C C Dynamical A L T E C S H Connecting orbits and invariant manifolds in the spatial three-body problem Shane D. Ross Control and Dynamical Systems, Caltech Work with G. Gómez, W. Koon, M. Lo, J. