AIAA 2002-4717 APPLICATION OF TISSERAND’S CRITERION TO THE DESIGN OF GRAVITY ASSIST TRAJECTORIES J. K. Miller* Jet Propulsion Laboratory California Institute of Technology Pasadena, California Connie J. Weekst Loyola Marymount University Los Angeles, California ABSTRACT ration of planets that otherwise would not be ac- Gravity assist involves the use of the gravita- cessible with current launch vehicle capability. tional attraction of an intermediate planet to in- Most interplanetary and planetary orbiter mis- crease the orbit energy of a spacecraft, enabling sion trajectories since the beginning of the space it to attain the target planet. The primary diffi- age have used Keplerian two-body motion in their culty in designing a gravity assist trajectory is the design. Missions requiring three-body transfers are determination of the encounter times of the launch generally limited to those involving the satellites planet, intermediate planet and target planet. of the major planets, for example missions to La- Once these times are determined, it is a relatively grange points in the Earth-Sun system, or the Voy- simple procedure to use Lambert’s theorem and nu- ager mission to the outer planets. Even though merical integration to complete the detailed design. gravity assist trajectories can be designed by re- In this paper, a procedure is described for finding peated application of two-body theory, they are in- these encounter times using Lambert’s theorem and cluded in the three-body classification because the a new criterion based on Tisserand’s criterion to gravity assist requires a simultaneous exchange of identify pairs of transfer orbits between the launch energy among three bodies. The three-body th& planet and intermediate planet and between the in- ory employed for the design of gravity assist tra- termediate planet and target planet. jectories involves the use either of vectors defin- ing the approach and departure hyperbolic asymp- INTRODUCTION totes with respect to the gravity assist planet, or of Gravity assist trajectories are an important Tisserand’s criterion, which pertains to the inter- class of trajectories that have been used by Voy- planetary Keplerian orbits connecting the launch, ager, Galileo, Cassini, and other missions to tour gravity assist and target planets. It will be shown the solar system. The design of interplanetary tra- that while both design techniques follow from the jectories involves finding an orbit that will trans- Jacobi integral, they yield significantly different re- fer a spacecraft from the vicinity of one planet to sults, since they represent different approximations the vicinity of another planet. The accessibility of of the true equations of motion. In this paper, a a target planet, particularly those beyond the or- criterion is developed that combines both of these bit of Jupiter, depends on finding a transfer orbit design techniques. with energy relative to the initial starting point, normally the Earth, within the capability of the JACOB1 INTEGRAL launch vehicle. The use of gravity assist to increase An important integral describing constraints the transfer orbit energy has opened up the explo- on energy transfer for the restricted three body problem was discovered by Carl Gustav Jacob Ja- * Member of the Navigation Systems SectionJet Propulsion cobi in the Nineteenth century. A point mass mov- Laboratory, Associate Fellow ing in the vicinity of two massive bodies in circular t Professor of Mathematics, Loyola Marymount University orbits about their barycenter will conserve a cer- Copyright 200mhe American Institute of Aeronautics and As- tain function of the state and gravitational param- tronautics Inc. All rights reserved eters of the massive bodies referred to as Jacobi’s 1 American Institute of Aeronautics and Astronautics integral. The constant of integration is called Ja- cobi's constant. The equations of motion for two massive bodies and a point mass are given by, 21 - x 22 - x X = GM1- i-GM2- rf 4 Equation 4 may be put into a form that can be in- The two massive bodies rotate around the barycen- tegrated by defining the function ter and the rotation rate is simply 27~divided by the period of the orbit, w= P3 \JGM1+GM2 and substituting into Equations 4. where p is the distance separating the two massive bodies. The geometry is illustrated on Figure 1. rY Adding Equations 6, The integral of Equation 7 given below is called the Jacobi integral. .2 .2 .2 2' +y' +z' =2u-c or Figure 1 Restricted Two Body Geometry .2 -2 .2 2' + y' + 2' = W2Xl2 + w2yt2+ GM1 GM2 The primed coordinate system (XI, y', 2') in Figure 2- + 2- -c (9) r1 r2 1 represents a rotating coordinate system in which the two massive bodies lie on the x' axis, with where C is the constant of integration. TISSERAND'S CRITERION x = x'coswt - y'sinwt Francois Felix Tisserand was a nineteenth cen- y = x' sin wt + y' cos wt tury astronomer who discovered a unique applica- tion of Jacobi's integral to identify comets. In the z = z' (3) restricted three body problem, a certain function of After differentiating Equations 3 twice, substitut- the orbit elements before and after a planetary en- ing into Equations 1 and eliminating the sine and counter is conserved. If this function is computed for two comet observations on different orbits and cosine terms, the following result is obtained as shown in Reference 1. the results are the same, one may conclude that the 2 American Institute of Aeronautics and Astronautics observations are of the same comet and the comet 14 is multiplied through by p and divided by GM1, has encountered a planet between the observations. Tisserand’s criterion in dimensionless coordinates This may be confirmed by propagating the orbits becomes forward or backward in time to see if they encoun- tered a planet. In the application of Tisserand’s criterion to gravity assist trajectory design, the procedure is If the first observation of a spacecraft or comet reversed. Transfer trajectories from the launch has orbit elements al, el and il and the second ob- planet to the intermediate planet and from the in- servation after a planetary encounter has orbit el- termediate planet to the target planet are com- ements a2, e2 and iz, then puted using Lambert’s theorem. These trajecto- ries are matched based on Tisserand’s criterion to identify viable launch and encounter opportunities. Tisserand’s criterion follows directly from Jacobi’s integral. Using Equations 3, the Jacobi integral is transformed back to inertial coordinates (the un- primed coordinates on Figure 1). GRAVITY ASSIST VECTOR DIAGRAM x2 + 7j2 + i2= 2w(xjr - yk)+ Figure 2 shows the encounter geometry in the GM1 GM2 2- +2--c vicinity of the intermediate planet that supplies r1 r2 the gravity assist energy boost to the spacecraft. For GMl much greater than GM2, the z component The incoming velocity of the spacecraft (V1)is sub- of the angular momentum vector is given by, tracted from the planet velocity (V,) to obtain the planet relative approach velocity (vi) as shown in xy - yx = h, = hcosi (11) the upper vector diagram on Figure 2. The lower vector diagram shows the same relationship for the h = dGM1 a(1- e2) outgoing velocity vectors. If the incoming and out- going velocities are computed far from the planet and from the vis viva integral the energy is given yet close enough to the planet that the heliocentric by, 21 energy may be assumed constant, the velocities k2 + y2+ t2= GMi(- - -) (12) r1 a Substituting Equations 11 and 12 into Equation 10 P gives 21 GM1(- - -) - 2wdGM1 a(1- e2)cosi = r1 a GMi GM2 2- +2--c r1 r2 Substituting Equation 2 for w and for small GM2 compared with GM1, a( 1 - e2) Cx- cosi (14) GM1a + 2GMl\/l P3 In the literature, Tisserand’s criterion is often de- veloped in dimensionless coordinates and the Ja- cobi constant modified to remove constant param- Figure 2 Gravity Assist Vector Diagram eters. If a is divided by p to define ii and Equation 3 American Institute of Aeronautics and Astronautics vi and v, are approximately the v, vectors as- The energy of the spacecraft relative to the planet, sociated with the two-body hyperbola about the the potential energy of the spacecraft relative to planet. In the limit of two-body motion assumed the Sun and the velocity of the planet relative to for patched conic trajectories, vi and v, are equal the sun may be regarded as constant. Collecting in magnitude. Since the planet velocity is also as- these “constant” terms on the left side gives, sumed to be constant during the relatively short time interval of the planet encounter, the outgoing 3GM, vector diagram may be superimposed on the incom- C=-- Vi2 ing vector diagram as shown on Figure 2. The out- P going heliocentric spacecraft velocity magnitude is N greater than the incoming velocity magnitude and N- GMs +2GMs al(l - COSil (22) a1 { P3 ea the spacecraft has acquired additional orbit energy ’ relative to the Sun. The energy acquired by the Equation 22 provides an interesting insight into spacecraft comes from the Sun and the planet. the geometrical meaning of Jacobi’s constant in the Consider the triangle formed by the spacecraft limit where one of the gravitating bodies is much and planet heliocentric velocity vectors and the in- more massive than the other. The terms in the coming velocity vector. From the law of cosines, Jacobi integral are related to the velocity vector diagram of the participating bodies and the rela- tionship to energy conservation is incidental. The orbit of the planet about the Sun may be CASSINI TRAJECTORY DESIGN approximated by a circle with velocity magnitude given by, The Cassini mission to Saturn provides an ex- ample of the application of Tisserand’s criterion to the design of a gravity assist trajectory.