AN ALTERNATIVE DESCRIPTION OF SWING-BY TRAJECTORIES IN TWO AND THREE DIMENSIONS

Transcription

1 ADVANCES IN SPACE DYNAMICS 4: CELESTIAL MECHANICS AND ASTRONAUTICS, H. K. Kuga, Editor, 1-14 (004). Instituto Nacional de Pesquisas Esaciais INPE, São José dos Camos, SP, Brazil. ISBN AN ALTERNATIVE DESCRIPTION OF SWING-BY TRAJECTORIES IN TWO AND THREE DIMENSIONS Liliane A. Maia Deartamento de Matemática, Universidade de Brasília, Brasília-DF , Brazil - lilimaia@unb.br Ricardo R. Fragelli Deartamento de Engenharia Mecânica, Universidade de Brasília, Brasília-DF , Brazil - fragelli@hotmail.com ABSTRACT In this aer we simulate and classify the Swing-by maneuvers of a sacecraft in close aroach with a celestial body, alternatively roosing an identification of the category of the orbits in the synodic coordinate system. The mathematical model used is the restricted circular three-body roblem in two and three dimensions. In general the classification of orbits are resented in letter-lots which consist of grahs of orbital categories reresented by letters. The coordinates are chosen from the initial conditions of the roblem. In our work we use the eriasis distance r and the sherical angles of the sacecraft. Henceforth, we are able to use satial rectangular coordinates instead and obtain a view of the sacecraft at its eriasis osition and the changes in the trajectory during the close aroach. The letters are substituted by oints in the new grah where the study of the sign of energy and angular momentum gives the right classification of the orbits. INTRODUCTION The gravity assisted maneuver is a technique used in sace missions to reduce fuel consumtion like in Weinstein 199, Swenson 199, Broucke and Prado 1993, Prado and Broucke 1995 and many others (see Broucke 1988 and reference therein). The standard maneuver uses a close aroach with a celestial body to modify the velocity, energy and angular momentum of a dimensionless body (a sacecraft). The model used in the resent study is the three-dimensional restricted circular tree-body roblem (Szebehely 1967). For fixed arameters V, r, α and β, to described later, the roblem consists of studying the motion of the sacecraft near the close encounter with a secondary body of the system. In articular, the energy and angular momentum of the sacecraft before and after the close aroach. Those 1

2 quantities are used to classify its orbit u to sixteen classes of orbits, according to the changes in the energy and angular momentum caused by the close encounter (see Prado and Broucke 000). The main goal is to simulate a large number of orbits, under some fixed arameters, classify them into those classes and use an alternative model to describe the regions of different classes of orbits. This alternative descrition of orbital regions, differently from the standard ones found in the literature, uses the satial rectangular coordinates in the rotating reference system. Henceforth, if is ossible to obtain a view of the sacecraft at its eriasis osition together with its class of orbit. Moreover, using this alternative model it is ossible to derive equations in order to obtain necessary conditions for similarity between two systems. MATHEMATICAL MODEL For the resent study the well known model of three-dimensional restricted circular three-body roblem (Szebehely 1967) is used. It consists of two bodies M 1 and M, called rimaries, of masses m 1 and m moving in circular orbits about their mutual center of mass, with constant angular velocity w, and a third body M 3 of negligible mass (the sacecraft) witch moves under the gravitational effect of the central and erturbing masses without affecting their motion. As usual, dimensionless units are used, in such way that the distance between the rimaries, the total mass of the system and the angular velocity w of the system are equal to one. The equations of motion of the third body in the rotating frame are given by && x x + µ x (1 µ ) y& = x (1 µ ) µ 3 3 r r y && y + x& = y (1 µ ) 3 r 1 z && z = (1 µ ) r z µ r y µ 3 r m where µ =, r 1 and r are the distances from M 1 and M to M 3, resectively. In the case m1 + m of lanar motion z = 0. It is also necessary to have equations to calculate the energy ( E ) and angular momentum (C ) of the sacecraft, which are resectively the following E = ( x + y& ) + ( x& + y) µ µ + and C = x r r + y + xy& x& y. In the restricted roblem neither the orbital energy nor the angular momentum are conserved since the third body does not affect the motion of the rimaries. However, the dynamical system still has na

