INVARIANT MANIFOLDS, THE SPATIAL THREE-BODY PROBLEM AND SPACE MISSION DESIGN

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1 AAS -3 INVARIANT MANIFOLDS, THE SPATIAL THREE-BODY PROBLEM AND SPACE MISSION DESIGN G. Gómez,W.S.Koon,M.W.Lo, J.E. Marsden,J.Masdemont, and S.D. Ross August 2 The invariant manifold structures of the collinear libration points for the spatial restricted three-bod problem provide the framework for understanding complex dnamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold tubes associated to libration point orbits are the phase space structures that provide a conduit for orbits between primar bodies for separate three-bod sstems. These invariant manifold tubes can be used to construct new spacecraft trajectories, such as a Petit Grand Tour of the moons of Jupiter. Previous work focused on the planar circular restricted three-bod problem. The current work extends the results to the spatial case. INTRODUCTION New space missions are increasingl more complex. The require new and unusual kinds of orbits to meet their scientific goals, orbits which cannot be found b the traditional conic approach. The delicate heteroclinic dnamics emploed b the Genesis Discover Mission dramaticall illustrates the need for a new paradigm: stud of the three-bod problem using dnamical sstems theor.,2,3 Furthermore, it appears that the dnamical structures of the three-bod problem (e.g., stable and unstable manifolds, bounding surfaces), reveal much about the morpholog and transport of materials within the solar sstem. The cross-fertilization between the stud of the natural dnamics in the solar sstem and applications to engineering has produced a number of new techniques for constructing spacecraft trajectories with desired behaviors, such as rapid transition between the interior and exterior Hill s regions, temporar capture, and collision. 4 The invariant manifold structures associated to the collinear libration points for the restricted three-bod problem, which exist for a range of energies, provide a framework for understanding Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona, Spain. gomez@cerber.mat.ub.es Control and Dnamical Sstems, California Institute of Technolog, MC 7-8, Pasadena, California 925, USA. koon@cds.caltech.edu Navigation and Mission Design Section, Jet Propulsion Laborator, California Institute of Technolog, Pasadena, CA 99, USA. Martin.Lo@jpl.nasa.gov Control and Dnamical Sstems, California Institute of Technolog, MC 7-8, Pasadena, California 925, USA. marsden@cds.caltech.edu Departament de Matemàtica Aplicada I, Universitat Politècnica de Cataluna, Barcelona, Spain. josep@barquins.upc.es Control and Dnamical Sstems, California Institute of Technolog, MC 7-8, Pasadena, California 925, USA. shane@cds.caltech.edu

2 these dnamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold tubes associated to libration point orbits are the phase space structures that provides a conduit for material to and from the smaller primar bod (e.g., Jupiter in the Sun- Jupiter-comet three-bod sstem), and between primar bodies for separate three-bod sstems (e.g., Saturn and Jupiter in the Sun-Saturn-comet and the Sun-Jupiter-comet three-bod sstems). 5 Furthermore, these invariant manifold tubes can be used to produce new techniques for constructing spacecraft trajectories with interesting characteristics. These ma include mission concepts such as a low energ transfer from the Earth to the Moon and a Petit Grand Tour of the moons of Jupiter. See Figures and 2. Using the invariant manifold structures of the 3-bod sstems, we were able to construct a transfer trajector from the Earth which executes an unpropelled (i.e., ballistic) capture at the Moon. 6 An Earth-to-Moon trajector of this tpe, which utilizes the perturbation b the Sun, requires less fuel than the usual Hohmann transfer. Inertial Frame Sun-Earth Rotating Frame V Earth V L Earth L 2 Moon's Orbit Sun Ballistic Capture Moon's Orbit Ballistic Capture x x Figure : (a) Low energ transfer trajector in the geocentric inertial frame. (c) Same trajector in the Sun-Earth rotating frame. (Jupiter-Ganmede rotating frame) L Jupiter Ganmede L 2 (Jupiter-centered inertial frame) Transfer orbit V Jupiter Europa s orbit Ganmede s orbit (Jupiter-Europa rotating frame) Europa Jupiter L 2 x (Jupiter-Ganmede rotating frame) x (Jupiter-centered inertial frame) x (Jupiter-Europa rotating frame) (a) (b) (c) Figure 2: The Petit Grand Tour space mission concept for the Jovian moons. In our previous stud, we showed an orbit coming into the Jupiter sstem and (a) performing one loop around Ganmede (shown in the Jupiter- Ganmede rotating frame), (b) transferring from Ganmede to Europa using a single impulsive maneuver (shown in the Jupiter-centered inertial frame), and (c) getting captured b Europa (shown in the Jupiter-Europa rotating frame). 2

