Celestial Mechanics, Vol., No. 4, 979, pp [5] Michalodimitrakis, M., «Hill s problem: families of three-dimensional periodic orbits (Part I)», Astrophysics and Space Science, Vol. 68, 98, pp [6] Howell, K. C. «Three-Dimensional, Periodic, Halo Orbits», Celestial Mechanics, Vol. 3, No., 984, p [7] Hamilton, D.P., Krivov, A.V., «Dynamics of Distant Moons of Asteroids», Icarus, Vol. 8, No., 997, pp [8] Gómez, G., Llibre, J., Martínez, R., Simó, C., Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The Case of Collinear Libration Points, World ScientiÞc Monograph Series in Mathematics, Vol., World ScientiÞc, Singapore New Jersey London Hong Kong,. [9] Scheeres, D.J., Guman, M.D., Villac, B.F., «Stability Analysis of Planetary Satellite Orbiters: Application to the Europa Orbiter», Journal of Guidance, Control, and Dynamics, Vol. 4, No. 4,, pp [] Lara, M., San Juan, J.F., «Dynamic Behavior of an Orbiter Around Europa», Journal of Guidance, Control and Dynamics, Vol. 8, No., 5, pp [] Gómez, G., Marcote, M, Mondelo, J.M., «The invariant manifold structure of the spatial Hill s problem», Dynamical Systems: An International Journal, Vol., No., 5, pp [] Lara, M., Palacián, J.F., «Hill Problem Analytical Theory to the Order Four. Application to the Computation of Frozen Orbits Around Planetary Satellites», Proceedings of the th International Symposium on Space Flight Dynamics, 7, reference number NASA/CP [3] Peláez, J., Sanjurjo-Rivo, M., Lara, M., Rodríguez, F., Bombardelli, C., «Dynamics and Stability of Tethered Satellites at Lagrangian Points: Equations of Motion», ESA/ESTEC Report, 8

112 94 Section 7.5: Relevant bibliography in chronological order

113 Chapter 8 Halo orbits and fast rotating tethers 8. Equations of motion When the third body is not a point mass, the attitude of the body is also part of the problem. For small structures, as standard artificial satellites, the attitude dynamics can be studied independently of the orbital motion of the center of mass, because the corresponding equations of motion decouple. On the contrary, in the case of large structures as long tethers or solar sails the attitude problem no longer decouples from the orbital one, and the rotational-translational motion must be studied as a whole. However, the rototranslational motion may admit some simplification for specific problems. One of these is the case of fast rotating tethers that is: when the rotation rate of the tethered satellite around its center of mass is much faster than the rotation rate of the system. In the next two chapters only inert tethers will be considered; electrodynamic tethers would be taken into account in future works. Moreover, since in a fast rotating tether the tension is provided by the centrifugal forces associated to the angular rate of the tether, we will assume that the tether is always under tension, i.e., it never becomes slack. For a fast rotating tether the value of Ω the non-dimensional angular velocity of the own tether is large, Ω. There are two time scales: ) the period of the orbital dynamics of both primaries and for which τ = ωt is of order unity this is the «slow time» and ) the period of the intrinsic rotation of the tether for which τ =Ω τ of order unity this is the «fast time». To obtain the governing equations we perform to operations: ) we average the original governing equations on the fast time scale, ) we introduce the Hill approach and 3) we take the limit Ω. The governing equations of the system 95

114 96 Section 8.: Equations of motion dynamics turn out to be: ξ η = 3ξ ξ ρ 3 + λ ρ 5 [ ξs (ñ/ρ) 3ñ sin φ ] η +ξ = η ρ 3 + λ [ ] ρ 5 ηs (ñ/ρ)+3ñcos φ sin φ ζ = ζ ζ ρ 3 + λ ρ 5 [ ζs (ñ/ρ) 3ñ cos φ cos φ ] (8.) (8.) (8.3) φ = cosφ tan φ (8.4) φ = sin φ (8.5) where S (ñ/ρ) = (3/) ( 5ñ /ρ ), ñ = ξ sin φ (η sin φ ζ cos φ )cosφ For given values of the tethered satellite s mass distribution and central body s gravitational constant, the parameter λ: λ =(L/l) a ν /3, depends on the tether s length L, the characteristics of the Hill system defined by the distance between the primaries l and the reduced mass of the central body ν, andthemass distribution of the tethered satellite defined by the coefficient a (/ a /4). We call it the tether s characteristic length and, in principle, it is assumed to be of order unity. Then, the attitude motion equations (8.4) (8.5) decouple from the motion of the center of mass, and it can be resolved to give sin φ = sinβ sin(α τ), cos φ = cos β +sin β cos (α τ), sin φ = sin β cos(α τ) cos β +sin β cos (α τ), cos φ = cos β cos β +sin β cos (α τ), where the integration constants α, β, may be expressed unambiguously as function of the initial conditions by making τ =in the equations above. Thus, β [,π/] and cos β =cosφ, cos φ,, sin α =sinφ, / sin β, cos α =sinφ, cos φ, / sin β For a rotating tether the angles (φ,φ ) determine the direction of the angular momentum of the system relative to the synodic frame; if the rotation of the tether is fast, these angles evolve in the same scale of time than the orbital motion of the center of mass. The above solution for (φ,φ ) correspond to a trivial rotation of the angular momentum around the Oζ axis of the synodic frame; note that in this model, the angular momentum is a constant vector of the inertial space.

115 Chapter 8: Halo orbits and fast rotating tethers Constant direction of the angular momentum One should note that the initial conditions φ, = φ, =correspond to an equilibrium of the attitude motion φ = φ =(and ñ = ζ). In such a case, the angular momentum of tethered system keeps a constant direction, and the motion of its center of mass is given from ξ η = 3ξ ξ ρ 3 3Λ ρ 5 ξ η + ξ = η ρ 3 3Λ ρ 5 η ζ = ζ ζ ρ 3 3Λ ρ 5 ζ ) ( 5 ζ ρ ) ( 5 ζ ρ (3 5 ζ ρ ) (8.6) (8.7) (8.8) wherewescaledtheparameter Λ= λ (8.9) to highlight that this problem is formally equal to the Hill-oblateness problem, which is a Hill problem perturbed by the oblateness of the central body (in the Hill-oblateness problem Λ=J α where J is the oblateness coefficient and α is the equatorial radius of the central body in Hill s non-dimensional units). Introducing the effective potential function given by V = ( 3 ξ ζ ) + [+ Λρ ( ρ 3 )] ζ ρ. the governing equations take the classical form ξ η = V ξ, η + ξ = V η, ζ = Vζ, In a direct approach to the problem it would be possible to proceed as with the Hill problem: we would study the perturbed problem by fixing a desired λ (or Λ) value and computing then the new positions of the collinear points, periodic orbits, etc. A more straightforward procedure is, however, to consider the Hill problem as a particular case of the tethered satellite problem the corresponding to a zero value of the non-dimensional tether length λ, i.e. λ = and then to investigate how the tether s length affects the characteristics of a given orbit among the known periodic orbits of the Hill problem. Thus, one can consider the orbit of the Hill problem as the orbit of a tethered satellite with vanishing tether length, and use it as starter for computing a new family of periodic orbits for tether s (characteristic) length variations. 8.. Lyapunov orbits When the starting orbit of the classical Hill problem is a Lyapunov orbit, in general, the tether length does not produce beneficial effects on the stability of the orbit; quite on the contrary, for increasing values of the tether length λ it may destabilize the orbits even more.

116 98 Section 8.: Constant direction of the angular momentum Horizontal stability curve 6.8 Stability index 64 6 Orbit period curve Vertical stability curve Period Η Critical points Tether's characteristic length Ξ Figure 8.: Left: stability-period diagram of the family of Lyapunov orbits with Jacobi constant C =.4895 for tether s length variations; the horizontal gray lines correspond to the critical values k = ± (in the arcsinh scale). Rigth: Hill s problem Lyapunov orbit (λ =, full line) and an orbit with a tehter s characteristic lenght λ =(dotted). As illustration, we focus on a Lyapunov orbit of the classical Hill problem (λ = ) with the minimum horizontal stability index k h = (see Fig. 7.). It is the «less unstable» Lyapunov orbit and it appears for C =.4895 (the vertical stability index is k v = 4.338). Fig. 8. summarizes the result of our analysis; the left plot of the figure shows the stability-period diagram of this family in which we observe the clear growth of the horizontal stability index (note the arcsinh scale used for presenting the stability curves). The right plot of Fig. 8. shows how tether s length variations affect not only stability but also the shape and size of the orbits. The stability diagram of Fig. 8. shows that the vertical stability index always remain with moderate values. Note also the three vertical bifurcation points at k v = ±. Besides, the period of the orbits grows when lengthening the tether until reaching a maximum. Then, a reflection occurs and the period reduces for increasing lengths of the tether. Initial conditions of the critical orbits are provided below, with the format: λ/, period, miss distance; position (ξ,η,ζ); velocity( ξ, η, ζ); (real indices), k h, k v, bound for the error in the stability indices.. Starting orbit (Hill s problem) For a numerically determined periodic orbit, the «miss distance» is a measure of how periodic is, given by the difference between initial conditions of and those after a fundamental period T. In our computations, the miss distance is determined as the maximum of [ξ ξ(t )] +[η η(t )] +[ζ ζ(t )] / ξ + η + ζ and [ ξ ξ(t )] +[ η η(t )] +[ ζ ζ(t )] / ξ + η + ζ

117 Chapter 8: Halo orbits and fast rotating tethers 99.E E+.5E E+.E+.E+.E E+.E E E+.4E-5. Period doubling, vertical bifurcation orbit e e+.e e+.e+.e+.e E+.E E+3 -.E+.E-5 3. Vertical bifurcation orbit e e+.3e e+.e+.e+.e E+.E E+4.E+.E-5 4. Maximum period (reflection in T ) E E+.8E E+.E+.E+.E E+.E E E+.E-5 5. Vertical bifurcation orbit e e+.e e+.e+.e+.e E+.E E+4.E+.3E-5 We computed the family of three-dimensional periodic orbits for tether s length variations that bifurcates at λ =.95445, but lengthening the tether further destabilizes the orbits and we do not present the results. Further trials choosing a Lyapunov orbit with a given tether s length λ as a starter and computing corresponding families of Lyapunov orbits for variations of the Jacobi constant did not show any qualitative changes, and provided similar stability results of those of the Hill problem Lyapunov family. Other example is provided in Fig. 8.. Now, we started from a Lyapunov orbit of the Hill problem very close to the collinear point L and compute the family for variations of the tether s characteristic length. The behavior is similar to the previous example, and the effect of lengthening the tether is increasing the instability of the orbits, as shown in the stability-period diagram of Fig. 8., as well as their size and period.

118 Section 8.: Constant direction of the angular momentum Horizontal stability curve Stability indices 86 6 Critical orbits Orbit period curve Period Η Tether's characteristic length Vertical stability curve Ξ Figure 8.: Left: stability-period diagram of a family of Lyapunov orbits close to L for tether s length variations; the horizontal gray lines correspond to the critical values k = ± (in the arcsinh scale). Rigth: starting orbit (λ =, full line) and an orbit with λ =. (dotted). Two sample orbits are provide in the right plot of Fig. 8., the small one corresponding to the starting Lyapunov orbit of the Hill problem (λ =), and the larger orbit corresponds to a tether s characteristic length λ =.. Again, bifurcations to three-dimensions occur, as noted in the stability diagram of Fig. 8.. In this case we found two simple period and two period-doubling bifurcation orbits. The computation of the simple-period bifurcations for tether s length variations produce Halo-type orbits close to the collinear L point and slightly reduces the instability. Due to the high instability of the critical orbits, and the high value of λ where they occur, we do not try to compute the period-doubling bifurcated branches. Initial conditions of the critical orbits are provided below, with the same format as before.. Starting orbit (Hill s problem).e e+.3e e+.e+.e+.e E-.E E E+.E-5. Vertical bifurcation orbit e e+.3e e+.e+.e+.e E+.E E+4.E+.6E-6

119 Chapter 8: Halo orbits and fast rotating tethers 3. Vertical bifurcation orbit e e+.e e+.e+.e+.e E+.E E+4.E+.7E-6 4. Period-doubling vertical bifurcation orbit e e+.7e e+.e+.e+.e E+.E E+5 -.E+.E-5 5. Period-doubling vertical bifurcation orbit e e+.4e e+.e+.e+.e E+.E E+5 -.E+.4E Eight-shaped orbits This subsection focuses on the new families of periodic orbits which appear when the starting orbit of the classical Hill problem is a eight-shaped orbit. Despite the clear instability of these orbits of the Hill problem that can be appreciated in Fig. 7.4, variations of the tether s length make it possible, in general, to find orbital stability for values of the Jacobi constant C below a critical value close to C 3. Besides the effects on stability produced by the tether s length variations, the shape and size of the orbit is also affected, since it moves and curves towards the origin. To illustrate the qualitative behavior, three examples follow for values of the Jacobi constant C =,.7 and 3. In each one of the first two examples (C =and C =.7) we find two stability regions separated by a region of complex instability. The last example (C =3) shows a different qualitative behavior in which tether s length variations never stabilize the orbits. Length variations with constant C = This family, which starts from a highly unstable orbit (λ =) of the Hill problem, exists for values of λ in the interval [,.883]. Lengthening the tether produces orbits of smaller size and period, which reduce their instability character (see Fig. 8.3). One of the stability indices, say k, decreases continuously while the other slightly increases. The period reduces for increasing tether s length until a minimum that occurs at λ/ =.945, very close to the maximum tether s characteristic length λ/ =.946 for which periodic orbits of this

120 Section 8.: Constant direction of the angular momentum k stability curve Orbit period curve 3.48 Stability indices : end : start k stability curve Tether's characteristic length Period Ζ Ξ.6 Figure 8.3: Left: stability-period diagram of a family of eight-shaped, periodic orbits with constant C =for tether s length variations; the horizontal gray lines correspond to the critical values k = ±. Rigth: stable orbits for λ =. (left) λ =.5 (center), and unstable orbit of the Hill problem (λ =, right). family exist (point in the left plot of Fig. 8.3). Periodicity is broken for higher values of λ, but the family can be continued for decreasing values of the tether s length. The reflected branch of the family is made of orbits that immediately enter a region of stability at λ/ =.93. Shortening further the tether produces stable orbits until λ/ =.675 (point 3 in the left plot of Fig. 8.3), where the tethered satellite enters a region of complex instability. At λ/ =.998 (point 4 in the left plot of Fig. 8.3) the satellite enters another stability region, and the stability character changes again to instability at λ/ = (point 5 in the left plot of Fig. 8.3) reaching high instability values for shorter lengths of the tether. The right plot of Fig. 8.3 shows three sample orbits of this family. The higher one (the rightmost, in red) is the starting orbit, a highly unstable eight-shaped orbit of the Hill problem with C =. The other two are stable orbits, one of each stability region, with tehter s characteristic lenght λ =. (left) and λ =.5 (center). Below we provide initial conditions of some orbits of this family with the format: λ/, period, miss distance; position (ξ,η,ζ); velocity( ξ, η, ζ); either (real indices), k, k, bound for the error in the stability indices, or (complex conjugate indices), IRe(k), IIm(k), bound for the error in the stability indices. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (; Hill s problem).e e+.3e e+.e+.e+.e E E E E+.5E-6

121 Chapter 8: Halo orbits and fast rotating tethers 3. Minimum relative period (reflection in T ) E E+.4E E+.E+.E+.E E E E E+.E-6 3. Maximum tether s length (; reflection in λ) e e+.3e e+.e+.e+.e E E E E+.E-6 4. Change to stability e e+.6e e+.e+.e+.e E E+.E E+.E-6 5. Change to complex instability (3) E E+.E E+.E+.E+.E E E E E-6.3E-6 6. Change to stability (4) E E+.7E E+.E+.E+.E E E E E-7.6E-6 7. Change to real instability (5) E E+.3E E+.E+.E+.E E E+.E E-.7E-6 8. Maximum relative period (reflection in T ) E E+.3E E+.E+.E+.E E E E E+.7E-6

122 4 Section 8.: Constant direction of the angular momentum Length variations with constant C =.7 The behavior of this family is similar to the previous one, although the stability occurs at different values of the tether s characteristic length. This new family exist for values of λ [,.387]. Figure 8.4 shows the stability-period diagram of this family in the left plot, to which we refer in what follows without explicit mention. The family starts from a highly unstable orbit of the Hill problem (λ =); for increasing lengths of the tether the instability reduces as well as the orbital period. At λ/ =.6436 a reflection occurs (point ) andthe periodicity is destroyed for higher lengths of the tether : end Orbit period curve Stability indices k stability curve k stability curve Period Ζ. : start Tether's characteristic length Ξ.4.6 Figure 8.4: Left: stability-period diagram of a family of eight-shaped, periodic orbits with constant C =.7 for tether s length variations; the horizontal gray lines correspond to the critical values k = ±. Right: stable orbits for λ =.7 (left, black) λ =.84 (center, blue), and unstable orbit of the Hill problem (λ =, right). The reflected branch exists for decreasing values of λ and starts with periodic orbits that are mildly unstable; in this region of mild instability the orbit period reaches the minimum after which it continuously grows. At λ/ = the family enters a region of stability until the tether s characteristic length is shortened to λ/ =.99389, where the tethered satellite changes from stability to complex instability. The tethered satellite enters a new region of stability at λ/ = Shortening further the tether pushes the satellite out of the stability region at λ/ =.8446 and the instability grows high for smaller values of λ. The right plot of Fig. 8.4 shows three sample orbits. The higher one (the rightmost, in red) is the starting orbit a highly unstable eight-shaped orbit of the Hill problem with C =.7. The other two are stable orbits, one of each stability region, with tehter s characteristic lenght λ =.7 (left) and λ =.84 (center).

123 Chapter 8: Halo orbits and fast rotating tethers 5 Below we provide initial conditions of some orbits of this family with the usual format. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (; Hill s problem).e e+.e e+.e+.e+.e E E E E+.5E-6. Maximum tether s length (; reflection in λ) e e+.3e e+.e+.e+.e E E E E+.E-6 3. Minimum relative period (reflection in T ) E E+.3E E+.E+.E+.E E E E E+.E-6 4. Change to stability e e+.8e e+.e+.e+.e E E+.E E+.E-6 5. Change to complex instability (3) E E+.6E E+.E+.E+.E E E E E-6.3E-6 6. Change to stability (4) E E+.7E E+.E+.E+.E E E E E+.4E-6 7. Change to real instability (5) E E+.4E E+.E+.E+.E E E+.E E+.4E-6

124 6 Section 8.: Constant direction of the angular momentum Length variations with constant C =3 For higher values of the Jacobi constant the behavior changes and the length of the tether can be continuously increased (with the limits for practical applications imposed by the simplifications of the model) without finding any reflection. As a consequence, there are no stable periodic orbits for variations of the tether length. This change in the qualitative behavior takes place for a critical value of the Jacobi constant in the interval [.7, 3] (closer to the upper limit). Figure 8.5 shows the stability-period diagram of this family. The family starts from a highly unstable orbit of the Hill problem (λ =); for increasing lengths of the tether the instability reduces as well as the orbital period. The minimum instability occurs for a tether s characteristic length λ = , wherek 38, and for higher lengths of the tether the instability grows continuously. The k stability index crosses twice the critical value +, the corresponding orbits are bifurcation orbits from which new families of periodic orbits originate. Specifically, the point λ = corresponds to the termination of the family of Halo orbits with constant C =3discussed below. Besides, k = at the point λ = where a period doubling bifurcation is expected Stability indices Bifurcation orbits k stability curve Period Orbit period curve k stability curve Tether's characteristic length.6.4 Figure 8.5: Stability-period diagram of a family of eight-shaped, periodic orbits with constant C =3for tether s length variations; the horizontal gray line corresponds to the critical values k =+. Below we provide initial conditions of some orbits of this family with the usual format (first field: λ/):. Starting orbit (Hill s problem).e e+.e e+.e E+.E E+.E E E+.E-5

125 Chapter 8: Halo orbits and fast rotating tethers Stability indices : end : start k stability curve Orbit period curve k stability curve Tether's characteristic length Period.5 Ζ Ξ.6.8 Figure 8.6: Left: stability-period diagram of a family of Lyapunov-to-eight-shaped orbits with constant C =.9 for tether s length variations; the horizontal gray lines correspond to the critical values k = ±. Right: sample orbits for λ =(red, unstable), λ =. (magenta, unstable), and λ =.4 (blue, stable).. Bifurcation orbit e e+.e e+.e E+.E E+.E E E+.E-5 3. Bifurcation orbit e e+.4e e+.e+.e+.e E E E+3.E+.E-6 4. Period doubling bifurcation orbit e e+.e e+.e E+.E E+.E E+ -.E+.9E-6

126 8 Section 8.: Constant direction of the angular momentum 8..3 Eight-shaped to Lyapunov orbits This family starts from a highly unstable orbit of the family linking Lyapunov and eightshaped orbits of the Hill problem with C =.9. The stability-period diagram of this family is presented in the left plot of Fig. 8.6, in which we note that increasing the tether s length reduces the instability and the orbit period, the orbits very soon turning to the eight shape. A relative minimum in the period occurs at λ/ = , and the family continues with unstable orbits until a maximum tether s length of λ/ =.3 after which the periodicity is broken. However, a different branch of the family can be computed by shortening the tether. The reflected branch of the family continues with unstable orbits that change to complex instability at λ/ = During the excursion over the complex instability region, the orbit period reaches a maximum at λ/ =.487 after which the period continuously reduces until the termination of the family. Then, at λ/ =.393 the tethered satellite enters a thin region of stability. Shortening further the tether immediately introduces orbit instability at λ/ =.839. The instability grows high for smaller values of λ. The right plot of Fig. 8.6 shows three sample orbits of this family. The larger one (kidneyshaped, in red) is the starting orbit, a highly unstable orbit with C =.9 pertaining to the family of periodic orbits of the Hill problem linking Lyapunov and eight-shaped orbits. The intermediate orbit (eight-shaped, in magenta) is the highly unstable orbit with the relative minimum period. The last one (in blue) is a stable orbit with a tether s characteristic lenght λ =.4. Below we provide initial conditions of some orbits of this family with the usual format. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (; Hill s problem).e e+.e e+.e+.e+.e E E E E+.4E-6. Minimum relative period (reflection in T ) E E+.3E E+.E+.E+.E E E E E+.5E-6 3. Maximum tether s length (; reflection in λ) e e+.3e e+.e+.e+.e E E E E+.5E-6

127 Chapter 8: Halo orbits and fast rotating tethers 9 4. Change to complex instability (3) E E+.3E E+.E+.E+.E E E E E+.5E-6 5. Maximum relative period (reflection in T ) E E+.3E E+.E+.E+.E E E E E+.7E-6 6. Change to stability (4) E E+.E E+.E+.E+.E E E E E+.9E-6 7. Change to real instability (5) E E+.E E+.E+.E+.E E E E E+.E Halo orbits This subsection focuses on the new families of periodic orbits which appear when the starting orbit of the classical Hill problem is a Halo orbit. Lengthening the tether of a tethered satellite in an Halo orbit tends to improve its stability characteristics, in general, even finding stability in some cases. Again, variations of the tether s length modify the size and shape of the tethered satellite s orbit and its relative position with respect to the origin. However, once a stable orbit has been found for a certain length of the tether, variations of the Jacobi constant might maintain stability while slightly changing other orbit s characteristics. This way more stable orbits can be found for a given tether length. Several examples follow for different values of the Jacobi constant (C =.7,.5,.,.5 and 3); note that the unstable character of the starting Halo orbits increases with the value of C. The first one starts from a «stable» Halo orbit of the Hill problem; lengthening the tether leads to a reflection and, then, shortening it the new family links the stable Halo orbit of the Hill problem with the «unstable» one that exists for the same value of the Jacobi constant (cf. Fig. 7.3). A similar behavior, linking two different orbits of the Hill problem with the same value of C by means of lengthening and shortening the tether is found for

128 Section 8.: Constant direction of the angular momentum Η Stability indices k stability curve Orbit period curve Period. Ζ.5 k stability curve Tether's characteristic length Ξ.4.6 Figure 8.7: Left: Stability-period diagram of the family of Halo orbits with C =.7 for tether s length variations. Right: stable (full line) and unstable (dashed) Halo orbits of the Hill problem. say C.5; the link-families find stability in-between for relatively small tether s characteristic lengths (examples below for C =.5, C =.). Starting from higher values of the Jacobi constant shows a different behavior, but stability regions are also found despite they occur for higher tether s characteristic lengths (examples for C =, C =.5, below). Finally, the more unstable Halo orbits of the Hill problem seem not to be amenable to stabilize, and lengthening the tether produces the termination of the family onto an orbit of the family of eight-shaped orbits (example below for C =3). Family of Halo orbits with C =.7 When starting from a stable Halo orbit, orbits that exist in a small region close to the left margin of the left plot of Fig. 7.3, the family of periodic orbits for tether s length variations exists only for very small characteristic lengths of the tether. It suffers a reflection at λ =7.4 5 with a change to instability. When shortening the tether after the reflection, the instability of the tethered satellite s orbits continuously grows until the termination of the family onto another Halo orbit of the Hill problem. Thus, by means of lengthening and shortening the tether we find a connection between two different Halo orbits of the Hill problem with the same Jacobi constant, which are very close in terms of size and shape. The left plot of Fig. 8.7 shows the stability-period diagram of this new family, where we also note that the period continuously grows. The right plot of the figure shows the starting (stable, full line) and ending (unstable, dashed) Halo orbits, both belonging to the

129 Chapter 8: Halo orbits and fast rotating tethers Hill problem (λ =). Below we provide initial conditions of the starting, reflection, and termination orbit of this family with the usual format (first field: λ/).. Starting orbit (Hill s problem).e e+.8e e-.e e+.e e+.e e E+.9E-6. Maximum tether s length (reflection in λ) e e+.e e+.e E+.E E+.E+.7479E E+.3E-6 3. Termination orbit (Hill s problem).e e+.e e+.e E+.E E+.E E E+.3E-6 Family of Halo orbits with C =.5 If we start from an unstable Halo orbit of the Hill problem close to the stability region, lengthening the tether reduces the instability until finding a maximum tether s characteristic length λ = Beyond this value the periodicity is destroyed. Shortening the tether continues the family by a different branch made of stable orbits until the value λ =.3545 where the tethered satellite orbit changes to instability until the termination of the family in a different Halo orbit of the Hill problem. The period of the family reduces in a continuous way. Figure 8.8 shows the stability-period diagram in the left plot. The right plot shows sample orbits of the family. Below we provide initial conditions of the starting, reflection, and termination orbit of this family with the usual format (first field: λ/). Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (: Hill s problem).e e+.e e-.e e+.e e+.e e E+.8E-6

130 Section 8.: Constant direction of the angular momentum.5 Η Stability indices 4 5 : start 4: end k stability curve : reflection Orbit period curve k stability curve Period.5. Ζ Tether's characteristic length... Ξ.4.6 Figure 8.8: Stability-period diagram of the family of Halo orbit with C =.5 for tether s length variations. Right: unstable Halo orbits of the Hill problem (red and magenta) and stable (blue) Halo orbits with a tethers characteristic length λ =.5.. Maximum tether s length (: reflection in λ) e e+.5e e+.e E+.E E+.E E E+.3E-6 3. Change to instability (3) E E+.6E E+.E E+.E E+.E E+ -.E+.4E-6 4. Termination orbit (4: Hill s problem).e e+.4e e+.e E+.E E+.E E E+.7E-6 Family of Halo orbits with C =. The starting orbit is now more unstable, but the behavior is analogous to that of the previous family. The reflection occurs for a higher length of the tether and the stability region occurs

131 Chapter 8: Halo orbits and fast rotating tethers 3.5 Η. 9 k stability curve : start : reflection.5 Stability indices 4: end Orbit period curve k stability curve.3.. Period. Ζ Tether's characteristic length.. Ξ.4.6 Figure 8.9: Stability-period diagram of the family of Halo orbit with C =. for tether s length variations. Right: unstable Halo orbits of the Hill problem (red and magenta) and stable (blue) Halo orbits with a tethers characteristic length λ =.9. in a lesser range of λ values. The stability-period diagram is shown in the left plot of Fig. 8.9 and the right plot shows three sample orbits. Below we provide initial conditions of the starting, reflection, and termination orbit of this family with the usual format (first field: λ/). Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (: Hill s problem).e e+.6e e-.e e+.e e+.e e E+.E-5. Maximum tether s length (: reflection in λ) e e+.e e-.e e+.e e+.e e E+.8E-6 3. Change to instability (3) E E+.E E+.E E+.E E+.E E+ -.E+.3E-6

132 4 Section 8.: Constant direction of the angular momentum 4. Termination orbit (4: Hill s problem).e e+.3e E-.E E+.E E+.E E E+.3E-5 Family of Halo orbits with C = Stability indices : end : start 6 k stability curve 5 Orbit period curve k stability curve : reflection Tether's characteristic length Figure 8.: Stability-period diagram of the family of Halo orbit with C =for tether s length variations. The horizontal, gray lines mark the critical values k = ± in the arcsinh scale. This case is illustrated in Fig. 8.. The family starts at with an unstable Halo orbit of the Hill (λ =)problem for C =;theinstability reduces when lengthening the tether until reaching a maximum tether s characteristic length at. The periodicity destroys if lengthening the tether above this limit value, and the family of periodic orbits reflects over itself for decreasing values of the tether s length entering a narrow stability region. At 3 the family enters a region of complex instability until 4, where the stability indices change to real instability. The family continues with unstable orbits when shortening the tether, with another area of complex instability between 5 and 6, until ending close 7. We do not continue the family until its termination, which apparently occurs with a highly unstable, planar, collision orbit of the Hill problem. The evolution of the orbital period is also shown in Fig. 8. with a dashed line. It generally reduces along the family, although after shortening the tether down to a minimum characteristic length λ., the period grows until the termination of the family. Figure 8. shows a magnification over the stability region in the left plot of Fig. 8., and two orbits of this family in the right plot: the dashed one is the starting orbit from the Hill problem Halo orbit, and the solid line corresponds to a stable orbit for λ =.9 after the reflection. If we now vary the Jacobi constant of one desired solution we can compute a new family of Halo orbits for a given tether s length. However, variations of the Jacobi constant soon change the stability characteristics and this procedure can be considered as a fine tuning of the desired solution. As illustration, we start from the orbit with λ =.9 of the family Period

133 Chapter 8: Halo orbits and fast rotating tethers 5.5 Η Stability indices Ζ Tether's characteristic length.. Ξ.4.6 Figure 8.: Left: Magnification on the stability region of Fig. 8.. Right: Halo orbits for λ =(dashed, unstable) and λ =.9 (solid line, stable); the black and red dots joined by a gray straight line mark the origin and L point, respectively. with constant C =, and compute the new family for variations of the Jacobi constant. Results are summarized in Fig. 8. in which we see that the stability region of the new family is very narrow, bounded by a reflection to the left (at C =.99988) and a change to complex instability to the right (C =.3). 4 3 Stability indices Reflection Change to complex instability Jacobi constant Figure 8.: Stability curves of the family of Halo orbits with tether s characteristic length λ =.9 and variations of the Jacobi constant. The dots correspond to the starting orbit. Changes in the orbit s shape inside the stability region are very small and hardly can be appreciated at the graphics resolution, so we do not provide illustrations of orbits. Below we provide initial conditions of some orbits of this family with the usual for-

134 6 Section 8.: Constant direction of the angular momentum mat. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in Fig Starting orbit (; Hill s problem).e e+.e e+.e e+.e e+.e e E+.4E-6. Maximum tether s length (; reflection in λ) e e+.5e e+.e e+.e e+.e+.6576e E+.9E-6 3. Change to complex instability (3) E E+.8E E+.E E+.E E+.E E E-5.8E-6 4. Change to real instability (4) E E+.E E+.E E+.E E+.E E E+.E-5 5. Change to comlex instability (5) E E+.7E E+.E E+.E E+.E E E+.5E-6 6. Minimum period (reflection in T ) E E+.E E-.E E+.E E+.E E E+.5E-5 7. Change to real instability (6) E E+.4E E-.E E+.E E+.E E E+.E-4

135 Chapter 8: Halo orbits and fast rotating tethers 7 Family of Halo orbits with C =.5 A more favorable situation is found when starting from a Halo orbit of the Hill problem with C =.5, as illustrated in the stability diagram of Fig Despite the initial stability is much higher than in the previous case, the instability reduces when lengthening the tether until finding a maximum tether s characteristic length λ =.5 beyond which the periodicity is destroyed (point of Fig. 8.3). Similarly to the previous case of C =, a reflection occurs and the tethered satellite enters a thin region of stability. Shortening the tether just a little bit brings the tethered satellite into a region of complex instability (point 3 of Fig. 8.3). However, contrary to the case of C =, shortening more the tether pushes the tethered satellite away the complex instability region towards a linear stability region for tethers with characteristic length.344 >λ>.488 (between points 4 and 5 of Fig. 8.3). Shorter tethers produce instability (k > ) and the family seems to end with a collision orbit of the Hill problem : start Orbit period curve k stability curve.9.6 Stability indices 3 3 6: end k stability curve..9.5 Period Tether's charcteristic length Figure 8.3: Stability-period diagram of the family of Halo orbits with C =.5 for tether s length variations. Figure 8.4 shows a detail in the stability region that exists after the reflection (left plot), in which we appreciate how extremely thin it is. The right plot of Fig. 8.4 presents three sample orbits: the starting Halo orbit of the Hill problem (dashed, highly unstable); a stable orbit of the thin stability region in the left plot of Fig. 8.4 which shrinks the Halo in the η axis direction, but still is broad enough to show Haloing characteristics. The third orbit in the right plot of Fig. 8.4 belongs to the wide region of stability of Fig. 8.3, and has notably shrunk in the η axis direction in practice probably loosing the characteristics required by real Halo applications. Below we provide initial conditions of some orbits of this family with the usual format. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram of Fig. 8.3.