3 1 integral of the motion, the Jacobi constant, given by J = V Ω, where 1 1 µ µ Ω = ( x + y ) + +. r r 1 ALGORITHM TO SOLVE THE PROBLEM In order to describe the maneuver it is standard to use the four indeendent arameters: i) V, the magnitude of the velocity of the sacecraft at eriasis; for our urosesly velocity vectors V r arallel to the x-y lane were considered; ii) r, the distance between the sacecraft and the second mass M during the closest aroach (eriasis); iii) α, the angle between the rojection of the eriasis line in the x-y lane and the line that connects the two rimaries; iv) β, the elevation angle between the eriasis line and the x-y lane. Figure 1 Initial conditions in two and three-dimensions. Once a system is established (i.e., µ is fixed at some value), the initial osition ( x i yi, zi ) velocity ( Vx Vy, Vz ), and i, i i are considered for the sacecraft at eriasis. With these considerations, a numerical algorithm is built using MaleV software to solve the roblem, in the following stes: i) arbitrary values for the arameters V, r, α and β are given; ii) with these values, the initial conditions in the rotating system are comuted x = 1 µ ) + r cos β cosα y = cos β senα z = sen β iii) i ( i r i r Vxi = V senα Vy i = V cosα Vz i = 0 with these initial conditions, the equations of motion are integrated forward in time until the distance r is equal to 0.5, half the distance between the rimaries in the dimensioless 3

4 iv) system. At this oint the numerical integration is stoed and the energy ( E + ) and the angular momentum ( C + ) after the close encounter are calculated; the initial conditions are considered again and the equations of motion are integrated backwards in time, until the secified distance is reached again. Then the energy ( E ) and angular momentum ( C ) before the encounter are obtained. For all simulations, a Runge-Kutta 4 th -5 th order was used, erformed by MaleV software. The energy determines whether the trajectory is ellitic ( E < 0 ) or hyerbolic ( E > 0 ) and the angular momentum indicates when it is retrograde ( C < 0 ) or direct ( C > 0 ). Parabolic trajectories are given by E = 0 and linear trajectories by C = 0. The goal is to identify the category of the orbit of the sacecraft before and after the close encounter for a large range of given initial conditions. The standard classification and reresentation according to the change in the orbit of the sacecraft is by assigning letters to orbits, the so called letter-lot diagram forend in the related literature (Broucke 1988). In the lanar case, letter-lots are reresented in two-dimensional diagrams that have α in the horizontal axis and the Jacobi constant J in the vertical axis, for fixed values of r and µ. In the threedimensional restricted circular three-body roblem, a letter-lot is made for fixed values of r, µ and V, in a diagram that has angle α in the vertical axis and angle β in the horizontal axis (see Felie 000). RESULTS In this study a velocity at eriasis V is fixed and the r is a variable taking values in an interval where the influence of M in the change of orbit of the sacecraft is significant. The value of r should not be so large such that the gravitational force of M on the sacecraft would hardly be noticed, neither it should not be so small to result in cature of the sacecraft for a long eriod. According to Broucke 1988, where a convenient interval for the Jacobi constant J is assumed 1.45 J 1.5, it is necessary to find the corresonding V s for J in this interval. The Earth-moon system ( µ = 0. 01) is initially fixed and for r s found in Broucke 1998, it is ossible to obtain lower and uer bounds for V s corresonding to 1.45 J Table 1 Ranges for V s corresonding to 1.45 J r Lower V Uer V In order to erform the simulations of orbits, the values chosen for V s are 1.5,,.5, 3, 4, 5, 10 and 14. 4

5 Considering J constant with resect to the angle α, it is ossible to calculate lower and uer bounds for r s for each V fixed, as it is shown in table. Note that for each V fixed it is ossible to find an r such that the calculated J is equal to 1.45, however the same does not haen for J = In the latter cases, or in case the calculated uer bound for r is suerior to the radius of influence, it is taken the uer bound r = for the study of orbital regions. Table Initial conditions for the study of orbital regions. V Lower r Uer r Lower r Uer r calculated calculated assumed assumed Não tem Não tem With µ = and V fixed for a value in table, a simulation of an orbit is erformed for each initial condition as defined before, for r in that interval and the orbit is classified according to the signs of E, E +, C and C +. An examle of the results obtained and reresented in this alternative diagram is shown below for V =1.5. 5

6 E- < 0 Figure Orbital regions for V = 1. 5 and r in Earth-moon system. In order to understand this reresentation, consider two oints P 1 and P which are indicated in the diagrams as initial conditions. The oint P 1 has olar coodinates α = 3º and r = and P has α = 300º and r = P 1 is found in a region of tye A (direct ellise to direct ellise) and P is located in a region K (direct hyerbola to direct hyerbola). E- < 0 Figure 3 Reresentation of oints P 1 and P in diagram of orbital regions. A simulation of the trajectories of the sacecraft in the rotating reference system, under the fixed arameters µ = 0. 01, V = 1. 5 and eriasis at P 1 and P is shown in figure 5. 6