3 Moreover, b decoupling the Jovian moon n-bod sstem into several three-bod sstems, we can design an orbit which follows a prescribed itinerar in its visit to Jupiter s man moons. In an earlier stud of a transfer from Ganmede to Europa, 7 we found our transfer V to be half the Hohmann transfer value. As an example, we generated a tour of the Jovian moons: starting beond Ganmede s orbit, the spacecraft is ballisticall captured b Ganmede, orbits it once and escapes, and ends in a ballistic capture at Europa. One advantage of this Petit Grand Tour as compared with the Voager-tpe flbs is the leap-frogging strateg. In this new approach to space mission design, the spacecraft can circle a moon in a loose temporar capture orbit for a desired number of orbits, perform a transfer V and become ballisticall captured b another adjacent moon for some number of orbits, etc. Instead of flbs lasting onl seconds, a scientific spacecraft can orbit several different moons for an desired duration. The design of the Petit Grand Tour in the planar case is guided b two main ideas. First, the Jupiter-Ganmede-Europa-spacecraft four-bod sstem is approximated as two coupled planar three-bod sstems. Then, the invariant manifold tubes of the two planar three-bod sstems are used to construct an orbit with the desired behaviors. This initial solution is then refined to obtain a trajector in a more accurate 4-bod model. See Figure 3. Intersection point Europa s L 2 p.o. stable manifold Spacecraft transfer trajector Europa s L 2 p.o. stable manifold Poincare section V at transfer patch point Jupiter Europa Jupiter Europa Ganmede s L p.o. unstable manifold Ganmede s L p.o. unstable manifold Ganmede Ganmede (a) (b) Figure 3: (a) Find an intersection between dnamical channel enclosed b Ganmede s L periodic orbit unstable manifold and dnamical channel enclosed b Europa s L 2 periodic orbit stable manifold (shown in schematic). (b) Integrate forward and backward from patch point (with V to take into account velocit discontinuit) to generate desired transfer between the moons (schematic). The coupled 3-bod model considers the two adjacent moons competing for control of the same spacecraft as two nested 3-bod sstems (e.g., Jupiter-Ganmede-spacecraft and Jupiter-Europaspacecraft). When close to the orbit of one of the moons, the spacecraft s motion is dominated b the 3-bod dnamics of the corresponding planet-moon sstem. Between the two moons, the spacecraft s motion is mostl planet-centered Keplerian, but is precariousl poised between two competing 3-bod dnamics. In this region, orbits connecting unstable libration point orbits of the two different 3-bod sstems ma exist, leading to complicated transfer dnamics between the two adjacent moons. We seek intersections between invariant manifold tubes which connect the capture regions around each moon. In the planar case, these tubes separate transit orbits (inside the tube) from non-transit orbits (outside the tube). The are the phase space structures that provide a conduit for orbits between regions within each three-bod sstems as well as between primar bodies for separate three-bod sstems. 4 The extension of this planar result to the spatial case is the subject of the current paper. 3

4 z z Extending Results from Planar Model to Spatial Model Previous work based on the planar circular restricted three-bod problem (PCR3BP) revealed the basic structures controlling the dnamics. 4,5,6,7 But future missions will require three-dimensional capabilities, such as control of the latitude and longitude of a spacecraft s escape from and entr into a planet or moon. For example, the proposed Europa Orbiter mission desires a capture into a high inclination polar orbit around Europa. Three-dimensional capabilit is also required when decomposing an n-bod sstem into three-bod sstems that are not co-planar, such as the Earth-Sun-spacecraft and Earth-Moonspacecraft sstems. These demands necessitate the extension of earlier results to the spatial model (CR3BP). See Figure 4. Maneuver performed Jupiter Ganmede's orbit Europa's orbit z.5.5 Close approach to Ganmede (a) -. 5 x.5.5 Injection into high inclination orbit around Europa x x (b) (c) Figure 4: The three dimensional Petit Grand Tour space mission concept for the Jovian moons. (a) We show a spacecraft trajector coming into the Jupiter sstem and transferring from Ganmede to Europa using a single impulsive maneuver, shown in a Jupiter-centered inertial frame. (b) The spacecraft performs one loop around Ganmede, using no propulsion at all, as shown here in the Jupiter-Ganmede rotating frame. (c) The spacecraft arrives in Europa s vicinit at the end of its journe and performs a final propulsion maneuver to get into a high inclination circular orbit around Europa, as shown here in the Jupiter-Europa rotating frame. In our current work on the spatial three-bod problem, we are able to show that the invariant manifold structures of the collinear libration points still act as the separatrices for two tpes of motion, those inside the invariant manifold tubes are transit orbits and those outside the tubes are non-transit orbits. We have also designed an algorithm for constructing orbits with an prescribed itinerar and obtained some initial results on the basic itinerar. Furthermore, we have applied the techniques developed in this paper to the construction of a three dimensional Petit Grand Tour of the Jovian moon sstem. B approximating the dnamics of the Jupiter-Europa-Ganmede-spacecraft 4-bod problem as two 3-bod subproblems, we seek intersections between the channels of tran- 4

5 sit orbits enclosed b the stable and unstable manifold tubes of different moons. In our example, we have designed a low energ transfer trajector from Ganmede to Europa that ends in a high inclination orbit around Europa. CIRCULAR RESTRICTED THREE-BODY PROBLEM The orbital planes of Ganmede and Europa are within.3 of each other, and their orbital eccentricities are.6 and., respectivel. Furthermore, since the masses of both moons are small, and the are on rather distant orbits, the coupled spatial CR3BP is an excellent starting model for illuminating the transfer dnamics between these moons. We assume the orbits of Ganmede and Europa are co-planar, but the spacecraft is not restricted to their common orbital plane. The Spatial Circular Restricted Three Bod Problem. We begin b recalling the equations for the circular restricted three-bod problem (CR3BP). The two main bodies, which we call genericall Jupiter and the moon, have a total mass that is normalized to one. Their masses are denoted b m J =µand m M = µ respectivel (see Figure 5(a)). These bodies rotate in the plane counterclockwise about their common center of mass and with the angular velocit normalized to one. The third bod, which we call the spacecraft, is free to move in the three-dimensional space and its motion is assumed to not affect the primaries. Note that the mass parameters for the Jupiter-Ganmede and Jupiter-Europa sstems are µ G = and µ E = , respectivel. Choose a rotating coordinate sstem so that the origin is at the center of mass and Jupiter (J) and the moon (M) are fixed on the x-axis at (µ,, ) and ( µ,, ) respectivel (see Figure 5(a)). Let (x,, z) be the position of the spacecraft in the rotating frame. Equations of Motion. There are several was to derive the equations of motion for this sstem A efficient technique is to use covariance of the Lagrangian formulation and use the Lagrangian directl in a moving frame. 9 This method gives the equations in Lagrangian form. Then the equations of motion of the spacecraft can be written in second order form as where ẍ 2ẏ =Ω x, ÿ2ẋ=ω, z=ω z. () Ω(x,, z) = x2 2 2 µ r µ r 2 µ( µ), 2 where Ω x, Ω,andΩ z are the partial derivatives of Ω with respect to the variables x,, and z. Also, r = (xµ) 2 2 z 2,r 2 = (xµ) 2 2 z 2. This form of the equations of motion has been studied in detail and are called the equations of the CR3BP. After appling the Legendre transformation to the Lagrangian formulation, one finds that the Hamiltonian function is given b H = (p x ) 2 (p x) 2 p 2 z 2 Therefore, Hamilton s equations are given b: ẋ = H p x = p x, ẏ = H p = p x, ż = H p z = p z, Ω(x,, z), (2) ṗ x = H x = p x Ω x, ṗ = H = p x Ω, ṗ z = H z =Ω z, 5