136 8 Section 8.: Constant direction of the angular momentum..5. Η. Stability indices Ζ Λ Ξ.6 Figure 8.4: Left: Magnification on the stability region of Fig Right: Halo orbits for λ = (dashed, unstable), λ =.534 (broad, solid line, stable), and λ =. (thin, solid line, stable); the black and red dots joined by a gray straight line mark the origin and L point, respectively.. Starting orbit (; Hill s problem).e e+.e e+.e e+.e e+.e e e+.6e-6. Maximum tether s length (; reflection in λ) e e+.5e e+.e e+.e e+.e e e+.5e-6 3. Change to complex instability (3) E E+.E E+.E E+.E E+.E E E+.4E-6 4. Change to stability (4) E E+.4E E+.E E+.E E+.E E E+.8E-6

137 Chapter 8: Halo orbits and fast rotating tethers 9 5. Change to real instability (5) E E+.4E E+.E E+.E E+.E+.4E E+.E-5 From thin-halo to thin-eight-shaped orbits: We choose one of the stable orbits in the wide stability region of Fig. 8.3, and continue it for variations of the Jacobi constant to show how increasing values of the Jacobi constant tend to symmetrize the orbit with respect to the orbital plane of the primaries. Specifically, we chose to continue the stable orbit with tether s characteristic length λ =. (C =.5). The stability diagram of the new family of periodic orbits is presented in the left plot of Fig For decreasing values of the Jacobi constant the family very soon enters a region of complex instability and we do not pursue the continuation of the family in that direction. The continuation towards higher values of the Jacobi constant shows better characteristics extending the stability region up to C.6; then, while one of the stability indices always remains between the critical values ±, the other get values higher than two, reaching its maximum (k =6.) atc 3.. This family ends at C 3.8 in a termination orbit of a family of eight-shaped orbits discussed below. For the computed orbits of this family we found that the evolution of the period is opposite to the Jacobi constant, continuously decreasing for increasing values of C. The rigth plot of Fig. 8.5 shows two sample periodic orbits with λ =.. The stable starting one (C =.5) and the termination orbit (k =, k =.56956) atc 3.8. Below we provide initial conditions of some orbits of this family with the usual format, except the first field is C/ instead of λ/. Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Starting orbit (Thin-Halo with C =.5 and λ =.) -.5E E+.7E E+.E E+.E E+.E E E+.E-5. Change to complex instability E E+.3E E+.E E+.E E+.E E E-5.E-5 3. Change to real instability

138 Section 8.: Constant direction of the angular momentum.. Ξ Stability indices 3 Change to complex instability Termination Orbit period curve Period Ζ Jacobi constant.6 Figure 8.5: Stability-period diagram of the family of periodic orbits derived from the thin- Halo orbit with λ =., C =.5, for variations of the Jacobi constant. Right: a stable orbit for C =.5 and the termination eight-shaped orbit; the black and red dots joined by a gray straight line mark the origin and L point, respectively E E+.6E E+.E E+.E E+.E+.3E E+.8E-6 4. Bifurcation orbit E E+.E E+.E+.E+.E E E+.E E+.E-6 Thin-eight-shaped orbits: The continuation of the eight-shaped termination orbit of the previous family for variations of the Jacobi constant gives rise to the a new family of eight-shaped periodic orbits with tether s characteristic length λ =.. The stability diagram of this new family is presented in Fig. 8.6, where it has been combined with the stability curves of the previous family to highlight the bifurcation. The family seems to start with an eight-shaped unstable orbit through the collinear point L. Increasing values of the Jacobi constant produce reducing values of the k stability index Note that this family cannot be obtained by choosing an eight-shaped orbit of the Hill problem, then computing the family for variations of the tether s characteristic length up to λ =., and then computing the family of periodic orbits with constant λ =. for variations of the Jacobi constant.

139 Chapter 8: Halo orbits and fast rotating tethers and increasing values of the period; the stability index k remain very close to the critical value +. At C =5.387 the family suffer a reflection (point in Fig. 8.6) after which the tethered satellite orbits enter a region of stability. After the reflection, decreasing values of the Jacobi constant produce stable orbits with increasing period. A small region of mild instability occurs between C = and C = , the last one the critical point from which the family in the previous paragraph bifurcates (point 3 in Fig. 8.6). Then, the family continues with stable orbits until C =3.544 where the orbits change to complex instability (point 4 in Fig. 8.6). After a short excursion by the complex instability region, the tethered satellites with characteristic length λ =. change again to stability at C = (point 5 in Fig. 8.6), but soon changes to instability at C = Decreasing values of the Jacobi constant produce orbits of increasing instability and period, and we do not continue the family until its termination Orbit period curve.7 Stability indices k stability curve k stability curve 3 4 k k Period Jacobi constant Figure 8.6: Stability-period diagram of the family of eigth-shaped periodic orbits with tether s characteristic length λ =. for variations of the Jacobi constant. The bifurcation branch is colored in red and blue, and the gray straight line marks the critical value k =. Figure 8.7 shows several sample orbits of this family, all of them stable. The eightshaped orbits of this family move along the ξ axis from the collinear point towards the origin with decreasing size and width. From a certain point, they «eight» is so narrow that the orbits seem as mere arcs in the (ξ,ζ)-plane. Below we provide initial conditions of some orbits of this family with the usual format (first field: C/). Numbers in typewriter style in the title make reference to corresponding points in the stability-period diagram in the left plot of Fig Maximum tether s length (; reflection in C) E E+.3E E+.E+.E+.E E E+

140 Section 8.: Constant direction of the angular momentum.5 Η.5. Ξ Ζ. Figure 8.7: Stable eight-shaped orbits with tether s characteristic length λ =. and, from the left to the right, C =5., C =4., andc =3.3; the black and red dots joined by a gray straight line mark the origin and L point, respectively..7754e e+.9e-6. Change to real instability E E+.6E E+.E+.E+.E E E+.E E+.E-6 3. Change to stability (3) E E+.E E+.E+.E+.E E E+.E E+.E-6 4. Change to complex instability (4) E E+.8E E+.E+.E+.E E E E E+.E-6 5. Change to stability (5)

141 Chapter 8: Halo orbits and fast rotating tethers E E+.E E+.E+.E+.E E E E E-6.E-6 6. Change to real instability E E+.E E+.E+.E+.E E E+.E E+.E-6 Family of Halo orbits with C =3 For higher values of the Jacobi constant, variations of the tether s characteristic length apparently never stabilize the tethered satellite s orbit. The stability-period diagram of this family is shown in the left plot of Fig Both the instability and the period decrease for increasing values of the tether s characteristic length. Lengthening the tether narrows and twists the Halo orbit until converting it on an eight-shaped orbit, as shown in the right plot of Fig Thus, the termination point at λ = is a bifurcation orbit of the family of eight-shaped orbits with constant C =3for tether length variations (see Fig. 8.5). Stability indices 65 k stability curve 7 47 Orbit period curve 3 3 k stability curve Tether's characteristic length Period.5 Η...4. Ζ Ξ.6 Figure 8.8: Left: Stability-period diagram of the family of Halo orbits with C =3for tether s length variations. Right: Sample orbits showing the transition from Halo to eightshaped. Initial conditions of the starting and termination orbit of this family are given below with the usual format (first field: λ/).. Starting orbit (Hill s case)

142 4 Section 8.: Constant direction of the angular momentum.e e+.e e+.e E+.E E+.E E E+.3E-6. Termination orbit e e+.4e e+.e+.e+.e E E E+3.E+.E-6 The same behavior is found for higher values of the Jacobi constant C > 3 and the corresponding families are not discussed Summary of results Results in the previous sections may be summarized as follows: In the case of Lyapunov orbits we did not find beneficial effects on orbit stability when using tethered satellites and, quite on the contrary, lengthening the tether further destabilizes the orbits. Lengthening the tether of a tethered satellite in an eight-shaped orbit, in general, reduces instability. Besides, stability regions are found for certain lengths of the tether when the Jacobi constant remains roughly below C =3. Other effect produced by tether s length variations is to change the shape and size of the orbits, which move and curve towards the origin. Stability regions of tethered satellites in Halo orbits are generally found, even when starting from highly unstable orbits of the Hill problem. The general effect of lengthening the tether is to narrow and twist the Halo, sometimes converting the Halo in a thin, eight-shaped orbit. However, depending on the Jacobi constant value of the starting Halo orbit, the required length of the tether to stabilize the orbit may be small and the stabilized orbit may retain most of the Haloing characteristics. We have to take into account two different motivations to use a tether: ) in order to stabilize an unstable Halo orbit and ) in order to take advantage of some characteristics of tethered systems. In this second case, the tether is not responsible for providing the dynamic stability to the system; quite the opposite, it should enjoy the stability properties of some Halo orbits which turns out to be stable without tether (for values of the Jacobi constant C close to the lower limit). Therefore, the most promising regions are close to the Halo family s stability region of the Hill problem, where tethers of few tens of kilometers may stabilize Halo orbits of the simplified model. For specific applications one needs to check that the tether length required for stabilization is feasible, say of few tens of km, and that the corresponding Halo orbit does not impact the central body.

143 Chapter 8: Halo orbits and fast rotating tethers 5 Figure 8.9 shows the evolution of the maximum and minimum distance to the origin 3 of every orbit of the Halo family (no tether) combined with the stability diagram on Fig We see on Fig. 8.9 a continuous reduction of the minimum distance from the beginning of the family as a bifurcation of the Lyapunov family to its end (not presented in the figure) with a collision in the origin. In order to provide a clearer insight into the problem, we show on Fig. 8. the evolution of the stability indices of the Halo family of the Hill problem as a function of the minimum distance to the origin, instead of the Jacobi constant. Figure also presents the equatorial radius of several celestial bodies in the Hill problem units, and highlight the stability region. Note on Fig. 8. that Halo orbits close to Enceladus are stable, and that stable Halo orbits may exist for many celestial bodies, their distances to Stability indices 6 Maximum distance curve k stability curve Minimum distance curve k stability curve Jacobi constant Figure 8.9: Stability diagram of the Halo family of the Hill problem (no tether) showing the maximum and minimum distance to the origin. Stability indices Jupiter The Moon Europa Io Deimos Enceladus k stability curve k stability curve Minimum radius Hill problem units. Phobos Figure 8.: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the minimum distance to the origin. the origin obviously depending on the particular body. Thus, for instance, we find stable Halo orbits in the sun-jupiter system with a minimum distance to Jupiter of Jupiter s equatorial radius; at a minimum distance to the Moon of Moon s equatorial radius, in the Earth-Moon system; or in the Jupiter-Europa system at a minimum distance to Europa of Europa s equatorial radius. However, in the Martian system stable Halo orbits about Deimos or Phobos are always impact orbits. Furthermore, the equatorial radius of Amalthea (.7 units) is larger than the Hill radius, and, therefore, the whole Halo family is made of impact orbits. Distance Hill units 3 Halo orbits not being perturbed Keplerian ellipses, it has no sense talking about the semi-major axis or any other orbital element

144 6 Section 8.: Constant direction of the angular momentum Figure 8.: Halo orbit close to Io, stabilized with a tether length of 7 km. The minimum distance to Io is about one half of it radius Figure 8.: Halo orbit about Eros, stabilized with a tether length of km.

145 Chapter 8: Halo orbits and fast rotating tethers 7 Stability indices 4 Eros Mercury The Moon k stability curve Europa Io Enceladus k stability curve Minimum radius Hill problem units Figure 8.3: Stability diagram of the Halo family of the Hill problem (no tether) as a function of the minimum distance to the origin. it just because it is depicted in the rotating frame. C =.9583 Closer orbits to selected bodies may be stabilized using fast rotating inert tethers. This is feasible for the mildly unstable orbits that live close to the stability region. Figure 8.3 shows a closer detail on Fig. 8.. An obvious trial from Fig. 8.3 is trying to stabilize Halo orbits close to the jovian moon Io. Figure 8. shows an example for this moon, where a tether length of 7 km can be used to stabilize Halo orbits. Remark that the orbit is not Keplerian. While the orbit resembles an ellipse (or Halo), Initial conditions of this orbit are:.7375e e+.e e e+.e e e E E E+.6E-6 with the format λ/, period, miss distance, position, velocity, stability indices (k,k ). A more challenging trial from Fig. 8.3 is to stabilize Halo orbits close to the asteroid Eros. Despite stability occurs at a minimum distance of 4 Eros radius, 4 this radius is only of about 6.5 km so that a tether of km may be used to stabilize Halo orbits as close as 56 km of minimum distance. Figure 8. shows an example for this minor planet. Initial conditions of the orbit are: C = E E+.7E E E+.E E E E E E+.8E-6 with the format λ/, period, miss distance, position, velocity, stability indices (k,k ) Initial conditions of some stable Halo-derived orbits For the convenience of interested readers, initial conditions of other stable orbits pertaining to the previously computed families are given below, with the usual format: C/, period, miss distance, position, velocity, stability indices (k,k ). 4 We choose an equatorial radius of 6.5 km for Eros, the largest dimension of this potato-shaped body of 33 km long, 3 km wide and 3 km thick.

146 8 Section 8.3: Constant direction of the angular momentum The first two correspond to the Halo-shaped stable orbits in the right plots of Fig. 8. and 8.4, respectively; the third one, to the stretched-halo stable orbit in the right plot of Fig. 8.5; and the last three sets of initial conditions correspond to the three stable eight-shaped orbits of Fig λ =.9, C = -.E E+.E E+.E E+.E E+.E E E+.3E-6. λ =.5344, C =.5 -.5E E+.7E E+.E E+.E E+.E E E+.E-6 3. λ =., C = E E+.7E E+.E E+.E E+.E E E+.E-5 4. λ =., C = E E+.E E+.E+.E+.E E E E E+.E-6 5. λ =., C =4. -.E E+.5E E+.E+.E+.E E E E E+.3E-6 6. λ =., C = E E+.E E+.E+.E+.E E E E E+.7E-6

147 Chapter 8: Halo orbits and fast rotating tethers Varying direction of the angular momentum Now we deal with the full equations of motion ξ [ ] η = 3ξ ξσ 3 + λσ5 ξs 3ñ sin β sin(α τ) η +ξ [ ] = ησ 3 + λσ5 ηs +3ñsin β cos(α τ) ζ [ ] = ζ ζσ 3 + λσ5 ζs 3ñ cos β (8.) (8.) (8.) where σ = ρ, ñ =sinβ [sin(α τ) ξ cos(α τ) η]+cosβ ζ, S = 5 (ñσ) 3. Again, we have a nonlinear differential system of three degrees of freedom, but now it has time dependent, periodic coefficients whose period is π. Therefore, should periodic solutions exist their period must be a multiple of π. Besides, we have three parameters: the tether s characteristic length λ, and the initial attitude angles α and β. Initial conditions of a periodic orbit can be computed from the previous case by finding π-multiple periodic solutions in the computed families. This solutions will occur for given λ and C. Then variations of either α or β will permit to find periodic orbits for the desired values of the parameters. Note, however, that the unique relevant new parameter is β. Since α always appears in the equations of motion, Eq. (8.) (8.), through circular functions of α = α τ, a change in α can be compensated with a change in the time origin. So, essentially, a variation of α produces a shift in time: the orbit remains the same but starting from a different point. This tangent displacement along the orbit results in a different set of initial conditions for τ =. Therefore, there is no loose of generality in fixing α =what makes φ, =and shows that the relevant parameter is β φ,. Thus, in what follows we will only perform sparse computations of families for variations of α just to check the above mentioned behavior, and do not provide any description of specific families Periodic Halo orbits for C =. Figure 8.4 shows the stability-period diagram of Fig. 8.9 with some resonant periods highlighted. Specifically, we only considered the terms 5, 3 8, 3, of the Farey sequence of order 9, which after multiplied by π give values between the maximum and minimum period of the family. 5 Only the orbit with period 3π/8 is stable. It is 3 π-resonant after 6 orbital periods. 5 The Farey sequence of order n is the sequence of completely reduced fractions between and that have denominators less than or equal to n, arranged in order of increasing size.

148 3 Section 8.3: Varying direction of the angular momentum 9 4 Π Stability indices Π 3 Π 4.3. Period Tether's characteristic length Figure 8.4: Stability of some resonant orbits of the family of periodic orbits with constant attitude for C =.. Vertical lines highlight the resonances and the two horizontal lines mark the critical values k = ± Periodic Halo orbits for C =. After searching the previous computed families, unfortunately we did not find any stable π-periodic solution. However, we select a variety of π-periodic unstable solutions and perform the continuation of the corresponding families for variations of the new parameter. Thus, Fig. 8.5 shows the stability-period diagram of Fig. 8. with some resonant periods highlighted. Specifically, we only considered the terms 3 8, 5, 3 7, 4 9, of the Farey sequence of order 9, which after multiplied by π give values between the maximum and minimum period of the family. The orbits with the resonant periods of Fig. 8.5 are clearly unstable, the less strong instability corresponding to the 4 π/5 resonance that occurs for λ/ = , and stability indices k = , k = The resonant orbit is π-periodic after five orbital periods. Initial conditions (τ =)ofthisorbitare(ξ,η,ζ, ξ, η, ζ) e e+.e e e E+ We use these initial conditions and compute the family of periodic orbits of the nonconstant attitude problem, Eqs (8.) (8.), for variations of β starting from β =. However, after the 5-cycle propagation the stability indices of the initial 4π-periodic orbit are k = 9.6, k =6.7898, the high instability making the propagation of the family 6 Or eigenvalues λ =6.544, λ =/λ, λ 3 =.4664, λ 4 =/λ 3,sincek i = λ i +/λ i, i =,

149 Chapter 8: Halo orbits and fast rotating tethers Π Π.8 Stability indices 3 3 Π 7 3 Π.6 4 Π 5.4 Period Tether's characteristic length. Figure 8.5: Stability of some resonant orbits of the family of periodic orbits with constant attitude for C =.. Vertical lines highlight the resonances and the two horizontal lines mark the critical values k = ±. really hard for this long period of 4π. 7 Thus, we are only able to continue the family up to small values of β,sayβ <, with poor periodicity, say miss distance < 8. Increasing values of β show a slight tendency to decrease instability, but it is not enough at all to find stable orbital motion. For these small values of β the 5-cycle repeat orbits do not distinguish from a single cycle orbit to the precision of the graphics and we do not provide plots Periodic Halo orbits for C =.5 Figure 8.6 shows the stability-period diagram of Fig. 8.3 with some resonant periods highlighted, which have been chosen with the same criteria as before. Despite all the resonances presented in Fig. 8.6 correspond to unstable orbits, we might try the less unstable orbits. For instance the periodic orbit with λ/ = and initial conditions e E+.E E E E+ has a period 3π/4 and stability indices k = (or λ =9.66, /λ =.3943), k =.5638; therefore, it is 3 π-periodic after eight orbits. However, the periodic orbit after eight periods has k = (= λ 8 +/λ8 ), a notable instability that, jointly with the long period of 6 π, in practice makes it impossible to propagate the family. A similar behavior is provided by the 4π/9 resonant orbit (λ/ = ) despite the resonance occurs in a shorter period of π. The stability index k =.57 (k =.35994) after nine periods converts in k =.5 9 making it impossible the continuation of the family. Initial conditions of the resonant orbit are 7 Due to the exponential grow of the eigenvalues the stability indices of a n-fold period periodic orbit are k i = λ n i +/λ n i, i =,, whereλ i are the eigenvalues of the simple period periodic orbit.

150 3 Section 8.3: Varying direction of the angular momentum Stability indices Π Π.9 Π 7 4 Π.5 Π Tether's charcteristic length 6 Π 7 4 Π 5 Π.6 3 Π Figure 8.6: Stability of some resonant orbits of the family of periodic orbits with constant attitude for C =.5. Vertical lines highlight the resonances and the two horizontal lines mark the critical values k = ± Period E E-.E E E E+ A more promising case is found with the π/-periodic orbit, which occurs for λ/ = Despite the orbit has complex stability indices k, =.745 ± i, the moduli of the eigenvalues are relatively small ( λ,3 =.964, λ,4 =.53384) so they are for the resonant orbit, which is π-periodic after four orbits ( λ,3 4 = 3.365). Initial conditions of the resonant orbit are e e-.e e e E+ The continuation of the corresponding family of (thin-halo) π-periodic orbits for variations of β shows negligible changes in the orbits stability character. However, we must say that we found difficulties in maintaining small values of the miss-distance and only continue de family up to β =.6 (.3 ), satisfying ourselves with reaching a miss distance less than 8 for the computed orbits. A sample orbit is presented in Fig. 8.7, in which we note the changes produced in the orbit shape Periodic, eight-shaped orbits for λ =. Having large regions of stability, the family of eight-shaped periodic orbits with a tether s characteristic lenght λ =. is the most promising family to investigate the effects of the tethered satellite s non-constant attitude in its orbital motion. Figure 8.8 shows a detail on the stability-period diagram of Fig. 8.6 with some resonant periodic orbits highlighted. We note that the resonances π/9, π/4, π/7, and π/5 occur with stable orbits; the resonances π/3 and 4 π/9 happen to slightly unstable periodic orbits; all the other resonances considered are found in regions of clear instability, although the π/

151 Chapter 8: Halo orbits and fast rotating tethers 33. Ζ Ξ.4.6 Figure 8.7: Unstable π-periodic orbit after four cycles, for λ =.866 and β =.56. The black (left) and red (right) dots joined by a gray line are the origin and L point, respectively. resonance may deserve attention due to its mild instability (k, =. ±.9 ßß) and strong resonant character (π-periodic after only four cycles). The π/4 resonance The starting orbit is π-resonant after eight cycles. Initial conditions are e+ -.E+.E+ -.E E E+ Increasing values of β produce negligible changes in stability, but the shape of the orbit notably changes as shown in Fig The stability changes to instability at β.33 ( 7 ) and then grows continuously. For no special reason, we stop the continuation at β =.557 where the modulus of the maximum eigenvalue of the periodic orbit is λ =.5. The π/5 resonance The starting orbit is π resonant after five cycles. Initial conditions are e+.e+.e+.e E E+

152 34 Section 8.3: Varying direction of the angular momentum Stability indices Π 3 Π 5 4 Π 3 k 4 Π 7 Π k 4 Π Π 9 5 Π 3 Π 7 Π 4 Π Period. k k Jacobi constant Figure 8.8: Stability of some resonant orbits of the family of eight-shaped periodic orbits with constant attitude for λ =.. Vertical lines highlight the resonances and the two horizontal lines mark the critical values k = ±.. Ζ.... Ξ.4.6 Figure 8.9: Stable eight-shaped periodic orbit after eight cycles with a tether s characteristic length λ =. and β =. The stability changes to instability at β ( 6 )andthengrows continuously. Figure 8.3 shows a sample stable orbit of this family for β. The π/3 resonance The starting orbit is π resonant after six cycles. Initial conditions are e+.e+.e+.e E E+

153 Chapter 8: Halo orbits and fast rotating tethers 35. Ζ.... Ξ.4.6 Figure 8.3: Stable eight-shaped periodic orbit after five cycles with a tether s characteristic length λ =. and β =5.7. and the orbit, with stability indices k =.687 (λ =.497, λ =/λ )andk =.4897, is slightly unstable. After six cycles, the resonant orbit is clearly unstable with k =.93 (= λ 6 +/λ6 ). Continuation of the family for increasing values of β shows a tendency to decrease the instability, and, eventually, the family enters a region of stability. However, we do not continue this family much farther because the numerical continuation experiences difficulties. Figure 8.3 shows a sample stable orbit of this family for β =.. Ζ..... Ξ.4 Figure 8.3: Stable eight-shaped periodic orbit after six cycles with a tether s characteristic length λ =. and β =.7..6 The π/ resonance Despite its complex instability with k, =.7594 ±.966 ßß, the starting orbit is π resonant after only four cycles. Initial conditions are e+.e+.e+.e E E+

154 36 Section 8.4: Appendix The higher modulus of the eigenvalues after the four cycles that make the orbit resonant is λ = Despite the continuation of the family for increasing values of β slightly decreases the instability, we are not able to find stable motion and experienced difficulties to continue the family for values higher than β =.77 (= 4.4 ). The sample orbit of Fig. 8.3 has two unstable modes with real eigenvalues greater than ; specifically λ = , λ 6 = Ζ.... Ξ.4.6 Figure 8.3: Unstable eight-shaped periodic orbit after four cycles with a tether s characteristic length λ =. and β = Periodic orbits of the migrating branch family for λ =. Finally, Fig shows the stability-period diagram in the left plot of Fig. 8.5 with some resonant orbits highlighted. Since these resonant orbits of the constant attitude problem are clearly unstable, we do not expect stable regions in the non-constant attitude problem, and just provide the figure for the sake of completeness. 8.4 Appendix Typically, the procedures used for the continuation of families of periodic orbits rest upon the computation of differential corrections that require the integration of the variational equations. So, for the benefit of programmers, below we provide the variational equations of the rotating tethered satellite problem, and describe the basics ideas on computing differential corrections.

155 Chapter 8: Halo orbits and fast rotating tethers Stability indices 4 Π 4 Π 9 Π Period Jacobi constant Figure 8.33: Some resonant orbits of the migrating (from thin-halo to eight-shaped) family of periodic orbits with constant attitude for λ =.. Vertical lines highlight the resonances and the two horizontal lines mark the critical values k = ± Variational equations In the general case, a first order expansion of Eqs. (8.) (8.) around (ξ+δξ, η+δη, ζ+δζ) gives δ ξ [ ] = 3δξ σ (σδξ+3ξδσ)+ 5 λσ4 ξs 3ñ sin β sin(α τ) δσ [ ] + λσ5 S δξ + ξδs 3 δñ sin β sin(α τ) +δ η ] δ η = σ (σδη+3ηδσ)+ 5 λσ4[ ηs +3ñsin β cos(α τ) δσ [ ] + λσ5 S δη + ηδs +3δñ sin β cos(α τ) δ ξ δ ζ = [ ] δζ σ (σδζ+3ζδσ)+ 5 λσ4 ζs 3ñ cos β δσ [ ] + λσ5 S δζ + ζδs 3 δñ cos β where δσ = σ 3 (ξδξ+ ηδη+ ζδζ) δñ = sinβ sin(α τ) δξ sin β cos(α τ) δη +cosβ δζ δs = 5ñσ(σδñ +ñδσ)

156 38 Section 8.4: Appendix Variations of λ: If the variations are produced by the variation of the tether s characteristic length λ, we need to complete the variational equations with the terms δ ξ = δ ξ [ ] + σ5 ξs 3ñ sin β sin(α τ) δ η [ ] = δ η + σ5 ηs +3ñ sin β cos(α τ) δ ζ = δ ζ [ ] + σ5 ζs 3ñ cos β Variations of α : δñ by If the variations are produced by the variation of α, we need to replace δñ = δñ +sinβ cos(α τ) ξ +sinβ sin(α τ) η and complete the variational equations with the terms δ ξ = δ ξ 3 λσ5 ñ sin β cos(α τ) δ η = δ η 3 λσ5 ñ sin β sin(α τ) δ ζ = δ ζ Variations of β: to replace δñ by Finally, if the variations are produced by the variation of β, we need δñ = δñ +cosβ [sin(α τ) ξ cos(α τ) η] sin β ζ and complete the variational equations with the terms δ ξ = δ ξ 3 λσ5 ñ cos β sin(α τ) δ η = δ η + 3 λσ5 ñ cos β cos(α τ) δ ζ = δ ζ + 3 λσ5 ñ sin β 8.4. Basics on differential corrections computation Let x be of dimension m and let 8 be an m-dimensional differential system to which a solution ẋ = F (x,t) (8.3) x = x(t) (8.4) is known for certain initial conditions. Assume also that y = x + ξ is another solution to Eq. (8.3). Therefore, ẏ = ẋ + ξ = F (x + ξ,t). 8 This section has been taken from: Lara, M., and Russell, R.P., «Fast Design of Repeat Ground Track Orbits in High-Fidelity Geopotentials,» The Journal of the Astronautical Sciences, 8, in press.

157 Chapter 8: Halo orbits and fast rotating tethers 39 If we assume y to be close enough to x for all t, the displacements ξ will be small. Then, a Taylor series development of the force function up to the first order brings the variational equations ξ = J ξ (8.5) where J is the Jacobian matrix of elements F i / x j,(i, j =,...,m). The variational equations, Eq. (8.5), are a linear homogeneous system of ordinary differential equations whose general solution is given by a linear combination of m independent solutions. That is, m ξ = C i ξ i (t), (8.6) i= where C i are m arbitrary constants and ξ i = ξ i (t) form a fundamental system to Eq. (8.5). Now, assume that after a time t = T the known solution x = x(t) is periodic or almost periodic, and that we want to determine the values of C i that make y also to be periodic with, probably, a new period T = T +ΔT. Then, for the theorem on existence and uniqueness of solutions to initial value problems, it is enough for the periodicity condition y(t) =y(t + T ) to be fulfilled that y() = y(t +ΔT ). (8.7) For close solutions to x the correction ΔT is small and, neglecting higher order quantities and noting that x/ T = F (x(t ),T ), the periodicity condition Eq. (8.7) is written or x() + ξ() = x(t )+F (x(t ),T )ΔT + ξ(t ) F (x(t ),T )ΔT + m [ξ i (T ) ξ i ()] C i = x() x(t ) (8.8) i= Therefore, the problem of computing a periodic solution close to a known periodic or almost periodic solution is reduced to, first, computing a fundamental system to Eq. (8.5) and, then, solving the linear system Eq. (8.8) for the unknowns C i and ΔT. Remarkably, when the fundamental system to Eq. (8.5) is computed using the identity matrix I m as initial conditions, the constants C i result to be the required corrections to the initial conditions x() that determine the initial conditions y() of the new periodic solution. There are several matters that may make solving Eq. (8.8) difficult, especially when the equations of motion Eq. (8.3) accept integrals. The linear system has fewer equations than unknowns; the matrix to invert may be singular or bad conditioned; etc. These are problems of linear algebra that may be approached in a variety of forms, and there is a wealth of algorithms for the continuation of families of periodic orbits based on the computation of differential corrections. The interested reader is referred to the literature. Recommended readings. Hénon, M., «Exploration Numérique du Problème Restreint. II.Ñ Masses égales, stabilité des orbites périodiques,» Annales d Astrophysique, Vol. 8, No., 965, pp Szebehely, V., Theory of Orbits The Restricted Problem of Three Bodies, Academic Press, New York, Broucke, R.A., «Stability of Periodic Orbits in the Elliptic Restricted Three-Body Problem,» AIAA Journal, Vol. 7, No. 6, 969, pp Siegel, C.L., Moser, J.K., Lectures on Celestial Mechanics, Springer-Verlag, 97.