7 Before Before After After Figure 4 - Trajectories of the sacecraft in regions A and K for a short time. Note that for a longer eriod, one trajectory is a siral around the center of mass, characteristic of a hyerbolic orbit, and the other is a trajectory comosed of small loos enclosing the bigger rimary, which characterizes an ellitic orbit. Before After Before After Figure 5 - Trajectories of the sacecraft in regions A and K for a long time. Simulations for other values of the arameter regions. V in table give the following diagrams of orbital 7

8 E- < 0 E- < 0 Figure 6 - Orbital regions for V = and r (left) and for V =. 5 and r 0.04 (right). E- < 0 E- < 0 Figure 7 - Orbital regions for V = 3 and r (left) and for V = 4 and r (right). In each grah a oint on the axis of the rimaries indicates the signs of the funcions E-, E+, C-, C+ at that oint. As one moves to the right hand side and crosses their zero level curves, then the signs of these functions change from negative into ositive at oints chosen in the ring domain. Note that the zero level curves of the energy ( E = 0 and E + = 0 ) look alike and are almost aralel for V small. When they intercet the horizontal axis they meet at α = 180º and α = 360º. Similarly for the zero level curves of the angular momentum ( C = 0 e C + = 0 ) which leave the ring domain for larger radii almost at α = 70º. If V > and r's are sufficiently large the energies ( E and E + ) are ositive and the regions are reduced to three classes ( retrograde hyerbola to retrograde hyerbola, retrograde hyerbola to direct hyerbola and direct hyerbola to direct hyerbola). Moreover, those regions are delimited by curves of zero angular momentum. As r increases, those curves aroach asymtotically a line at M of constant angle α, which for C + is close to α = 70º. 8

9 SIMILARITY RESULTS The objective now is to develo a technique that may ermit associating two systems which have similar diagrams of orbital regions, under different arameters and initial conditions. Broucke in (Broucke 1988) verified that letter-lot diagrams of orbital regions, in the coordinate system of variables α and J in the axis, are similar for given arameters µ = C1 and r = C C1, if airs of values are taken in the sets C 1=0.01, 0.001, and and C =0.01, 0.1, 1, 10, 100. In fact, if there is similarity of diagrams then µ r = r 1, µ 1 (1) where ( r 1, µ 1 ) and ( r, µ ) are eriasis radii and the coefficients of mass for systems 1 and, resectively. Once µ = 0. 01, r = and α = 180º are fixed, the velocities V are calculated corresonding to 1.5 J 1.5, and the values obtained are V Then, a value of V is fixed and the deendence of J with resect to the indeendent variable α can be measured. Figure 8 - Relation between the Jacobi constant and α for fixed V and r. The Jacobi constant J may be considered a constant function of the variable angle α, with an error smaller than 10-4, for r = It is assumed that the initial conditions µ = 0. 01, r = and 1.5 J 1.5 may be relaced by µ = 0. 01, r = and V , giving the same diagrams of orbital regions. Analogously, for µ = and r = , again assuming J constant with resect to α, the corresonding velocity values are 9

10 V , and so on. The following table shows the same calculations for other airs ( µ, r ). Table 3 - Ranges of velocities corresonding to 1.5 J 1. 5 Initial Conditions Calculated Interval of Velocities ( µ = 0. 01, r = 0. 01) V ( µ = , r = ) V ( µ = , r = ) V ( µ = , r = ) V These airs of initial conditions are equivalent in the sense that for the corresonding intervals of velocities V s their diagrams of orbital regions should be similar. Now, fixing V = (case µ = 0. 01) and calculating an interval of r 's such that J takes its standard values, it results in r Analogously, if V = (case µ = ) then the resulting interval of r 's is r Therefore, V = ( µ = 0. 01, r = 0. 01) is equivalent to V = ( µ = , r = 0.001) and both reresent a horizontal line in the diagrams of orbits in coordinates J, α. Other simulations for those initial conditions and other choices of r 's show a similarity between these two systems. Figure 9 - Study of J for different values of r in two systems. However, for lower velocities like V = ( µ = 0. 01) it is not ossible to find an interval of r s which would give the standard interval of J 's. Moreover, here J is not a constant function of the variable α, as the following examles show. The left hand side shows from to to bottom the 10

11 cases µ = and r varying in the intervals [0.01,0.] and [0.01,0.5], and the right hand side shows in the same order, cases µ = and r taking values in the intervals [0.001,0.0] and [0.001,0.05]. Figure 10 - Study of J for different intervals of r in two systems Given two systems with eriasis r 1 and r and coefficients of mass µ 1 and µ, resectively, for the urose of this study if there is similarity then equation (1) is satisfied. This imlies that if r 1 is taken equal to µ 1 then r is equal to µ. Moreover, if their Jacobi constants J 1 and J are assumed constant functions of the angle α, then α may be taken equal 180 º. With these initial conditions, in order to obtain similarity it is assumed J 1 = J and it follows that their resective eriases velocities V 1 and V in the rotational reference system must satisfy the condition: = V 1 ( 1 µ 1) + (1 µ ) V. () Some examles of similar diagrams of orbital regions are given below for high and low velocities. Their resective initial conditions satisfy equations (1) and (). 11

12 E- < 0 E- < 0 Figure 11 - Similar diagrams of orbital regions for two different systems V 1 =. 814 and µ 1 = 0.01 (left) and V =. 877 and µ = (right). E- < 0 E- < 0 Figure 1 - Similar diagrams of orbital regions for two different systems V 1 =. 0 and µ 1 = 0.01 (left) and V = and µ = (right). ORBITAL REGIONS IN THREE DIMENSIONS The study of three-dimensional orbital regions involves one more variable: the angle beta. Simulations are erformed for a sherical shell taking V fixed and 0 β 90º, since there is symmetry for 90º β 0. Each of the sixteen classes of regions is viewed searately. For a fixed angle β, the oints of the boundary of a region are determined. Then, the angle β takes different values between 0 and 90 in order to generate different surfaces. Finally, these surfaces are ut together in order to generate the desired region. A better view of a region is obtained using MaleV to roduce animations of the three-dimensional icture. The following ictures ilustrate the tye of view obtained as outut of the rograms for V =

13 E- < 0 F B A I K Figure 13 - Orbital regions for V = 1. 5 including the letters of classes of orbits. Figure 14 - Region K (direct hyerbola to direct hyerbola). CONCLUSIONS The effects of a close aroach of a sacecraft with a celestial body were simulated and classified, using a numerical algorithm and the lotting resources of MaleV software. An alternative descrition of diagrams of orbital regions in the rotational coordinate system is roosed. The 13

14 mathematical model used is the restricted circular three-body roblem in two and three dimensions. In general the classification of orbits are resented in letter-lots which consist of grahs of orbital categories reresented by letters. In this work the eriasis distance r and the sherical angles α and β of the sacecraft are used as variables, and the eriasis velocity V is fixed. Henceforth, it is ossible to use satial rectangular coordinates instead and obtain a view of the sacecraft at its eriasis osition and a classification of the changes in the trajectory during the close aroach, simultaneously at the same diagram. Moreover, a rocedure was develoed to study the similarity of two systems, or equivalently, the similarity of two diagrams of orbital regions for different initial conditions. Equations involving the eriasis radii, the coefficients of mass and the velocities of two systems, were derived as necessary conditions for similarity of two systems. REFERENCES BROUCKE, R.A.; The celestial mechanics of gravity assist. AIAA/AAS Astrodynamics Conference, Minneaolis, MN, 1988 (AIAA Paer 88-40). BROUCKE, R.A. & A.F.B.A. PRADO (1993). "Juiter Swing-By Trajectories Passing Near the Earth", Advances in the Astronautical Sciences, Vol. 8, No, BROUCKE, R.A.; PRADO, A.F.B.A.; A study of the effects of juiter in sace trajectories. IX Colóquio Brasileiro de Dinâmica Orbital, 000. FARQUHAR, R.W. & D.W. DUNHAM (1981). "A New Trajectory Concet for Exloring the Earth's Geomagnetic ", Journal of Guidance, Control and Dynamics, Vol. 4, No., FELIPE, G.; PRADO, A.F.B.A.; Otimal maneuvers in three dimensional swing-by trajectories. IX Colóquio Brasileiro de Dinâmica Orbital, 000. FRAGELLI, R. R..; Estudo da manobra assistida ela gravidade em duas e três dimensões, Tese de mestrado, Deartamento de Engenharia Mecânica, Universidade de Brasília, 00. FRAGELLI, R. R.; MAIA, L. A.; The owered swingby in three dimensions. X Colóquio Brasileiro de Dinâmica Orbital, Guaratinguetá, 000. PRADO, A.F.B.A. & R.A. BROUCKE (1995). "A Classification of Swing-By Trajectories using The Moon". Alied Mechanics Reviews, Vol. 48, No. 11, Part, November, PRADO, A.F.B.A.; An analytical descrition of the close aroach maneuver in three dimensions. IAF-00-A5.05, 000. SZEBEHELY, V.; Theory of orbits, Academic Press, New York, SWENSON, B.L. (199). "Netune Atmosheric Probe Mission", AIAA Paer WEINSTEIN, S.S. (199). "Pluto Flyby Mission Design Concets for Very Small and Moderate Sacecraft", AIAA Paer