6 z S/C M L 2 r L r 2 x J µ µ (a) (rotating frame) Forbidden Region Interior Region J L Exterior Region M L 2 NT T T A L 2 B NT (rotating frame) x (rotating frame) (b) Moon Region x (rotating frame) (c) Figure 5: (a) Equilibrium points of the CR3BP as viewed, not in an inertial frame, but in the rotating frame, where Jupiter and a Jovian moon are at fixed positions along the x-axis. (b) Projection of the three-dimensional Hill s region on the (x, )-plane (schematic, the region in white), which contains a neck about L and L 2.(c)Theflowin the region near L 2 projected on the (x, )-plane, showing a bounded orbit around L 2 (labeled B), an asmptotic orbit winding onto the bounded orbit (A), two transit orbits (T) and two non-transit orbits (NT), shown in schematic. A similar figure holds for the region around L. Jacobi Integral. The sstem () have a first integral called the Jacobi integral, which is given b C(x,, z, ẋ, ẏ, ż) =(ẋ 2 ẏ 2 ż 2 )2Ω(x,, z) =2E(x,, z, ẋ, ẏ, ż). We shall use E when we regard the Hamiltonian as a function of the positions and velocities and H when we regard it as a function of the position and mementa. Equilibrium Points and Hill s Regions. The sstem () has five equilibrium points in the (x, ) plane, 3 collinear ones on the x-axis, called L,L 2,L 3 (see Figure 5(a)) and two equilateral points called L 4 andl 5. These equilibrium points are critical points of the (effective potential) function Ω. The value of the Jacobi integral at the point L i will be denoted b C i. The level surfaces of the Jacobi constant, which are also energ surfaces, are invariant 5- dimensional manifolds. Let M be that energ surface, i.e., M(µ, C) ={(x,, z, ẋ, ẏ, ż) C(x,, z, ẋ, ẏ, ż) =constant} 6

7 The projection of this surface onto position space is called a Hill s region M(µ, C) ={(x,, z) Ω(x,, z) C/2}. The boundar of M(µ, C) is the zero velocit curve. The spacecraft can move onl within this region. Our main concern here is the behavior of the orbits of equations () whose Jacobi constant is just below that of L 2 ;thatis,c<c 2. For this case, the three-dimensional Hill s region contains a neck about L and L 2, as shown in Figure 5(b). Thus, orbits with a Jacobi constant just below that of L 2 are energeticall permitted to make a transit through the two neck regions from the interior region (inside the moon s orbit) to the exterior region (outside the moon s orbit) passing through the moon (capture) region. INVARIANT MANIFOLD AS SEPARATRIX Studing the linearization of the dnamics near the equilibria is of course an essential ingredient for understanding the more complete nonlinear dnamics. 4,,2,3 Linearization near the Collinear Equilibria. We will denote b (k,,,,k,) the coordinates of an of the collinear Lagrange point. To find the linearized equations it, we need the quadratic terms of the Hamiltonian H in equation (2) as expanded about (k,,,,k,). After making a coordinate change with (k,,,,k,) as the origin, these quadratic terms form the Hamiltonian function for the linearized equations, which we shall call H l H l = 2 {(p x ) 2 (p x) 2 p 2 z ax2 b 2 cz 2 }, where, a, b and c are defined b a =2ρ,b=ρ, and c = ρ and where ρ = µ k µ 3 (µ) kµ 3. A short computation gives the linearized equations in the form ẋ = H l p x = p x, ṗ x = H l x = p x ax, ẏ = H l p = p x, ṗ = H l = p x b, ż = H l = p z, ṗ z = H l p z z = cz. It is straightforward to show that the eigenvalues of this linear sstem have the form ±λ, ±iν and ±iω, whereλ,νand ω are positive constants and ν ω. To better understand the orbit structure on the phase space, we make a linear change of coordinates with the eigenvectors as the axes of the new sstem. Using the corresponding new coordinates q,p,q 2,p 2,q 3,p 3, the differential equations assume the simple form and the Hamiltonian function becomes q = λq, ṗ = λp, q 2 = νp 2, ṗ 2 = νq 2, q 3 = ωp 3, ṗ 3 = ωq 3, (3) H l = λq p ν 2 (q2 2 p2 2 )ω 2 (q2 3 p2 3 ). (4) 7

8 Solutions of the equations (3) can be convenientl written as q (t) =qe λt, p (t)=p e λt, q 2 (t)ip 2 (t) =(q2p 2)e iνt, q 3 (t)ip 2 (t) =(q3ip 3)e iωt, (5) where the constants q,p,q 2 ip 2,andq 3 ip 3 are the initial conditions. These linearized equations admit integrals in addition to the Hamiltonian function; namel, the functions q p, q2 2 p 2 2 and q3 2 p2 3 are constant along solutions. The Linearized Phase Space. For positive h and c, the region R, which is determined b H l = h, and p q c, is homeomorphic to the product of a 4-sphere and an interval I, S 4 I; namel, for each fixed value of p q between c and c, we see that the equation H l = h determines a 4-sphere λ 4 (q p ) 2 ν 2 (q2 2 p2 2 )ω 2 (q2 3 p2 3 )=hλ 4 (p q ) 2. The bounding 4-sphere of R for which p q = c will be called n, and that where p q = c, n 2 (see Figure 6). We shall call the set of points on each bounding 4-sphere where q p = the equator, and the sets where q p > orq p < will be called the north and south hemispheres, respectivel. q A NT p q =c p q = T q p =h/λ p q =c p A NT p q = p 2 q 2 p 3 q 3 n n T 2 A q p =h/λ A planar oscillations projection vertical oscillations projection saddle projection Figure 6: The flow in the equilibrium region has the form saddle center center. On the left is shown the projection onto the (p,q )-plane (note, axes tilted 45 ). Shown are the bounded orbits (black dot at the center), the asmptotic orbits (labeled A), two transit orbits (T) and two non-transit orbits (NT). The Linear Flow in R. To analze the flow in R, one considers the projections on the (q,p )- plane and (q 2,p 2 ) (q 3,p 3 )-space, respectivel. In the first case we see the standard picture of an unstable critical point, and in the second, of a center consisting of two uncoupled harmonic oscillators. Figure 6 schematicall illustrates the flow. The coordinate axes of the (q,p )-plane have been tilted b 45 and labeled (p,q ) instead in order to correspond to the direction of the 8