158 4 Section 8.4: Appendix 5. Deprit, A., Henrard, J., «Natural families of periodic orbits,» Astronomical Journal, Vol. 7, No., 967, p Deprit, A.. «Intrinsic Variational Equations in Three Dimensions,» Celestial Mechanics, Vol. 4, No., 98, pp Karimov, S. R., Sokolsky, A. G., «Periodic motions generated by Lagrangian solutions of the circular restricted three-body problem,» Celestial Mechanics and Dynamical Astronomy, Vol. 46, No. 4, 989, pp Allgower, E.L., Georg, K., Numerical continuation methods, Springer, Simó, C., «On the analytical and numerical approximation of invariant manifolds,» ind. Benest and C. Froeshlé (eds.), Modern methods in Celestial Mechanics, Editions Frontières, 99.. M. Lara, J. Peláez, «On the numerical continuation of periodic orbits. An intrinsic, 3- dimensional, differential, predictor-corrector algorithm,» Astronomy and Astrophysics, Vol. 389,, pp

159 Chapter 9 Stability at the Collinear Lagrangian Points 9. Introduction The Lagrange equilibrium solutions of the Circular Restricted Three Body Problem (CRTBP) have turned out to be much more important than they seemed at a first sight. There have been many space applications in the past, and at present, which have used these libration points as essential elements of space missions (see Ref. [7]). They will likely play a much more important role for future space exploration missions. Particularly attractive are the collinear points (L, L and L 3 ) given their location and accessibility. Unfortunately all of them are unstable, which means a spacecraft to be kept at or orbiting around them will require correction manoeuvres typically to be performed at the expense of propellant mass. Colombo, one of the pioneer about the use of space tethers showed in [8] the feasibility of controlling the unstable nature of the collinear Lagrangian points exploiting a varying-length tether system. Such a concept was later studied more deeply by Farquhar in Refs. [9, ] where he set out the following problem: «Consider two satellites of equal mass that are connected by a light cable of adjustable length. Is it possible to stabilize the position of the mass center of this configuration in the vicinity of a collinear libration point by simply changing the length of the cable with an internal mechanism?». This control scheme, which exploits the capability of dumbbell systems of shifting the centre of gravity position with respect to the centre of mass in a controlled manner, allows keeping the position of an artificial satellite close to the Lagrangian points without using propellant. Misra et al. [] later studied the problem with a different approach and included the possibility of using rotating tethered system of constant length to make easy the stabilization process. In the latter study the dynamics of a passive (constant-length) rotating dumbbell system were investigated without addressing possible control strategies. The need for further, more in-depth dynamic and control analysis of the system was highlighted. In general, tether dynamics is complex (see [4]) and close to the collinear points it is influenced by many different factors. One of them is the reduced mass ν of the primary around which the tether is moving. Usually this parameter is small and the Hill approximation al- 4

160 4 Section 9.: Previous Analysis lows a much more simple description of the dynamics which permits to gain an insight into the complex evolution of the tethered system. For these cases the Hill formulation directly provides an excellent approximation to the problem. Moreover, it gives significant clues for the analysis of the general case in which ν is of order unity; this way the approximation makes easier the indispensable numerical analysis associated with a detailed description of the dynamics. Most of the analysis performed previously neglected the tether mass. However, some cases considered in the literature involve very long tethers; for instance, Misra et al. in [] described the dynamics of a 5 km long tether. Even with very light materials the mass of the tether could be important; the SEDS tether had a mass of.33 kg/km and using the same tether (Spectra,.7 mm of diameter) the mass would reach, approximately, 5 kg. Farquhar in [9] consider a tether km long made of Aluminum and with a mass about 54.5 kg; assuming a wire the diameter of the tether would be.5 mm. Even with such a very fine tether the mass start to be significant. Therefore, it seems appropriate to include the mass of the tether in the formulation, specially when some control strategies with variable-length tethers required to increase the tether length in a significant way. In this report we investigate the dynamics of a tethered system near the collinear libration points exploiting the benefits of the Hill approach. We try a simple strategy that permits, using a feedback control law, to stabilize the system around equilibrium positions which are basically unstable. Rotating tethers, with constant or variable length, are investigated. More recently Peláez and Scheeres [8, 9] proposed to place electrodynamic tethers for permanent power generation at points in the neighborhood of the Lagrangian points of the inner Jupiter moonlets (Metis, Adrastea, Amalthea and Thebe). The electrodynamic tether will be deorbiting the moonlet by using its gravitational attraction; in doing so it converts the mechanical energy of the moonlet into electrical energy that can be used onboard. As a consequence, a continuous power can be extracted from the orbital energy of the moonlet. In that case, current control was proposed as the sole mean to stabilize both the position and the attitude of a constant-length non-rotating electrodynamic tether system placed in the vicinity of the unstable Lagrangian points of the moonlets. The dynamical analysis of [9] shows that there exist equilibrium positions where the tether could be operated appropriately. Some of these equilibrium positions are stable and other unstable. In this report electrodynamic tethers are not considered; but in the near future, the effects of the electrodynamics forces on the stability of the equilibrium positions close to the Lagrangian points would be taking into account for tethers made of conductive materials. 9. Previous Analysis Firstly, we consider the article [] by Farquhar. He obtained the governing equations of a tethered system close to a collinear point: ) taking the cartesian coordinates of the end masses as generalized coordinates (the tether length is a constraint), ) assuming a massless tether with two equal end masses, 3) expanding the equations around the small primary and neglecting terms higher than second order, and 4) assuming no additional perturbations on

161 Chapter 9: Stability at the Collinear Lagrangian Points 43 the system. Farquhar found unstable equilibrium positions close to the collinear points. He studied first the one dimensional motion of the tether, neglecting the coupling terms. Then, he tackle the tridimensional analysis and introduces a control law involving the orientation of the cable; the out-of-plane motion turns out to be not controlable with this law, and a detailed analysis is carried out to obtain the gains which ensure stability for the in-plane motion. The approach gathered in the article [] by Misra el al. is devoted to the same purpose, having one s eye on the Earth-Moon system. However, there are differences regarding Ref. []; some of them are listed in what follows: ) the paper only consider the in-plane motion and it takes as generalized coordinates the cartesian coordinates of the system center of mass and the tether in-plane libration angle, ) they use a massless tether, but the end masses can be different. In a classical analysis, similar to Farquhar s one, they linearize the equations of motion in the neighborhood of a collinear Lagrangian point and present some numerical experiments to avoid the instability; they are based on the control of the tether length and on the effect of rotation on tether motion. Accordingly, a control law is provided showing its stabilizing efficacy for specific values of the control gains. Moreover, the possibility of stabilize the dynamics of a tethered system by means of its rotation is presented in a particular case. More complex analysis have been carried out by Wong and Misra in papers [, 3], where the three-dimensional controlled dynamics of a spinning tether connected three spacecraft system near the second libration point of the Sun-Earth system is examined. The center of mass follows a predefined trajectory (Lyapunov orbit, Lissajous trajectory or halo orbit); in order for this spinning formation to be useful as an interferometer, the tether librations must be controlled in a way such that the observation axis of the system can be pointed in a specified direction, while the center of mass follows one of these trajectories. In this report a new approach facing the general treatment of a two point tethered system moving in the neighborhood of a collinear libration point is presented. We extend the analysis previously cited by accommodating: ) a three dimensional analysis allowing different mass configurations and considering the effect of the tether mass, ) a more solid linearization process which avoid, in some extension, the problems associated with several small magnitudes, 3) a reduced number of parameters describing the motion of the system, and 4) a more tractable set of equations which make easy the understanding of the main feature of the dynamics. Our approach is based on the Hill approximation; this formulation uses a spatial scale in the neighborhood of the small primary where the gravity gradient of the main primary, the inertia Coriolis force and the gravitational attraction of the small primary are of the same order of magnitude. Such an approach allows to analyze the dynamics of the system in the Lagragian points with the suitable accuracy for the majority of the binary systems of the Solar System and permits to present the equations of motion in an understandable way even when a general case is considered. Moreover, there is no need of carrying out a linearization

162 44 Section 9.3: Equations of Motion to achieve the set of equations since the simplification of the original equations is due to an asymptotic expansion in the parameter ν, i.e., the reduced mass of the small primary which is quite small in a great number of cases. At last but not least, this formulation provides a closed model involving only one non-dimensional parameter which sheds light into the importance and the organization of each and every element implicated in the dynamics. 9.3 Equations of Motion In this Chapter we analyze the equilibrium positions of an inert tether in the neighborhood of one collinear Lagrangian point. The starting point are the equations ( ) that we collect here for convenience. The set of equations is as follows: { } ξ η (3 ρ 3 )ξ = λ ρ 5 3Ñ cos ϕ cos θ ξs (Ñ ρ ) (9.) θ +(+ θ) { η + ξ + η ρ 3 = λ ρ 5 3Ñ cos ϕ sin θ ηs (Ñ ρ ) { } ζ + ζ( + ρ 3 )= λ ρ 5 3Ñ sin ϕ ζs (Ñ ρ ) [ ] Is ϕ tan ϕ +3cosθsin θ = 3Ñ ( ξ sin θ + η cos θ) I s ρ 5 cos ϕ I ϕ + [ s ϕ +sinϕcos ϕ ( + I θ) ] +3cos θ s } (9.) (9.3) (9.4) = 3Ñ ( sin ϕ[ξ cos θ + η sin θ]+ζcos ϕ) (9.5) ρ5 where ρ = ξ + η + ζ and the quantity Ñ and the function S (x) are given by Ñ = ξ cos ϕ cos θ + η cos ϕ sin θ + ζ sin ϕ, S (x) = 3 (5x ) (9.6) In these equations the new parameter that captures the influence of the tether length is λ = ɛ a, andweassume,inprinciple,that λ O() ν/3 Note that for a tether of varying length the parameter ɛ = L d is a function of time since the l tether length L d (t) is changing. Moreover, some terms of these governing equations involve the ratio: I s d = L { Λ d ( + 3 cos φ) ( I s L d 3sin )} (9.7) φ Λ d Note that in this analysis the mass of the tether is included. In order to neglect the tether mass, we only have to introduce the condition Λ d =in the above expression. For a tether of constant length the parameter ɛ is also constant and the quotient I s /I s vanishes, that is, I s /I s =. These equations have been derived under the Hill assumption (limit ν ) for an inert, non-rotating tether of varying length.

163 Chapter 9: Stability at the Collinear Lagrangian Points 45 In the search for equilibrium positions we will consider the tether tension given by ) T m m ω L d (Ñ ϕ +cos ϕ{( + m + m θ) +3cos θ} + ρ 3 {3 } L d ρ L d (9.8) This expression has been derived under the following assumptions: ) the tether is inert, )the mass of the tether has been neglected and 3) the Hill approach has been taken into account. Obviously, at any equilibrium position the tether tension must be positive, because a cable does not support compression stress. The above expression, when particularized for an equilibrium position, takes the form ) (Ñ T T c cos ϕ{+3cos θ} + ρ 3 {3 }, T c = m m ω L d (9.9) ρ m + m We will use expression (9.9) to check the tension which appears in a given equilibrium position; if the tension is positive the equilibrium position will exist; if negative, the equilibrium position will not exist. 9.4 Equilibrium equations The equations determining the equilibrium positions, that is, the steady solutions of the system (9.-9.6) are: ξ (3 ρ ) { )} (Ñ 3 = λρ 5 3 Ñ cos θ cos ϕ ξs (9.) ρ { )} η (Ñ ρ 3 = λ ρ 5 3 Ñ sin θ cos ϕ ηs (9.) ρ ζ (+ ρ ) { )} (Ñ 3 = λρ 5 3 Ñ sin ϕ ζs (9.) ρ ρ 5 sin θ cos θ cos ϕ = Ñ (η cos θ ξ sin θ) (9.3) ρ 5 sin ϕ cos ϕ [ +3cos θ ] =3Ñ ( ξ sin ϕ cos θ η sin ϕ sin θ + ζ cos ϕ) (9.4) We could perform a selective search of equilibrium positions, starting by detecting the equilibrium positions which exist on the coordinates planes Oξη, Oξζ and Oηζ. However, the most important lie on the orbital plane of primaries Oξη and, instead of a selective search, the analysis will be focused on these positions. First of all, an approximation for the gravitational potential of both primaries has been made in the equations (9.-9.4). For example, the gravitational potential of the small primary has been expanded in powers of the ratio L/r,wherer is the distance r = OG (see figure B.4), and terms of order equal or greater than 3 have been neglected in such an expansion.

164 46 Section 9.5: Equilibrium positions on the plane Oξη L r 4 3 λ Figure 9.: Ratio L/r vs. λ, fora =/4. This approach works fine when L/r. Most of the times, parameter λ is small for feasible cable lengths in the binary systems of interest in the Solar System. At the most interesting equilibrium positions on the Oξ axis we disclose here this result that relation can be expressed in terms of non dimensional variables as follows: L 3 ρ = 3 e r 3 a and it is plotted as a function of λ in figure 9. for a =/4. Note that the approach used in the deduction of the equations of motion is fine when λ is small but it fails when λ is of order unity. Thus, for values of λ of order unity it would be necessary to improve the description of the gravitational actions of both primaries. 9.5 Equilibrium positions on the plane Oξη When the center of mass is on this plane ζ =and equation (9.) becomes: Ñ sin ϕ = As a consequence there are only two possible cases: ) Ñ =and ) sin ϕ =. Let us assume, for a moment, that Ñ =. Equations (9.-9.4) and the condition Ñ =take the form: ( ξ 3+ ρ 3 3 ) λ ρ 5 = (9.5) ( η ρ 3 3 ) λ ρ 5 = (9.6) sin θ cos θ cos ϕ = (9.7) sin ϕ cos ϕ [ +3cos θ ] = (9.8) cos ϕ(ξ cos θ + η sin θ) = (9.9) Observe that the solution cos ϕ =leads, via (9.9), to a negative tension what is impossible for a tether; therefore we will assume that cos ϕ. But in such a case equation (9.8) provides the solution sin ϕ = ϕ =(remember ϕ [ π, π ]). As a consequence we always have sin ϕ =and there is two different possibilities depending on the value of Ñ.

165 Chapter 9: Stability at the Collinear Lagrangian Points 47 Case Ñ =: When Ñ =the equations ( ) reduce to ( ξ 3+ ρ 3 3 ) λ ρ 5 = (9.) ( η ρ 3 3 ) λ ρ 5 = (9.) sin θ cos θ = (9.) ξ cos θ + η sin θ = (9.3) These equations provide two possible equilibrium positions parametrized with ρ and summarized in the following table: E.P. ξ η ζ θ ϕ λ Tension T Existence ±ρ,π ±ρ ±π/ 3 ±ρ,π 3 ρ T c (3 ρ 3 ) 3ρ3 > 3 ρ5 ( ρ 3 3) T c( ρ 3 ) Never ρ 5 3 (3 ρ 3 ) T c(3 + ρ 3 ) 3ρ3 > 4 ±ρ ±π/ 3 ρ T c ( ρ 3 ) Never 5 ±ξ ±η ( ) (9.34) (9.36) 3ρ 3 > Table 9.: Possible equilibrium positions on plane Oξη Case Ñ : When Ñ, equations (9.-9.4) take the form: ξ (3 ρ ) { )} (Ñ 3 = λρ 5 3 Ñ cos θ ξs (9.4) ρ { )} η (Ñ ρ 3 = λ ρ 5 3 Ñ sin θ ηs (9.5) ρ ρ 5 sin θ cos θ =sinθcos θ ( ξ + η )+ξη (cos θ sin θ) (9.6) By eliminating the term S between equations ( ) we obtain: ξη= λ ρ 5 (ξ cos θ + η sin θ)(ξ sin θ η cos θ) = λ ρ 5 [(ξ η )sinθcos θ + ξη(sin θ cos θ)] and taking into account equation (9.6) this relation becomes: ξη= λ sin θ cos θ (9.7) By introducing relation (9.7) in equation (9.6) we obtain the relation: sin θ cos θ ( ρ 5 + ξ η + λ cos θ ) =

166 48 Section 9.5: Equilibrium positions on the plane Oξη In summary we arrive to the following system: ξ (3 ρ ) { 3 = λρ 5 3 Ñ cos θ ξs { η ρ 3 = λ ρ 5 3 Ñ sin θ ηs )} (Ñ ρ )} (Ñ ρ (9.8) (9.9) ξη= λ sin θ cos θ (9.3) sin θ cos θ ( ρ 5 + ξ η + λ cos θ ) = (9.3) where only three equations are independents (one equation is linear combinations of the other three). Equation (9.3) provides three subcases, namely:.- sin θ =. In this subcase equation (9.3) indicates that ξ =or η =.Butthecase ξ =leads to the solution summarized in Table 9. and already known. Therefore we only have to study the situation η =which leads to the solution 3 summarized in Table cos θ =. In this subcase equation (9.3) indicates newly that ξ =or η =. But the case η =leads to the solution summarized in Table 9. and already known which is not an equilibrium due to the tether tension. Therefore we only have to study the situation ξ =which leads to the solution 4 summarized in Table 9. which is not an equilibrium because the parameter λ cannot be negative. 3.- ρ 5 + ξ η + λ cos θ =. In this subcase we can form the following equations ξ + η = ρ ξ η = ρ 5 λ cos θ } ξ = (ρ ρ 5 λ cos θ) η = (ρ + ρ 5 + λ cos θ) from which we get 4ξ η = ρ 4 (ρ 5 + λ cos θ) This relation, taking into account (9.3), becomes λ sin θ = ρ 4 (ρ 5 + λ cos θ) cos θ = ρ4 ρ λ ρ 5 λ (9.3) and it permits to express (ξ,η ) as follows ξ = λ ρ +ρ 7 ρ 4 4ρ 5, η = λ + ρ +ρ 7 + ρ 4 4ρ 5 (9.33) Using these expressions it is possible to eliminate the variables (ξ,θ) in equation (9.9) which becomes: η (5ρ ρ 7 +5ρ 4 +6ρ λ 5λ )=

167 Chapter 9: Stability at the Collinear Lagrangian Points 49 T T c Figure 9.: Tension vs. the nondimensional tether length λ λ Assuming η the only positive root of this equation turns out to be: ( ) ρ λ = ρ 3 ρ (9.34) 5 By introducing this value of λ in relations (9.33) the values of (ξ,η) can be obtained; they turn out to be (ξ = ±ξ,η = ±η ) where (ξ,η ) are given by: 45ρ ξ = ρ 6 8ρ 3, 5ρ 5ρ η = ρ 6 8ρ 3 (9.35) 5ρ Notice that for each one of the solutions for (ξ,η) the inclination θ of the tether can be obtained by the expressions ( ). Finally, the tension at the equilibrium position, for a massless tether, takes the value T = 5ρ3 + T c ρ 6 8ρ 3 ρ 3 (9.36) Thus, we arrive to a parametric description, taking ρ as the parameter, of the equilibrium position which is summarized as 5 in Table 9.. Figure 9.3 shows the situation of the equilibrium positions in the plane Oξη for varying values of ρ. The minimum value of ρ =(/3) /3 takes place at two symmetric positions on the Oη axis (the closets to the small primary) and for them the θ angle vanishes. The arrows show the inclination of the tether in the corresponding equilibrium position η - ρ ξ λ Figure 9.3: On the left: sketch in the plane (ξ,η) of the equilibrium position 5 in Table 9.; the arrows indicate the value of θ. On the right: distance ρ vs. the non-dimensional tether length λ

168 5 Section 9.6: Stability analysis for constant length 9.6 Stability analysis for constant length In this section we analyze the stability properties of the equilibrium positions detected in previous sections. First of all, we consider the equilibrium position corresponding to the solution 3 summarized in Table 9.. In this analysis the tether length is constant. Notice that the governing equations (9.-9.5) can be expressed like an autonomous dynamical system y = F ( y,λ) where the vector y is y =(ξ,η,ζ,θ,ϕ, ξ, η, ζ, θ, ϕ) T. At the equilibrium position y takes the value y e given by: y e =(ξ e = ±ρ e,η e =,ζ e =,θ e =,ϕ e =, ξe =, η e =, ζ e =, θ e =, ϕ e =) T and the value of the parameter λ is λ e = ρ5 e 3 (3 ρ 3 ) > where ρ e > e ( ) 3 3 We introduce the variations y = y e + δ y which are governed by the following system of linear equations:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, dδ y = M δ y M(χ) =,,,,,,,,, dτ k, k, k 3, k 4, k 5,,,,, k, k, k 3, k 4, k 5,,,,, k 3, k 3, k 33, k 34, k 35,,,,, k 4, k 4, k 43, k 44, k 45,,,,, k 5, k 5, k 53, k 54, k 55,,,,, The coefficients k,...,k 55 involved in the square matrix M are functions of the equilibrium solution (ξ e,η e,ζ e,θ e,ϕ e ) that we are analyzing here; as a consequence, they are functions of ρ e (or λ e ). The eigenvalues of M provide the stability properties of the equilibrium positions. In general there are five pairs of conjugate complex numbers which are functions of ρ e (or λ e ). In this case the coefficients k [i,j] are given by K = 5ρ 3 e ρ 3 e,,,,, 6ρ 3 e ρ, 3 e, ± 3ρ3 e ρ, e,, 7ρ 3 e ρ, 3 e, ± 3ρ3 e ρ e, ± 3 ρ, 4 e, 3 ρ3+ e ρ, 3 e,, ± 3 ρ 4 e,, 4ρ3 e +3 ρ 3 e

169 Chapter 9: Stability at the Collinear Lagrangian Points 5 The upper sign corresponds to the equilibrium position emerging from the collinear point L and the lower sign to the equilibrium position emerging from L. The secular equation of the matrix M is (s +4) (s + +7ρ3 e ρ 3 ) (s 6 + p(ρ e )s 4 + q(ρ e )s + r(ρ e ) ) = e where the polynomials p(ρ e ), q(ρ e ) and r(ρ e ) are p(ρ e )= (ρ3 e ) ρ 3 e q(ρ e )= 5 ρ6 e 6ρ3 e 4 ρ 6 e r(ρ e )= 6(45 ρ6 e +9ρ 3 e ) ρ 6 e r(ρ e) The eigenvalues ρ 3 e ρ s = ±i and s = ± e ρ 3 i e Figure 9.4: Polynomial r(ρ e ) vs. ρ e are associated with the generalized coordinates (ζ,ϕ) and they provide dynamic stability. The sources of possible instabilities should be found in the other generalized coordinates (ξ,η,θ). The eigenvalues associated with (ξ,η,θ) depends on the roots of the cubic equation f(x, ρ e )=x 3 + p(ρ e ) x + q(ρ e ) x + r(ρ e )= (9.37) Any positive root of (9.37) gives place to a pair of real eigenvalues one of.7 them positive: the equilibrium position turns out to be unstable in such acase. Anynegative root of (9.37). f(x, ρ e) - gives place to a pair of pure imaginary eigenvalues which do not destabilize the system. Figure 9.4 describes -4-6 the values of the polynomial r(ρ e ) vs. ρ e showing that it always takes negative values. Figure 9.5 plots the left x hand side of (9.37), that is f(x, ρ e ), vs. x for several values of ρ Figure 9.5: Polynomial f(x, ρ e ) vs. x for several e. As a values of ρ e =.7,.8,...,. consequence, in an acceptable range of values of ρ e a root of the cubic equation (9.37) is always positive. The other two roots are negative. In summary, there are four eigenvalues which are pure imaginary numbers and two eigenvalues which are real numbers: one of them positive and the other one negative. Here i represents the imaginary unit

170 5 Section 9.6: Stability analysis for constant length Figure 9.6 shows the unstable eigenvalue s as a function of the nondimensional tether length λ. Thevalue s = which corresponds to λ = is typical from the unstable character of the collinear Lagrangian point L in the Hill approach. What this figure shows is that the presence of the tether does not improve the stability properties of the equilibrium position. Contrary to expectations, the instability increases clearly with the tether length. s λ Numeric Asymptotic Figure 9.6: The real unstable eigenvalue s vs. λ This important fact can be described with more detail using an asymptotic solution for the cubic equation (9.37). In effect, the non-dimensional tether length λ is usually small. In such a case, the equilibrium position can be expressed by the approximated formulae (.). Retaining only the terms of order λ, equation (9.37) becomes x 3 +( 36 3 /3 λ) x +( /3 λ) x /3 λ and this equation has three roots that can be approximated by x /3 λ, x /3 λ, x /3 λ The roots (x,x ) are always negative and they do not introduce instability on the system. On the contrary, the root x 3 is always positive and it is the source of instability which characterize this equilibrium position. Notice that for increasing values of λ the value of x 3 also increases. Thus long tethers give place to a stronger instability. In the paper [] by Misra et al. there are some comments and figures that, incorrectly interpreted, could introduce some confusion on this subject. Figure 9.7 is a reproduction, based in our formulation, of the figure in reference []. It corresponds to an inert tether km long taking the system Earth Moon as primaries. Tether mass is neglected and both end masses are equal. In this figure we show the time evolution of the coordinates of the center of mass G and the end masses; initially the tether is placed along the Ox axis and its center of mass G is situated km away from the Lagrangian point L. Figure shows the separation of the different elements from the Lagrangian point L. From a qualitative point of view the agreement between this figure an figure of [] is excellent; from a quantitative point of view there exist small differences which should be attributed to the Hill approach used in our formulation. However, for this example the value of the non-dimensional tether length λ is quite small: λ Thus, if we consider a point mass with the same initial conditions as the center of mass G of the tethered system, its time evolution turns out to be identical to the time evolution of the system center of

171 Chapter 9: Stability at the Collinear Lagrangian Points x of G m free m tethered 5 m free 5 y of G x y - m tethered m free - -5 m free t (in hours) t (in hours) Figure 9.7: Reproduction of figure in []. Here x is the distance from L and not the distant from the center of mass of the Moon, as in our formulation (distances in km). mass. That is, for a free material particle initially placed at the position of G, the time evolution of its distance x(t) will be given also by the red line on the left picture of figure 9.7. Thus, the effects of the tether length will appear when the parameter λ takes greater values; in the Earth-Moon system due to the high value of the characteristic length L c = l ν / km would be necessary very long tethers. However, in systems where L c is smaller the effect of the tether length would be more remarkable.

172 54 Section 9.6: Stability analysis for constant length

173 Chapter Stabilization of the Collinear Lagrangian Points with variable length tethers. Introduction In this Chapter we face a control problem. Since the equilibrium position is unstable, the following question raises: is it possible to reach an stable system by changing the tether length in an appropriate way? Now the parameter λ(t) becomes a function of the time. In the governing equations (9.-9.5) the influence of a varying tether length appears in the terms I s /I s ; for a massless tether this term turns out to be: I s d = L = λ(t) I s L d λ(t) We focus the analysis in the three unstable coordinates (ξ,η,θ).. One-dimensional analysis To clarify the stability analysis we follow the Farqhuar scheme and we begin with the study of the tether motion when it is restricted to move on the Ox axis. Thus we neglect the coupling due to the Coriolis force an we focus on the one dimensional motion of the tether. This situation can be understood assuming that there are forces normal to the Ox axis that keep the tether on this line. Such a movement is described by the equation: ξ = ξ (3 ρ ) 3 3ξ λ ρ 5 with ρ = ξ. For the sake of simplicity we will assume the tether in the neighborhood of the external collinear point L ;thusξ> and ρ = ξ. This equation takes the form ξ =3ξ ξ 3 λ ξ 4 (.) 55

174 56 Section.: One-dimensional analysis and provides a first integral: the conservation of the total energy of the system: T + V = E where the kinetic T and the potential V energies turn out to be 5 V (ξ) dv dξ d V dξ T = ξ, V(ξ,λ) = 3 ξ ξ λ ξ 3-5 E Here E, the total energy of the system, is constant and its value is fixed by the initial conditions (ξ, ξ ). Figure. shows an sketch of the potential curve V = V (ξ) for the particular value λ =.. Once the initial conditions (ξ, ξ ) ξ Figure.: Potencial energy V (ξ,λ) and its derivatives for the particular value λ =. have been fixed, the value of E is known and a further integration provides the solution ξ dξ τ = ± ξ (E V (ξ,λ)) where the sign (+) or ( ) is determined by the initial conditions. The motion takes place in the interval of values of ξ where the potential curve is below the line of the total energy E. Notice that, for a given value of the non-dimensional tether length λ, the potential curve has a maximum just at the equilibrium position given by λ e = ξ e (ξ3 e 3 ) ξ e = ξ e (λ e ) This expression can be expanded when the tether length is small and provide the asymptotic solution ξ e ( 3 ) λ 9 λ + O(λ 3 ) (.) The convergence of this solution is bad and we need to take many terms in order to get a precise value for ξ e ; however, when λ is really small the two first terms give a first approximation that can be useful. In any case, note that such an equilibrium position is unstable. Thus, if ξ >ξ e the center of mass fall down to the right along the potential curve; if ξ <ξ e the center of mass fall down to the left. Just on the equilibrium position E = V (ξ e ) and ξ e =. However, if we separate the particle a small amount Δξ from the equilibrium position its velocity is no longer zero and takes the value ξ d V dξ ξ=ξe Δξ =Δξ 3+ ξ 3 + λ ξ 5 (.3)

175 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 57.. Tether control It is important to underline that the analysis above is only true for a tether of constant length, that is, when λ is constant. For a tether of varying length the energy equation takes the form: de dτ = V λ λ de dτ = ξ λ 3 As a consequence, to change the tether length causes a change in the total energy of the system, in the potential energy V and also in the position of the equilibrium point (the maximum of the potential curve). The problem that we face here is the following: is it possible to choose a law for λ(t) which permits to gain control on the tether motion? In this one-dimensional case this problem seems simple. In effect, let us assume that the initial conditions (ξ, ξ ) are fixed. We can use equation (.) to determine the function λ = λ(τ) that provides a previously fixed motion for the center of mass of the tether. For example, we can impose the following law for the coordinate ξ: ξ ξ(τ) =ξ + sin ω s τ ω s In such a case the tether length which is determined by equation (.) is given by λ(τ) = ( 3 ω s +) 5 ξ ω s +( ω sξ 3 ξ 3 ) ξ ω s sin 5 ω s τ ( 4 3 ω s +5) ξ ξ ω s 4 sin 4 ω s τ ( + ωs) ξ 3 ξ sin 3 ω s τ+ sin ω s τ +( 3 3 ω sξ 3 5ξ 3 ) ξ ξ ω s sin ω s τ ξ ξ Once the value of λ(τ) is known we can determine the tether tension which is given by: ( ) T = T λ 3+ ρ 3 λ λ + λ, T = m m ω lν/3 (.4) 4 λ m + m a From this point of view the most general motion for the coordinate ξ would be ξ(τ) =ξ A + ξ B e αωsτ cos(ω s τ + φ A ) (.5) wherewehavefiveparametersfree: (ξ A,ξ B,ω s,α,φ A ). By adjusting appropriately the values of these parameters we can provide useful solutions to the tether control problem. Notice that two of them are fixed by the initial conditions (ξ, ξ ) from which the motion starts. The relationships are } ξ cos φ A ξ = ξ A + ξ B cos φ A ξ A = ξ + ω S (sin φ A + α cos φ A ) ξ = ξ B ω s (α cos φ A +sinφ A ) ξ ξ B = ω S (sin φ A + α cos φ A ) ω s

176 58 Section.: One-dimensional analysis O Dim. x L =( 3 ) 3 L c N. Dim. ξ L =( 3 ) 3 L 3 3 λl c Δx 3 3 λ () () Figure.: Distances from the small primary and from the collinear point L when λ (L c = lν /3 ) However and before to continue with the analysis we review briefly the stabilization technique proposed by Farquhar in [9, ] and considered by Misra et al. in []. Such a technique is based in the following reasoning. Let us assume that the tether length is L and for that length we have the equilibrium position labeled with () in figure.. If the center of mass G of the tether is situated on the right of this position is acted by a force that impulses it toward the right. By increasing the tether length up to L = L +ΔL we move the equilibrium position up to the point labeled with () in figure.; now the force acting on G impulses it toward the left. Then we decrease the tether length in order to move the equilibrium position on the left side of G... Thus, by changing the tether length in appropriate way the center of mass G can be stabilized andkeptinthe neighborhood of the collinear lagrangian point L. The point here is that for many cases.3 the value of λ is quite small even for very long tethers. For example, we can consider a massless tether with two equal.5. masses at its ends a =/4 inthe λ.5 system Earth-Moon for which we have:..5 l km, G ν.8 ω.66 6 rad/s Figure.3 shows the values of λ for Tether length L (km) different tether lengths; all of them are Figure.3: Values of the non-dimensional parameter λ vs. tether length L (in km) at the L = km the non-dimensional tether very small in this case (for example, when system Earth-Moon length is λ ). We ask the following question: if the equilibrium position moves an amount Δx, how much is the increase ΔL that provides such a displacement? Taking into account the small values of λ we can use the relation ξ e ( 3 ) λ obtained from (.) which take the following dimensional form: x e =( 3 ) 3 L c +3 3 L L c a dx e dl =3 3 a L L c From this relation, assuming small values for Δx and ΔL we obtain: ΔL = dl Δx Δx ΔL = Δx (.6) dx e λa λ

177 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 59 A more detailed analysis shows that the required ΔL is given by ( ΔL L ) + ΔL L = Δx L 3 3 (.7) λa As a consequence, the variation of the tether length turns out to be very high since λ. For example, for a tether km long in the system Earth-Moon, a displacement of the equilibrium position about Δx =km (we are moving the equilibrium position not the center os mass G) requires a change of the tether length about ΔL 47 km (first approximation (.6)) or ΔL 87 km (second approximation (.7)). These reasons show that the strategy proposed by Farqhuar can not be directly implemented easily when the parameter λ is very small. Moreover, there is another risk involved in the problem in such cases. In effect, this strategy only can be used on the right of the collinear point L ;itcannotbeusedon the left of L because there are no equilibrium positions on the left. Thus, if the center of mass of the spacecraft crosses the L (L )point toward its left (right) the tether can not be used to control the position of the center of mass. But the equilibrium position is very close to L when λ is small; for example, in the Earth-Moon system and for a tether km long the distance between L and the equilibrium position is about 4 km only!.. Simple control strategies Despite these difficulties in what follows we design some simple strategies to control the tether indirectly based in the Farquhar ideas. Let us assume that we have a massless tether and let L be its nominal deployed length; let λ be the value of the parameter λ which corresponds to the nominal length L. We perform the following change of variables: ξ = ξ L +3 /3 λ ( + u(τ)), λ = λ ( + v(τ)) Assuming that λ the equation (.) becomes d u dτ 9u(τ)+9v(τ)+O(λ )= (.8) which should be integrated starting from the initial conditions First strategy u() = u, u() = u From the equation (.8) some simple strategies can be designed. For example, we can select the value of the function v(τ) as follows: This way equation (.8) becomes v(τ) =u(τ)+ u(t) ü + u + u =

