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AIAA 2002-4717 APPLICATION OF TISSERAND’S CRITERION TO THE DESIGN OF

J. K. Miller* Jet Propulsion Laboratory California Institute of Technology Pasadena, California

Connie J. Weekst Loyola Marymount University Los Angeles, California

ABSTRACT ration of that otherwise would not be ac- Gravity assist involves the use of the gravita- cessible with current capability. tional attraction of an intermediate to in- Most interplanetary and planetary orbiter mis- crease the energy of a , enabling sion trajectories since the beginning of the space it to attain the target planet. The diffi- age have used Keplerian two-body motion in their culty in designing a gravity assist is the design. Missions requiring three-body transfers are determination of the encounter times of the launch generally limited to those involving the 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 - 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 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 . 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 , depends on finding a 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 mov- Laboratory, Associate Fellow ing in the vicinity of two massive bodies in circular t Professor of Mathematics, Loyola Marymount University orbits about their 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 . 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 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. The seg- ments of the Cassini trajectory that are of interest are from Earth to Jupiter and from Jupiter to Sat- urn. For the purpose of this analysis, the encounter The of the spacecraft may be re- garded as a two-body conic. The velocity magni- time and initial conditions at Earth relating to en- tude is given by ergy are given. The first step is to determine the encounter times at Jupiter and Saturn. An initial guess of the encounter times of Jupiter and Saturn is made based on the approximate flight times associated In the plane of the orbit, the angle A is simply the with a Hohmann transfer. Point to point conic so- flight path angle (7). For the general case, the an- lutions for the trajectory segments from Earth to gle A is a function of y and the inclination of the Jupiter and from Jupiter to Saturn are computed spacecraft orbit plane with respect to the planet using the solution of Lambert’s theorem discovered orbit plane il and by Lagrange. A point to point conic solution as- sumes zero mass for the planets and only the grav- ity of the sun is included. The solution of Lam- bert’s theorem gives the two-body conic connect- ing two position vectors where the flight time is known. The two position vectors are obtained from the planetary ephemerides and the conic trajectory is computed from planet center to planet center Making these substitutions into Equation 15 gives, as shown on Figure 3. 2GM, GM, GM, The next step is to compute the velocity vec- +-- tors relative to Jupiter, one for the incoming tra- .+---P a1 P jectory segment (vi) and one for the outgoing tra- jectory (v,). If the Jacobi constants for the two COSil (21)

4 American Institute of Aeronautics and Astronautics and Saturn are permitted to vary over a suitable range of times. For each pair of Earth-Jupiter en- counter times and Jupiter-Saturn encounter times, a Lambert solution is computed and the Jacobi con- stant is computed from the orbit elements. The Jacobi constants for the two interplanetary trajec- tory legs are matched and the results cross plot- ted on Figure 4. Several approximations may be used for computing the Jacobi constant. Results for Tisserand's criterion and the Jupiter energy cri- sunLO terion are shown on Figure 4 as dashed lines. The Jupiter Jupiter energy criterion (Equation 22) is equiva- lent to matching the incoming and outgoing veloc- -) ity magnitudes relative to Jupiter. For this paper, Interplanetaryleg 1 CI=f(a,e,i) a criterion is used that matches the average of Tis- serand's criterion and the Jupiter energy criterion and is shown on Figure 4 as the solid line. The equation for this papers criterion, after simplifica- tion to remove constant parameters, is given by, Earth/ o/

Figure 3 Earth-Jupiter-Saturn Encounter trajectory segments do not match, then the follow- ing procedure can be used to find potentially viable encounter time solutions. The encounter time of Jupiter is fixed and the encounter times of Earth

..."...I....."._ ...... ".""...,

, Tisscrand locuz

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Jupiter encounter Dec 30,2000 i i 2003 IIIl4.8 June 1 Jun 29 Jul27 Aug 24 Sep 21 Fhth Encounter Date - 1999 Figure 4 Earth-Jupiter-Saturn Loci 5 American Institute of Aeronautics and Astronautics For a given pair of Earth launch and Saturn en- ephemerides at the appropriate times. The patch counter times indicated on Figure 4, the approach point times are selected such that hte spacecraft and departure velocity vectors at Jupiter are ob- position is near the sphere-of-influence of the plan- tained and the hyperbolic conic relative to Jupiter ets. The planetary ephemerides may be computed is computed. A preliminary assessment of the vi- from two-body orbit elements with respect to the ability of the Jupiter centered hyperbola is per- Sun. This procedure is repeated several times for formed. A trajectory that intersected the surface of the new patch points until a ballistic trajectory is Jupiter, for example, or hits one of Jupiter's satel- obtained from Earth to Saturn. It will be neces- lites would not be viable. Next, the encounter con- sary to allow the Saturn encounter time to vary a ditions at Earth and Saturn are examined for vi- small amount from the point to point solution. The ability. If the energy at Earth or Saturn is unac- results, shown on Figure 4 for three launch dates, ceptable, the trajectory is not viable. If a viable compare favorably with the point to point solutions trajectory is not found for all the launch date en- from this papers criterion. Also, the Cassini design counter date pairs indicated by Figure 4, the above point, obtained by numerical integration, is shown procedure is repeated for another Jupiter encounter on Figure 4 for comparison. time. The patched conic solution is used as a start- Once a viable set of encounter times has been ing point for targeting an integrated trajectory. A determined, a patched conic trajectory is designed comparison of the Cassini integrated trajectory and that connects Earth Jupiter and Saturn. the patched conic solution is shown on Figure 6. State vectors are computed from the patched conic trajectory and differenced with state vectors ob- tained from the integrated Cassini . The magnitude of the position difference is plotted a Encounter hyperbola as Saturn centerad function of time and the heliocentric range of the spacecraft is also plotted for comparison. The max-

Point imum error is less than one percent of the heliocen- I " sun I tric range. Since the period of the Saturn orbit is Interplanetary Leg 2 t 29 years, an error of several months in the predicted encounter time at Saturn from the point to point ., ., J Jupiter Patch mint 3 conic solutions should be expected. This error in computing the encounter times is exacerbated by "unto= hypsrbola (Jupiter centered) accelerations from the third body that has been ig- Interplanetary Leg 1 nored for the two-body computations. However, a design error of only one percent enables a fairly accurate assessment of mission design constraints

atch Point 1 from the conic solution. -wBlmtmter hyperhola (Earth centersd) IE+OID

E x IEWOB Figure 5 Earth-Jupiter-Saturn Encounter The procedure involves computing the approach 2 1E-7 and departure velocity vectors at the patch points "6 o[ conic Position srror shown on Figure 5 from the point to point conic so- 0 g IE+OW lution. The two-body is then 0) computed with respect to each of the participating 3eo loowo planets. For Earth and Saturn, the departure and 1wM) approach target plane positions are given. A new -ZE+W7 2EW EEW IEMO8 1.4EMO8 set of patch point positions relative to the Sun are Time -seconds from Jan 1,2000 12OO:OO computed. The states relative to Earth, Jupiter and Saturn are added to the respective planetary Figure 6 Cassini Conic Design Error 6 American Institute of Aeronautics and Astronautics ACKNOWLEDGEMENT Bristol,UK., 1982 The work described in this paper was carried 2. Strange, N. J. and J. A. Sims,“Methods for the out by the Jet Propulsion Laboratory, California Design of V-Infinity Leveraging Maneuvers”, Institute of Technology, under a contract with the AAS paper 01-437., 2001. National Aeronautics ans Space Administration. 3. Strange, N. J. and J. M. Longuski, “Graphi- cal Methods for Gravity-Assist Trajectory De- REFERENCES sign”, Journal of Spacecraft and Rockets, Vol. 1. Roy, A, E., Orbital Motion, Adam Hilgar Ltd., 39, No. 1, January-February 2002.

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