9 flow in later figures which adopt the NASA convention that the larger primar is to the left of the small primar. With regard to the first projection we see that R itself projects to a set bounded on two sides b the hperbola q p = h/λ (corresponding to q2 2 p2 2 = q2 3 p2 3 =, see (4)) and on two other sides b the line segments p q = ±c, which correspond to the bounding 4-spheres. Since q p is an integral of the equations in R, the projections of orbits in the (q,p )-plane move on the branches of the corresponding hperbolas q p = constant, except in the case q p =, where q =orp =. Ifq p >, the branches connect the bounding line segments p q = ±c and if q p <, the have both end points on the same segment. A check of equation (5) shows that the orbits move as indicated b the arrows in Figure 6. To interpret Figure 6 as a flow in R, notice that each point in the (q,p )-plane projection corresponds to a 3-sphere S 3 in R given b ν 2 (q2 2 p2 2 )ω 2 (q2 3 p2 3 )=hλq p. Of course, for points on the bounding hperbolic segments (q p = h/λ), the 3-sphere collapses to a point. Thus, the segments of the lines p q = ±c in the projection correspond to the 4-spheres bounding R. This is because each corresponds to a 3-sphere crossed with an interval where the two end 3-spheres are pinched to a point. We distinguish nine classes of orbits grouped into the following four categories:. The point q = p = corresponds to an invariant 3-sphere Sh 3 of bounded orbits (periodic and quasi-periodic) in R. This 3-sphere is given b ν 2 (q2 2 p 2 2) ω 2 (q2 3 p 2 3)=h, q = p =. It is an example of a normall hperbolic invariant manifold (NHIM). 4 Roughl, this means that the stretching and contraction rates under the linearized dnamics transverse to the 3-sphere dominate those tangent to the 3-sphere. This is clear for this example since the dnamics normal to the 3-sphere are described b the exponential contraction and expansion of the saddle point dnamics. Here the 3-sphere acts as a big saddle point. See the black dot at center the (q,p )-plane on the left side of Figure The four half open segments on the axes, q p =, correspond to four clinders of orbits asmptotic to this invariant 3-sphere Sh 3 either as time increases (p = ) or as time decreases (q = ). These are called asmptotic orbits and the form the stable and the unstable manifolds of Sh 3.Thestable manifolds, W s (Sh 3 ), are given b ν 2 (q2 2 p 2 2) ω 2 (q2 3 p 2 3)=h, q =. The unstable manifolds, W s (Sh 3 ), are given b ν 2 (q2 2 p2 2 ) ω 2 (q2 3 p2 3 )=h, p =. Topologicall, both invariant manifolds look like 4-dimensional tubes (S 3 R). See the four orbits labeled A of Figure The hperbolic segments determined b q p = constant > correspond to two clinders of orbits which cross R from one bounding 4-sphere to the other, meeting both in the same hemisphere; the northern hemisphere if the go from p q =cto p q = c, andthe southern hemisphere in the other case. Since these orbits transit from one region to another, we call them transit orbits. See the two orbits labeled T of Figure 6. 9

10 4. Finall the hperbolic segments determined b q p = constant < correspond to two clinders of orbits in R each of which runs from one hemisphere to the other hemisphere on the same bounding 4-sphere. Thus if q >, the 4-sphere is n (p q = c) and orbits run from the southern hemisphere (q p < ) to the northern hemisphere (q p > ) while the converse holds if q <, where the 4-sphere is n 2. Since these orbits return to the same region, we call them non-transit orbits. See the two orbits labeled NT of Figure 6. McGehee Representation. As noted above, R is a 5-dimensional manifold that is homeomorphic to S 4 I. It can be represented b a spherical annulus bounded b two 4-spheres n,n 2,asshown in Figure 7(b). Figure 7(a) is a cross-section of R. Notice that this cross-section is qualitativel ω a r d d 2 a 2 r S r h 3 2 b b 2 r 2 n d a n 2 S 3 h r 2 r 2 d 2 n 2 a 2 b 2 n b r a a 2 d 2 r d 2 a 2 a d d (a) (b) Figure 7: (a) The cross-section of the flow in the R region of the energ surface. (b) The McGehee representation of the flow in the region R. the same as the illustration in Figure 6. The following classifications of orbits correspond to the previous four categories:. There is an invariant 3-sphere Sh 3 of bounded orbits in the region R corresponding to the point l. Notice that this 3-sphere is the equator of the central 4-sphere given b p q =. 2. Again let n,n 2 be the bounding 4-spheres of region R, and let n denote either n or n 2.We can divide n into two hemispheres: n, where the flow enters R, andn, where the flow leaves R. There are four clinders of orbits asmptotic to the invariant 3-sphere Sh 3. The form the stable and unstable manifolds to the invariant 3-sphere Sh 3. Let a and a (where q =and p = respectivel) be the intersections with n of the stable and unstable manifolds. Then a appears as a 3-sphere in n,anda appears as a 3-sphere in n. 3. Consider the two spherical caps on each bounding 4-sphere given b d = {(q,p,q 2,p 2,q 3,p 3 ) p q =c, q < }, d = {(q,p,q 2,p 2,q 3,p 3 ) p q =c, p > }, d 2 = {(q,p,q 2,p 2,q 3,p 3 ) p q =c, q > }, d = {(q,p,q 2,p 2,q 3,p 3 ) p q =c, p < }.

11 If we let d be the spherical cap in n bounded b a, then the transit orbits entering R on d exit on d 2 of the other bounding sphere. Similarl, letting d be the spherical cap in n bounded b a, the transit orbits leaving on d have come from d 2 on the other bounding sphere. 4. Note that the intersection b (where q p =)ofn and n is a 3-sphere of tangenc points. Orbits tangent at this 3-sphere bounce off, i.e., do not enter R locall. Moreover, if we let r be a spherical zone which is bounded b a and b, thennon-transit orbits entering R on r exit on the same bounding 4-sphere through r which is bounded b a and b. Invariant Manifolds as Separatrices. The ke observation here is that the asmptotic orbits form 4-dimensional stable and unstable manifold tubes (S 3 R) to the invariant 3-sphere Sh 3 in a 5-dimensional energ surface and the separate two distinct tpes of motion: transit orbits and non-transit orbits. The transit orbits, passing from one region to another, are those inside the 4- dimensional manifold tube. The non-transit orbits, which bounce back to their region of origin, are those outside the tube. In fact, it can be shown that for a value of Jacobi constant just below that of L (L 2 ), the nonlinear dnamics in the equilibrium region R (R 2 ) is qualitativel the same as the linearized picture that we have shown above. 5,6,7 This geometric insight will be used below to guide our numerical explorations in constructing orbits with prescribed itineraries. Construction of Orbits with Prescribed Itineraries in the Planar Case In previous work on the planar case, 4 a numerical demonstration is given of a heteroclinic connection between pairs of equal Jacobi constant Lapunov orbits, one around L, the other around L 2. This heteroclinic connection augments the homoclinic orbits associated with the L and L 2 Lapunov orbits, which were previousl known. 2 Linking these heteroclinic connections and homoclinic orbits leads to dnamical chains. X Stable Unstable Stable U 3 Unstable U 4 J U 3 J M U Stable U 2 L L 2 Unstable Stable U 2 Unstable Figure 8: Location of Lagrange point orbit invariant manifold tubes in position space. Stable manifolds are lightl shaded, unstable manifolds are darkl. The location of the Poincaré sections (U,U 2,U 3,andU 4 )arealsoshown. We proved the existence of a large class of interesting orbits near a chain which a spacecraft can follow in its rapid transition between the inside and outside of a Jovian moon s orbit via a moon encounter. The global collection of these orbits is called a dnamical channel. We proved a theorem which gives the global orbit structure in the neighborhood of a chain. In simplified form, the theorem essentiall sas:

12 .. For an admissible bi-infinite sequence (...,u ;u,u,u 2,...) of smbols {I,M,X} where I, M, and X stand for the interior, moon, and exterior regions respectivel, there corresponds an orbit near the chain whose past and future whereabouts with respect to these three regions match those of the given sequence. For example, consider the Jupiter-Ganmede-spacecraft 3-bod sstem. Given the bi-infinite sequence (...,I;M,X,M,...), there exists an orbit starting in the Ganmede region which came from the interior region and is going to the exterior region and returning to the Ganmede region. Moreover, we not onl proved the existence of orbits with prescribed itineraries, but develop a sstematic procedure for their numerical construction. We will illustrate below the numerical construction of orbits with prescribed finite (but arbitraril large) itineraries in the three-bod planet-moon-spacecraft problem. As our example, chosen for simplicit of exposition, we construct a spacecraft orbit with the central block (M,X;M,I,M). See Figures 9(a) and 9(b). (Jupiter-Moon rotating frame) Forbidden Region L M Stable Unstable L 2 Manifold Manifold Forbidden Region x (Jupiter-Moon rotating frame) (a) (Jupiter-Moon rotating frame) (;M,I) (X;M) Intersection Region M = (X;M,I) Stable Manifold Cut Unstable Manifold Cut (Jupiter-Moon rotating frame) (b).6 P ( I )=(;M,I,M).5 (M,X,M,I,M) Orbit (Jupiter-Moon rotating frame) (M,X;M,I,M) (Jupiter-Moon rotating frame) Jupiter Moon P( X )=(M,X;M) (Jupiter-Moon rotating frame) (c) x (Jupiter-Moon rotating frame) (d) Figure 9: (a) The projection of invariant manifolds W s,m u,m L,p.o. and WL 2,p.o. in the region M of the position space. (b) A close-up of the intersection region between the Poincaré cuts of the invariant manifolds on the U 3 section (x =µ, > ). (c) Intersection between image of X and pre-image of I labeled (M, X;M, I,M). (d) Example orbit passing through (M, X;M, I,M) region of (c). Example Itinerar: (M,X;M,I,M). For the present numerical construction, we adopt the following convention. The U and U 4 Poincaré sections will be ( =,x <,ẏ<) in the interior 2

13 region, and ( =,x <,ẏ>) in the exterior region, respectivel. The U 2 and U 3 sections will be (x =µ, <, ẋ>) and (x =µ, >, ẋ<) in the moon region, respectivel. See Figure 8 for the location of the Poincaré sections relative to the tubes. A ke observation for the planar case is a result which has shown that the invariant manifold tubes separate two tpes of motion. The orbits inside the tube transit from one region to another; those outside the tubes bounce back to their original region. Since the upper curve in Figure 9(b) is the Poincaré cut of the stable manifold of the periodic orbit around L in the U 3 plane, a point inside that curve is an orbit that goes from the moon region to the interior region, so this region can be described b the label (;M,I). Similarl, a point inside the lower curve of Figure 9(b) came from the exterior region into the moon region, and so has the label (X; M). A point inside the intersection M of both curves is an (X; M,I) orbit, so it makes a transition from the exterior region to the interior region, passing through the moon region. Similarl, b choosing Poincaré sections in the interior and the exterior region, i.e., in the U and U 4 plane, we find the intersection region I consisting of (M; I,M) orbits, and X, which consists of (M; X, M) orbits. Flowing the intersection X forward to the moon region, it stretches into the strips in Figure 9(c). These strips are the image of X (i.e., P ( X )) under the Poincaré mapp, and thus get the label (M,X;M). Similarl, flowing the intersection I backward to the moon region, it stretches into the strips P ( I ) in Figure 9(c), and thus have the label (; M,I,M). The intersection of these two tpes of strips (i.e., M P ( X ) P ( I )) consist of the desired (M,X;M,I,M) orbits. If we take an point inside these intersections and integrate it forward and backward, we find the desired orbits. See Figure 9(d). Extension of Results in Planar Model to Spatial Model Since the ke step in the planar case is to find the intersection region inside the two Poincaré cuts, a ke difficult is to determine how to extend this technique to the spatial case. Take as an example the construction of a transit orbit with the itinerar (X; M,I) that goes from the exterior region to the interior region of the Jupiter-moon sstem. Recall that in the spatial case, the unstable manifold tube of the NHIM around L 2 which separates the transit and non-transit orbits is topologicall S 3 R. For a transversal cut at x =µ(a hperplane through the moon), the Poincaré cut is a topological 3-sphere S 3 (in R 4 ). It is not obvious how to find the intersection region inside these two Poincaré cuts(s 3 ) since both its projections on the (, ẏ)-plane and the (z,ż)-plane are (2-dimensional) disks D 2. (One eas wa to visualize this is to look at the equation: ξ 2 ξ 2 η 2 η 2 =r 2 =rξ 2r2 η. that describes a 3-sphere in R4. Clearl, its projections on the (ξ, ξ)-plane and the (η, η)-plane are 2-disks as r ξ and r η var from to r and from r to respectivel.) However, in constructing an orbit which transitions from the outside to the inside of a moon s orbit, suppose that we might also want it to have other characteristics above and beond this gross behavior. We ma want to have an orbit which has a particular z-amplitude when it is near the moon. If we set z = c, ż = where c is the desired z-amplitude, the problem of finding the intersection region inside two Poincaré cuts suddenl becomes tractable. Now, the projection of the Poincaré cut of the above unstable manifold tube on the (, ẏ)-plane will be a closed curve and an point inside this curve is a (X; M) orbit which has transited from the exterior region to the moon region passing through the L 2 equilibrium region. See Figure. Similarl, we can appl the same techniques to the Poincaré cut of the stable manifold tube to the NHIM around L and find all (M,I) orbits inside a closed curve in the (, ẏ)-plane. Hence, b using z and ż as the additional parameters, we can appl the similar techniques that we have developed for the planar case in constructing spatial trajectories with desired itineraries. See Figures, 2 and 3. What follows is a more detailed description. 3

14 γ z z.. z (z,z ) z Figure : Shown in black are the ẏ (left) and zż (right) projections of the 3-dimensional object C u2, the intersection of W u (M2 h ) with the Poincaré section x =µ. The set of points in the ẏ projection which approximate acurve,γ z ż, all have (z,ż) values within the small box shown in the zż projection (which appears as a thin strip), centered on (z, ż ). This example is computed in the Jupiter-Europa sstem for C = C u C s2. z C s2 C u z Figure : The ẏ (left) and zż (right) projections of the 3-dimensional objects C u2 and C s. This example is computed in the Jupiter-Europa sstem for C = Finding the Poincaré Cuts. We begin with the 5th order normal form expansion near L and L 2. 8,9,2 The behavior of orbits in the coordinate sstem of that normal form, (q,p,q 2,p 2,q 3,p 3 ), are qualitativel similar to the behavior of orbits in the linear approximation. This makes the procedure for choosing initial conditions in the L and L 2 equilibrium regions rather simple. In particular, based on our knowledge of the structure for the linear sstem, we can pick initial conditions which produce a close shadow of the stable and unstable manifold tubes (S 3 R) associated to the normall hperbolic invariant manifold (NHIM), also called central or neutrall stable manifold, in both the L and L 2 equilibrium regions. As we restrict to an energ surface with energ h, there is onl one NHIM per energ surface, denoted M h ( S 3 ). The initial conditions in (q,p,q 2,p 2,q 3,p 3 ) are picked with the qualitative picture of the linear sstem in mind. The coordinates (q,p ) correspond to the saddle projection, (q 2,p 2 ) correspond to oscillations within the (x, ) plane, and (q 3,p 3 ) correspond to oscillations within the z direction. Also note that q 3 = p 3 =(z=ż= ) corresponds to an invariant manifold of the sstem, i.e., the planar sstem is an invariant manifold of the three degree of freedom sstem. 4

15 ...5 γ2 z'z'. Initial condition corresponding to itinerar (X;M,I).5. γ z'z' Figure 2: On the (, ẏ)-plane are shown the points that approximate γz 2 ż and γz ż, the boundaries of int(γz 2 ż ) and int(γz ż ), respectivel, where (z, ż )=(.35, ). Note the lemon shaped region of intersection, int(γz ż ) int(γ 2 z ż ), in which all orbits have the itinerar (X; M,I). The appearance is similar to Figure 9(b). The point shown within int(γz ż ) int(γz 2 ż ) is the initial condition for the orbit shown in Figure 3. The initial conditions to approximate the stable and unstable manifolds (W s ±(M h ),W u ±(M h )) are picked via the following procedure. Note that we can be assured that we are obtaining a roughl complete approximation of points along a slice of W s ±(M h )andw u ±(M h ) since such a slice is compact, having the structure S 3. Also, we know roughl the picture from the linear case.. We fix q = p = ±ɛ, whereɛis small. This ensures that almost all of the initial conditions will be for orbits which are transit orbits from one side of the equilibrium region to the other. Specificall corresponds to right-to-left transit orbits and corresponds to left-to-right transit orbits. We choose ɛ small so that the initial conditions are near the NHIM M h (at q = p = ) and will therefore integrate forward and backward to be near the unstable and stable manifold of M h, respectivel. We choose ɛ to not be too small, or the integrated orbits will take too long to leave the vicinit of M h. 2. Beginning with r v =, and increasing incrementall to some maximum r v = rv max,welook for initial conditions with q3 2 p 2 3 = rv, 2 i.e. along circles in the z oscillation canonical plane. It is reasonable to look along circles centered on the origin (q 3,p 3 )=(,) on this canonical plane since the motion is simple harmonic in the linear case and the origin corresponds to an invariant manifold. 3. For each point along the circle, we look for the point on the energ surface in the (q 2,p 2 ) plane, i.e., the (x, ) oscillation canonical plane. Note, our procedure can tell us if such a point exists and clearl if no point exists, it will not used as an initial condition. After picking the initial conditions in (q,p,q 2,p 2,q 3,p 3 ) coordinates, we transform to the conventional CR3BP coordinates (x,, z, ẋ, ẏ, ż) and integrate under the full equations of motion. The integration proceeds until some Poincaré section stopping condition is reached, for example x =µ. We can then use further analses on the Poincaré section, described below. Example Itinerar: (X; M,I). As an example, suppose we want a transition orbit going from outside to inside the moon s orbit in the Jupiter-moon sstem. We therefore want right-to-left 5

16 z x x z z x Figure 3: The (X, M, I) transit orbit corresponding to the initial condition in Figure 2. The orbit is shown in a 3D view and in the three orthographic projections. Europa is shown to scale. The upper right plot includes the z = section of the zero velocit surface (compare with Figure 5(b)). transit orbits in both the L and L 2 equilibrium regions. Consider the L 2 side. The set of rightto-left transit orbits has the structure D 4 R (where D 4 is a 4-dimensional disk), with boundar S 3 R. The boundar is made up of W s (M2 h )andwu (M2 h ), where the means right-to-left, M 2 h is the NHIM around L 2 with energ h, and 2 denotes L 2. We pick the initial conditions to approximate W(M s 2 h )andwu (M 2 h ) as outlined above and then integrate those initial conditions forward in time until the intersect the Poincaré section at x = µ, a hperplane passing through the center of the moon. Since the Hamiltonian energ h (Jacobi constant) is fixed, the set of all values C = {(, ẏ, z, ż)} obtained at the Poincaré section, characterize the branch of the manifold of all Lagrange point orbits around the selected equilibrium point for the particular section. Let us denote the set as C uj i, where denotes the right-to-left branch of the s (stable) or u (unstable) manifold of the L j, j =,2Lagrange point orbits at the i-th intersection with x =µ. We will look at the first intersection, so we have C u2. The object C u2 is 3-dimensional ( S 3 ) in the 4-dimensional (, ẏ, z, ż) space. For the Jupiter- Europa sstem, we show C u2 for Jacobi constant C =3.28 in Figure. Thus, we suspect that if we pick almost an point (z, ż )inthezżprojection, it corresponds to aclosedloopγ z ż ( S )intheẏprojection (see Figure ). An initial condition (, ẏ,z,ż ), where (, ẏ ) γ z ż will be on W (M u 2 h ), and will wind onto a Lagrange point orbit when integrated backwards in time. Thus, γ z ż defines the boundar of right-to-left transit orbits with (z,ż) = (z,ż ). If we choose (, ẏ ) int(γ z ż ) where int(γ z ż ) is the region in the ẏ projection enclosed 6

17 b γ z ż, then the initial condition (, ẏ,z,ż ) will correspond to a right-to-left transit orbit, which will pass through the L 2 equilibrium region, from the moon region to outside the moon s orbit, when integrated backward in time. Similarl, on the L side, we pick the initial conditions to approximate W(M s h )andwu (M h ) as outlined above and then integrate those initial conditions backward in time until the intersect the Poincaré section at x =µ, obtaining C s. We can do a similar construction regarding transit orbits, etc. To distinguish closed loops γ z ż from L or L 2, let us call a loop γ j z ż if it is from L j,j =,2. To find initial conditions for transition orbits which go from outside the moon s orbit to inside the moon s orbit with respect to Jupiter, i.e. orbits which are right-to-left transit orbits in both the L and L 2 equilibrium regions, we need to look at the intersections of the interiors of C u2 and C s. See Figure. To find such initial conditions we first look for intersections in the zż projection. Consider the projection π zż : R 4 R 2 given b (, ẏ, z, ż) (z,ż). Consider a point (, ẏ,z,ż ) π zż (C u2 ) π zż (C s ), i.e. a point (, ẏ,z,ż )where(z,ż ) is in the intersection of the zż projections of C u2 and C s. Transit orbits from outside to inside the moon s orbit are such that (, ẏ,z,ż ) int(γz ż ) int(γ2 z ż ). If int(γ z ż ) int(γ2 z ż )=, then no transition exists for that value of (z, ż ). But numericall we find that there are values of (z, ż ) such that int(γz ż ) int(γ2 z ż ). See Figures and 2. In essence we are doing a search for transit orbits b looking at a two parameter set of intersections of the interiors of closed curves, γzż and γ2 zż in the ẏ projection, where our two parameters are given b (z,ż). Furthermore, we can reduce this to a one parameter famil of intersections b restricting to ż =. This is a convenient choice since it implies that the orbit is at a critical point (often a maximum or minimum in z when it reaches the surface x =µ.) Technicall, we are not able to look at curves γ j zż belonging to points (z,ż) inthezżprojection. Since we are approximating the 3-dimensional surface C b a scattering of points (about a million for the computations in this paper), we must look not at points (z,ż), but at small boxes (z ±δz, ż±δż) where δz and δż are small. Since our box in the zż projection has a finite size, the points in the ẏ projection corresponding to the points in the box will not all fall on a close curve, but along a slightl broadened curve, a strip, as seen in Figure 2. For our purposes, we will still refer to the collection of such points as γ j zż. Transfer from Ganmede to High Inclination Europa Orbit. Petit Grand Tour. We now appl the techniques we have developed to the construction of a full three dimensional Petit Grand Tour of the Jovian moons, extending an earlier planar result. 7 We here outline how one sstematicall constructs a spacecraft tour which begins beond Ganmede in orbit around Jupiter, makes a close flb of Ganmede, and finall reaches a high inclination orbit around Europa, consuming less fuel than is possible from standard two-bod methods. Our approach involves the following three ke ideas:. treat the Jupiter-Ganmede-Europa-spacecraft 4-bod problem as two coupled circular restricted 3-bod problems, the Jupiter-Ganmede-spacecraft and Jupiter-Europa-spacecraft sstems; 2. use the stable and unstable manifolds of the NHIMs about the Jupiter-Ganmede L and L 2 to find an uncontrolled trajector from a jovicentric orbit beond Ganmede to a temporar capture around Ganmede, which subsequentl leaves Ganmede s vicinit onto a jovicentric orbit interior to Ganmede s orbit; 3. use the stable manifold of the NHIM around the Jupiter-Europa L 2 to find an uncontrolled trajector from a jovicentric orbit between Ganmede and Europa to a temporar capture 7