178 6 Section.: One-dimensional analysis and the solution starting from the above initial conditions is Obviously, the function v(τ) turns out to be u(τ) =(u +(u + u ) τ) e τ v(τ) =(u + u +8(u + u ) τ) e τ Figure.4 shows the results obtained with this strategy in the Earth-Moon system and for two different cases: ) for a tether km long and, ) for a tether 5 km long. In both cases we considered two equal end masses m = m = 5 kg and a massless tether (a =/4). Notice that the initial conditions are: u =.5 and u =.. They correspond, for the tether km to an initial separation about 4.6 km; for the tether 5 km long the initial separation is about.65 km. We can see on the pictures of figure.4 some characteristics that can be underlined. First of all, the nominal tether length ( km in the first case and 5 km in the second case) are the limiting values of this strategy but the initial tether length is quite different (58 km for the first case and 43 km for the second one). Thus, this strategy require a previous preparation of the tether whose length should be appropriately tuned before the beginning of the control manoeuver. One interesting feature is that the most of the manoeuver takes place clearly on the right of the equilibrium position; this fact decreases the risk to fall down on the left of the collinear point L where the tether can not be used to control the position of the center of mass. Another interesting feature can be found in the small retrieval velocities which appear during the manoeuver; in effect, most of the time such a velocities are smaller than m/s, which are quite convenient. However, associated with this fact is the duration of the manoeuver: it takes one month, more or less, to reach a stabilized situation. However, the most disappointing result in this example is related with the very small tether tension which appears during the process: of the order of hundredth of a Newton. With these quite small values the operation of the tether is quite difficult or almost impossible. Second strategy We can design strategies which start from the nominal value of the tether length, that is, strategies which do not require a previous preparation of the tether length. In effect, notice that the solution of (.8) has the form u(τ) =Ce 3τ + De 3τ + u p (τ) for any function v(τ) that we use. We require for the control function v(τ) the following form: v(τ) =e βτ (A cos Ωτ + B sin Ωτ)

179 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers x(km) L(km) Time (hours) Time (hours) L(m/s) T (Nw) Time (hours) Time (hours) Figure.4: Control manoeuver. Massless tether with two equal (5 kg) end masses in the Earth-Moon system. The initial conditions selected for this run are: u =.5 and u =.. Upper pictures L = km. Lower pictures L = 5 km.8 5 x(km).6.4. L(km) Time (hours) Time (hours) L(m/s) T (Nw) Time (hours) Time (hours)

180 6 Section.3: Linear Approximation for small values of λ where (A, B, β, Ω) are free parameters that should be fixed someway. In order to determine the values of (A, B) we impose the following conditions: ) initially v() =, thatis,we start the control stage with a tether deployed length equal to L and ) for the solution of u(τ) the integration constant D should be zero. These conditions provide the following values for (A, B) A = Ω +(β +3) 9(β +3) (3u + u ) βω β +3 B = Ω +(β +3) 9Ω (3u + u ) and the control function v(τ) becomes v(τ) = Ω +(β +3) 9Ω (3u + u )e βτ sin Ωτ Figure.5 summarizes the results obtained with this second strategy in the same cases considered previously: ) for a tether km long and, ) for a tether 5 km long. In these runs the values of the parameters β and Ω have been chosen arbitrarily: β =.5, Ω=.5; therefore it could be possible to perform some kind of optimization on them. Note that the initial conditions are also the same as in the previous cases: u =.5 and u =.. On the pictures of the figure we can see a quite similar trend. The main difference should be found in the initial condition for the tether length that now is equal to the nominal tether length L ; notice that the deployment velocities are now a little higher but always lower than m/s. The maximum deployed length is now lower. However, the tether tension is also of the same order: hundredths of a Newton..3 Linear Approximation for small values of λ Since λ is small, in general, and it plays a relevant role in the dynamics, our first analysis assumes that the distance between the system center of mass G and the equilibrium position is small and of order λ (in non-dimensional variables). Thus we try to control deviations from the equilibrium position which are of the order of λ. In order to do that we carry out the following change of variable: ξ = ξ L +3 /3 λ ( + u(τ)), η =3 /3 λ v(τ), λ = λ ( + s(τ)) (.9) where ξ L =(/3) /3 is the distance from the lagrangian point L to the center of mass of the small primary. The next step is to linearize the governing equations (9.-9.7) neglecting terms of order

181 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers x(km) L(km) Time (hours) Time (hours) L(m/s) T (Nw) Time (hours) Time (hours) Figure.5: Control manoeuver. Massless tether with two equal (5 kg) end masses in the Earth-Moon system. The parameters arbitrarily selected for this run are: β =.5, Ω=.5, u =.5 and u =.. Upper pictures L = km. Lower pictures L = 5 km x(km).5 L(km) Time (hours) Time (hours) L(m/s).5 T (Nw) Time (hours) Time (hours)

182 64 Section.3: Linear Approximation for small values of λ λ in front of unity. The Hill equations linearized in λ lead to this system of equations: ( + s(τ)) d θ dτ + ds dτ d u dτ dv dτ 9 u(τ)+ 9 ( 3cos θ cos ϕ ) s(τ)+ 7 ( cos θ cos ϕ ) = (.) d v dτ +du dτ +3v(τ) 9cos ϕ cos θ sin θ ( + s(τ)) = (.) d w dτ ( dθ dϕ ( + ) tanϕ + dθ dτ dτ dτ ( + s(τ)) d ϕ dτ + ds dϕ +(+s(τ)) sin ϕ cos ϕ dτ dτ +4w(τ) 9cosϕ cos θ sin ϕ ( + s(τ)) = (.) )( + ds ) +cosθ sin θ ( + s(τ)) = (.3) dτ [ ( + dθ ) +cos θ] = (.4) dτ which should be integrated starting from the appropriate initial conditions. In what follows we analyze two different approaches trying the control of this linearized system. The first one is an «ad hoc» control scheme which emerges from the analysis of the one dimensional case and later on it has been extended to the bidimensional problem. The second one is a classical control approach for the tridimensional problem that we call «proportional control»..3. «Ad hoc» control In this approach, the solution of the above system of equations is forced to be bounded, avoiding directly the instability. Firstly, the one dimensional problem is considered and then we will tackle the full in-plane analysis. One dimensional. With the tether along the Ox axis the variables v(τ),w(τ),θ(τ),ϕ(τ) vanish. The equation giving the evolution of the system and its analytical solution turn out to be: d u dτ 9 u(τ)+9s(τ) = u(τ) =Ce 3 τ + De 3 τ + u p (τ) (.5) The solution of (.5) involves a particular solution u p (τ) for any control function s(τ). We select a control function s(τ) which fulfills the following requirements: ) it should be fitted to the general form s(τ) =e βτ (A cos Ωτ + B sin Ωτ) where the parameters (A, B, β, Ω) satisfy the next conditions, ) the particular solution u p (τ) must cancel the contribution of De 3 τ, and 3) initially the tether length is the one which corresponds to the equilibrium position around which we are linearizing. These conditions provide the following values for the parameters (A, B): A = B = Ω +(β +3) (3 u + u ) 9Ω Figure.6 shows the response of the system for the values Ω =.5 and β =.5 arbitrarily selected. The initial conditions are u =.5 and u =and a massless tether has been selected. Figures show how the system tends asymptotically to the equilibrium position keeping the corresponding equilibrium tether length.

183 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 65 x(km) L(m/s) t (hours) t (hours) L(km) T (N) t (hours) t (hours) Figure.6: One dimensional control scheme. Massless tether with two equal end masses (5 kg) in the Earth-Moon system. The selected parameters are β =.5, Ω=.5, forthe initial conditions u =.5 and u =. The nominal tether length is L = km The maximum length is roughly a 5 % larger than the final one. The control manoeuvre involves small deployment and retrieval tether velocities of the order of m/s. Notice that the tether tension is small, of the order of 6 mn. Bidimensional. In this paragraph we use the same ideas to attack the in plane problem since the instability takes place within it. Additionally, we assume the system attitude variations small, i.e., θ and θ. Hence, the system of equations (.)-(.4) yields to: d u dτ dv 9 u(τ)+9s(τ) dτ = (.6) d v dτ +du +3v(τ) 9 θ dτ = (.7) d θ dτ + ds +θ dτ = (.8) In this case, the general solution for (u(τ),θ(τ),v(τ)) of the system of differential equations can be expressed as: u(τ) = C sin γ τ + C e γτ + C 3 cos γ τ + C 4 e γτ + C 5 3cos( 3τ) C6 3sin( 3τ)+up(τ) θ(τ) = C 5 sin 3τ + C 6 cos 3τ 7+4 3( 7) 7+4 3( 7) v(τ) = C cos γ τ + C e γτ C 3 sin γ τ C 4 e γτ γ γ γ γ + 7, γ = C sin 3τ C cos 3τ + v p(τ) where γ = + 7

184 66 Section.3: Linear Approximation for small values of λ x(km) t (hours) L(km) t (hours) L(m/s) T (N) t (hours) t (hours) Figure.7: Bidimensional control scheme. Massless tether with two equal masses (5 kg) at both ends in the Earth-Moon system. The selected parameters are β =.5, Ω=.5, for the initial conditions u =.5 and u =. The nominal tether length is L = km As in the previous paragraph, we select a control function s(τ) which fulfills the following requirements: ) it should be fitted to the general form s(τ) =e βτ (A cos Ωτ + B sin Ωτ) where the parameters (A, B, β, Ω) satisfy the next conditions, ) the particular solutions of u(τ),v(τ),θ(τ) must cancel the contribution of the term C e γτ,and3) initially the tether length is the one which corresponds to the equilibrium position around which we are linearizing. With θ( o ) t (hours) Figure.8: Evolution of the attitude angle θ when the bidimensional control scheme is used. this requirements, we obtain, A = and an additional relation B = B(β,Ω). Figure.7 shows the evolution of the system for the bidimensional motion in the same case considered in figure.6, that is, the response of the system for the values Ω=.5 and β =.5, with the initial conditions u =.5 and u =. The tether length and its variation are similar to the previous case. The differences which appear in the position of the center of mass and in the tension are due to the impossibility of canceling, simultaneously, the contribution of the unstable and oscillatory terms. The residual oscillation in position, attitude

185 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 67 and tension is, therefore, due to those terms. Figure.8 shows the solution for the angle θ during this control phase; the values of θ are small consistently with the hypothesis which has been made..3. Proporcional control Keeping the linear approximation, it is possible to set out more general control strategies for the tridimensional problem which match the general form s(τ) = K i x i and where the constants K i are the gains and x i the state variables of the system. A simple analysis provide indications about the options provided by this kind of control. As previously, we suppose that θ, ϕ and its derivatives are small (θ, ϕ, θ, ϕ ). Consequently, the system can be expressed as: d u dτ dv 9 u(τ)+9s(τ) dτ = (.9) d v dτ +du +3v(τ) 9 θ(τ) dτ = (.) d w +4w(τ) 9 ϕ(τ) dτ = (.) d θ dτ +θ(τ)+ds(τ) dτ = (.) d ϕ +3ϕ(τ) dτ = (.3) (.4) Taking into account the above control function, the system of equations takes the following linear form: where y = d y dt = A y (.5) { u, v, w, θ, ϕ, u, v,ẇ, θ, ϕ} T and the matrix A is given by 9 9 K u 9 K u( Ku) +K θ 9 K 9 K v 3 uk v+3k v +K θ 9 K 9 K w 4 uk w+4 Kẇ +K θ 9 K 9 K θ uk θ +K θ A T 9 K ϕ 9 +K θ = 9 K u Ku+ K v 9 K u +K θ 9 K v +K θ 9 K 9 Kẇ ukẇ K w +K θ 9 K 9 K θ uk θ K θ +K θ 9 K ϕ 9 K uk ϕ 9 Kẇ+3 K ϕ Kv+K u( 9 K v) 9 K uk ϕ K ϕ +K θ 3

186 68 Section.4: General analysis. Full Problem. Proportional control Firstly, we analyzed the stabilization possibilities when only one gain is different from zero, i.e., when s(τ) =K i x i. The analysis of the eigenvalues of A shows that only the particular case s(τ) =K u u(τ) with K u > leads to a stabilized system since in this case all the eigenvalues are pure imaginary and the system would oscillate around the equilibrium position. In a second step, we include a second variable with a non-vanishing gain; the idea is to obtain an asymptotically stable behavior for the in-plane motion, studying control schemes which correspond to this more general expression s(τ) =K u u(τ) +K i x i. However, the analysis of the eigenvalues of A shows that there is no successful combinations of gains which provide asymptotic stability. In the most favorable case, when s(τ) =K u u(τ) +K u u(τ) with K u >, K u >, the matrix A has two complex conjugate imaginary eigenvalues associated to the in plane motion: therefore, we have no asymptotic stability with this more elaborated control function..4 General analysis. Full Problem. Proportional control Around the main equilibrium position along the Ox axis, the variational equations for the full problem take the form: ( δ ξ = 5 ) ξe 3 δξ +δ η 3 δλ ξe 4 (.6) ( δ η = 6 ± ) ξe 3 δη + (3 ξ3 e ) ξe δθ δξ (.7) ( δ ζ = 7 ) ( 3 ξ 3 ξe 3 δζ + e ) ξe δϕ (.8) δ θ = ± 3 ( ξe 4 δη 3 ± ) ξe ( 3 4 ± 3 ξe 3 δ ϕ = ± 3 ξe 4 δζ δθ 3 J g δ λ λ e (.9) ) δϕ (.3) where δλ stands for the variations of λ from its equilibrium value λ e. Here the parameter J g given by J g = Λ d ( + 3 cos φ) ( 3sin ) φ Λ d is a function of the tether mass configuration; for a massless tether J g =. These equations show that coordinates (ζ,ϕ) associated to the out-of-plane motion are decoupled from the other coordinates. That is, the out-of-plane motion cannot be corrected by the variation of the tether length as it has been pointed out by Farquhar []. Fortunately, the out-of-plane motion is stable (oscillatory). From now on, we focus the analysis on the in-plane motion which is the source of the instability exhibited by the system. Thus taking the vector y =(ξ,η,θ, ξ, η, θ) T and introducing the variations: y = y e + δ y, λ(t) =λ e + δλ(t)

187 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 69 These equations can be written as follows: dδ y = M δ y + b δλ + b δ dτ λ where the 6 6 square matrix M is given by: M =,,,,,,,,,,,,,,, b, b, b 3,,, b, b, b 3,,, b 3, b 3, b 33,,, and the vectors ( b, b ) turn out to be: B = 5ρ 3 e ρ 3,, e 6ρ 3 e 3ρ 3 e, ρ 3, ξ e e ρ 3 e 3, ξ e ρ 5, ρ3 e + e ρ 3 e b =(,,, 3ξ e ρ 5,, ) T, b =(,,,,, ) T e λ e We introduce control on the system acting on the tether length δλ. Let k be the vector of constant gains: k =(Kξ,K η,k θ,k ξ,k η,k θ) which allow to express the variation of the non-dimensional tether length as follows: δλ = k δ y δ λ = k δ y That is, in this case the length variation λ(τ) is governed by a proportional control law. The equations that govern the time evolution of y turn out to be K ξ λ e Figure.9: Sufficient condition for K ξ to stabilize the system as a function of λ e. Values over the curve (K ξ >Kξ ) provide stability. dδ y dτ = N δ y where N =(I [ b, k]) (M +[ b, k]) I is the unit matrix of size six and [ a, b] stands for the dyadic product of vectors a and b. The detailed stability analysis of this full system can be cumbersome. Nevertheless, taking into account the results of the preceding section, we firstly consider the simpler case in which only one gain K ξ is different from zero. In that situation, the characteristic polynomial takes the form: [ ( 3 s 3 Kξ + ξe 4 4 )] [( 6 ξe 3 s Jg +7 + ξe ) ( ξe 7 K ξ 5 6 ξe )] ξe 6 s+ [( 9 +6 ξe 4 3 ) ( ξe 7 K ξ ξe 3 )] (.3) ξe 6 =

188 7 Section.5: Control Drawbacks The Descartes rule of signs provides a sufficient condition to be fulfilled by K ξ in order to stabilize the system; such a condition is drawn in figure.9 as a function of the equilibrium position λ e for a massless tether (J g =). This control scheme provides stability but not asymptotic stability. In order to do that, it seems convenient to include gains which affects the velocity variables. Considering the analysis of the precedent section it is appropriate to include the two control gains: K ξ and K ξ, even if they are not able to stabilize the system in the linear approximation. For the full problem considered here, the Routh-Hurwitz stability criterion (see Ref. [?]) has been used in order to establish whether that pair of gains can provide stability. The result indicate that they must fulfil the following conditions K ξ >, (+J g )ξe 4 ((Jg ξ e 3K ξ <, +J g 6)ξe 3 +) ξe 3(( + J g)ξe 4 ξ e 3K ξ ) K ξ > (.3) Unfortunately, they are incompatible; in effect, for a massless tether (J g =), the numerator of the last condition is positive for all values of ξ e and the denominator negative! Therefore we can conclude that the system can not be stabilized using values of K ξ and K ξ different from zero at the same time..5 Control Drawbacks In previous sections we have deduced control laws which permit stabilize the system within certain limits. However, these control strategies exhibit some difficulties which should be underlined. For example, if we are working in the neighborhood of L (the external Lagrangian point) the above strategies only can be used when the center of mass G is on the right of L (x G >x L, see Fig..). If G becomes on the left side of L (x G <x L ) there is not possible control based on the tether line. And there exist more unsuitable characteristics that we will comment in what follows..5. Length Variations The philosophy underneath the control strategies is based on the Farquhar ideas. In general, the length of the tether is changed in such a way that the system center of mass G approach continuously the equilibrium position. An important question arise in this matter: if the position of the equilibrium point must be moved an amount Δx e, what is the minimum tether length variation ΔL required to produce such a translation? In dimensional values, the equation which link both variations is: ΔL = (.33) Δx e 3 3 a λ In the general case, this ratio can be expressed as a function of the non-dimensional equilibrium position: dl dx e = 5 ρ 3 e 3 a 3 ρ 3 e (.34)

189 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 7 dl dx e dl dx e λ General. eq. (.34) Linear. eq. (.33) Figure.: Ratio between the variation of length needed and the deviations of the center of mass of the system vs. λ. Comparison of linear and general expressions. 6 Sun-Earth Earth-Moon Mars-Phobos Jupiter-Amaltea 4 Jupiter-Io Saturn-Enceladus Both relations are shown in figure.; note that the linear approximation fits well till values of λ of the order of. The important point is that for small values of λ (λ < 6 )thevariationδl is very high; thus, to move the equilibrium position km require deploy (or retrieve) km of tether line. The presence of a minimum suggest that it would be worth working near that point where dl/dx e is approximately 4.5. That implies searching for values of λ.3. Nevertheless, in order to be coherent with the approximations made in the model (remember figure 9.) the values of λ should be smaller than λ<.asa result, values of λ of the order of keep the coherence with our approximation and provide feasible tether length variations. Figure. shows the ratio dl dx e as a function of the tether length (km) for the equilibrium position in different binary systems of interest. The part of the curve above the minimum and on the right should not 4 be considered since it corresponds to too L (km) high values of λ. Figure. permits to Figure.: Ratio between the variation of establish which systems of primaries would length needed and the deviations of the center of mass of the system as a function of the strategies carried out in this pages. On be more suitable to be controlled with the tether length (km) the right side of the plot the tether length needed to work in the vicinity of the minimum of dl/dx e is extremely high. However, the values on the left side of the figure lead to more reasonable values of the tether length, with the ratio dl/dx e near its minimum. Hence, this analysis help us to evaluate «a priori» the suitability of the control strategies based in the variation of the tether length for each one of the binary systems of the Solar System. There exist other troubles in the proportional control regarding the tether length. As we stated before, there exist a limit due to the fact that the equilibrium positions are on only one side of the Lagrangian point (on the right for L, on the left for L ) and not on both sides. Therefore, it is necessary to ensure that the tether is in the correct side. For the L case, for example, this condition can be expressed in dimensional and non dimensional

190 7 Section.5: Control Drawbacks values as follows x x e =Δx< x e x L, ξ ξ e =Δξ< ξ e 3 /3 (.35) Δξ λ Secondly, the tether length is obviously positive L> and this leads to a restriction on the control strategy. Due to the definition of λ, the tether length L is proportional to λ and this variable changes according to the control law and, in general, it depends on the variations Δ y with respect to the equilibrium position: λ = λ e + k Δ y Hence, the constraint can be written as: λ e + k Δ y >. An upper bound for λ can be obtained considering: λ e + k Δ y λ e k Δ y Δξ C max Δξ C max Figure.: Upper picture: Δξ max in both cases, Δξ C max (blue line) and Δξ C max (red line). Lower picture, ratio between the maximum allowable perturbations for both constraints λ And, this is a sufficient condition on the modulus of the control gains vector in order to fulfil the constraint L>: λ e k < (.36) Δ y max We should note that the vector of gains must fulfil the condition related to the stability of the system. It is not easy to determine, in a general manner, the value Δ y max since it However, it seems reasonable to as- depends on the dynamical evolution of the system. sume that in a controlled system the maximum perturbation coincides with the initial one since the control scheme bring the system closer to the unstable equilibrium point. As a consequence, Δ y max = Δ y in many situations. For the particular case of the control law exposed in the previous section: λ = λ e + K ξ (ξ ξ e ), the constraint (.36) can be expressed as: K ξ <λ e /(ξ ξ e ) max. Assuming that the maximum takes place at the initial instant, the condition (.36) in K ξ takes the following form: K ξ < K ξ < λ e ξ ξ e λe a x x e non-dimensional (.37) dimensional (.38)

191 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 73 We obtained two requirements for the control strategy: ) positive tether length L and ) center of mass G on the correct side of the Lagrangian point. As we show in what follows the first one is much more restrictive than the second. In order to compare them, both restrictions can be measured in terms of the maximum admisible perturbation; this one is defined as the maximum initial perturbation which can be controlled fulfilling the required constraint () or )). For the second constraint, once the equilibrium position for the nominal operation of the tether has been selected, we have to ensure that the initial position belongs to an interval of length ξe 3 /3 and centered in ξ E. Hence, the maximum allowable perturbation is the semi-interval = ξe 3 /3 and this value is fixed for each equilibrium position. For the first constraint, we can obtain the maximum allowable perturbation from the relation (.37). Δξ (C) max Δξ C max = λ E K ξ Δx max 5 4 L Sun-Earth Earth-Moon Mars-Phobos Jupiter-Amaltea Jupiter-Io Saturn-Enceladus Figure.3: Maximum perturbation admisible vs. tether length L (both in km) for several binary systems. Since the value of Δξ C max depends not only on λ E but also on the control gain, we will use the minimum value of K ξ which guarantees the stability of the system, i.e., Kξ (see figure.9), in order to provide a real maximum of the perturbation. Accordingly, the value of the maximum allowable perturbation for each constraint can be drawn as a function of the equilibrium position as it is shown in the upper picture of figure.. As a consequence, the requirement L> is more restrictive than the necessity of placing the center of mass on the correct side of the Lagrangian point. Figure.3 shows the maximum perturbation admisible in the position of the center of mass G (in km) versus the tether length L (in km) for several binary systems of interest in the Solar System. This figure has been elaborated taking into account the above results, that is, the constrain ) is the real limit..5. Tension Tethers require positive tension since a cable can not withstand compression. As a consequence there exists a natural limit for tether operation which will be explored in this paragraph. The expression of the tension in a massless cable within the Hill approximation is collected in equation (9.8). Hence, the tension will depend on the tether control law through the last term: ( L d /L d ). We analyze the stabilizing proportional control studied in the previous section (λ =

192 74 Section.5: Control Drawbacks λ e + K ξ (ξ ξ e )) in order to determine the influence of the different elements involved in the control law: initial perturbation, equilibrium position and mass configuration. Figure.4 shows the maximum and the minimum tensions for a massless tether as a function of the initial perturbation, for a given 5 equilibrium position (in this case the corresponding to λ e = ). T As it can be seen, there exists T c 5 a maximum admisible initial perturbation over which the tension becomes negative. Moreover, the maximum value of the initial perturbation in the figure has been cho Δξ sen as the maximum Δξ which can be controlled with the minimum Figure.4: Maximum and minimum tether tension gain K as a function of the initial perturbation Δξ for a given ξ of this equilibrium position. equilibrium position: λ e =. That means that the positive tension constraint is more restrictive than the requirement L > exposed in the previous paragraph. This way, it is possible to associate a critical value of Δξ to each 4 equilibrium position. Figure.5 summarizes such a critical value as Δξ a function of λ e. Along the blue 5 line, the zero tether tension condition is reached; as a consequence, the admisible values of Δξ must be below this line. Here the red line represents the requirement L>. It is clear, from the figure, that this last requirement is the most restrictive; therefore, the last figure constitutes also the maximum perturbation allowed which can be controlled with feasible tether length variations maintaining the tether tighten λ e Figure.5: Δξ which provides zero tension as a function of the equilibrium position described by the parameter λ e (blue line). Δξ which provide zero tether length when the minimum value of K ξ needed to stabilize the system is considered (red line).

193 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 75.6 Rotating Tethers The Attitude Dynamics of a rotating tether can be analyzed more intuitively using the Newton-Euler formulation. The core of the analysis is the angular momentum equation: H G u 3 m d H G dt = M G G u Here M G is the resultant of the external torques applied to the center of mass G of the tethered system and H G is the angular momentum of the system, at G, in the motion relative to the center of mass. In the extended Dumbbell Model which we use here the angular velocity of the tether and its angular momentum H G are: Ω = u u + u ( u Ω ), H G = I Ω = I s ( u u) m φ φ u 3 u u z Attached to the tether we take a reference frame Gu u u 3 where the unit vectors are given by: u = u, u = u u, u 3 = u u In this body frame the angular momentum is: x G u u u φ 3 φ y s H G = I s Ω u 3, where Ω = u u = u and the angular momentum equation takes the form: d Ω dt u 3 + Ω d u 3 dt = I s M G This way we obtain the following equations: Figure.6: Gu u u 3 frame attached to the tether in the Dumbbell Model (upper picture). Definition of the Tait-Bryant ( or Cardan) angles (lower picture). d u dt d u 3 dt d Ω dt = Ω u = M Ω I s u = M 3 I s From a mathematical point of view, the orden of the system of differential equations is four. For rotating tethers the attitude dynamics is described in a better way using Euler angles

194 76 Section.6: Rotating Tethers in sequence 3 (Tait-Bryant or Cardan angles) instead of the classical libration angles (θ, ϕ). In terms of the Bryant angles the unit vectors u and u 3 are given by u =(cosφ cos φ 3, cos φ sin φ 3 +sinφ sin φ cos φ 3, sin φ sin φ 3 cos φ sin φ cos φ 3 ) u 3 =(sinφ, sin φ cos φ, cos φ cos φ ) The equations governing the time evolution of the Bryant angles are: dφ dτ = M ω cos φ 3 dφ, I s Ω cos φ dτ = M ω sin φ 3 I s Ω dφ 3 dτ =Ω + M ω cos φ 3 tan φ, I s Ω dω dτ = M 3 ω I s where Ω = Ω /ω is the non-dimensional form of Ω. These equations should be integrated from the initial conditions: at τ =: φ = φ,φ = φ, φ 3 = φ 3, Ω =Ω For a fast rotating tether the value of Ω is large, Ω. There are two time scales: ) the period of the orbital dynamics of both primaries and for which τ is of order unity this is the «slow time» and ) the period of the intrinsic rotation of the tether for which τ =Ω τ of order unity this is the «fast time». To obtain the governing equations we perform to operations: ) we average the original governing equations, and ) we introduce the Hill approach. For the averaged process we introduce a stroboscopic frame (see figure.7). For example, consider one of the governing equations dφ dτ = f(φ,φ,φ 3, Ω ) Its averaged form is: < dφ dτ >= π f(φ,φ,φ 3, Ω )dτ π G u 3 v 3 φ 3 u v v u u To integrate the function f(φ,φ,φ 3, Ω ) the slow variables (φ,φ, Ω ) take constant values and the fast variable φ Figure.7: Stroboscopic frame 3 is approximated by φ 3 τ + φ 3. After averaged the governing equations and introducing the Hill approach, for a fast rotating tether the evolution of the center of mass and tether attitude is the same as in the case of constant tether length; the differences between both cases are: ) now λ = λ(τ) and ) the time evolution of Ω (τ) is different in φ 3

195 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 77 both cases. The governing equations are collected in ( ), providing that the rotation rate of the tether keeps a high value (Ω (τ) ). They are written here for convenience { } ξ η =(3 ρ 3 )ξ λ ρ 5 3Ñ sin φ ξs (Ñ ρ ) (.39) { } η + ξ = η ρ 3 + λ ρ 5 3Ñ cos φ sin φ + ηs (Ñ ρ ) (.4) { } ζ = ζ( + ρ 3 ) λ ρ 5 3Ñ cos φ cos φ ζs (Ñ ρ ) (.4) dφ dτ =cosφ tan φ (.4) dφ dτ = sin φ (.43) They only have a free parameter λ, defined in (.49), and where the quantity Ñ is given by: Ñ = ξ sin φ (η sin φ ζ cos φ )cosφ (.44) The equations ( ) should be integrated from the initial conditions: at τ =: ξ = ξ, η = η, ζ = ζ, ξ = ξ, η = η, ζ = ζ,φ = φ,φ = φ (.45) When the initial conditions are φ = φ =the solution for the angles φ and φ is φ (τ) =φ (τ), that is, if initially the tether rotates in a plane parallel to the orbital plane of both primaries, the direction of the angular momentum keeps a constant value. In these equations the time evolution of angles (φ,φ ) is decoupled from the other variables and it can be integrated separately. The solution for them has been obtained in ( ) and we reproduce here for convenience: sin φ = cos φ = cos α sin β cos β +cos α sin β, sin φ =sinβ sin α (.46) cos β cos β +cos α sin β, cos φ = cos β +cos α sin β (.47) Obviously the constant values (β,α ) are given by the initial conditions: cos β =cosφ cos φ, β [, π ] (.48) sin α = sin φ sin β, cos α = sin φ cos φ sin β (.49) Finally, equations ( ) are exactly the same for tethers of constant length and for varying length tethers; the differences between both cases are: ) λ is constant for a tether of constant length and it is a function of time, λ = λ(τ), for a varying length tether, and ) the time evolution of Ω (τ) is different in both cases but for a fast rotating tether the particular evolution of Ω (τ) is irrelevant to a large extent, provided that it takes large values, that is Ω (τ).

196 78 Section.6: Rotating Tethers.6. Rotating tethers. Equilibrium positions The steady solutions of equations ( ) provide the equilibrium positions of a rotating tether. Equations ( ) only have the steady solution φ = φ =. Therefore, in order to have an equilibrium position for the tethered system the rotation of the tether should take place in a plane parallel to the orbital plane of both primaries. As a consequence Ñ takes the value Ñ = ζ, and the stationary equations become: { ( ξ 3 ρ ( ) λ ζ 4 ρ 5 5 )} = (.5) ρ { ( η ρ 3 3 ( ) λ ζ 4 ρ 5 5 )} = (.5) ρ { ( ζ + ρ 3 + ( ) )} λ ζ 4 ρ = (.5) ρ Moreover, the length must be constant, that is, λ =constant. These equations have several solutions; two of them are on the Oz axis and they are spurious, that is, in fact they are due to the expansions carried out in the model but they are not real equilibrium positions. However, the equations provide two real equilibrium positions for which the center of mass G is on the Ox axis; one of them is close to the external collinear point (L ) and the other one close to the internal Lagrangian point (L ). For the sake of simplicity we focus on the first one which is given by: ( ξ e = ρ e, η e = ζ e =, λ e = 4ρ5 e 3 ) 3 ρ 3, ρ 3 e > /3 (.53) e For small values of λ the above solution can be expanded in power of λ and provide the asymptotic solution ξ e ( 3 ) λ λ 6 + O(λ3 ) (.54) Comparing these expressions with their equivalent in the non-rotating case, (see (.)), an important conclusion can be deduced: in order to place the center of mass G in an equilibrium position at a given distance from the Lagrangian point L the real length of the rotating tether must be twice the real length of the non-rotating tether. In other words, for two tether of the same length, the equilibrium position of the rotating tether is four times closer to L than the non-rotating tether..6. Linear stability analysis of the collinear equilibrium positions In previous sections, it was found that the motion of a tethered system with constant length was slightly more unstable than the motion of the particle. Now the effect of a fast angular velocity will be studied and compared with the non-rotating tether. In order to study the linear stability, the variational equations are deduced by introducing the following variations in the dynamical system ( ): ξ = ξ e + δξ η = δη ζ = δζ φ = δφ φ = δφ

197 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 79 The resulting linear equations can be written in terms of the state vector dδ y dτ = M e δ y M e (ρ e )= y =(ξ,η,ζ,φ,φ, ξ, η, ζ, φ, φ ) T k, k, k,3 k,4 k,5 k, k, k,3 k,4 k,5 k 3, k 3, k 3,3 k 3,4 k 3,5 k 4, k 4, k 4,3 k 4,4 k 4,5 k 5, k 5, k 5,3 k 5,4 k 5,5 The k sub-matrix takes the form: 5ρ 3 e ρ 3 e 3 k(ρ e )= 5ρ3 e ρ ± 3ρ3 3 e e ρ e (.55) (.56) The stability properties of the equilibrium position are given by the eigenvalues of the M e ; they turn out to be: ρe (5ρ s, = ±i, s 3,4 = ±i, s 5,6 = ± 3 e ) i ρ e ρ e ( 4ρ 3e + 6ρ 6e 4ρ3e + ) s 9, = ± ρ i e ( ρ e +4ρ 3 e + ) 6ρ 6 e 4ρ 3 e + s 7,8 = ±, ρ e s 7 s 7 (non-rotating) ρ e Figure.8: Unstable eigenvalue Except s 7,8, all of them are pure imaginary, so they do not destabilize the system. However, s 7,8 summarizes a pair of real eigenvalues; one of them is always positive and makes the equilibrium position unstable. Figure.8 shows that for larger tethers, the system becomes more unstable, as in the non-rotating case. In summary, the presence of the tether does not improve the stability properties of the equilibrium position. As in the case of non-rotating tethers, the instability increases with the tether length; however, the increase is slower than in the non-rotating tether case (see figure.8).

198 8 Section.6: Rotating Tethers.6.3 Varying Length Tether. Linear control strategy In spite of the instability, the variation of the tether length brings the chance of obtain a stable motion around the equilibrium position. In the Hill approximation, the governing equations for fast rotating tethers are given in ( ). The time evolution of the angular velocity Ω (τ), now it is governed by the new equation: dω dτ =sinφ sin φ cos φ Ω Is I s (.57) The admisible equilibrium positions appear when φ = φ =. Insuchacase,byintegrating equation (.57) we obtain the conservation of the modulus of the angular momentum: Ω I s = constant (.58) Therefore, for a rotating tether at an equilibrium position, the fast rotation Ω is determined by the variation of the moment of inertia I s due to the variation of the tether length. The three variational equations describing the attitude and the motion of the center of mass normal to the orbital plane are the same that in the constant length tether case: δ ζ = δ φ = δφ δ φ = δφ { 5ρ3 e + ρ 3 e } δζ + { 3ρ3 e + ρ e } δφ They are uncoupled from the other degree of freedom and they have pure imaginary eigenvalues associated with, so this motion is bounded. However, since these equations do not depend on the variation δλ(t), it will not be possible to obtain asymptotic stability for the out-of-plane motion with the control schemes carry out in this paper. The remaining two variational equations, which govern the motion of the center of mass in the orbital plane, can be expressed: [ δ ξ δ η ] [ = ][ δ ξ δ η ] [ 5ξ 3 e + ξe 3 or, by using the new state vector y =(ξ,η, ξ, η): 3 ] [ δξ δη ] [ 3 + 4ξe 4 ] δλ where M(ρ e )= 5ξ 3 e ξ 3 e 3 dδ y dτ = Mδ y + b δλ (.59) and b =(,, 3 4ξe 4, ) T (.6) From equation (.54) can be deduced that a differential increment Δλ e takes associated a variation of the equilibrium position of the same order Δξ e (as it will be seen later, with

199 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 8 dimensional variables does not happen the same). This justify the use of a proportional control law, based on the variations: y = y e + δ y, λ(t) =λ e + δλ(t) where y is the state vector described before. Therefore we select the following proportional control law: δλ = K δ y where K =(K ξ,k η,k ξ,k ξ) is the gain vector. By replacing δλ with the above control law, the system becomes: takes the form dδ y dτ = Nδ y where the matrix N = M +[ b, K] (.6) N = 5ξ 3 e ξ 3 e and its characteristic polynomial is: 3Kξ 4ξ 3Kη 3K ξ e 4 4ξ e 4 4ξ e 4 3 3K η 4ξ 4 e s 4 + 3K ξ 4ξ 4 e s 3 + 3ξ4 e 6K η +8ξ e +3K ξ 4ξ 4 e s 3 K η 3K ξ 4ξ 4 e s +3 8ξ e 6ξ 4 e +3K ξ 4ξ 4 e = (.6).6.4 Searching oscillatory stability In this initial stability analysis, control gains that remove the hyperbolic instability of the equilibrium point and replace it with oscillatory stability will be searched. Once this level of stability is achieved, the center of mass G of the system will undergoes an oscillatory motion about the equilibrium position. General assignment of the gains results in a 4th order characteristic polynomial (see (.6)) in s with all the coefficients different from zero. To simplify the analysis, coefficients of s 3,ands are made null by taken: K ξrot K ξnon rot ρ e Figure.9: Plot of stability condition for K ξ gain K ξ = K η = K η = (.63) The resulting characteristic polynomial is then: a(ξ e,k ξ ) x 4 + b(ξ e,k ξ ) x + c(ξ e,k ξ )= (.64)

200 8 Section.6: Rotating Tethers a = (.65) b = 3ξ4 e +8ξ e +3K ξ 4ξe 4 c =3 6ξ4 e +8ξ e +3K ξ 4ξe 4 (.66) (.67) The discriminant of this equation is stated as: Δ(ξ e,k ξ )=b 4ac If Δ > then the polynomial has three distinct, real roots in s. From Descartes Rule of Signs, if the coefficients a, b and c are all positive, then the roots all have negative parts. If both of these conditions are satisfied, then the roots all have the form s j = w j,and therefore all the eigenvalues of the system are pure imaginary, s j = ± w j, and the oscillatory stability is achieved. Figure.9 shows the lower limit for K ξ needed for fulfill all the requirements mentioned. This limit can be stated explicitly: K ξ 4 3 ρ e ( 5ρ 3 e ) (.68) Figure.9 also shows gain requirement for the non-rotating tether (which were studied in previous sections); another important conclusion is remarked: for the rotating tether gain is larger than the gain for the non-rotating one. Note that this was expected; the proposed control law is proportional to displacements, and, as can be seen from equations (.,.54), in order to vary the equilibrium position the same quantity, the rotating tether requires a Δλ larger than the non-rotating one..6.5 Asymptotic stability Although a small increment of the nondimensional variable Δξ e requieres an increment of the tether length Δλ e of the same order: Δξ e = 3/3 4 Δλ e, when this relation is expressed in term of dimensional variables: ΔL = 3 /3 Δx e, λe a it es found that, when λ e, anincrementδx e of a few kilometers requires a variation of the tether length too high. Therefore, since the variations of L are limited by the tether deployment capacity, it is necessary to reduce the variations of Δx e by searching asymptotic stability. Through the Routh-Hurwitz theorem, it has been found that asymptotic stability is guaranteed if the gains of the control law satisfy the relations K ξ 4 3 ρ e ( 5ρ 3 e ) K ξ > K η = K η = (.69)

201 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers 83 Note that the condition for K ξ is the same that the one required for oscillatory stability. Just adding the dissipation K ξ > the asymptotic stability is achieved. Consider the linearized one-dimensional motion, that is, when the center of mass G is limited to the Oξ axis by neglecting the other coordinates and the coupling effects (Coriolis terms). Such motion is described by the equation: δ ξ + 3 4ξ 4 e where the natural frequency w n is given by w n = w n (ξ e,k ξ )= K ξ δ ξ + w nδξ = (.7) 3K ξ 4ξ 4 e 5ξ3 e ξ 3 e Equation (.7) has two stable eigenvalues when K ξ > given by s, = 3K ξ 8ξ 4 e ± [ 3K ξ 8ξe 4 ] wn (.7) The gain K ξ is introducing damping in the system; the critical value which produce critical damping is given by K ξcr = 8 3K ξ 3 ξ4 e 4ξe 4 5ξ3 e ξe 3 (.7) The analysis performed here for the one-dimensional case can be translated almost directly to the bi-dimensional case as the simulations collected in what follows highlight. Figure. shows the results obtained in two simulations of a rotating tether with different values of K ξ and K ξ. The characteristics of the simulations are: λ e =.4 (ξ e =.77); ξ ξ e = 3, η e = 3 ; Ω =5. Pictures on the left side corresponds to a control law without dissipation (K ξ =)andforwhichk ξ =4. Pictures on the right side of the figure corresponds to the same case but now with a control law including dissipation; two different values of K ξ has been taken (K ξ =.5 subcritical and K ξ =3supercritical). As a consequence, the fast rotating tethers are able to be asymptotically controlled; and this is a significative difference with the non-rotating tethers which are not able to be controlled in that way. From this point of view, figure. is good example. Related with these simulations figure. describes the time evolution of other important parameters: ) the angular velocity Ω of the rotating tether (the initial value has been selected close to Ω =5) and ) the evolution of the parameter λ. Note that without dissipation both parameters oscillate around some averaged values with small amplitudes (roughly the % of the initial values). However, when the control law includes dissipation both parameters tend asymptotically to constant values. Note that the time evolutions of both parameters are not independent since they are linked by the relation (.58).

202 84 Section.7: Rotating Tethers δ ξ 3 x 3 K ξ =4 δ ξ 5 5 x 4 K ξ =3 K ξ = δξ δξ δ η x 3 K ξ = x 3 δη δ η.5.5 x 3 K ξ =3 K ξ = x 4 δη.5 x 3 K ξ =4 x 4 K ξ =3 K ξ =.5 η.5 η ξ ξ Figure.: Nondimensional simulation; λ e =.4 (ξ e =.77); ξ ξ e = 3, η e = 3 ; Ω =5

203 Chapter : Stabilization of the Collinear Lagrangian Points with variable length tethers K ξ = K ξ =3 K ξ = Ω Ω τ τ.5. K ξ =4.5. K ξ =3 K ξ =.5 λ.5 λ τ τ Figure.: Nondimensional simulation; λ e =.4 (ξ e =.77); ξ ξ e = 3, η e = 3 ; Ω =5.7 Summary and future research lines From the analysis carried out in this paper some significant conclusion can be drawn. First of all, we deduced a robust formulation including in the analysis the Hill approach which gives place to simpler expressions and simultaneously keeps the necessary accuracy on the results. With this new formulation, we found a new non-dimensional parameter which captures the influence of the tether length and play a central role in the dynamics. We also found several equilibrium positions close to the Lagrangian collinear points where a tethered satellite can be placed to exploit the benefits of these special positions. These results agree with the previous ones given in the literature. Since the equilibrium positions are unstable we obtained several control laws which permit the operation of the tethered system in the neighborhood of the Lagrangian point, using rotating and non-rotating tethers. This stabilization techniques open the door to interesting applications of tethered systems in different fields of the Astrodynamics and the Space Exploration. In general, the use of tethers is more appropriate in binary systems whose secondary primary is small, because the tether length is smaller. We also underline some difficulties associated with the stabilization techniques carried out in this paper; for example, the

204 86 Section.7: Summary and future research lines stabilization proposed here only can be used on one side of the selected Lagrangian point, since on the other side there is no equilibrium positions. Therefore, it is forbidden to cross the collinear point. However, we show that there is a very high sensitivity of the tether length with the deviation from the equilibrium position; from this point of view the most restrictive requirement is to keep a non-vanishing tether length. Rotating tethers exhibit a better behavior from the control point of view; however, its operation require more accuracy because the ranges of operations are narrower for them. In the future this analysis will be extended in order to take into account the power needed to operate the tether when the tether length is varied to provide control on the system.

205 Chapter Io exploration with electrodynamic tethers. Introduction The Galileo s fly-by of Io during its approach to Jupiter on 7 December 995 confirmed the strong volcanic activity of this Jovian moon, detected years before by the Voyagers when, they discovered nine simultaneously active volcanic sites. The scientific relevance of this discovery is highlighted in ref. [5]: «Io has represented the greatest challenge of Galileo s extended missions and arguably its greatest success. Images, spectra, and fields and particles data have begun to elucidate the broad variety of phenomena ranging from styles of ultrabasic volcanism and the SO cycle to the structure and dynamics of Io s core and mantle to the varied and complex interactions between Io and the Jovian magnetosphere». Another interesting comment that underlines the importance of this Galilean moon can be found in [53]: «This system, i.e., Io and its atmosphere, the Io plasma torus, and Jupiter with its magnetosphere, is very strongly coupled with a number of feedback mechanisms. In the history of space science, this system has also played an important role in the progress of understanding satellite plasma interactions in general». Regarding Io, however, many questions are still open as a paradoxical consequence of our better understanding of its crucial role in many interesting feature of the Jovian world. Some of these questions are of dynamic character, for instance there is a resurgence of interest in the rotation of Io or in the long term behavior of its orbital dynamics (see [54, 55]); other are of specific interest for some segment of the scientific space exploration field (see [56]). The interest stirred up by Io is reflected in some of the missions proposed to visit this moon as, for example, the VOLCAN project described in [57]. The Io Community Panel addressed a report to the USA-National Research Council (NRC) (see [59]) underlining that: «Io is the most dynamic body in the Solar System. The only place beyond Earth where we can watch large-scale geology in action». Any mission to the Jovian world must face some hindering factors. The scientific payload is usually a small fraction of the whole spacecraft mass due to launcher limitations and the low dry/wet mass fraction characteristic of chemical propulsion. Moreover, solar panels 87

206 88 Section.: Introduction become rapidly ineffective further from the Sun. The solar intensity at Jupiter, 5 AU distant from the Sun, is only one twenty-fifth of its value at Earth. As a consequence, energy is a scarce commodity in this kind of missions and the total energy which will be consumed by the spacecraft should be transported onboard. The power source used in all missions to the outer planets, Radioisotope Thermoelectric Generators (RTGs), are relatively weak, require large masses which at the end strongly penalize the mission scientific payload and turn out to be very expansive. Arguably, tackling the power availability issue is the first priority towards advancing Jupiter exploration. If power availability is sufficient the mission would enjoy the following benefits: ) fuel mass would be reduced by resorting to electric propulsion, ) more powerful scientific instruments and communications could be used which would permit to carry out high throughput science with shorter timescales hence reducing radiation exposure. This last point is interesting because, in addition to fuel needs and power availability issues, radiation survivability represents a serious technological challenge. This is especially true when missions to the inner Galilean moons, Io and Europa, are considered. The prolonged exposure to Jupiter strong radiation environment implies heavy radiation shielding of sensitive devices besides the use of radiation-hard and fault tolerant electronics. Moreover high power availability is key to enable active radiation protection techniques (e.g. magnetic and/or electrostatic radiation shielding) which could become practical farther in the future. For an Io exploration mission the power needs are particularly high. The extreme temporal variability of Io surface requires high data rate to adequately monitor surface changes in addition to high spatial resolution. Because of that and due to the harsh radiation environment of Io large data storage systems and high bandwidth communications are required which are power demanding. Recent studies [6]-[9] have addressed the power generation capability of passive bare electrodynamic tethers in the Jupiter environment. Sanmartin et al. [6, 6] have investigated power generation and storage for a spinning electrodynamic tether performing a tour of the different Galilean moons. Bombardelli et al. [6] have studied the behavior of electrodynamic tethers in polar orbits obtaining power generation without affecting orbital energy. Peláez and Scheeres [8, 9] have proposed to place electrodynamic tethers in the neighborhood of the Lagrangian points of the inner Jupiter moonlets (Metis, Adrastea, Amalthea and Thebe) for power production. In all these analysis, and also in this paper, it is assumed that the collection of electrons takes place in the Orbital Motion Limited (OML) regime. In the concept proposed by Peláez and Scheeres the gravitational attraction of the moonlet is exploited simultaneously to the electrodynamic interaction with the Jupiter plasma so that orbit drift is prevented and it becomes possible to continuously extract power from the orbital energy of the moonlets (for the case of Metis, Adrastea and Amalthea) or from Jupiter fast rotating plasmasphere (for the case of Thebe). Current control was used as the sole mean to stabilize both the position and the attitude of the constant-length nonrotating electrodynamic tether system. The dynamical analysis of [9] shows that there exist equilibrium positions where the tether could be operated appropriately. Some of these equi-

207 Chapter : Io exploration with electrodynamic tethers 89 librium positions are stable and other unstable. The operation of the probe in an stable equilibrium position would be preferable, but the probe would be operated in an unstable equilibrium position by using a feedback control law. A similar mission to Io would be a perfect complement for the JUNO project (see [58]).. Non rotating electrodynamic tether at equilibrium In the first part of this paper we extend the analysis of [8, 9] to the case of the Io moon. We analyze the pros and cons of this option. Table. summarizes the power and mass of RTG s used in past missions to Jupiter. The main result of [8, 9] is that an EDT operated in Metis is able to produce a larger level of electrical power than the one provided by the classical RTG s. Mission Thermal W Electrical W Mass (kg) Pionner, Voyager y 7, 47 7 Ulysses 4, Galileo 8,8 57 Cassini 3, Table.: Power and mass of RTG s in past missions A pair of examples will help us to grasp the scale of the problem. We are speaking about two bare Aluminum tapes with mass around kg providing 4 5 watts of electrical power continuously; roughly, one order of magnitud larger than the power provided by RTG s. Table. summarizes the characteristics of both tethers. Length L (km) Width d w (mm) Thickness h (mm) Power W u (w) Tether mass m T (kg) Table.: Power and mass of EDT at Metis The performances of the bare tether as electron collector depend on two external fields: the magnetic field and the electronic plasma density. Since the orbital radius of Io (4,8 km) is larger than the radius of Metis (8, km) the Jovian magnetic field will be weaker and the performances of the bare tether will be smaller. However, the tether will be operated in the neighborhoods of the peak exhibited by the plasma density we follow the model of Divine & Garret described in the Ref. [67] which is associated with the iogenic plasma source, that is the rate of supply of ions, mass, momentum, and energy to the plasma torus created by neutrals from Io. In addition the relative velocity of Io with respect to the local

208 9 Section.: Non rotating electrodynamic tether at equilibrium plasma is as high as km/s. The hight plasma density and velocity compensate for the weaker magnetic field providing relative high power generation capability. The bare self-balanced tether would be placed at an equilibrium position relative to the synodic frame associated to the binary system Jupiter Io. Since the orbit of Io is outside of the stationary orbit, the Jovian plasmasphere at Io is faster than the moon. As a consequence, the equilibrium position is in front of Io, at a distance given by d e = R Ioν 3 χ, where χ = I avb L mr Io ω ν 3 Here R Io = 4, 8 km is the Io s orbital radius, ω = rad/s is the angular velocity of the synodic frame, L the tether length, B =.79 6 T the Jovian magnetic field at the Io s orbit, ν = is the reduced mass of Io and I av the averaged tether current. Finally, m is the total mass of the spacecraft. In the configuration considered here the tether has a load of impedance Z C at the cathodic end. By tuning appropriately the value of Z C the useful power provided by the tether reaches an optimum value (a relative maximum). d w = cte L = cte W u (kw) 4 3 d w =mm L =35km L =5km d w =5mm d w =5mm d w =36mm L =5km L =km L =8km. 7 3 m T (kg) Figure.: Different optimized configurations in Io (h =.5 mm) Figure. summarize the results of the calculation performed following closely the theory developed in [8, 9] for the Io case and using an Aluminum tape.5 mm thick; the electronic plasma density considered taken from ref. [67] is about n =.8683 m 3. In the abscissa-axis figure shows the tether mass m T (in kg); in the ordinate-axis

209 Chapter : Io exploration with electrodynamic tethers 9 the useful energy provided by the tether (in kilowatts); in both axes we use a logarithmic scale. Two families of curves have been drawn on the figure: the green lines show the variation of the useful power W u with the tether mass m T when the tether length L is fixed; along these lines the only parameter which changes is the tether width d w. The red lines show the variation of the useful power W u with the tether mass m T when the tether width d w is fixed; along these lines the only parameter which changes is the tether length L. For the red lines, the slope is larger than for the green lines: the useful energy is more sensitive to the variations of the tether length L than to the variations of the tether width d w. Obviously, changing the tether length and width simultaneously is possible to increase W u keeping the tether mass m T constant. From figure. we select three tether designs which are summarized in Table 3. The useful power provided for the designs A and B is about watts, and for the configuration C is 3 watts. In all cases the tether dimensions, width and length are feasible for the Aluminum tape.5 mm thick. In order to calculate the nondimensional parameter χ and the distance d e between the center of mass of the spacecraft and the center of mass of Io, we assume a total mass m = 5 kg for the spacecraft, that is, the mass of the payload and the rest of devices ranges in the interval [33, 383] kg. A B C L (km) d w (mm) m T (kg) Z C (Oh) W u (w) 3 I sc (A) I av (A) φ (deg) χ d e (km),5 98,9 65,83 T (mn) Table 3. Three possible designs h =.5 mm Note that the small values of the parameter χ assures the stable character of the equilibrium positions where the spacecraft is situated; in effect, in [9] is proved that for values of χ in the range χ [,.5] the corresponding equilibrium position is stable. Therefore, the control needs are strongly reduced which at the end allows some increase of the scientific payload. However, the distance d e is large, compared with the equatorial radius of Io (r Io = 8 km). This is due to the strong gravitational field of this Jovian satellite, which requires to place the spacecraft far away from the moon if the electrodynamic thrust has to be balanced with the gravitation of Io. A first sight these equilibrium positions are not appropriated for the direct exploration of Io, due to the value of d e ; nevertheless, the spacecraft lies just inside the Io plasma torus, one of the most interesting element of the Jovian world. In effect, the Jovian plasmasphere has the iogenic plasma source which is now the subject of a detailed investigation (see, for example, refs. [68, 69]). The interaction between the plasma and the distribution of energetic heavy ions near the Io orbit, the mechanisms governing the mass loss at Io, the stable character of the plasma source, the role of Io volcanic plumes and

210 9 Section.3: Rotating electrodynamic tether orbiting Io the supply of dust to the magnetosphere are some open questions closely related with the Io plasma torus. From this point of view, a spacecraft at rest and placed in this mentioned equilibrium position could becomes a valuable source of scientific data on the Jovian world. Finally, the tension supported by the tether in these equilibrium positions is small, of the order of mn (see Table 3). These values of the tension are due to the smallness of the Jovian gravity gradient which is associated to the small averaged density of this planet. This is a weak point of this proposal which deserve to be studied in detail in future works, since it has implications in the deployment of the tether and also in the station keeping maneuvers. In particular, the use of rotating tethers, instead of steady tethers, could be an appropriate solution to get the tension needed to assure the structural stability of the tether. Another interesting alternative is to use some of the electrical energy produced by the tether to try to increase the tension; it should be taken into account that the tether will balanced and therefore the component of the forces normal to it will be globally zero and only locally could appear small non vanishing lateral forces. In the next sections we try to obtain energy from the Jovian plasmasphere with a rotating tether not at rest at the equilibrium position but orbiting around Io..3 Rotating electrodynamic tether orbiting Io Since gravitational attraction in the vicinity of Io is considerably higher than for the case of the Jovian inner moonlets it becomes possible to have the tether orbiting the moon while perturbed by the tether electrodynamic force. Similarly to the concept above Io will provide a«gravitational anchor» to keep the tether gravitational energy constant while continuous power is generated. In the rest of this paper we study an EDT on a retrograde elliptic equatorial orbit around Io with the tether rotating in the orbital plane. First of all, these orbits are stable against Jupiter gravitational pull (when eccentricity and semimajor axis are not too high) therefore the only critical part of the orbital design problem is to ensure the control of the Lorentz force in such a way not to destabilize the orbit while providing as much power as possible. An additional advantage of these orbits is that the Lorentz force can act as thrust or drag depending on the position on the orbit relatively to the Jupiter direction. This means that, in addition to power generation, the tether can be used to raise or lower the orbit around Io by judiciously controlling the tether current. The outline of the rest of the paper is as follows: first we derive basic relations for maximum power generation and maximum thrust of an EDT system in the Jupiter environment. Next we study the power production capability at Io and the impact on orbital stability. Finally we introduce a very simple control algorithm for propellantless orbit maneuvering inside the Io environment whose effectivness is evaluated numerically. There were some speculation about the convenience to avoid, in a mission to Io, the use of orbiters similar to the Europa Orbiter (EO) proposed by the NRC in 999. For example, ref. [59] states «the experience gained with the EO permits to consider as unrealistic an Io

211 Chapter : Io exploration with electrodynamic tethers 93 Orbiter for the next decade». Instead of the Io orbiter some repeated flyby missions to Io, appears more realistic although clearly insufficient. We will shows in this paper that an Io orbiter based in a rotating electrodynamic tether moving in an equatorial retrograde elliptic orbit is a quite interesting alternative which is free of the main problems afflicting EO while provides many other benefits. In the next section we will develop the basic theory for the electron collection of an electrodynamic bare tether; this theory can be used with rotating and no-rotating tethers..4 Optimum Power Generation of an EDT with no ohmic effects a Let us consider a bare electrodynamic tether (Fig..) of length L operating in the generator mode and flying along a generic orbit in the magnetosphere of a planet. Let the tether cross section be a tape of width d w and thickness h with h d w. Let us assume the tether maintains a rectilinear shape with ϕ (see fig..3) the angle between the tether line and the local motional electric field E along the orbit where the latter is defined as: E =( v sc v pl ) B (.) where B is the local magnetic field while v sc and v pl are the orbital velocity of the center of mass of the tethered system and the local velocity of the planet co-rotating plasma. Let there be a load of impedance Z C inserted just before the cathodic end for power extraction purposes. Due to the relatively low electronic plasma density of Jupiter magnetosphere the current flowing into the electrodynamic tether is small and ohmic effects can be neglected for tethers of practicable size. Similarly, I V A A e Most part of this section has been taken from Ref. [6] V t Imax = IC B z = z B z = z I V p Z C C C e z ZC IC z EmL Figure.: Schematic of a bare electrodynamic tether working in generator mode with negligible ion collection on the cathodic segment. I indicates the conventional (positive) current

212 94 Section.4: Optimum Power Generation of an EDT with no ohmic effects a the ion current collection in the negatively biased portion of the tether can be neglected as well as the losses at the anodic and cathodic ends of the tether. Under these assumptions and neglecting the magnetic field and plasma density variation along the tether the outside motional electric field can be assumed constant as well as the potential inside the conductive tether. Hence the potential difference between the tether surface and the plasma is linear with respect to the abscissa z from anodic to cathodic end (see Figure.) and will go to zero at a point B where z = z is given by: z = L Z CI C (.) E m where E m is the modulus of the motional electric field component along the tether line and I C is the tether current at the cathodic end. Under OML validity conditions and following the simplifying assumptions above current profile along the tether obeys: di dz = d w π en ee m (z z) m e, ( <z<z ) (.3) di dz =, (z <z<l) (.4) where e =.6 9 C is the electron charge and m e =9. 3 kg is the electron mass. Here n is the electronic density in the planet co-rotating plasma. The current profile is obtained by solving equation (.3), with the initial condition I() = : I(z) = 4d w 3π en eem [(z ) 3/ (z z) 3/ ], <z<z (.5) m e After introducing the non-dimensional zero-bias abscissa: ζ = z L = Z C I C E m L (.6) The maximum current along the tether follows from Eqs.(.5-.6): I max = I C = I ζ 3/ where I = 4d w 3π en ee m L 3 (.7) m e The average current along the tether and the power at the load can now be computed as: I av = L L I(z)dz = I ( 5 ζ)ζ3/, Ẇ = Z C I C = I E m L( ζ)ζ 3/ (.8) The power Ẇ has a maximum which can be obtained by imposing zero first derivative relative to ζ; the optimum value of ζ and the corresponding load impedance which is derived from equations (.6-.7) are given by ζ opt = 3 5, Zopt C = 3π 8 ζ opt ζ 3/ opt d w n me E m e 3 L = π 3 d w n me E m e 3 L (.9)

213 Chapter : Io exploration with electrodynamic tethers 95 Equation (.9) shows that the impedance needs to be continuously controlled to follow the variation of the motional electric field E m along the orbit. In addition, for the general case of a rotating electrodynamic tether, the impedance is also modulated by the variation of E m due to the tether rotation..5 Power Generation in Ionian Orbit ϕ z E x ωt Figure.3: Sketch of the equatorial retrograde orbit and angles y For the present analysis we will consider spinning electrodynamic tethers along retrograde equatorial orbits around Io, which offer the benefit of dynamical stability as it is explained in the next section. High inclination orbits, which are preferred for science, are more critical from the stability point of view and will be considered in future studies. A quick estimation of the maximum power which can be generated in ionian orbit can be done after realizing that orbital velocity around Io, whose escape velocity is about.5 km/s, has a small influence on the motional electric field which is mostly governed by the orbital velocity of Io around Jupiter (v IO = 7.3 km/s ). After neglecting the effect of the velocity around Io and considering constant electron plasma density of about n = cm 3 and constant magnetic field intensity of μt, the motional electric field is approximated with a constant vector aligned with the Io-Jupiter line: E ( v IO v pl ) B E π (cos ωt, sin ωt, ), E π. V/m (.) where ω is Io orbital rate and v pl the plasma velocity at Io. The latter reaches about 74 km/s. In equation (.) the vector E is expressed in an inertial frame whose first axis coincides with the Io-Jupiter line at the initial time (t =). As the tether spins around an axis orthogonal to the orbital plane the motional electric field component along the tether line is: E m = E π cos ϕ (.) After plugging equation (.) into equations (.7-.8) and assuming constant the value of ζ the rotation-averaged power for a single tether arm of length L yields: Ẇ av = π π/ π/ cos 3/ ϕdϕ 4d w 3π n E 3 π L5 e 3 m e [ ( ζ) ζ 3/] (.) In the averaging process we assume two sets of plasma contactors at the tether ends switching the role of the tethers end each half turn: it is a dual cathode spinning electrodynamic tether, for best performance.

214 96 Section.6: Orbit Stability P (Kw) Tether length (km) Figure.4: First order approximation of averaged generated power for a 5 cm wide tape tether of different lengths in Io orbit. Equation (.) is plotted in Fig..4 considering a 5 cm wide tape tether. When the influence of the orbital velocity around Io is taken into account on the orbital motion together with the effect of the Lorentz force and Jupiter gravitational pull the power will fluctuate around a mean value. Numerical simulations have been performed to account for all these effects considering a spacecraft of 5 kg equipped with a 5 km long and 5 cm wide EDT and flying along different orbits around Io (see Fig..5). Figure.5: Power generated with a 5 km long 5 cm wide tape tether along circular retrograde orbits of different radii: r =. r Io (grey solid line), r =. r Io (dark solid line), and r =3. r Io (dark dotted line)..6 Orbit Stability Among the Galilean satellites Io is the most critical from the point of view of orbit stability due to the strong gravitational pull of Jupiter. Although there is no detailed analysis of orbit stability focused on Io the case of Europa has received considerable attention in the literature (see for example Refs. [[63, 64, 65, 66]]) and many considerations can be extended to the Io case. In particular, following a similar analysis to the one performed by Lara and Russell [66] for the case of Europa, it can be seen that retrograde equatorial orbits around Io are stable as long as the eccentricity and the semimajor axis are smaller than a critical value. After running a montecarlo simulation of different orbit conditions and including both Jupiter third body perturbation and the effect of J it has been seen that orbits with apocenter radius less than 3.5 Io radii can be safely considered stable for at least a year. For an electrodynamic tether the effect of the Lorentz force on the orbit stability needs also to be investigated. The rotation-averaged Lorenz force on a tether arm of length L obeys: F = [ π/ ] LI av ( u B)dϕ = π/ π π L I av u dϕ B (.3) π/ π/

215 Chapter : Io exploration with electrodynamic tethers 97 where u is the tether line unit vector. When the load impedance is controlled for maximum power generation the non-dimensional zero-bias length ζ is constant and the following relation holds: π/ π/ I av u dϕ = E π π/ π/ E I av cos ϕ E π dϕ = π/ π I av cos ϕdϕ (.4) E π π/ Taking into account equations (.7-.8) and (.4), equation (.3) yields F = k Eπ ( E π B) (.5) where [ ] k π/ = cos 3/ ϕdϕ 4d w π π/ 5π n L 5 e 3 (5 ζ)ζ 3/ L.667 d w n 5 e 3 (5 ζ)ζ 3/ m e m e (.6) Similarly to what was done in the previous section an approximated expression for the Lorentz force can be derived after neglecting the influence of the orbital velocity around Io on the motional electric field and assuming a non-tilted dipole magnetic field. In this way the Lorentz force vector projected on an inertially fixed Iocentric reference system with x along the Jupiter-Io line and y along the Io velocity vector at t =(see Fig..3) yields: E π L F F ( sin ωt,cos ωt,) with F =.667 d w n B 5 e 3 (5 ζ) ζ 3/ (.7) m e As the direction of the force rotates around the orbit with angular rate ω the effect on orbital energy is, on average, zero. Several numerical simulations were carried out to assess the influence of the Lorentz force on the orbit stability. Figure.6 shows the evolution of the orbital radius after a month starting from circular retrograde orbits of different initial semimajor axes and having the Lorentz force acting continuously for power generation purposes. Although not visible in the plots a small drift r r Io time (days) Figure.6: Evolution of orbital radius for circular retrograde orbits of different radii (.,.,.5 and 3 Io radii) under the effect of the Lorentz force of a 5 km long 5 cm wide tape tether with power-optimized impedance control. in orbital energy appears with the orbits below Io radii getting slowly deorbited; when the initial radius is larger the orbit tends to gain energy. The small drift can be easily controlled by switching off the tether current at convenient points.

216 98 Section.8: Orbit Control.7 Orbit Control Because, as seen above, the main contribution to the motional electric field comes from Io orbital velocity rather than the spacecraft orbital velocity around Io the action of the Lorentz force results mainly into one thrust arc and one drag arc along each Ionian orbit as represented in figure.7. A simple control strategy has been implemented for a preliminary test of the propulsion capability of the EDT. The value control parameter ζ is switched to (maximum Lorentz force) on thrust arcs and switched to (no force) on drag arcs. A numerical simulation has been conducted assuming a 5 kg spacecraft equipped with a 5 km long and 5 cm wide electrodynamic tether starting from a low altitude retrograde circular orbit (r =. Io radii). Figure.8 reports the trajectory evolution during the spiral out phase until escape from Io gravitational field which require less than 5 months. Note that considering an aluminium tape tether of.5 mm thickness (which canbeachievedwithcurrent technology) the mass of the 5 km tether is 7 kg, that is, less than kg. Thrust arc v IO E π Io F Drag arc Figure.7: Schematic of electrodynamic thrust and drag arc for a generic Ionian orbit. We neglect the orbital velocity around Io in the determination of motional electric field r r Io y r Io x 4 6 r Io time (days) Figure.8: Spiral-out trajectory (upper) and orbital radius evolution (lower) for a 5 kg spacecraft propelled by a 5 km long 5 cm wide tape tether with simple current control strategy

217 Chapter : Io exploration with electrodynamic tethers 99.8 Summary and future research lines The use of bare electrodynamic tethers in scientific missions to Io have been investigated from the point of view of the generation of power and thrust. By using non-rotating self-balanced tethers situated in stable equilibrium positions of the synodic frame Jupiter Io a power generation of 3 kw can be achieved with tethers of reasonable length (5 35 km) and mass (5 7 kg). This configuration is well appropriated for the exploration of the Io plasma torus, since due to the large distance between the Jovian moon and the spacecraft (,, km for a total mass of 5 kg) the direct exploration of Io could be problematical. Future analysis will be focused on the stability issue related to the tether tension which in the present model turns out to be small (of the order of mn). The alternatives to be explored are: ) due to the high values of the energy generated, would be possible to use a part of it to increase the tether tension, and ) to change from a non-rotating to a rotating tether which provide, by centrifugal forces, the tension needs to assure the structural stability of the system. We also investigated the use of electrodynamic tethers along retrograde, equatorial orbits around Io for power generation and propellantless propulsion. Modest size tethers (5 km long) can provide more than one kw of continuous power on Ionian orbit without compromising the orbital stability. A very simple non-optimized control strategy has been tested which allows reaching escape velocity after less than 5 months starting from low Ionian orbit. Future studies will be focused on the definition of an optimum control strategy to reach escape velocity in minimum time. The possibility of using the Lorentz force to stabilize high inclination Ionian orbit will also be a matter for future analysis.

218 Section.8: Summary and future research lines

219 Part II Triangular Lagrangian Points

220

221 Chapter Circular Restricted Three Body Problem: a Commented Review In this section a fast overview of main results of the classical circular restricted problem of three bodies will be given. The problem has been deeply investigated and a huge amount of literature can be found about the argument. Despite the forty-years of distance, and the progresses happened in the meantime, a good introduction to the argument remains the classical book of Szebehely [6], published in 967. He made a glistening and ordered summary of main works published till the 967 and described in simple form a list of topics inherent to the argument: the derivation of motion equations, the adimensionalization of equations, the problem of order reduction, the treatment of singularities by means of regularization methods, the description of qualitative aspects of the problem, the derivation of the motion near the equilibrium points using the linear approximation, the Hamiltonian techniques for periodic orbits investigation, the summary of main numerical explorations, and the main modifications to the restricted problem (the 3D case, the elliptical problem and the Hill formulation of the problem). In this chapter we derive the equations of CRTBP both in Cartesian and spherical coordinates, we recall how calculate the Lagrangian point positions and the expression of the classical Jacobi integral. The following chap. 4 and chap. 5 will make a review of important concepts regarding the dynamics of bodies placed at triangular points.. Equations of CRTBP in Cartesian Coordinates The classical problem of three bodies can be simply stated as follows: calculate the trajectories of three point particles m, M and M subjected to reciprocal gravitational attractions. The general problem can t be solved in close form, so it is usually simplified in its circular restricted form, where the motion of masses M and M (the two primaries ) is supposed to be known, and the third mass is supposed to be infinitesimally small, so that it doesn t affect the motion of primaries. The usual reference frame where equations are obtained is the synodic reference frame. It is centered in the center of mass of the two primaries, has the X-axis aligned from M to M (with M >M ), the Z-axis perdendicular to the 3

222 4 Section.: Equations of CRTBP in Cartesian Coordinates revolution plane of the primaries and is co-rotating with them at the same angular velocity. Since the two primaries constitutes a classical kepler problem, they rotates with a pulsation givenbythethirdkeplerlaw: G(M + M ) Ω= d 3 (.) where d is the distance between the two primaries. The center of mass of the primaries is given by the barycenter relation d M = d M,fromwhich: d = d M M +M = dν and d = d M M +M = dν,andtherelationν = ν exists. The synodic reference frame is not an inertial one, so Newton equation of motion must be written using kinematic theorem of relative accelerations: where: R + Ω R + Ω Ω R = F gr + F gr = μ R R 3 μ R R 3 (.) x S R = y (.3) z x dν x + dν S R = R d = y = y (.4) z z x dν x dν S R = R d = y = y (.5) z z are respectively the position vector of mass m in Synodic Reference Frame, and the two distance vectors of mass m from the two primaries. The modules of distance vectors from primaries are: R =(x + dν ) + y + z (.6) R =(x dν ) + y + z (.7) The equations of motion of circular restricted three body problem result given by: ẍ Ωẏ =Ω x μ R 3 ÿ +Ωẋ =Ω y μ R 3 z = μ R 3 y μ R 3 y (x + dν ) μ R (x dν 3 ) y μ R y 3 (.8) The right hand sides of.8 can be thought as the gradient of the following potential function: J(x, y, z) = Ω (x + y )+ μ + μ (.9) R R Equation.8 can be conveniently adimensionalized by imposing:

223 Chapter : Circular Restricted Three Body Problem: a Commented Review 5 G(M + M )=Masses are non-dimensionalized with the total mass of the system, d =Lenghts are non-dimensionalized with the distance between primaries, /Ω =Time is non-dimensionalized with the inverse of pulsation of two primaries. Adimensionalization is convenient because the equations can be written as a function of a unique parameter (usually it is ν, the mass ratio of the mass of the second primary over the total mass of the primaries). By calling the new adimensional coordinates with the greek symbols ξ = x/d, η = y/d and ζ = z/d and with a minimal algebraic work we can obtain the non-dimensional form of the CRTBP equations: where: ξ η = ξ ν ρ (ξ + ν 3 ) ν η + ξ = η ν ζ = ν ρ 3 ρ 3 ζ ν ρ 3 ζ η ν ρ 3 η ρ 3 (ξ ν ) (.) ( ) ρ R = =(ξ + ν ) + η + ζ (.) d ( ) ρ R = =(ξ ν ) + η + ζ (.) d are the new adimensional distance from primaries. The right hand sides of. can be thought as the gradient of the following potential function: J(ξ,η,ζ) = (ξ + η )+ ν + ν ( ρ = ν ρ ρ + ) ( ρ + ν ρ + ) (.3) ρ Both the dimensional equations.8 and the adimensional equations. must be used with a simple but important caveat : we must assure the X-axis of the Synodic Reference Frame is pointed from the major primary M to the minor primary M,(withM >M ). For example, Szebehely adopts the opposite convention pointing the X-axis from the smallest primary M to the greatest primary M, so he obtain equations with different signs. Here we report the adimensional form of Szebehely to clear the differences: where: ξ η = ξ ν ρ (ξ ν 3 ) ν η + ξ = η ν ζ = ν ρ 3 ρ 3 ζ ν ρ 3 ζ η ν ρ 3 η ρ 3 (ξ + ν ) (.4) ρ =(ξ ν ) + η + ζ (.5) ρ =(ξ + ν ) + η + ζ (.6) In the present work the Szebehely convention won t never be used.

224 6 Section.: Equations of CRTBP in Spherical Coordinates. Equations of CRTBP in Spherical Coordinates Here we derive the equations of the Circular Restricted Three Body Problem in polar coordinates. The plane version of the problem will be considered, because for future considerations it will be sufficient to consider the motion within the revolution plane of two primaries. The D non-dimensional equations are: where: { ξ η = ξ ν ρ (ξ + ν 3 ) ν ρ (ξ ν 3 ) η + ξ = η ν η ν ρ η 3 ρ 3 (.7) ρ =(ξ + ν ) + η (.8) ρ =(ξ ν ) + η (.9) The new set of coordinates (r, α) and its derivatives with resptect to the time are (cosine and sine are indicated with the single initial letter c and s for brevity): { ξ = r cos(α) η = r sin(α) { ξ =ṙcα αrsα η =ṙsα + αrcα { ξ = rcα αrsα α rcα ṙ αsα η = rsα + αrcα α rsα +ṙ αcα (.) (.) (.) where the r = ξ + η is the adimensional radial coordinate, and α is the angular coordinate measured from the X-axis of Synodic Reference Frame. By substituting expressions in Eq..7, we obtain: where: { rcα αrsα = α rcα +ṙ αsα +ṙsα +r αcα + rcα + GRAV rsα + αrcα = α rsα ṙ αcα ṙcα +r αsα + rsα + GRAV (.3) GRAV = nu ρ 3 (rcα + nu ) nu ρ 3 (rcα nu ) (.4) GRAV = nu ρ 3 rsα nu ρ 3 rsα (.5) (.6) and ρ = r + ν +ν rcα (.7) ρ = r + ν ν rcα (.8) As can be seen from Eq..3, the two equations in r and α results to be coupled, but they can be decoupled with the following procedure:

225 Chapter : Circular Restricted Three Body Problem: a Commented Review 7 [ ] cα rsα ][ r = sα rcα α ] [ r = α and it finally results: [ α ] rcα +ṙ αsα +ṙsα +r αcα + rcα α + rsα ṙ αcα ṙcα +r αsα + rsα [ α r +r α + r ( ) ( ṙ r α ṙ r ] ) + [ ][ ] cα sα GRAV sα r cα r GRAV [ ] GRAV GRAV (.9) (.3) { r = α r +r α + r ν ρ (r + ν 3 cα) ν ρ (r ν 3 cα) α = ( ) ( ṙ r α ṙ ) ( ) r + sα r ν ρ 3 ν ρ (.3) 3 ρ 3 ρ 3 Equations.3 are the equations of the D Circular Restricted Three Body Problem in polar coordinates (r,α)..3 Lagrangian Points Lagrangian points are the equilibrium points of the CRTBP dynamical system. From definition of equilibrium point, the positions of Lagrangian points can be easily found imposing equal to zero the acceleration and the velocity vector in the motion equation. The solution gives five equilibrium positions within the plane of revolution of the two primaries, and they are conventionally grouped in 3 collinear points L, L and L 3,andtriangular points L 4 and L 5. The three collinear points lie in the X-axis of the Synodic Reference Frame connecting the primaries, and their adimensional positions can be found from the three roots of the following non-linear equation: ν f(τ) = τ + ν 3 (τ + ν ν )+ τ ν 3 (τ ν ) τ (.3) The roots of equation.3 can be found using numerical techniques. The two triangular points are placed at vertexes of two equilateral triangle formed with the segment connecting the primaries, and so their position have a simple explicit expression given by (adimensional coordinates): ν L 4,5 = ± 3 (.33) A list of useful techniques to solve the non-linear equation of collinears (Eq..3) can be found in [6]..4 Jacobi Integral The famous Jacobi integral of motion in the synodic reference frame is:

226 8 Section.4: Jacobi Integral ( ) ( R μ = Ω R + + μ ) + const. (.34) R R No other analytic integrals with global validity exist, as Poincaré demonstrated in 896.

227 Chapter 3 Equations of motion of a Tether at the Triangular Lagrangian Points In this chapter the basic equations necessary to describe the motion of a variable-length tether in the proximity of triangular Lagrangian points have been explicitly derived. The tether will be modeled like a rigid body capable of varying its length (Extended Dumbbell Model). Both orbital and attitude equations will be derived using Lagrange analytical formulation. At triangular points the ratio (L/R) of tether length over the distance of center of mass from the primaries can be considered infinitesimally small for all practical cases, and effects due to tether dimension are neglectable. 3. Reference Frames The Inertial Reference Frame I, G(î, ĵ, ˆk) is defined as: G: OrigininthecenterofmassG of the planetary system, X I axis (unit vector î): pointing toward a fixed inertial direction defined at t = t, Z I axis (unit vector ˆk): perpendicular to the orbital plane of two primaries, aligned with the angular momentum vector of the two primaries. Y I axis (unit vector ĵ): as a consequence. The Synodic Reference Frame S, G(ŝ, ŝ, ŝ 3 ) (non-inertial frame) is defined as: G: OrigininthecenterofmassG of the planetary system X S axis (unit vector ŝ ): aligned with the line joining M with M Z S axis (unit vector ŝ 3 ): coincident with the Z I axis of the Inertial Reference Frame, Y S axis (unit vector ŝ ): as a consequence. The Orbital Reference Frame O, M (ˆp, ˆq, ˆr) (non-inertial frame) is defined as: 9

228 Section 3.: Reference Frames Z I m T Y I Z I C ϕ θ m T Y I X I Z SYN Y SYN Z O Y O M G X SYN x L M x X O L β t) = Ω ( t t ) ( X ( t = t ) I Figure 3.: Reference Frames used in the dynamical analysis of the tether. M : origin coincident with the second primary M, X O axis (unit vector ˆp): aligned with the line joining M and M, Y O axis (unit vector ˆq): belonging to the orbital plane of primaries, Z O axis (unit vector ˆr): as a consequence. The Body Reference Frame B, C(û, ˆv, ŵ) (non-inertial frame) is defined as: C: origin in spacecraft Center of Mass, X B axis (unit vector û): pointing from C toward m, (Y B,Z B ) plane (unit vectors ˆv,ŵ): perpendicular to X-axis, passing through the point C. The Planar Attitude Reference Frame P, C(ˆι, ˆς,ˆκ) (non-inertial frame) is defined as: C: origin in spacecraft Center of Mass, X P axis (unit vector ˆι): rotated of the in-plane attitude angle θ w.r.t. the Inertial, (Y P,Z P ) plane (unit vectors ˆς,ˆκ): parallel to the (X I,Y I ) plane of the Inertial.

229 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 3. Roto-translations between Reference Frames In this section the transformations between reference frames defined in the previous paragraph have been reported. Given two arbitrary reference frames A and B, and an arbitrary vector P with components expressed in the B reference frame, the transformation which takes B P in A P is: A P = A P OB + A B [R]B P (3.) where P OB is the position vector of the origin of the B reference frame with respect to the origin of A, A B [R] is the rotation matrix which projects the components expressed in B in the components expressed in A. The transformation consists in a translation plus a rotation. The following graphic convention has been used with vectors: a left-apex has been posed to indicate the frame where the components of the vector are expressed (ex.: B P means the components of vector P are expressed in the B reference frame. The complete list of rotation matrices has been reported below: cos β sin β I S [R] = sin β cos β (3.) I O [R] =I S [R] = cos β sin β sin β cos β (3.3) cos θ sin θ I P [R] = sin θ cos θ (3.4) B P [R] = I B[R] = I P [R] P B[R] = cos ϕ sinϕ (3.5) sin ϕ cosϕ cos θ cos ϕ sin θ cos θ sin ϕ (3.6) sin θ cos ϕ cos θ sin θ sin ϕ sin ϕ cosϕ O S [R] =[] (3.7) P S [R] = P I [R] I S[R] (3.8) B S [R] =B I [R]I S [R] (3.9) P O [R] =P I [R]I O [R] (3.)

230 Section 3.4: Extended Dumbbell Model B O[R] = B I [R] I O[R] (3.) The inverse rotations are equal to the transposed matrices. The rotation matrix I S [R] between the Synodic and the Inertial Reference Frame and the rotation matrix I O [R] between the Orbital and the Inertial Reference Frame are identical, since they are parallel and have the same angular velocity by definition. This matrix is a function of the angle β(t) expressing the angular motion of the the planetary system w. r. t. the inertial space (see Fig. 3.) where: β(t) =Ω (t t ) (3.) with Ω is equal to the angular velocity of the planetary system, given by Eq... The rotation between the Inertial Reference Frame and the Body Reference Frame involves the attitude angles of the tether. Since the tether is approximated as a monodimensional rigid structure, only angles are sufficient to describe the body orientation w. r. t. the inertial space. The angles (see Fig. 3.) are chosen in the following manner: θ :is the in-plane attitude angle (plane parallel to the X-Y plane of Inertial Reference Frame) ϕ :is the out-of-plane attitude angle. Two rotation matrices, P I [R(θ)] and B P [R(ϕ)], are thus needed for the complete rotation matrix: ˆι î cos θ sin θ î ˆς = I [R(θ)] ĵ = sin θ cos θ ĵ (3.3) ˆκ ˆk ˆk û ˆι cos ϕ sinϕ ˆι ˆv = B [R(ϕ)] ˆς = ˆς (3.4) ŵ ˆκ sin ϕ cosϕ ˆκ The rotation matrix between the Orbital and the Synodic reference frame is trivially the unitary matrix: the vector transformation between the two systems is only given by a translation. 3.3 Extended Dumbbell Model Tether has been modeled assuming the Extended Dumbbell Model. The detailed description of such a model, regarding the mass geometry analysis, the evaluation of its angular velocity, angular momentum and kinetic energy, are reported in the Appendix F.

231 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 3 Z I m T κˆ ςˆ Y I C û ϕ θ ιˆ m T X I Figure 3.: Reference Frames involved in the attitude definition of EDM. From To I S O P B I [] S I [R] O I [R] P I [R] B I [R] S I S [R] [] O S [R] P S [R] B S [R] O I O [R] S O [R] [] P O [R] B O [R] P I P [R] S P [R] O P [R] [] B P [R] B I B [R] S B [R] O B [R] P B [R] [] Table 3.: Rotation matrices between reference frames.

232 4 Section 3.4: Gravitational Actions 3.4 Gravitational Actions In this section the resultant of gravitational forces and torques acting on the tether are evaluated. The newtonian force acting on an infinitesimal mass element has been integrated over the whole body mass, considered of arbitrary shape. Approximate relations valid for L/R << have been also derived Gravitational Potential The potential of gravitational forces over the tether is classically given by: μ V gr = dm (3.5) m R dm where μ = GM is the gravitational parameter of the attracting body (with G universal gravitational constant, and M mass of the primary), dm is the mass element of the body, and R dm is the scalar distance of dm from the attracting center. The ratio /R dm which compares in 3.5 can be conveniently rewritten observing that R dm = R + r = R ˆR + rû = R( ˆR + ηû) (3.6) where R is the position vectors of the CoM of the body from the center of the primary ( R is the position vector from the first primary and R from the second primary), r is the position vector of the mass element dm from the CoM of the body, and η = r/r, thus: R dm = R +η cos α + η = R ( ) n η n P n [cos α] (3.7) where P n [cos α] are the Legendre polynomials and cos α = ˆR û. Substituting Eq. 3.7 in 3.5 the potential is given by: V gr = μ R n= ( ) n P n [cos α] n= m η n dm (3.8) An explicit formulation of Eq. 3.8 has been derived by Peláez, and it is reported below: V gr = mμ R ( ( ) L n + ( ) R) n a n P n [cos α] n= (3.9) where:

233 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 5 P [x] = P [x] =x are the first Legendre Polynomials, P [x] = 3x P 3 [x] = 5x3 3x P 4 [x] = 35x4 3x +3 8 (3.) a = (3 sin φ Λ T ) a 3 = 4 cos φ(sin φ Λ T ) (3.) are coefficients independent on the size of the tether, and Λ T = mt m is the ratio of the mass of the tether over the total mass of the spacecraft, φ is the mass angle defined by: sin φ = m m + ΛT, cos φ = m m + ΛT. corresponding to the potential of a point mass, and a power series of the ratio L/R, wherel is the tether length and R the distance of center of mass C from the attracting primary. The ratio L/R is usually small and the corresponding terms negligible, but this condition does not hold for an extremely long tether or for a tether orbiting very close to one of the primaries. Equation 3.9 shows the potential is given by the sum a zeroth-order term mμ R 3.4. Resultant of Gravitational Forces Let be μ the gravitational parameter of the gravitational attractor. The gravitational force acting of the tether is given by: μdm F gr = R m Rdm 3 dm (3.) where μ is the gravitational parameter of the attracting primary, R dm is the distance vector of dm from the center of the primary, and R dm is the module of the distance vector R dm. Substituting the vector decomposition of Eq. 3.6 in Eq. 3.: dm F gr = μr m Rdm 3 ( ˆR + ηû) (3.3) The ratio /Rdm 3 can be rewritten similarly to Eq. 3.7: R 3 dm = R 3 ( + η cos α + η ) 3/ = R 3 where the S n [cos α] are the polynomials of the series: ( ) n η n S n [cos α] (3.4) n=

234 6 Section 3.4: Gravitational Actions S [x] = S [x] =3x S [x] = 3 (5x ) S 3 [x] = 5 (7x3 3x) Substituting Eq. 3.4 in Eq. 3.3 gives: F gr = μ R S 4 [x] = 5 8 (x4 4x +) n= ( ) n S n [cos α] m (3.5) η n ( ˆR + ηû)dm (3.6) An explicit formulation of Eq. 3.6 has been derived by Peláez and it has been reported below: ( F gr = mμ R R 3 + ( L ( ) n R n= ) n a n ( S n [cos α] ˆR S n [cos α]û) ) (3.7) The gravitational force over the whole body, given by Eq. 3., can also be expressed in an alternative way, using direct Taylor series expansion. Ratio R dm R can be rewritten as: 3 dm R 3 dm = ( R dm R dm ) = 3/ [( R + r) ( R + r)] = 3/ R 3 [+ R r R + r R ] 3/ (3.8) Calling χ = R r R where: R 3 dm + r R, Eq. 3.8 becomes: = R 3 n= ( 3 )χ n = R ( n 3 3 χ χ 35 ) 6 χ (3.9) ( ) α = ) = ( α n + ( α n ) α n n + (3.3) are the Newton binomial coefficients, recursively defined. The convergence of the series 3.9 is guaranteed when χ <. Substituting Eq. 3.9 in Eq. 3. and using the position of Eq. 3.6 gives: F gr = μ R m ( ˆR + ηû) n= ( 3) χ n dm (3.3) n

235 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 7 and remembering that η = r R,andthat m rdm =(from definition of center of mass), gives: F gr = μm R R 3 μ R ˆR m n= ( 3) χ n dm μ n R 3 û m The two integrals in Eq. 3.3 have the form: m r m n= n= ( 3 n ( 3 n r n= ) χ n dm = 3 m χ χ 35 ) χ n dm = m ( 3) χ n dm (3.3) n 6 χ3 +...dm (3.33) 3 5 rχ + 8 rχ 35 6 rχ3 +...dm (3.34) and they can be simplified with the approximation r R, sothatχ = R r R + r R R r R : m r n= m n= ( 3) χ n dm 3 n m ( 3) χ n dm 3 r n m ( ) R r R R r R dm + 5 dm + 5 m r m ( R r R ( R r R ) dm 35 ) dm 35 m m r ( ) 3 R r dm R (3.35) ( ) 3 R r dm R (3.36) Substituting the two approximate expressions of Eq and 3.36 into 3.3, the following approximate expression has been obtained: F gr μm R R 3 μ R 3 ( R + r) m Equation 3.37 takes the form: ( 3) ( R ) n r n R dm (3.37) F gr μm R R 3 μ ( ) ( ) ( ) 3 R r R 3 ( R + r) 3 m R + 5 R r R 35 R r R +... dm (3.38) The approximate expression of Eq shows that the resultant of gravitational forces acting over an extensive body in a newtonian field can be evaluated as the sum of the classical expression of Newton law of gravitation F gr = μm R R and a series of higher order 3 terms proportional to the scalar product ( R r). High order terms are all identically equal to zero when R r =, that is when R and r are perpendicular. High order terms can be neglected supposing R>>r, corresponding to the condition that the body dimension is much smaller than its distance from the attracting body. n=

236 8 Section 3.4: Gravitational Actions m r C dm R R R dm, R dm, M M Figure 3.3: Vectors used in the gravitational actions modeling. When we deal with different primaries of gravitational constants μ and μ,theapproximate expression of Eq. 3.38, becomes the linear superimposition of the forces of the two primaries; at zero-th order the expression is given by: F gr μ m R R 3 μ m R R 3 (3.39) where R is the distance vector of the satellite center of mass from the first primary and R is the distance vector of the satellite center of mass from the second primary. Projecting the force of Eq in the Synodic and in the Orbital Reference Frame the components of the force can be explicitly obtained: S F gr = S F gr m μ S R R 3 μ S R R 3 = μ R 3 O F O gr = F gr m μ O R R 3 μ O R R 3 = μ R Torque around the center of mass x + dν y z μ R 3 x + d y z μ R 3 x dν y z (3.4) x y (3.4) z The gravitational torque acting on the tether is given by the integral of all infinitesimal torques due to the gravitational attraction of the primaries: M C,gr = m μdm Rdm 3 r R dm (3.4)

237 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 9 where: r is the position vector of the mass element dm from the CoM C of the body. Torque is calculated using the body center of mass C as a pole for moments. Expressing R dm with Eq. 3.6, the cross product of Eq. 3.4 becomes r R dm = r ( R + r) = r R, and so Eq. 3.4 is: M C,gr = m μdm Rdm 3 r R (3.43) Using Eq. 3.4 and r = rû = Rηû, equation 3.43 can be rewritten as: M C,gr = μ ( ) R ( ) n S n [cos α] η n+ dm (û R) (3.44) n= m And explicit formulation of Eq has been derived by Peláez and it has been reported below: M C,gr = μm R (û R) ( L n ( ) R) n a n S n [cos α] (3.45) n= An alternative formulation can be derived using direct Taylor expansion. Substituting Eq. 3.9 in Eq gives: M C,gr = μ R R 3 r m n= ( 3) χ n dm (3.46) n where the series starts from and not from zero because the first term is identically equal to zero by definition of center of mass m rdm =. With the approximation r R, so that χ = R r R + r R R r R, equation 3.46 becames: ( M C,gr μ n 3) R n R n+3 r( R r) n dm (3.47) n= m The first terms of series 3.47 are: M C,gr 3μ R R 5 r( R r)dm + 5 m μ R R 7 r( R r) dm 35 m μ R R 9 r( R r) 3 dm m (3.48) A useful expression for estimation of gravitational torque can be derived from Eq considering only the first term, from which a simplified formulation can easily be obtained. Taking only the linear term of Eq. 3.48: M C,gr = 3μ R 5 r( R r)dm R (3.49) m a zero quantity m r dm R R can be subtracted inside the integral without changing the result:

238 Section 3.5: Gravitational Actions M C,gr = 3μ R 5 r( R r)dm R 3μ m R 5 r dm R R (3.5) m in order to obtain: M C,gr = 3μ [ ] R 5 r( R r) r R dm R m M C,gr = 3μ R 5 R m [ ] r R r( R r) dm M C,gr = 3μ R R 5 [( r r)[] r r] Rdm (3.5) m In Eq. 3.5 the definition of tensorial product can be introduced, ã( b c) :=(ã b) c, in order to obtain the inertia tensor definition m ( r r)[] r rdm := [I C]. The resulting approximate expression for the gravitational torque is given by the following expression: M C,gr = 3μ R 5 R [I C ] R (3.5) Equation 3.5 is the expression of gravity torque exerted by a point-mass attractor over an extended satellite with inertia tensor [I C ]. It is a function of both the inertia tensor of the body [I C ] evaluated w.r.t. the center of mass C of the body, and the distance R of the tether center of mass from the primaries. When we deal with different primaries of gravitational constants μ and μ, the expression of Eq. 3.5 becomes the linear superimposition of the torques of the two primaries: M C,gr = 3μ R 5 R [I C ] R + 3μ R 5 R [I C ] R (3.53) where R is the distance vector of the satellite center of mass from the first primary and R is the distance vector of the satellite center of mass from the second primary. The inertia tensor of the EDM tether (given by Eq. F.7) has a simple structure, and it allows to simplify Eq Calling B R = [R () R () R (3)], B R = [R () R () R (3)], and projecting Eq in the Body Reference Frame: B M C,gr = 3μ I s R 5 R ()R (3) + 3μ I s R ()R () R 5 R ()R (3) R ()R () The unit vectors of Body Reference Frame are (û, ˆv, ŵ), so Eq. rewritten as: B M C,gr = 3μ I s R 5 ( R û)( R ŵ) + 3μ I s R ( R û)( R ˆv) 5 ( R û)( R ŵ) ( R û)( R ˆv) (3.54) 3.54 can also be (3.55)

239 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 3.5 Bare Electrodynamic Tethers 3.5. Motional Electric Field Analysis When the tether moves with a non-zero velocity with respect to the environment plasma and in presence of a magnetic field B, it is subjected to a motional electric field Ẽ given by: Ẽ =(ṽ s/c ṽ pl ) B (3.56) The motional electric field in SI units is measured in Volts per meter [V/m]. In Eq ṽ s/c is the inertial velocity of the spacecraft, ṽ pl is the inertial velocity of the plasma and B is the local magnetic field vector. For the velocity ṽ pl we can assume the environment plasma co-rotates with the planetary magnetic field, which is fixed to the rotating planet. The effective motional electric field must be obtained by projecting the vector Ẽ along the tether versor û: E t = Ẽ û = E π cos φ (3.57) The condition of greatest motional electric field happens when the tether is aligned with the vector Ẽ. Assuming the relative velocity of the system is a vector contained in the plane of revolution of the two primaries, and assuming a non-tilted dipolar magnetic field, Eq shows that the vector Ẽ is contained on the revolution plane of the two primaries and it is perpendicular to the velocity vector: ThemoduleofẼ in this simplified case is: v r,x v r,y B z Ẽ v r,y = v r,x B z (3.58) v r,z B z E π = B z (v r,x + v r,y) (3.59) 3.5. Tether Circuit Equation The circuit of the tether has been represented in Figure The equation of the circuit is given by: E t L =ΔV C +ΔV L +ΔV tether +ΔV A (3.6) The nominal motional electric field E t L [V] is equal to the sum of the following potential drops: the potential difference ΔV C between the environmental plasma and the cathode, the potential drop in the load ΔV L, the potential drop of the tether ΔV tether, and the potential drop between the anode and the plasma.

240 Section 3.6: Bare Electrodynamic Tethers I ΔV C ΔVL ΔVtether ΔVA PLASMA PLASMA CATHODE ANODE LOAD TETHER L E t Figure 3.4: Schematics of the circuit of the tether Current and Collection Model Assuming the ohmic resistance of the tether be negligible, the averaged current collected by a bare tether of tape section (width thickness, w t) can be estimated using the SanMartìn formula: I avg (OML) = 5 ( wl π ) q e N e qe E t L m e (3.6) where E t, given by Eq. 3.57, is the projection of the external motional electric field along the tether Power Extraction The ideal power P id that can be extracted using a bare electrodynamic tether is given by the projection of the Lorentz force F el along the velocity vector ṽ rel of the tether with respect to the plasma: P id = F el ṽ rel = F el (ṽ s/c ṽ pl ) (3.6) The useful power at the load if a fraction of P id ; the maximum generator efficiency is (Bombardelli): η max = I maxe t L = 5.3 (3.63) P max 5

241 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points Electrodynamic Actions 3.6. Resultant on the center of mass The electrodynamic force acting on an infinitesimal length ds of the cable is given by: d F el = n e Ads F L (3.64) where n e [electrons/m 3 ] is the electron volumetric density in the conducting cable, Ads [m 3 ] is the infinitesimal volume of the cable, and F L [N] is the Lorentz force F L = q e ṽ d B. Using the definition of current vector j = n e q e ṽ d, and supposing the current is parallel to the line element dˆl, ds j = jdˆl, Eq becomes: The total electrodynamic force is thus given by: d F el = Ajdˆl B = idˆl B (3.65) F el = L ( ) i(l)dˆl B = I avg L û B The magnetic field B can be regarded constant along the tether. (3.66) 3.6. Torque on the center of mass The infinitesimal electrodynamic torque, considering the center of mass C as a pole for moments, is given by: M C,el = r d F el = r i(l)dˆl B (3.67) Since the tether is approximated as an EDM, it is simply r = lû, and the total electrodynamic torque is given by: M C,el = û L i(l)ldˆl B (3.68) 3.7 Equations of motion with Lagrange approach The equations of motion have been derived using the classical Euler-Lagrange equation: d dt ( L q ) L q = F where q is the vector of generalized coordinates, q =[x y z θ φ L], and F is the vector of generalized forces F =[Q x Q y Q z Q θ Q φ Q L ], given from the principle of virtual work as: δw = F wδt (3.69)

242 4 Section 3.7: Equations of motion with Lagrange approach The first three components x, y, z are the position of tether center of mass, θ, φ are the attitude angles (in-plane and out-of-plane respectively), and L is the tether length, governed by an over-imposed control law. The Lagrange function L = T V of the spacecraft is given by: where: L = T V = T CoM + T rot V gr (3.7) T CoM = m (v rel) m( Ω ṽ rel ) ( R d ) m ( R d ) [Ω] [Ω]( R d ) (3.7) T rot = ω[i C] ω (3.7) ( V gr = mμ ( ) L n + ( ) n a n P n [cos α] (3.73) R R) n= The resulting Lagrange function projected in the appropriate reference frame is: L = m (ẋ +ẏ +ż )+mω((x + d )ẏ yẋ)+ mω ( (x + d ) + y ) + I s θ cos ϕ+ I s ϕ V gr (3.74) ( ) d L L = mẍ mωż dt ẋ x = mωż + mω (x + d ) V gr (3.75) x ( ) d L L = mÿ dt ẏ y = V gr (3.76) y ( ) d L L = m z + mωẋ dt ż z = mωẋ + mω z V gr (3.77) z ( ) d L dt θ = θi s c L ϕ + θ ϕi s sϕcϕ = (3.78) θ ( ) d L = I s ϕ + I dt ϕ L s ϕ ϕ = I s θ cϕsϕ (3.79) The dynamical equations governing the motion of the EDM tether are: mẍ mωż mω (x + d) =F gr,x + F el,x (3.8) mÿ +mωẋ mω y = F gr,y + F el,y (3.8) m z = F gr,z + F el,z (3.8) θ + θ L L θ ϕ tan ϕ = ϕ + ϕ L L + θ cos ϕ sin ϕ = M θ m r L cos ϕ (3.83) M ϕ m r L (3.84)

243 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 5 The first three equations the evolution of center of mass position in the Orbital Reference Frame, and the fourth and fifth equations describe the tether attitude as a function of the two orientation angles θ and ϕ with respect to the inertial frame. Inside attitude equations it has I been used s I = L s L,sinceI s = m r L and I s =m r L L. Attitude is completely determined by two orientation angles, because tether is a one-dimensional rigid body moving in the space. In the Synodic Reference Frame the specific equations of motion are: where: ẍ Ωż Ω x = f gr,x + f el,x (3.85) ÿ +Ωẋ Ω y = f gr,y + f el,y (3.86) z = f gr,z + f el,z (3.87) θ + L L θ θ ϕ tan ϕ = ϕ + L L ϕ + θ cos ϕ sin ϕ = M θ m r L cos ϕ (3.88) M ϕ m r L (3.89) fgr = F gr fel = F el (3.9) m m are the specific external forces [N/kg]. An arbitrary additional external force can be considered by adding its components on the right side of equations. For example, to simulate the navigation of tether with an additional thruster, the three components of the propulsion force F prop =(F prop,x,f prop,y,f prop,z ) must be added to the right side of the first three orbital equations. A vectorial form of equations is (Newton approach confirms it): R + Ω R + Ω Ω R = f gr, + f gr, + f el (3.9) [I C ] ω +[I C ] ω + ω [I C ] ω = M gr, + M gr, + M el (3.9) where the gravitational force (Eq. 3.7) and electrodynamical force (Eq. 3.66) in Eq. 3.9 are: ( fgr = μ ( ) R R L n ( 3 + ( ) n a n S n [cos α] R ˆR S n [cos α]û) ) (3.93) n= fel = I ( ) avgl û B (3.94) m and the gravitational torque (Eq. 3.45) and electrodynamical torque (Eq. 3.68) in Eq. 3.9 are: M gr = μ R (û R) ( L n ( ) R) n a n S n [cos α] (3.95) n=

244 6 Section 3.9: Motion when L/R is Infinitesimally Small M el = û L i(l)ldˆl B (3.96) The series comparing in the gravitational actions converge when the ratio L/R is small, that is when the maximum length of the tether is much less than the distance from the attracting body. This condition is true for all practical cases. 3.8 Motion when L/R is Infinitesimally Small At triangular points the ratio L/R can be considered infinitesimally small, and the expressions of gravitational force and torque simplify considerably, since only the first term of the series holds, all the others being zero. Equations of motion 3.9 and 3.9 take the following simplified form: R + Ω R + Ω Ω R = μ R R 3 μ R R 3 + f el (3.97) [I C ] ω +[I C ] ω + ω [I C ] ω = 3μ I s R R 5 [I C ] R + 3μ I s R R 5 [I C ] R + M el (3.98) Equation 3.97 of orbital position has been projected in the Synodic Reference Frame, and the attitude equation 3.98 has been projected in the Body Reference Frame. θ + L L θ θ ϕ tan ϕ = [ 3μ cos ϕ R 5 ϕ + L where: ẍ Ωẏ =Ω x μ R 3 (x + dν ) μ R 3 (x dν )+f el,x (3.99) ÿ +Ωẋ =Ω y μ R 3 y μ R 3 y + f el,y (3.) z = μ R 3 z μ R 3 z + f el,z (3.) L ϕ + θ cos ϕ sin ϕ = 3μ R 5 R ()R () + 3μ R 5 R ()R () + M θ,el I s ] (3.) R ()R (3) 3μ R 5 R ()R (3) + M ϕ,el (3.3) I s R () R () B R = R () = B S [R]S B R R = R () = B S [R]S R (3.4) R (3) R (3) and S R and S R are expressed by Eq..4 and Eq..5. Equations of orbital motion (Eq. 3.99, 3. and 3.) coincides with the classical CRTBP equations; this set of equations have been described in the Chapter. Attitude equations (Eq. 3. and 3.3) are the equation of a variable length tether in presence of two gravity gradient terms (due to the presence of the two primaries) and of the electrodynamic term.

245 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points Equations of motion in non-dimensional coordinates Equation of orbital motion of the limiting case L/R have been adimensionalized following the standard procedure of adimensionalization of the CRTBP literature: d =: Lenghts are adimensionalized w.r.t. the distance d between primaries; M + M =: Masses are adimensionalized w.r.t. the total mass of the planetary system; Ω =: Times are adimensionalized w.r.t. the inverse of mean angular velocity of primaries. Adimensionalization is convenient since it allows to obtain a form of equations independent from the particular planetary system, and all parameters of the system can be reduced to one single parameter. Procedure of adimensionalization is straightforward when motion equation are rewritten substituting the universal gravitational constant with the relation obtained from Kepler law G = d 3 Ω /(M +M ), remembering that M /(M +M )=ν = ν and M /(M +M )= ν = ν and then multiplying both sides for /(dω ): ẍ ẏ x ν ÿ +Ω ẋ Ω y dω = d3 Ω x + d M z R 3 y d3 Ω x f M el,x M + M z R 3 y + f el,y M + M z f el,z (3.5) ẍ ẏ x ν x + d ÿ +Ω ẋ Ω y dω = Ω ( ν) ( ) 3 y z R d z Ω ẍ dÿ d z d + ẏ x d ẋ d ν + y Ω d d = ( ν) ( ) 3 R d x+d d y d z d ( ν ) 3 R d Ω ν ( ) 3 R d The resulting adimensional equations in the Orbital Reference Frame are: x d y d z d x y + z + dω f el,x f el,y f el,z (3.6) f el,x f el,y f el,z (3.7) ξ η = ξ + ν ν ρ 3 (ξ +) ν ρ 3 ξ + F el,x mdω (3.8) η + ξ = η ν ρ 3 η ν ρ 3 η + F el,y mdω (3.9) ζ = ν ρ 3 ζ ν ρ 3 ζ + F el,z mdω (3.)

246 8 Section 3.: Hill approximation where ξ = x/d, η = y/d and ζ = z/d are the three adimensional coordinates, and R /d = ρ, R /d = ρ. When projected in the Synodic Reference Frame, adimensional equations coincides with the classical CRTBP equations (Eq..): where: ξ η = ξ ν ρ 3 (ξ + ν ) ν ρ 3 (ξ ν )+ F el,x mdω (3.) η + ξ = η ν ρ 3 η ν ρ 3 η + F el,y mdω (3.) ζ = ν ρ 3 ζ ν ρ 3 ζ + F el,z mdω (3.3) ρ =(ξ + ν ) + η + ζ (3.4) ρ =(ξ ν ) + η + ζ (3.5) The adimensionalization of attitude equations follows an identical procedure, and it results: θ + L L θ θ ϕ tan ϕ = [ 3ν cos ϕ ρ 5 ( ρ û)( ρ ˆv)+ 3ν ρ 5 ( ρ û)( ρ ˆv)+ M ] θ,el I s (3.6) ϕ + L L ϕ + θ cos ϕ sin ϕ = 3ν ρ 5 ( ρ û)( ρ ŵ) 3ν ρ 5 ( ρ û)( ρ ŵ)+ M ϕ,el I s (3.7) 3. Hill approximation Hill approximation of motion equations has been executed. When the parameter ν involved in the equations is small (M << M ), it can be removed from the equations. It is possible to arrive at the Hill formulation by performing the following change of variables: ξ a η = ν /3 b (3.8) ζ c Multiplying both sides of the adimensional form of equations for /ν /3 : ξ/ν /3 η/ν /3 ξ/ν /3 η/ν /3 + ξ/ν ( ν)/ν/3 /3 + η/ν /3 = ζ/ν /3 = ν (ξ +)/ν /3 ρ 3 η/ν /3 ν ξ/ν /3 ζ/ν /3 ρ 3 η/ν /3 F el + ζ/ν /3 mdω ν /3 (3.9)

247 Chapter 3: Equations of motion of a Tether at the Triangular Lagrangian Points 9 ä b + ḃ a ẋ = b + c ä b + ḃ a + ȧ = c ν ν /3 ν /3 ν ν /3 ν ρ 3 ( b ( ρ 3 ν ρ 3 ρ 3 ν ρ 3 ( ρ 3 ν ρ 3 ) c ξ+ ν /3 b c )( ξ ) b ν a ρ 3 b + c ) ν + /3 ν /3 It is possible to rewrite the terms ν/ρ 3, /ρ 3 and ν/ρ3 : ν ρ 3 = ( ρ ) 3 = ν /3 F el mdω ν /3 (3.) ν a ρ 3 b + c F el mdω ν /3 (3.) ( a + b + c ) 3 = υ 3 (3.) where the quantity υ = a + b + c has been defined. Similarly: ( ρ ) 3 ( ) 3 ( ρ 3 = = ( + ξ) d + η + ζ = +ξ + ξ + η + ζ ) ( ) 3/ 3/ = +ξ + ν /3 υ (3.3) Expanding this expression with a Mac-Laurin series and executing the limit for ν the Hill-approximated equations of motion have been obtained: ä b + ḃ 3a ȧ = a b F el + υ c c c mdω ν /3 (3.4)

248 3 Section 3.: Hill approximation

249 Chapter 4 Dynamics at the Triangular Lagrangian Points: Linear Analysis 4. Classical Linear Analysis In order to investigate the motion of a satellite in the proximity of a lagrangian point, the linear techniques can be used. Despite its simplicity, the study of the linearized system can provide several informations, as the proper frequencies of the system (eigenvalues), the independent directions of eigenspaces (eigenvectors) and linear stability considerations. 4.. Procedure of Linear Analysis The standard methodology consist in the following steps:. Coordinates transformation,. Taylor series expansion, 3. Linear variational equation of motion, 4. Analysis of associated eigenproblem (eigenvalues and eigenvectors), 5. Stability analysis. With the coordinates transformation we conveniently place the reference frame centered in the particular equilibrium point we are considering: { ξ = a + δ (4.) η = b + ɛ where (a, b) are the coordinates of the particular lagrangian point, and (δ, ɛ) are the new variational coordinates measured with respect to the equilibrium point. The Taylor series expansion is necessary to reduce the non-linear system to a linear one by neglecting all terms but the zero-order and first-order term of the expansion. The non-linear function J can be expanded with Taylor series around the equilibrium point x =(a, b): 3

250 3 Section 4.: Classical Linear Analysis J = J(a, b)+j ξ (a, b)δ + J η (a, b)ɛ +! J ξξ(a, b)δ + J ξη (a, b)δɛ +! J ηη(a, b)ɛ + O(3) (4.) where with the symbol J ξ we mean the partial derivative of J with respect to ξ. The equations of motion can thus be rewritten in a form that usually is called the linear variational equation of motion. At equilibrium points we have J ξ (a, b)δ = J η (a, b)ɛ =,so the linear variational equation takes the form: { δ ɛ = Jξξ (a, b)δ + J ξη (a, b)ɛ + O() (4.3) ɛ + δ = J ξη (a, b)δ + J ηη (a, b)ɛ + O() Equations 4.3 are the two second order linear variational equations, describing the planar motion around the equilibrium point. The associated system of first-order ordinary differential equations is: δ x =[A] x = ɛ J ξξ (a, b) J ξη (a, b) δ (4.4) J ξη (a, b) J ηη (a, b) ɛ The matrix [A] is the characteristic matrix of the linear system, and can be analyzed using standard eigen-problem methodologies. The associated characteristic equation is: det ([A] λ[i]) = (4.5) resulting in the following fourth-order equation in λ: λ 4 +(4 J ξξ (a, b) J ηη (a, b))λ + J ξξ (a, b)j ηη (a, b) (J ξη (a, b)) = (4.6) From the analysis of the roots of characteristic equations (eigenvalues of the linear system) the considerations on linear stability can be inferred. 4.. Application to Triangular Points The application of the procedure at triangular libration points allows to obtain in a straightforward way the motion of the satellite in the proximity of the equilibrium point. The result is classical, and it can be found for example in [6]; here we report the main results for convenience, because they will be useful in the present analysis since they will be often recalled. The values of J at triangular points are functions of the mass ratio ν only: J ξξ (L 4 )=J ξξ (L 5 )= 3 4 J ηη (L 4 )=J ηη (L 5 )= 9 4 (4.7) (4.8) J ξη (L 4,5 )=±3 3 (ν ) (4.9)

251 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 33 The characteristic equation 4.6 becomes: and the solutions are (Λ =λ ): λ 4 + λ ν ( ν )= (4.) Λ, = ± 7ν ( ν ) (4.) Motion around L4 and L5 is discrimined by the critical value ν cr =( 69/9)/ of the mass parameter, corresponding to the zero value of the discriminant under the square root: 7ν ( ν )=. Three cases can thus be revealed:. <ν<ν cr : Motion is bounded and is given by the superposition of harmonic oscillations with different frequencies s, s ;. ν cr <ν</ : Motion is unstable; 3. ν = ν cr : Solution contains secular terms. Stable solution Roots of characteristic equations are pure imaginary: λ, = ±i Λ = ±is λ 3,3 = ±i Λ = ±is (4.) where s and s are the two angular frequencies (or mean motions) that characterize the motion around triangular points. Frequencies s and s are the eigen-frequencies of the dynamical system and they don t depend on the coordinate system used. The solution of the motion is given by: [ ] [ ] [ ] [ ] [ ] ξ C S C S = cos(s η C t)+ sin(s S t)+ cos(s C t)+ sin(s S t) (4.3) The motion is a combination of short-period terms, associated with s : s = Im + 7ν ( ν ) Ω [ ] rad s (4.4) and long-period terms, associated with s : s = Im 7ν ( ν ) Ω [ ] rad s (4.5) The corresponding periods are T =π/s and T =π/s, measured in seconds.

252 34 Section 4.: Tether at Triangulars For values of ν occurring in the Solar System ( 3 <ν< 6 ) long-period terms (T =π/s ) are between (about) - 5 times the period of revolution of the primaries, and short period terms (T =π/s ) are approximately the same as the period of revolution of the primaries (s ); Short-period solution and long-period solution can be studied separately, but in both cases, the orbit results to be an ellipse with rotated principal axes: ξ Γ [ 4s + Jxy ] + η +ξηγ J xy =4s Γ (S + C ) (4.6) where: Γ i = s i + J > (i =, ) (4.7) yy(a, b) It is convenient to rotate the reference frame to align with principal axes of the ellipse: [ ] ξ = η [ cos(α) sin(α) sin(α) cos(α) ][ ξ η ] (4.8) where the angle depends on the planetary system only: tan(α) = 3( ν ) (4.9) Unstable solution When the mass parameter is ν cr <ν /, that is when it is comprised in the range between the critical value ν cr = and the condition ν =/ of equal primaries, the linear analysis reveals 4 complex eigenvalues, with positive real part. The motion result unbounded, the equilibrium unstable and orbits become spirals. Secular solution In the limiting critical case ν = ν cr there are couples of eigenvalues, each one with multiplicity equal to. The double roots give secular terms and the equilibrium is unstable. 4. Tether at Triangulars The non-dimensional equations of motion of the tether in the limiting case when the ratio (L/R) is infinitesimally small (ratio between tether length L and distance R from the gravitational attractor) are:

253 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 35 ξ η = ξ ν ρ 3 (ξ + ν ) ν ρ 3 (ξ ν )+ F el,x mdω (4.) η + ξ = η ν ρ 3 η ν ρ 3 η + F el,y mdω (4.) ζ = ν ρ 3 ζ ν ρ 3 ζ + F el,z mdω (4.) θ + L L θ θ ϕtanϕ = cosϕ [ 3ν ρ 5 ( ρ û)( ρ ˆv)+ 3ν ρ 5 ( ρ û)( ρ ˆv)+ M θ,el I s (4.3) ϕ + L L ϕ + θ cosϕsinϕ = 3ν ρ 5 ( ρ û)( ρ ŵ) 3ν ρ 5 ( ρ û)( ρ ŵ)+ M ϕ,el (4.4) I s where: ] ρ =(ξ + ν ) + η + ζ (4.5) ρ =(ξ ν ) + η + ζ (4.6) They constitute a non-linear system of Ordinary Differential Equations that don t posses an explicit analytical solution. A quantitative investigation of the system can easily be done by means of classical numerical algorithms for ODEs integration, like the Runge Kutta scheme or the Adams Bashforth Moulton algorithm. The numerical investigation by means of computer simulations allows to obtain informations about a particular solution of the ODEs system, starting from preestablished initial conditions and for a limited set of the time. A qualitative investigation of the non-linear ODEs system follows a more elegant way, and it allows to obtain more deep informations about the system, for example the stability of solutions around equilibrium points. A standard procedure in the analysis of a nonlinear system of ODEs is to find its equilibrium points, linearize around them,and obtain informations about motion near equilibrium points. 4.3 Inert tether When we deal with an inert tether the electrodynamic force is zero: in this case the three orbital equations coincide with the classical CRTBP equations, and the two attitude equations can be treated similarly to the Robinson solution [] Orbital motion: classical CRTBP When the Lorentz force is equal to zero, equations of orbital motion (Eq. 4., Eq. 4. and 4.) turn out to be equal to the classical CRTBP equations:

254 36 Section 4.3: Inert tether ξ η = ξ ν ρ 3 (ξ + ν ) ν ρ 3 (ξ ν ) (4.7) η + ξ = η ν ρ 3 η ν ρ 3 η (4.8) ζ = ν ρ 3 ζ ν ρ 3 ζ (4.9) Equilibrium positions of this set of ODEs are the five classical Lagrangian points within the synodic reference frame. Motion at triangular points L 4,5 =( ν, ± 3, ) has been accurately described in literature [6]. When evaluated at triangular points, the characteristic equation of the linear variational equation has the following simple bi-quadratic form: λ 4 + λ ν ( ν )= (4.3) When the mass parameter ν = M /(M + M ) is < ν < ν cr, with ν cr = ( 69/9)/ , equation 4.3 has two pure imaginary roots. In this case the linear analysis reveals that the motion at triangular points is stable, and in the planar-restricted case it owns two typical eigen-frequencies s and s, associated to a long period motion and a short period motion respectively (see expressions of Eq. 4.4 and Eq. 4.5). In general, the motion in the neighborhood of triangular points is given by the superimposition of short and long period motion. The linear analysis also reveals that each eigen-solution lies on a rotated ellipse with respect to the principal axis of the synodic frame, the angle being dependent only on the planetary system parameter: tan(α tr )= 3( ν ).Angleα tr isverycloseto3degfor most of planetary systems within the Solar System. The out-of-plane motion along z-axis is decoupled from (x,y) planar motion, and in the linear approximation it is simply expressed by an undamped harmonic oscillator. Figure 4. and 4. shows two examples of motion around triangular points in two different planetary systems, the Earth-Moon system, and the Jupiter-Io system. From figures the composition of short-period and long period motion can clearly be recognized Attitude: Robinson solution Equilibrium positions of the two attitude equations of the tether Eq. 4.3 and 4.4 can be found imposing θ = θ =and φ = φ =: = [ 3ν cosϕ ρ 5 ( ρ û)( ρ ˆv)+ 3ν ] ρ 5 ( ρ û)( ρ ˆv) = 3ν ρ 5 (4.3) ( ρ û)( ρ ŵ) 3ν ρ 5 ( ρ û)( ρ ŵ) (4.3) When tether center of mass coincides with one of the triangular points L 4,5 the position vectors ρ and ρ have the following explicit coordinates (in synodic reference frame):

255 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 37 S ρ (L 4,5 )= ( S ρ (L 4,5 )= (, ± ) 3, ), ± 3, (4.33) (4.34) (4.35) and their modules are unitary ρ =and ρ =. The following equilibrium positions can be revealed []:. (θ, ϕ) =( θ, π ) : Instable position tether is vertically oriented with respect the (ξ,η) revolution plane;. (θ, ϕ) =( π α, ) : Stable position tether versor û lies within the revolution plane (ξ,η), and is oriented of a θ = π α angle with respect to the ξ axis; 3. (θ, ϕ) =( α, ) : Instable position tether versor û lies within the revolution plane (ξ,η), and is oriented of a θ = α angle with respect to the ξ axis; Figure 4.3 shows the two equilibrium orientation of the inert librating tether within the revolution plane. When the two primaries have equal masses M = M, the stable solution is parallel to the η axis of synodic reference frame. When the secondary mass tends to zero M, the stable solution tends to point M, and the angle α tr 6deg. The unstable orientation is always at 9ř with respect the stable one. Robinson [] has executed the linearized analysis of the motion of a dumbbell system around attitude equilibrium positions when its center of mass is placed at triangular points locations. The procedure of the analysis follows the classical steps: to choose the equilibrium position as the zero of coordinate by means of a coordinate transformation, to obtain the variational equation and to analyze the main features of the associated linear variational equation. The analysis revealed that the in-plane motion of libration within the revolution plane of primaries is periodic, with an infinitesimal period T θ given by: T θ = 4π 6 + ( ν) = π 3(3ν 3ν +) 4 (4.36) where ν is the mass parameter of the planetary system, ν = The out-of-plane motion is periodic too, with period T ϕ given by: M M +M. 4π T ϕ = ( = ν) The corresponding frequencies of librations [rad/s] are: 4π + 6 3ν 3ν + (4.37)

256 38 Section 4.4: Active Electrodynamic Tether ω θ = Ω ( ) 6 + ν (4.38) ω ϕ = Ω ( ) ν (4.39) where Ω is the angular velocity of revolution [rad/s] of the two primaries. Figures 4.4 and 4.5 show respectively the periods and the frequencies of libration calculated with Eq and 4.39, both for in-plane and out-plane motion around angular equilibrium position. The extremal values are evidenced, both for periods and frequencies. When the mass parameter is zero, ν =, the mass of the secondary vanishes and the frequencies of oscillation coincides with the frequencies of a librating dumbbell around a single central body: ω θ (ν =)= 3Ω (4.4) ω ϕ (ν =)=Ω (4.4) When the mass parameter ν is increased, both frequencies decrease, meaning that the more the secondary is massive, the more libration of the dumbbell is slowed down. When the mass parameter ν =, the two primaries have the same mass, M = M and the frequencies reach the minimum values: ( ω θ ν = ) 6 = Ω (4.4) ( ω ϕ ν = ) 3 = Ω (4.43) For small displacements from the stable equilibrium position the two attitude angles describe Lissajous figures, as can be seen from an example in Fig In fact, the two variational coordinates γ and ψ, obtained from attitude angles angles θ and ϕ, evolve with two harmonic motions of the form: γ = A sin(pt+ α)ψ = B cos(qt + β) (4.44) and the two pulsations are not arbitrary but depends each other with the relation: P =Q Active Electrodynamic Tether 4.4. New equilibrium positions When the electrodynamic tether is active, the F el terms in Eq. 4., 4. and 4. are non-zero. The classical CRTBP solution results perturbed by electrodynamic force and the

257 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 39 dynamical system exhibits new equilibrium positions. The perturbed CRTBP adimensional equations are: ξ η = J ξ + F ξ (4.45) η + ξ = J η + F η (4.46) ζ = J ζ + F ζ (4.47) where J is the adimensional potential function given by Eq..3 and F =(F ξ,f η,f ζ ) is the adimensional perturbing force. Developing the derivatives of potential function J, the motion equations are: ξ η = ξ ν ρ 3 (ξ + ν ) ν ρ 3 (ξ ν )+Fel,ξ (4.48) η + ξ = η ν ρ 3 η ν ρ 3 η + Fel,η (4.49) ζ = ν ρ 3 ζ ν ρ 3 ζ + Fel,ζ (4.5) The condition of equilibrium position is expressed by: ξ = ξ = (4.5) η = η = (4.5) ζ = ζ = (4.53) and substituting in Eq we obtain the equations that the perturbing force must satisfy to have equilibrium: = J ξ + F ξ = ξ ν ρ 3 = J η + F η = η ν ρ 3 = J ζ + F ζ = ν ρ 3 (ξ + ν ) ν ρ 3 (ξ ν )+F ξ (4.54) η ν ρ 3 η + F η (4.55) ζ ν ρ 3 ζ + F ζ (4.56) By solving Eq the equilibrium positions in presence of electrodynamic force can be found. When the adimensional planar CRTBP is considered, the system is a -dimensional dynamical system with one parameter ν (the mass parameter). The configuration space is the D space of the adimensional coordinates (ξ,η), and the corresponding phase space is the 4D space (ξ,η, ξ, η). In the planar case the system is reduced to the first two equations:

258 4 Section 4.4: Active Electrodynamic Tether = J ξ + F ξ (4.57) = J η + F η (4.58) Using Eq we can estimate the value of adimensional external force capable to maintain the orbital position fixed in the synodic frame. The value of such a force depends only on the location, identified with the three coordinates (ξ,η,ζ), and on the mass parameter ν of the planetary system. Each point of the space (ξ,η,ζ) can be thought as an equilibrium point by applying the appropriate value of external force. As an example, Eq has been used to calculate the value of external force along the orbital path of the second primary. Figure 4.7 shows such calculation. The figure report the module of the force F mod = Fξ + F η + Fζ as a function of angle α =,,, 36 deg. Triangular lagrangian points are on the orbital path of the second primary, and they are encountered at α =6deg (L 4 )andα = 3 deg (L 5 ). As we expected, at these locations the external force is equal to zero, indicating that they are two points of stable equilibrium, and no external force is required to maintain the spacecraft fixed with respect the synodic Frame. The plot has been done for five different values of mass parameter, ν = 6, 5, 4, 3,, covering the most interesting planetary systems of our Solar System. The profile shapes are all identical when ν changes, evidencing a linear dependence (in logarithmic scale) of the equilibrium force with the mass parameter. The effect of the mass parameter ν is only to scale such a profile.

259 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 4 Figure 4.: Orbital motion (D) at triangular point of Earth-Moon system. Figure 4.: Orbital motion (D) at triangular point L5 of Jupiter-Io system.

260 4 Section 4.4: Active Electrodynamic Tether η 3 L 4 STABLE ORIENTATION π αtr α tr UNSTABLE ORIENTATION M M G ν ξ ν π + αtr UNSTABLE ORIENTATION 3 α tr L 5 STABLE ORIENTATION Figure 4.3: Equilibrium solutions of attitude equations at triangular points L 4 and L 5 of an inert librating tether.

261 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis Attitude Linear Analysis Libration around Stable Position T in plane : In plane period T out plane : Out plane period 5 4 π / sqrt(6) Periods of Libration IN PLANE PERIOD 3.5 π / sqrt(3) OUT PLANE PERIOD 4 π / sqrt(3) Mass parameter nu = M / (M + M ) Figure 4.4: Period of libration of the dumbbell as a function of the mass parameter of the planetary system..9 Attitude Linear Analysis Libration around Stable Position OUT PLANE FREQUENCY ω in plane : In plane freq ω out plane : Out plane freq Frequencies of Libration [Ω rev ].8 sqrt(3) IN PLANE FREQUENCY sqrt(3)/.3 sqrt(6)/ Mass parameter nu = M / (M + M ) Figure 4.5: Frequencies of libration of the dumbbell as a function of the mass parameter of the planetary system.

262 44 Section 4.4: Active Electrodynamic Tether theta [deg] t [T rev] theta dot [deg/s] theta dot [deg/s].5 x t [T rev].5 x theta [deg] phi [deg] phi dot [deg/s] phi dot [deg/s] t [T rev] x t [T rev] x phi [deg] L [km] phi [deg] t [T rev] theta [deg] Figure 4.6: Evolution of attitude angles θ,ϕ for a small displacement (of deg) of both in-plane and out-plane angles around the stable position of libration at L5. Angles describe Lissajous figures, which can be analytically derived from linear analysis [].

263 Chapter 4: Dynamics at the Triangular Lagrangian Points: Linear Analysis 45 Module of Adimensional Force for Equilibrium in Synodic Frame, for 5 values of mass parameter nu Module of Adimensional Force ν = - ν = -3 ν = -4 ν = -5 ν = L Angle α [deg] L5 Figure 4.7: Module of the adimensional force necessary to maintain a spacecraft fixed to the synodic Frame along the orbital path of the second primary. Triangular lagrangian points are on the orbital path of the second primary, and they are encountered at α =6deg (L 4 ) and α = 3 deg (L 5 ): at these locations the external force is equal to zero, indicating that they are two points of stable equilibrium, and no external force is required to maintain the spacecraft fixed with respect the synodic Frame.

264 46 Section 4.4: Active Electrodynamic Tether

265 Chapter 5 Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects For large amplitude motions around equilibrium points the linear theory loses its validity, and other techniques must be adopted to investigate dynamics. Due to the strong non-linear nature of the ODE dynamical system describing the large amplitude motion at triangular points, numerical integration of full-equations is necessary to study the motion of the tether. A dedicated MATLAB simulator has been developed, and a campaign of numerical simulations has been executed. Equations of motion have been integrated using classical Adams-Bashforth-Moulton algorithm for ODEs, and the simulator has been used to investigate and to infer the dynamical behaviour and features of the ODEs system. 5. Inert Tether 5.. Orbital Motion: Zero-Velocity Orbits Zero-velocity orbits at triangular libration points have been investigated. With zero-velocity orbits we mean orbital paths starting with zero or small initial velocity with respect the synodic frame. A classical study of non-linear motion around the triangular libration points has been conducted by McKenzie and Szebehely [98] in 98; here we summarize the most important results:. Because of the stable nature of triangular points L 4 and L 5, when a particle is initially in the neighborhoods of one of this points with zero or small initial velocity, it is expected to remain near the point and to librate around it (librational motion);. When the particle leaves the vicinity of L 4 or L 5, the motion is called unstable; 3. The transition between librational motion and unstable motion is expected to be smooth; 47

266 48 Section 5.: Inert Tether 4. However, the phase space around triangular libration points exhibits a considerable complexity, and a stability criterion is needed; 5. The following empirical stability criterion has been proposed: the motion is considered unstable when the orbit of the particle crosses the X-axis connecting the two primaries; 6. When the stability criterion is not satisfied, the librational motion is not preserved, and the third body is captured by one or both primaries or circulatory or random motion take place; 7. The region of stability surrounding the triangular points of Earth-Moon system has been calculated: the configuration space in the vicinity of L 4 has been divided into a 5 8 grid with a mesh size of.5 non-dimensional units and trajectories have been integrated for 48 time units. The result is the classical figure reported in Fig. 5., where black cells are the stable zones and white cells are the instable ones. Note that the banana-shaped region represents initial conditions and not the envelope of motions; 8. The complexity of the motion is evidenced near the extremities by the presence of isolated islands of stability. McKenzie and Szebehely concluded saying that discontinuities in the librational regions are not surprising because the restricted problem of three bodies is a non-integrable dynamical system, with no other analytic integrals with global validity than the Jacobian integral (as Poincaré demonstrated in 896). Using the MATLAB simulator developed for tether dynamics, a set of simulations has been conducted to characterize the in-plane and out-of-plane motion. Zero-velocity orbits starting on the circular orbit of the secondary have been characterized. All these orbits are planar and lie within the plane of revolution of the two primaries. Figure 5. shows six selected simulations in the Jupiter-Io planetary system, for six different initial points on the orbital path of the secondary. The six initial points are taken at six angles φ = 6, 5, 4,... deg. The simulation with angle φ = 6 correspond to the L 5 point as initial position, and in this condition no orbital motion of libration happens. The body remain in the equilibrium point for an arbitrary long time: a numerical simulation covering a time span of years has been executed, showing no deviation from equilibrium position. Each of the six figure shows the planar trajectory on the plane of revolution of the two primaries, in adimensional coordinates (ξ,η). Both planets Jupiter and Io have also been reported in scale. In the first four simulations φ = 6, 5, 4, 3 deg the trajectory do not intersect the ξ axis, thus it is stable in the McKenzie Szebehely sense. The last two simulations cross the ξ axis, so they are unstable. The periods of libration of the six simulations are respectively: T NaN,58,58,64,8,7 T rev,wheret rev is the period of revolution of the two primaries T rev.77 days. For small deviations from the triangular point the period of libration is in accordance with the period predicted from linear theory.

