A classification of swing-by trajectories using the Moon

Transcription

1 A classification of swing-by trajectories using the Moon Antonio Fernando Bertachini de Almeida Prado Research Engineer, Department of Space Mechanics and Control Instituto Nacional de Pesquisas Espaciais - INPE - Brazil Av. dos Astronautas 1758 São José dos Campos SP Brazil Roger Broucke Professor, Department of Aerospace Engineering and Engineering Mechanics University of Texas at Austin, Austin TX 7871 In the present paper we study and classify the swing-by maneuvers that use the Moon as the body for the close approach. The goal is to simulate a large variety of initial conditions for those orbits and classify them according to the effects caused by the close approach in the orbit of the spacecraft. Special attention is given to identify the regions where the captures and escapes occur. The classical three parameters (Jacobian constant, pericenter distance and angle of approach) used to identify a Swing-By maneuver are varied in large intervals to cover a large range of possibilities for the maneuver. Letter-plots figures are made to show the results obtained in a compact form. The theoretical prediction that for 0 ψ 180 the spacecraft losses energy and for 180 ψ 360 the spacecraft gains energy is confirmed. Regions containing trajectories that are candidates to generate Belbruno- Miller trajectories are identified. The well-known planar restricted circular threebody problem is used as the mathematical model. The equations are regularized (using Lamaître's regularization), so it is possible to avoid the numerical problems that come from the close approach with the Moon. INTRODUCTION The importance of the gravity-assist (or swing-by) trajectories (Öpik, 1976; Dowling et. al., 1990 and 1991; Minovich, 1961) can be very well understood by the number of missions that flew or are scheduled to fly using this technique (Dunham and Davis, 1985; Farquhar et. al., 1985; Marsh and Howell, 1988; Nock and Upholf, 1979; Weinstein, 199). A very successful example is the Voyager missions (Flandro, 1966) that flew to the outer planets of the Solar System with the use of successive swing-bys in the planets visited to gain energy. In this paper, the same study done for the maneuvers involving the planet Jupiter in Broucke and Prado (1993) and Prado (1993) is repeated for maneuvers involving the Moon. The maneuvers involving the Moon have many important practical applications. Those applications have potential for utilization in a shorter time than those involving the planet Jupiter. Among the several sets of initial conditions that can be used to identify uniquely one trajectory, the same one used in the papers written by Broucke (1988) and Broucke and Prado (1993) is used here. It is composed by the following three variables: 1) J, the Jacobian constant of the spacecraft (an integral of the restricted three-body problem); ) The angle ψ, that

2 is defined as the angle between the line M 1 -M (Earth-Moon) and the direction of the periapse of the trajectory of the spacecraft around the Moon; 3) r p, the distance from the spacecraft to the center of the Moon in the moment of the closest approach to the Moon (periapse distance). Note that the Jacobi constant is essentially equivalent to the hyperbolic excess velocity V, since they can be related by one single expression. For a large number of values of these three variables, the equations of motion are integrated numerically forward and backward in time, until the spacecraft is at a distance that can be considered far enough from the Moon, such that the Moon's effect is neglected and the system formed by the Earth and the spacecraft can be considered a two-body system. At these two points, two-body celestial mechanics formulas are valid to compute the energy and the angular momentum before and after the close approach. Those quantities are used to identify up to sixteen classes of orbits, accordingly to the changes in the energy and angular momentum caused by the close encounter. They are named with the first sixteen letters of the alphabet. The results are shown in letter-plots, where one letter describing the effects of the swing-by is plotted in a two-dimensional graph that has in the horizontal axis the angle ψ (the angle between the periapse vector and the Earth-Moon line) and in the vertical axis the Jacobian constant of the spacecraft. There is one plot for each value of the parameter r p. their mutual center of mass; the Earth and the Moon have a finite mass and the spacecraft is massless. With those assumptions, the problem consists in studying the motion of the spacecraft near the close encounter with the Moon. It is necessary to study its motion only near this point, because when the spacecraft is far from the Moon the system is governed by the two-body (Earth + spacecraft) dynamics, that has no change in energy or angular momentum. In particular, the energy and the angular momentum of the spacecraft before and after this close encounter are calculated, to detect the changes in the trajectory during the close approach. The orbits are classified in four categories: elliptic direct (negative energy and positive angular momentum), elliptic retrograde (negative energy and angular momentum), hyperbolic direct (positive energy and angular momentum) and hyperbolic retrograde (positive energy and negative angular momentum). The problem now is to identify the category of the orbit of the spacecraft before and after the close encounter with the Moon. Fig 1 explains the geometry involved in the close encounter. Earth Y Rotating System B Moon Rp ψ P X DEFINITION OF THE PROBLEM The primary objective of this paper is to simulate and classify swing-by trajectories passing near the Moon. To solve this problem, it is assumed the existence of three bodies: the Earth, the Moon and a third particle of negligible mass (the spacecraft). It is also assumed that the total system (Earth + Moon + spacecraft) satisfies the hypothesis of the planar restricted circular three-body problem: all the bodies are point masses; the Earth and the Moon are in circular orbits around Fig 1. Geometry of the Close Encounter. The spacecraft leaves the point A, crosses the horizontal axis (the line between the Earth and the Moon), passes by the point P (the periapse of the trajectory of the spacecraft around the Moon) and goes to the point B. Points A and B are chosen in a such way that the influence of the Moon at those points are negligible and, consequently, the energy is constant after B and before A. Two of the initial A

3 conditions are clearly identified in this figure: the periapse distance r p (distance measured between the point P and the center of the Moon) and the angle ψ, measured from the horizontal axis in the counterclock-wise direction. The distance r p is not to scale, to make the figure easier to understand. The third initial condition is the Jacobian constant J of the spacecraft (also called Jacobi Energy - see equation 4). Under those assumptions, the procedure involved to solve this problem is: i) To specify particular arbitrary values for the Jacobian constant, the periapse distance (r p ) and the angle ψ; ii) Starting with the spacecraft at the periapse (P), integrate numerically its orbit forward in time until it reaches the point B; iii) Starting again at the periapse (P), integrate its orbit backward in time until the spacecraft reaches the point A; iv) At the points A and B the energy and the angular momentum of the spacecraft are calculated, and the classification of both segments in the above defined categories is made. MATHEMATICAL MODEL AND ALGORITHM The equations of motion for the spacecraft are assumed to be the ones valid for the well-known planar restricted circular three-body problem. They are: V && x - y& = x - = x V && y+ x& = y - = y Ω x Ω y where Ω is the pseudo-potential function given by: ( x + y ) 1 (1- µ ) µ Ω = + + (3) r1 r (1) () The equations of motion given by (1-) are right, but they are not suitable for numerical integration in trajectories passing near one of the primaries. The reason is that the positions of both primaries are singularities in the potential V (since r 1 or r goes to zero, or near zero) and the precision of the numerical integration is affected every time this situation occurs. The solution for this problem is to use regularization, that consists of a substitution of the variables for position (x-y) and time (t) by another set of variables (ω 1, ω, τ), such that the singularities are eliminated in these new variables. Several transformations with this goal are available in the literature (see Szebehely (1967), chapter 3), like Thiele-Burrau, Lamaître and Birkhoff. They are called "global regularization", to emphasize that both singularities are eliminated at the same time. The case where only one singularity is eliminated at a time is called "local regularization". For the present research the Lamaître's regularization is used. This system of equations is non-integrable, and numerical integration is required to solve any problem that involves this model. One of the most important reasons why the rotating frame is more suitable to describe the motion of M 3 in the three-body problem is the existence of an invariant, that is called Jacobi integral (or energy integral). There are many ways to define the Jacobi integral and the reference system used to describe this problem (see Szebehely (1967), page 449). In this paper, the definitions used by Broucke (1979) are followed. Under this version, the Jacobi integral is given by: 1 J = ( x ) + y& - Ω ( x, y) = Const & (4) This is a very important result and it gives two of the most important properties of the three-body problem: 1) The existence of five equilibrium points (called the Lagrangian points L 1, L, L 3, L 4 and L 5 ), that are the points where Ω Ω = = 0. A particle located at x y one of those points with zero initial velocity remains in its position indefinitely. The first three (L 1, L, L 3 ) are called collinear points (because they are in the