18 around Europa. Once the spacecraft is temporaril captured around Europa, a propulsion maneuver can be performed when its trajector is close to Europa ( km altitude), taking it into a high inclination orbit about the moon. Furthermore, a propulsion maneuver will be needed when transferring from the Jupiter-Ganmede portion of the trajector to the Jupiter- Europa portion, since the respective transport tubes exist at different energies. Ganmede to Europa Transfer Mechanism. The construction begins with the patch point, where we connect the Jupiter-Ganmede and Jupiter-Europa portions, and works forward and backward in time toward each moon s vicinit. The construction is done mainl in the Jupiter-Europa rotating frame using a Poincaré section. After selecting appropriate energies in each 3-bod sstem, respectivel, the stable and unstable manifolds of each sstem s NHIMs are computed. Let Gan W u (M ) denote the unstable manifold of Ganmede s L NHIM and Eur W s (M2 )denotethe stable manifold for Europa s. L 2 NHIM. We look at the intersection of Gan W(M u )and Eur W(M s 2 ) with a common Poincaré section, the surface U in the Jupiter-Europa rotating frame, defined earlier. See Figure 4. x (Jupiter-Europa rotating frame) Gan γ zż Eur γ2 zż Transfer patch point x (Jupiter-Europa rotating frame) Figure 4: The curves Gan γzż and Eur γzż 2 are shown, the intersections of Gan W u (M )and Eur W s (M2 )with the Poincaré section U in the Jupiter-Europa rotating frame, respectivel. Note the small region of intersection, int(ganγzż ) int(eur γzż 2 ), where the patch point is labeled. Note that we have the freedom to choose where the Poincaré section is with respect to Ganmede, which determines the relative phases of Europa and Ganmede at the patch point. For simplicit, we select the U surface in the Jupiter-Ganmede rotating frame to coincide with the U surface in the Jupiter-Europa rotating frame at the patch point. Figure 4 shows the curves Gan γ zż and Eur γ 2 zż on the (x, ẋ)-plane in the Jupiter-Europa rotating frame for all orbits in the Poincaré section with points (z,ż) within (.6 ±.8, ±.8). The size of this range is about km in z position and 2 m/s in z velocit. From Figure 4, an intersection region on the xẋ-projection is seen. We pick a point within this intersection region, but with two differing velocities; one corresponding to Gan W u (M ), the tube of transit orbits coming from Ganmede, and the other corresponding to Eur W s (M 2 ), the orbits heading toward Europa. The discrepanc between these two velocities is the V necessar for a propulsive maneuver to transfer between the two tubes of transit orbits, which exist at different energies. 8

19 Four-Bod Sstem Approximated b Coupled PCR3BP. In order to determine the transfer V, we compute the transfer trajector in the full 4-bod sstem, taking into account the gravitational attraction of all three massive bodies on the spacecraft. We use the dnamical channel intersection region in the coupled 3-bod model as an initial guess which we adjust finel to obtain a true 4-bod bi-circular model trajector. Figure 4 is the final end-to-end trajector. A V of 24 m/s is required at the location marked. We note that a traditional Hohmann (patched 2-bod) transfer from Ganmede to Europa requires a V of 2822 m/s. Our value is onl 43% of the Hohmann value, which is a substantial savings of on-board fuel. The transfer flight time is about 25 das, well within conceivable mission constraints. This trajector begins on a jovicentric orbit beond Ganmede, performs one loop around Ganmede, achieving a close approach of km above the moon s surface. After the transfer between the two moons, a final additional maneuver of 446 m/s is necessar to enter a high inclination (48.6 ) circular orbit around Europa at an altitude of km. Thus, the total V for the trajector is 66 m/s, still substantiall lower than the Hohmann transfer value. Conclusion and Future Work. In our current work on the spatial three-bod problem, we have shown that the invariant manifold structures of the collinear libration points still act as the separatrices for two tpes of motion, those inside the invariant manifold tubes are transit orbits and those outside the tubes are non-transit orbits. We have also designed a numerical algorithm for constructing orbits with an prescribed finite itinerar in the spatial three-bod planet-moon-spacecraft problem. As our example, we have shown how to construct a spacecraft orbit with the basic itinerar (X; M,I) and it is straightforward to extend these techniques to more complicated itineraries. Furthermore, we have applied the techniques developed in this paper towards the construction of a three dimensional Petit Grand Tour of the Jovian moon sstem. Fortunatel, the delicate dnamics of the Jupiter-Europa-Ganmede-spacecraft 4-bod problem are well approximated b considering it as two 3-bod subproblems. One can seek intersections between the channels of transit orbits enclosed b the stable and unstable manifold tubes of the NHIM of different moons using the method of Poincaré sections. With maneuvers sizes ( V ) much smaller than that necessar for Hohmann transfers, transfers between moons are possible. In addition, the three dimensional details of the encounter of each moon can be controlled. In our example, we designed a trajector that ends in a high inclination orbit around Europa. In the future, we would like to explore the possibilit of injecting into orbits of all inclinations. Acknowledgments. We would like to thank Steve Wiggins, Laurent Wiesenfeld, Charles Jaffé, T. Uzer and Luz Vela-Arevalo for their discussions. This work was carried out in part at the Jet Propulsion Laborator and the California Institute of Technolog under a contract with the National Aeronautics and Space Administration. In addition, the work was partiall supported b the NSF grant KDI/ATM , JPL s LTool project, AFOSR Microsat contract F and Catalan grant 2SGR-27. References:. Howell, K., B. Barden and M. Lo, Application of Dnamical Sstems Theor to Trajector Design for a Libration Point Mission, Journal of the Astronautical Sciences, 45, No. 2, April- June 997, pp Gómez, G., J. Masdemont and C. Simó, Stud of the Transfer from the Earth to a Halo Orbit around the Equilibrium Point L, Celestial Mechanics and Dnamical Astronom 56 (993) and 95 (997),