267 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 49 As we go away from that point the period increases, and non-linear effects on motion become relevant. Despite the unstable nature of last two simulation, a quasi-periodic behavior can be inferred from the analysis of the phase space. In fact, long time simulations demonstrate a periodic orbital motion also in presence of ξ-axis crossing. As an example,fig. 5.3 shows the results of a simulation for T= T r ev 4.6 years of orbit in the Jupiter-Io system with an initial angle φ = deg. The figure shows the orbital state vector as a function of time, presented in 9 plots within a 3 3 matrix. In the first row of the matrix there are the three components (ξ,η,ζ) of the adimensional position vector as a function of time, in the second row there are the three components of the velocity vector (V x,v y,v z ) in km/s as a function of time, and in the last row there are the phase-spaces of each component. The phase spaces exhibit a bounded motion of periodic nature, even if the motion crosses the ξ-axis. The trajectory followed by the particle is showed in Fig When the initial conditions have in addition an out-of-plane component (ζ ), the situation is only apparently more complicate. In fact, out-of-plane motion is a pure harmonic oscillation with pulsation equal to the revolution pulsation Ω of the planetary system. Figures 5.5 and 5.6 show the results of a simulation starting at φ = 3 deg, with an initial out-of-plane component of ζ =.474, integrated for a time T =64T rev. The first figure shows the orbital state vector component, and the second figure shows a 3D plot of the trajectory. 5.. Orbital Motion: Sensitivity to Velocity Errors The non-linear motion around triangular libration points starting with a non-zero orbital velocity has been investigated. This analysis is an extension of the previous zero-velocity analysis. The analysis for non-zero velocities is of great interest to test the sensitivity of the system to velocity errors, and consequently it is often referred in literature as the sensitivity analysis to velocity errors. A classical reference work is the 98 paper of Szebehely and Premkumar [4]. The main results of their analysis are here summarized:. They numerically estimated the maximum initial velocity a body can have to preserve stability when it is placed exactly at the triangular point of the Earth-Moon system and it departs from the point in the direction defined by an initial angle θ;. Stability is intended as in [98]: motion around triangular point is stable when librational motion is preserved; 3. The main output is the graph reported in Fig. 5.7: it shows in a polar diagram the velocity values as a function of angle θ around the triangular point. The line connecting L 4 to the Earth is at 3 deg; 4. The largest velocity errors can happen along the line connecting L 4 to the Earth, and they respectively have the values of.5 km/s in the direction pointing from L 4 toward

268 5 Section 5.3: Active Electrodynamic Tether the Earth and.7 km/s in the opposite direction; 5. Minimum velocity errors are allowed in the direction perpendicular to the line connecting L 4 to the Earth, along the arc centered at Earth with radius. Here the sensitivity limits are only. and.5 km/s; 6. Fig. 5.8 reports both the stability regions obtained from the configuration space (the banana-shaped region) and from the velocity space. It can be seen that their extension is along two perpendicular directions. From the sensitivity analysis it can be inferred the boundaries of velocities that a body can have to librate around triangular points. 5. Active Electrodynamic Tether A numerical investigation with an active EDT must be referred to a particular planetary system, because electrodynamic force depends on environmental quantities such as the magnetic field and the environmental electron density. The chapter 7 is entirely devoted to the analysis of the active EDT in the Jupiter-Io planetary system. 5.3 Non-linear Stability The linear analysis described in the previous section is representetive of the satellite motion only in the proximity of the lagrangian point. In order to describe the motion at more distant locations from the point, non-linear methods become necessary. Several non-linear methods, both analytical (es.: Lindstedt-Poincaré, ecc.) and numerical (es.: invariant-manifold-based methods)areavailableforthepurpose. The qualitative results of linear stability analysis are partially affected in the non-linear case. Figure 5.9 is a schematic summary of the analysis that can be found in literature regarding stability issue at triangular points. In the following section a resume of the main studies of non-linear stability at triangular libration points will be given Non-linear Stability at Triangular Points The non-linear stability analysis of libration points and orbits around libration points has been in the previous century a deeply investigated topic, and produced a relevant development of analytical and experimental (computational) Dynamics. In 96 Leontovich [95] proved that the triangular libration points of the CRTBP were stable for all ν values satisfying the condition <ν<ν cr,withν cr = , but a set of measure zero. Five years later in 967 this set was determined with the work of Deprit and Deprit-Bartholomé [8], showing the set is constituted of only three points, corresponding to:

269 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 5 ν =.4... ν = (5.) ν 3 =.9... The first two cases ν and ν correspond to the resonance : and 3: between the frequencies of the linearized system describing the motion in the neighborhood of the libration points, and the third case ν 3 correspond to a case when a specific algebraic combination of the system frequencies and the coefficients of the 4 th order normal form of the Hamiltonian of the considered problem is equal to zero. Calling ω and ω the two linearized system frequencies, and γ = ω ω, which in their turn can be expressed through ν, the algebraic expression found by Deprit for the computation of ν 3 is the following: D 4 = 644γ4 54γ +36 6(4γ )(5γ (5.) 4) With the two subsequent works of Markeev in 969 [96] and in 97 [97] approximate floating point computations of the normal forms for the values of ν, ν and ν 3 has been given, and the stability of these three cases was estimated. Markeev showed that the resonant values ν and ν are unstable, and the third ν 3 is stable. His results was given with an accuracy of 3-4 significant digits. An analytical expression for the stability of the ν 3 case was found only in 986 by Meyer and Schmidt [99], confirming the stability found by Markeev. The analytical computation require the coefficients of normal form till the 6 th order. The exact numerical expression for the stability criterion was firstly calculated by Schmidt [9] three years after his previous work. He accomplished this task designing a specialized algebraic processor. Exact numerical expressions for the coefficients of the normal forms of two resonant cases ν and ν were obtained between 99 and 993 ([3] and []) with the works of Sokolsky and Shevchenko, by means of an application of a computer algebra package. Their accurate work confirmed the previous work of Markeev of lower accuracy. Stability of the triangular libration points in the CRTBP for the critical ratio of the masses ν cr = was firstly analyzed in 975 by Sokolsky [], who proved the system was formally stable. He evaluated the normal form coefficients in approximate floating point computations. The normal form of ν cr was analitically calculated by Schmidt [] in 994, and a discrepancy of one order of magnitude with respect the approximate results of Sokolsky was encountered over a coefficient of the normal form. However, the sign of the coefficient was right and the qualitative result of stability obtained by Sokolsky was confirmed. A confirm of all previous results was made in 6 by Bruno and Petrov [79], who executed a symbolic computation with Maple of the normal forms for the two resonant cases ν and ν, for the non-resonant case ν 3 and for the critical case ν cr ; they confirmed results of Markeev and Sokolsky: instability for ν, ν and stability for ν 3 and ν cr.

270 5 Section 5.3: Non-linear Stability Figure 5.: Non-linear stability around the L 4 triangular libration point of Earth-Moon system, taken from [98]. Black cells of the grid are the stable zones taking to librational motion around the triangular point and white cells are the unstable ones. Isolated islands of stability compare at extremities of the zone.

271 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 53 Figure 5.: Zero-velocity orbits for six selected simulations in the Jupiter-Io planetary system (ν = ); orbital trajectories are the blue curves, they are always within the revolution plane (ξ,η) of the two primaries. Starting point has been marked with a red circle, and the end point of the trajectory has been marked with a green x.

272 54 Section 5.3: Non-linear Stability.5 ξ = R x /d t [T rev].4 η = R y /d t [T rev].4 ζ = R z /d t [T rev]...5 Vx [km/s] Vy [km/s] Vz [km/s] t [T rev] t [T rev] x t [T rev]..5.5 Vx [km/s] Vy [km/s] Vz [km/s] ξ η ζ Figure 5.3: Zero-velocity orbit in the Jupiter-Io system long time simulation T = T rev 4.6 years, initial angle φ = deg. The first two rows are the position and velocity components as a function of time, the last row reports the phase spaces of each component. The render of the trajectory is showed in the next figure.

273 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 55 Figure 5.4: Render of the trajectory of previous figure. This orbit is unstable in the McKenzie-Szebehely sense, because it crosses the ξ axis. However, a periodic motion can be recognized.

274 56 Section 5.3: Non-linear Stability ξ = R x /d η = R y /d -.8 ζ = R z /d t [T rev] t [T rev] t [T rev]...5 Vx [km/s] Vy [km/s] Vz [km/s] t [T rev] t [T rev] 5 x t [T rev] x -6. Vx [km/s] Vy [km/s] Vz [km/s] ξ η ζ Figure 5.5: Zero-velocity orbit in the Jupiter-Io system with an initial out-of-plane component T =64T rev, φ = 3 deg. Out-of-plane motion is a pure harmonic oscillation with pulsation equal to the revolution pulsation Ω of the planetary system. The render of the trajectory is showed in the next figure.

275 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 57 Figure 5.6: Render of the trajectory of previous figure. Out of plane harmonic motion is evident.

276 58 Section 5.3: Non-linear Stability Figure 5.7: Sensitivity analysis to initial velocities errors for librational motion at L 4 triangular point of Earth-Moon system, as calculated from Szebehely and Premkumar [4].

277 Chapter 5: Dynamics at the Triangular Lagrangian Points: Investigations of Non-Linear Effects 59 Figure 5.8: Comparison of stability regions for the configuration space and for the velocity space [4]. Figure 5.9: Stability at triangular points.

278 6 Section 5.3: Non-linear Stability

279 Chapter 6 Io environment and Plasma Torus This section gives a brief survey on environment encountered by the electrodynamic tether (EDT) at Lagrangian points of Jupiter-Io system. Three major topics will be treated:. Io s gravitational field,. magnetic environment and 3. plasma environment (electron density of thermal population), since all of them influence the dynamics of the EDT. The estimation of involved parameters and the development of simplified models has been possible by means of in-locus exploration with automatic space probes (mainly using Voyager flyby and seven close-flybys of Galileo orbiter), and with the aid of earth-observations (mainly using spectroscopic techniques). A brief review on the mostly interesting models required to characterize the dynamics of the EDT will be given. The gravitational field of the moon and its shape parameters are reported in paragraph 6.., together with the estimation of Hill and Laplace sphere of influence of the moon. The gravitational force of the moon has been compared with Jupiter gravitational force. Io doesn t possess an internal magnetic field, but at locations near the moon the background magnetic field of 85 nt (of Jupiter) is considerably modified by Io-plasma local interactions. A simplified description of the magnetic environment will be given in section 6... Io is the unique moon in the Solar System exhibiting volcanic activity and the existence if Io s plasma torus is a consequence of this activity. Since the five lagrangian points of Jupiter- Io system lie all within Io s plasma torus, a brief introduction of the torus morphology will be given in paragraph 6..3, together with a simplified model describing the electron density as a function of location (the Divine and Garrett model). 6. Environment 6.. Gravitational field and shape parameters Using Galileo geodetic data of seven close flybys (I, I4, I5, I7, I3, I3 and I33) with the jovian moon, Io s gravitational parameters has been estimated (Anderson et al. [7], 996); most interesting parameters are reported in table 6.. 6

280 6 Section 6.: Environment Parameter Symbol Unit Value Gravitational parameter μ Io = GM km 3 /s ±. Average radius R km 8.6±.5 Mean density ρ kg/m ±.9 Angular velocity (orbital = rotational) ω rad/s Equatorial buldge J ±.7 6 Second degree potential Love number k -.94±.7 Second sectorial Stokes coefficient C ±.8 6 Dimensionless mean moment of inertia I/(MR ) ±. Io principal axes of inertia (c <b<a) a km 83.±.5 b km 89.±.5 c km 85.6±.5 Table 6.: Io gravitational and shape parameters. The mass of Io is similar to the Earth s Moon, and the J parameter of equatorial buldge is of the order of 3, one order of magnitude larger than Moon s one ( 4 ). Due to internal dissipation of mechanical energy, the Jupiter-Io system tends to a rigid-body motion: as a consequence Io s orbital and rotational periods have reached the common values of.769 days. Io exhibits a 4:: resonance with Europa and Ganymede, and the severe tidal heating due to this resonance is the principal cause of its active volcanism. The famous prediction that Io is a volcanically active body, made by Peale et al. [] in 979 was sensationally confirmed by the two Voyager spacecrafts. Using gravitational parameters it has been possible to infer models for the internal structure of the moon. The two most famous models describing Io s internal structure have been developed during the 8s: the Ross et al. model ([5], 99), based on a deep mantel assumption, and the Schubert et al. model([8], 98), based on an asthenosphere model; the second model has revealed to be the better one. In Zhang et al. ([8], ) can be found the derivation of Io s figure and dynamical parameters from internal structure models (both on mantel-assumption-based and asthenosphere-based models), together with a short but effective description of the theory of synchronous satellites. They also suggest that Io may have a large core, with a relative radius of.5. Further contributions to the theory of equilibrium figure and gravitational field of Io can be found in subsequent studies conducted by Zharkov ([9], 4). The sphere of influence of Io can be calculated using Hill s and Laplace s formulae: R Hill = R Jup Io = 47 km 3 3μ Io km 5.8R Io (6.) μjup

281 Chapter 6: Io environment and Plasma Torus 63 Lagrangian Points Position vector in Synodic Reference Frame [km] L [43.43; ; ] L [4344.; ; ] L3 [ ; ; ] L4 [88.64; 3659;.] L5 [88.64; -3659;.] Table 6.: position of Jupiter-Io lagrangian points position vectors in a synodic reference frame. ( ) ( ) μjup R Laplace = R Jup Io = 47 km 7834 km 4.3RIo (6.) μ Io In the Jupiter-Io system the Laplace sphere of influence gives approximatively the boundary of the zone where the motion of a body is influenced by Io s attraction. However, inside this zone the Jupiter gravitational attraction is still relevant and it can t be simply modeled as a perturbing force. Figure 6. shows the static gravitational field in the neighborhood of Io on its plane of revolution, due to the superimposition of jovian field and the moon field. Figure 6. is a slice of the 3D gravitational field along a line connecting the center of mass of the two bodies, and it shows comparison between Io and Jupiter gravitational forces exerted on an unitary mass. The gravitational acceleration at Io surface level is of the order of.8 m/s, and it reduces to <.5 m/s over an altitude of 3 km over surface. The gravitational attraction of Jupiter is of the order of.7 m/s and it is approximatively constant in the near-io environment ( half Io s surface acceleration). In the circular-restricted case the Jupiter attraction is compensated by the centrifugal force due to revolution of the moon around the planet. Table 6. reports the position of Jupiter-Io lagrangian points position vectors in a synodic reference frame with origin in the center of mass of the planetary system. The position of this points is reported in Fig The position of L and L lagrangian points fall inside the zone of interaction between Io s ionosphere and co-rotating plasma flow (the most difficult zone to model). The L 3, L 4 and L 5 fall inside the plasma torus. In this region the electron density is more homogeneous than in the near-io locations, and the Divine-Garrett model is a good approximation of the electron density distribution. In particular, the most attracting locations are the stable equilateral points L 4 (upstream Io) and L 5 (downstream Io). 6.. Magnetic environment The magnetic environment in the near-io zone is ideally given by the sum of three magnetic fields:

282 64 Section 6.: Environment B = B + B + B ext (6.3) where B is the magnetic field given by the first primary (Jupiter), B is the magnetic field given by the secondary (Io) and B ext is the magnetic field from other external currents (for example from other sources like plasma environment, ecc.). The jovian magnetosphere is traditionally subdivided into three regions,. the inner region (< R J ),. the middle ( 4 R J )andouter(> 4 R J ) region, consequently Io and Europa lie within the inner magnetosphere, and Ganimede and Callisto within the middle one. Magnetic field of primary: inner magnetosphere of Jupiter Magnetic field B of the first primary coincides at Io locations with the inner region of Jupiter magnetic field. The inner region of magnetosphere can be modeled as a potential field, similarly to the Earth s magnetic field. In this case the field can as usual be written as the gradient of a scalar potential B = Φ satisfying the Laplace equation B =;the solution is classically given by summation of spherical harmonics. Once the number of useful harmonics has been chosen, a numerical evaluation of the magnetic field can readily be done, for example using the classical algorithm of Roithmayr [4]. A detailed description of inner magnetosphere due to Khurana et al. can be found in the Bagenal s Jupiter monograph [53]. Magnetic field of secondary: Io s Magnetic field Khurana et al. ([9], ) analyzed the Galileo flybys at Io, and they stated that Io does not possess an appreciable internal magnetic field. The two polar passes I3 and I3 played a key role to accomplish this result, since from the previous four flybys the Io s magnetic field question was still open. The magnetic field the tether experience in the near- Io environment is not equal to the background magnetic field of Jupiter ( 85 nt), since it is locally modified by the Io-plasma interaction. Local variations due to plasma local phenomena affects the background field up to 4 % of nominal value. On the upstream side the magnitude is increased of 3 nt (revealed by the I4 Galileo flyby), and on the downstream side it is diminished of 6 nt (revealed by the I Galileo flyby). In fact, analysing Galileo magnetometer data of the I flyby occurred on 7 December 995 across the downstream wake ([93], 996), a magnetic field depression was encountered and the magnetic field decreased by 4% with respect to the local jovian field of about 835 nt. The depression was compatible both with an internal field of the jovian moon and with a variation in the energy density of local plasma. With subsequent close encounters between Galileo probe and the moon it has been showed that the energy densities of the plasma in the vicinity of Io plays a central role, and the presence of an Io dipole moment was excluded ([8], 996).

283 Chapter 6: Io environment and Plasma Torus 65 Assumptions for a simplified model In order to have a simple model of the magnetic field for analytical calculations the following assumptions has been done:. The magnetic field of Io and the magnetic field of Io-plasma interactions has been neglected: B = B ext = ;. The field B of Jupiter is dipolar, due to the first harmonic coefficient l =, m =, ; The resulting field is simply given by the classical dipolar model: B( r) = m [3( r 3 ˆm ˆ r)ˆ r ] ˆm (6.4) where m = μ m R 3 jup ˆm is the dipole moment vector of the planet, ˆm is its unit vector, μ m istheintensityofthedipole[tesla],r jup is the planet equatorial radius, r is the position vector of the spacecraft Plasma environment Plasma environment plays a fundamental role in the operations of the electrodynamic tether, since environmental electron density is one of the essential elements required by the system to operate. In fact, electrical current along a bare EDT is a function of plasma electron density N e, tether geometry, and electromotive field []. Io offers a plasma environment unique in the whole Solar System, since its volcanic activity acts as a source of mass that is injected in the Jovian magnetosphere, where it is subsequently ionized and taken to a state of a cold plasma, creating a dense toroidal structure called Io s plasma torus. The plasma environment at Io can be subdivided into three regions:. Near-Io region ( within Io s Lapalce SoI) : Io s ionosphere determines the electron density distribution;. Plasma torus : it is extended all along Io s orbit; plasma dynamics is governed by rotation of Jupiter magnetic field; 3. Interaction zone : the interaction between Io s ionospheric current system and the co-rotating plasma of the torus create an asymmetric zone which should be modeled using MHD or PIC simulations; it won t be considered within the frame of this work. In the following subsections a brief description of the three regions will be given; finally the simplified Divine and Garrett model describing the electron density of jovian plasma will be reported.

284 66 Section 6.: Environment Io ionosphere The Io ionosphere is identified by a dense, cool plasma that is at rest with respect to Io. It has been described in [8] (996). The composition of the ionospheric plasmas includes O ++,O + and S ++,S +, and SO + ions. The distribution of ionospheric electron densities changes from equatorial latitudes to Io s poles. Due to its peculiar volcanic activity Io acts as a source of neutral material within the inner jovian magnetosphere, at a rate of about 3 kg/s (see [9] and [77]). The dominant constituents of ejected materials are sulfur dioxide (SO ), molecular sulfur (S ), molecular oxygen (O ) and traces of sodium (Na ). A cloud of neutral sodium Na upward the orbital path of Io and co-rotating with it has been revealed by means of spectroscopic techniques [78]. The injected materials are ionized by processes of photoionization, electron impacts and charge exchange by the environmental plasma, adding ions to the magnetosphere. Heavy ions modify the magnetosphere structure (a description of their influence can be found in [9]), influence its dynamics and produce radiation visible from Earth. Molecular ions like SO + and SO + are more abundant near Io s position and they become more and more rare in regions distant from the moon. Molecular ions are gradually dissociated into atomic ions S + and O + by the magnetospheric co-rotating plasma. Io own also a very-low density atmosphere of about 9 bar, which is a consequence of its volcanic activity. Atmospheric density is patchy, disomogeneous and greater at the locations of active volcanic plumes. The main constituent is SO, supplied largely by volcanic plumes, with a lesser amount coming from evaporation of the SO frost deposit on the surface. The atmosphere formation on Io is a direct consequence of volcanic activity. When an eruption take place, a fraction of the plume escapes from the moon and the remainder part fall into the surface. The mean surface temperature is 3 K, so that the volcanic gases rapidly froze. The atmosphere is then constantly replenished by the surface sublimation, the process being driven by the strong Jupiter radiation (in the form of plasma) and by the direct sunlight exposure. As a consequence the atmosphere density is strongly variable from the Jovian and anti-jovian hemispheres, and from the Sun and anti-sun hemispheres, and it is most abundant to the equator where surface is warmest and most volcanic sites reside. A model for the calculation of Io s atmosphere at eastern and western elongations has been described by Wong and Smyth ([7], ). Plasma density at near-io locations has been probed by the Galileo spacecraft with 7 very close flybys, revealing a profile of electron density variable with Io s latitude (equator/poles) and with respect the direction of Io s orbital motion (upstream / downstream / flanks). From Bagenal s book we read: In the vicinity of Io local electron densities ranging from 35 cm 3 to 4 cm 3 have been observed [88], [83]. The polar passes revealed large increases of the plasma density over the polar cap, that have been interpreted as entry into ionosphere or into a flux tube magnetically linked to Io itself [94] on which the flow is greatly slowed and the pickup plasma is exceptionally dense. Electron density at equator [8] [87] shows peak densities over > 5 cm 3.

285 Chapter 6: Io environment and Plasma Torus 67 Plasma torus Along Io orbit the plasma is a dense and large toroidal region called the Io s plasma torus. An accurate description of the plasma torus structure, the Io s plasma interaction, and how it dynamically interacts with the Jovian magnetosphere can be found in the Bagenal [53]. The most abundant ion species that constitute the plasma of the torus are the ions of the SO molecule: S +,S ++,O + and O ++. The composition of the plasma torus has been investigated using spectroscopy techniques, revealing the principal constituents of the torus are S ++,S +,O +,NaandCl +. The fast Jupiter rotation (rotational period of hours) creates the confining dipole field and creates the centrifugal force which compresses the plasma to the equator. It also enforces the co-rotation of the plasma and is the main cause of pickup energy (between 8-56 ev). An electric field imposed across the system shifts the torus eastward. The Io torus exhibits a three dimensional structure, with thermal plasma oscillating about the centrifugal equator of the magnetic field. Ions moving in the torus are thus subjected to three components of motion: () corotation with planet, () gyration around field line, and (3) bounce along field line. Because of the jovian magnetic equator is tilted of about ř with respect to the spin rotation axis, the farthest point from rotation axis (defining the centrifugal equator) of a magnetic field line is different from points belonging to magnetic equator. The fastest plasma within the torus results to be the plasma at the centrifugal equator. The centrifugal equator of the magnetic field is defined as the locus of points belonging to the magnetic field that are the farthest from the spin axis. A simplified azimuthally-symmetric model of the plasma torus has been made by Bagenal [74] using plasma measurements of Voyager. It predicts variations between 7 cm 3 when Io is at the N-S center of the plasma torus (SIII ř and 9ř) and < cm 3 when Io is farthest from the torus center (SIII ř and ř). Outside of the local ionosphere in the Io plasma torus the electron density measured by Galileo varied within the range from about cm 3 to about 36 cm 3 ([87], [88], [8], [8], [83], [84]). Previous Voyager measurements revealed a maximum plasma torus density of about cm 3, differences due to the variable volcanic activity. Interaction zone The flow of corotating plasma is perturbed by the presence of Io. Elastic and inelastic collisions of plasma with Io s atmosphere and photoionization process modify the plasma physical parameters (density, momentum and energy). A fraction of the plasma flow impacts over Io s surface and is absorbed from the main stream. Remainder plasma is slowed by elastic collisions and by pickups in Io s ionosphere, is redirected around Io and then reaccelerated in the wake. The electrodynamic interaction between Io and the jovian magnetosphere produce an electric current of a total of about 7 amperes, with strong consequences on the near-io environment, on Jupiter s upper atmosphere, Jupiter radio emissions and Jupiter auroral emissions.

286 68 Section 6.: Orbits around Io Divine and Garrett model Since we are interested in a simplified description of the magnetic and plasma environment around Io we address to the Divine and Garrett model. An incisive summary of this model can be found in [85]. The model provides the densities of protons, electrons and six positive ion species. Here we only report the electron density, since it is the plasma parameters influencing the tether behavior. The Divine - Garrett model: electron density Electron density of co-rotating plasma in Divine and Garrett model is given by relations reported in table 6.3. The model allows to obtain the electron density as a function of the position vector of the spacecraft, with functions in the form: N e = N exp [ ( ) ] r r r ν H (6.5) where r =7.68R j (Jupiter radii) is the reference radius, H = R j is the reference height, N = m 3 is the reference electron density, and π <ν<πis the latitude. 6. Orbits around Io 6.. Electrodynamic tether around Io: numerical simulations A preliminary analysis of natural (non-electrodynamic) orbits of the tether at Io has been conducted, with the aim to study the gravitational environment at the Io location. As we expected, the major perturbation is due to the third body effect of Jupiter, as we obtained from several numerical simulations of the motion evolution has been conducted. Long simulations demonstrate that the main perturbation of Jupiter makes the orbits around Io become instable. Electrodynamic force can be used to obtain a quasi-stable trajectory when tether is placed in a retrograde equatorial orbit around the moon. Figure 6.5 shows an example of an inert tether of polar circular orbit of h=3 km of altitude over Io surface. Figures from 6.6 to 6.9 show an example of retrograde equatorial orbit of the EDT around Io. 6.. Jovian-synchronous orbits around Io In this section the idea of a jovian-synchronous orbit around Io has been considered. Like in the familiar sun-synchronous orbits, Io s equatorial bulge can be used in order to make a controlled precession of the right ascension of the ascending node Ω of the tether orbit around Io. The J coefficient of Io is of the order of 3 (see [[8]]). The classical first-order formula for the central body perturbation gives (Vallado, [[6]]): ( ) dω dt = 3 n Req a( e J cos(i) (6.6) )

287 Chapter 6: Io environment and Plasma Torus 69 Jovian Zone Range Formulae and Parameters [ ( ) ] r Inner plasmasphere <r<3.8 R J N e = N exp r r H (λ λc ) N =4.65 cm 3 r =7.685 R J H =. R J λ c =tan(α)cos(l l ) α =7deg l =deg [ ( ) ] Cool Torus 3.8 <r<5.5 R J N e = N exp rλ z H N and kt from table 6.4 H = H kt/e H =. R J E =. ev z = r(tan α)cos(l l ) α =7deg l =deg [ ( ) ] Warm Torus 5.5 <r<7.9 R J N e = N exp rλ z H N and kt from table 6.4 Parameters: same as Cool torus [ ( ) ] Inner Disc 7.9 <r< R J N e = N exp rλ z H N from table 6.4 H =(.8.4r)R J E = ev z = ( 7r 6 3 R J ) cos(l l ) l =deg Table 6.3: Electron Density at Jupiter - Divine and Garrett model.

288 7 Section 6.: Orbits around Io Jovecentric distance Electron Density Temperature R J log(n), cm 3 log(kt), ev Table 6.4: Equatorial parameter values for Jupiter s Divine and Garrett model. Equation [6.6] hes been rewritten grouping all physical parameters and rewriting a and e as a function of two more useful parameter, the apogee and perigee radii r a = a( + e) and r p = a( e): [ dω dt = 3 ] μio (R eq) ra + r p J cos(i) (6.7) (r a r p ) To keep locked the orbital plane of the spacecraft with the Io s revolution motion around Jupiter we need a dω dt equal to the angular velocity of Io around the planet: the condition of Jupiter-synchronous orbit happens when the precession of the right ascension of the ascending node is equal to the angular motion of Io around Jupiter: dω dt = n Io = μjup R 3 jup,io = rad/s (6.8) Getting the cos(i) quantity from Eq. (6.7) and substituting the condition of Jupitersynchronous orbit (6.8), we obtain: [ cos(i) = ( ) ] μjup ra + r p 3 μ Io Rjup,Io 3 Req J (r a r p ) (6.9) ra+r The cos(i) can be considered as a function of the fraction p (r ar p). The square brackets collect all the physical parameters involved in the analysis; substituting the numerical values (μ jup = km 3 /s, μ Io = 596 km 3 /s, R jup,io = 48 km, R eq = 8.3 km, J = 3 ), the numerical value of the bracket is of the order of.

289 Chapter 6: Io environment and Plasma Torus 7 Figure [6.] shows the values of cos(i) as a function of r p and r a : abscissa reports the periapsis radius r p, and curves are parametrized with respect to the apoapsis radius r a. Each curve in the figure gives the values of cos(i) as a function of r p for a given r a and since e =(r a r p )/(r a +r p ), the eccentricity changes along a single curve. A red curve has been also over imposed to highlight the circular orbits (when r a = r p ). The Jupiter-synchronous orbits must be chosen between the obtained curves, but three constraints restrict the allowable choices:. Io equatorial radius (at r p 8 km): it has been marked by a blue vertical line, all orbits at the left of this line fall within the moon body;. Laplace radius of influence (at r p 7834 km): it has been marked by a blue vertical line, all orbits at the right of this line don t experience Io s gravitational attraction; 3. The cosine of the inclination i must be a number <cos(i) < : it has been marked by a green horizontal line at cos(i) =. The allowable region has been evidenced with a blue rectangle. It can be seen that all orbits fall outside the rectangular region, so that no Jupiter-synchronous orbits can be obtained. The physical meaning of this result is that Io revolves too fast around Jupiter, and the temporal variation of the right ascension of the ascending node isn t capable to follow the orbital revolution of the moon.

290 7 Section 6.: Orbits around Io Figure 6.: Static gravitational field in the neighborhood of Io on its plane of revolution; dashed line is the direction connecting Io and Jupiter centers; extremes of the dashed line are L and L lagrangian points ( x ticked).

291 Chapter 6: Io environment and Plasma Torus 73 Figure 6.: Comparison between Io and Jupiter gravitational accelerations at near-io environment. Figure 6.3: Lagrangian point of Jupiter-Io system.

292 74 Section 6.: Orbits around Io Figure 6.4: Profiles of electron density and temperature - Divine and Garrett model.

293 Chapter 6: Io environment and Plasma Torus 75 Figure 6.5: Orbital path of Io orbit of altitude h=3 km over Io surface, synodic reference frame.

294 76 Section 6.: Orbits around Io.. x 7 ξ = R x /d. η = R y /d.5.5 ζ = R z /d.99.5 t [T rev]...5 t [T rev] x 6 t [T rev] Vx [km/s].5.5 Vy [km/s].5.5 Vz [km/s]..5 t [T rev]...5 t [T rev] x 6 t [T rev] Vx [km/s].5.5 Vy [km/s].5.5 Vz [km/s].. ξ... η 4 4 ζ x 7 Figure 6.6: Electrodynamic tether in equatorial orbit around Io L=5 km, Width = 5 cm, m = 6 kg, simulation time: T =T rev.769 days. State vector components: the first column shows the adimensional position R x of the CoM of the EDT, the orbital velocity V x and the phase space (R x,v x ) as a function of time; second and third columns are analogous for the Y and Z component. Tether initial position is marked with a red circle, final position with a green x. [SYM]

295 Chapter 6: Io environment and Plasma Torus x 3 6 θ [deg] 4 φ [deg] L [km] t [T rev] x 6 t [T rev] 4.5 t [T rev] θ dot [deg/s].4.3 φ dot [deg/s] 5..5 t [T rev] x 6 t [T rev] 5 x 3 θ dot [deg/s].4.3 φ dot [deg/s] 5 φ [deg]. 4 6 θ [deg] φ [deg] x θ [deg] Figure 6.7: Electrodynamic tether in equatorial orbit around Io L=5 km, Width = 5 cm, m = 6 kg, simulation time: T =T rev.769 days. Attitude angles (θ, φ) and tether length L: the first column shows the in-plane angle θ, the in-plane angular velocity θ, and the phase space (θ, θ); second column reports the out-of-plane angle φ, φ, and(φ, φ). The third column reports the length as a function of time (it is constant) and the Lissajous plane (θ, φ) of the two angles. [SYM]

296 78 Section 6.: Orbits around Io Figure 6.8: Electrodynamic tether in equatorial orbit around Io - 3D view [SYM]

297 Chapter 6: Io environment and Plasma Torus 79 6 x 3 4 Y syn X syn Figure 6.9: Electrodynamic tether in equatorial orbit around Io - top view [SYM]

298 8 Section 6.: Orbits around Io Figure 6.: Jupiter-synchronous orbits and the allowable region.