4 horizontal axis) and they are unstable for any value of µ. The last two (L 4 and L 5 ) are called triangular points (because they form an equilateral triangle with M 1 and M ) and they are stable for µ < , what means that they are stable in the present case (Earth-Moon), since µ = ) The curves of zero velocity. They are the curves given by the expression: ( x, y) J0 = -Ω (5) that is the equivalent of (4) in the special case where x& = y& = 0. Those curves are important, because they determine forbidden and allowed regions of motion for M 3 based in its initial conditions. More detail about these and other properties of the three-body problem can be found in Szebehely (1967) and Broucke (1979). The standard canonical system of units is used and it implies that: unit of distance = km, unit of time = days, unit of velocity = 1.03 km/s. It is also necessary to have equations to calculate the energy and the angular momentum of the spacecraft. Those calculations can be performed with the following formulas: + & & x y + x y 1 µ µ E = ( ) ( ) r1 r C = x + y + xy& yx& (7) + (6) With those equations, it is possible to build a numerical algorithm to solve the problem. It has the following steps: i) Arbitrary values for the three parameters: r p, J, ψ are given; ii) The initial conditions in the rotating system are computed with these values. The initial position is the point (X i, Y i ) and the initial velocity is (V Xi, V Yi ), where: ( µ ) Xi = R p cos( ψ) + 1 (8) Y = R sin ( ψ ) (9) i p VXi = V sin ( ψ ) (10) VYi = + V cos( ψ ) (11) where V = x& + y& is obtained from equation (4); iii) With those initial conditions, the equations of motion are integrated forward in time until the distance between the Moon and the spacecraft is bigger than a specified limit value d SM. At this point the numerical integration is stopped and the energy (E + ) and the angular momentum (C + ) after the encounter with the Moon are calculated from equations (6) and (7). Remember that it is assumed that the energy and the angular momentum is constant after this point, due to the fact that the perturbation from the Moon is too small to disturb significantly the two-body character of the dynamics; iv) Then the initial conditions are returned to the point P, and the equations of motion are integrated backward in time, until the distance d SM is reached again. Equations (6) and (7) are used to calculate the energy (E - ) and the angular momentum (C - ) before the encounter with the Moon; v) With those results, all the information required to calculate the change in energy (E + - E - ) and angular momentum (C + - C - ) due to the close approach with the Moon are available. With this algorithm available, the given initial conditions (values for r p, J and ψ) can be varied in any desired range to study the effects of the close approach with the Moon in the orbit of the spacecraft. RESULTS The results consist of plots that show the change of the orbit of the spacecraft, due to the close encounter with the Moon, for a large range of given initial conditions. First of all, it is necessary to classify all the close encounters between the Moon and the spacecraft, according to the change obtained in the orbit of the spacecraft. The letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P are used for this classification. They are assigned to the orbits according to the rules showed in Table 1. TABLE 1. Rules to assign letters to orbits

5 Before: After: Direct Ellipse Retrog Ellipse Direct Hyp Retrog Hyp Direct Ellipse A E I M Retrograde Ellipse B F J N Direct Hyperbola C G K O Retrog Hyperbola D H L P With those rules defined, the results consist of assigning one of those letters to a position in a twodimensional diagram that has the parameter ψ in the horizontal axis and the parameter J in the vertical axis. There is one plot for each desired value of the periapse distance. The same range for the variables ψ (180 ψ 360 ) and J (-1.45 J 1.55) used in Broucke (1988) and Broucke and Prado (1993) are used here. They are very adequate in showing the main characteristic of the plots. The interval 180 ψ 360 is used, and not the full range (0 ψ 360 ), because there is a symmetry between the chosen interval and the complementary interval 0 ψ 180. This symmetry comes from the fact that an orbit with an angle ψ = θ is different from an orbit with an angle ψ = θ only by a time reversal. It means that there is a correspondence between these two intervals. This correspondence is: I C, J G, L O, B E, N H, M D. The orbits A, F, K and P are unchanged. To decide the best range of values for the third parameter (periapse distance) several exploratory simulations have to be made. On the basis of those simulations, it was decided to make plots for the values: 1.1,.0, 5.0, and 10.0 Moon's radius. They span a useful range of values and they are able to show very well the evolution of the effects. Fig shows a series of diagrams covering the desired range for all the three variables. From a detailed study of those plots and the numerical values not showed here, it is possible to confirm the theoretical predictions (see more explanations of this topic in Broucke, 1988) that: i) When the fly-by is in front of the Moon (0 ψ 180 ), there is a loss of energy. This loss is maximum when ψ = 90 ; ii) When the fly-by is behind the Moon (180 ψ 360 ), there is an increase of energy. This increase is maximum when ψ = 70. A possible application is to obtain some saving in V for missions involving an escape from the Earth. A more interesting application is to obtain some saving for missions to the Moon itself, using the basic principles used by the Belbruno-Miller trajectories (Belbruno, 1987 and 1990; Miller and Belbruno, 1991; Yamakawa et. al., 199; Yamakawa et. al., 1993). In this type of maneuver, the spacecraft is launched from the Earth in a trajectory that makes it to pass close to the Moon (to make a swing-by to achieve a near-parabolic escape trajectory from the Earth), travel a long distance from this point and then return to be captured by the Moon (after using a three-dimensional fourth-body iteration with the Sun). This maneuver has a longer transfer time (several months) but it has a lower V (about 00 m/s in saving) than a standard Hohmann transfer. This is a very modern approach to send a spacecraft to the Moon that already proved its value in the Japanese mission Muses-A/Hiten. For this particular application, the trajectories that have more potential for use are the ones that transform an elliptic into a near-parabolic orbit. Those orbits are close to the frontier (in the letter-plots shown in this research) between the ones that transform an elliptic into another elliptic orbit (A, B, E, F) and those that transform an elliptic into a hyperbolic orbit (I, M, J, N). In the present research, the detected frontiers between those regions are A-I, B-J and F-N.

6 CONCLUSIONS A numerical algorithm to calculate the effects of a close approach with the Moon in the trajectory of a spacecraft is developed. Many trajectories are classified according to those effects. Special attention is given to identify the regions where the captures and escapes occur. The classical three parameters (Jacobian constant, pericenter distance and angle of approach) used to identify a Swing-By maneuver are varied in large intervals to cover a large range of possibilities for the maneuver. Letter-plots figures are made to show the results obtained in a compact form. The theoretical prediction that for 0 ψ 180 the spacecraft losses energy and for 180 ψ 360 the spacecraft gains energy is confirmed. Regions containing trajectories that are candidates to generate Belbruno-Miller trajectories are identified. ACKNOWLEDGMENT The authors are grateful to Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Post-Graduate Education Federal Agency) - CAPES - Brazil and INPE (National Institute for Space Research - Brazil) for funding this research. REFERENCES Belbruno EA (1987), Lunar Capture Orbits, a Method of Constructing Earth Moon Trajectories and the Lunar Gas Mission, AIAA In: 19th AIAA/DGLR/JSASS International Eletric Propulsion Conference, Colorado Springs, Colorado. Belbruno EA (1990), Examples of the Nonlinear Dynamics of Ballistic Capture and Escape in the Earth-Moon System, AIAA In: AIAA Astrodynamics Conference, Portland, Oregon. Broucke RA (1979), Traveling Between the Lagrange Points and the Moon, Journal of Guidance, Control, and Dynamics (4) Broucke RA (1988), The Celestial Mechanics of Gravity Assist, AIAA paper In: AIAA/AAS Astrodynamics Conference, Minneapolis, MN, Aug.. Broucke RA, and Prado AFBA (1993), Jupiter Swing-By Trajectories Passing Near the Earth, Advances in the Astronautical Sciences 8() Dowling RL, Kosmann WJ, Minovitch MA, and Ridenoure RW (1990), The Origin of Gravity-Propelled Interplanetary Space Travel. In: 41 st Congress of the International Astronautical Federation, Dresden, GDR, 6-1 Oct.. Dowling RL, Kosmann WJ, Minovitch MA, and Ridenoure RW (1991), Gravity Propulsion Research at UCLA and JPL, , In: 4 nd Congress of the International Astronautical Federation, Montreal, Canada, 5-11 Oct.. Dunham D, and Davis S (1985), Optimization of a Multiple Lunar-Swingby Trajectory Sequence, Journal of Astronautical Sciences 33(3) Farquhar RW, Muhonen D, and Church LC (1985), Trajectories and Orbital Maneuvers for the ISEE-3/ICE Comet Mission, Journal of Astronautical Sciences 33(3) Flandro G. (1966), Fast Reconnaissance Missions to the Outer Solar System Utilizing Energy Derived from the Gravitational Field of Jupiter, Astronautical Acta 1(4) Marsh SM, and Howell KC (1988), Double Lunar Swingby Trajectory Design, AIAA paper Miller JK, and Belbruno EA (1991), A Method for the Construction of a Lunar Transfer Trajectory Using Ballistic Capture, AAS In: AAS/AIAA Space Flight Mechanics Meeting, Houston, Texas. Minovich MA (1961), A Method for Determining Interplanetary Free-Fall Reconnaissance Trajectories, JPL Tec. Memo , Aug , 47 pp.. Nock KT, and Upholf CW (1979), Satellite Aided Orbit Capture, AAS/AIAA paper Öpik E (1976), Interplanetary Encounters, Elsevier, New York. Prado AFBA (1993), Optimal Transfer and Swing-By Orbits in the Two- and Three-Body Problems, PhD Thesis, University of Texas, Austin, Texas, USA. Szebehely VG (1967), Theory of Orbits, Academic Press, New York. Weinstein SS (199), Pluto Flyby Mission Design Concepts for Very Small and Moderate Spacecraft, AIAA paper In: AIAA/AAS Astrodynamics Conference, Hilton Head, SC, Aug Yamakawa H, Kawaguchi J, Ishii N, and Matsuo H (199), A Numerical Study of Gravitational Capture Orbit in the Earth-Moon System, AAS In: AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, Colorado, Feb. 4-6.

7 Yamakawa H, Kawaguchi J, Ishii N, and Matsuo H (1993), On Earth-Moon transfer trajectory with gravitational capture, AAS paper , In: AAS/AIAA Astrodynamics Specialist Conference, Victoria, CA. Rp = 1.1 Moon s Radius Rp =.0 Moon s Radius Rp = 5.0 Moon s Radius Rp = 10.0 Moon s Radius Fig. - Results for r p = 1.1,.0, 5.0, 10.0.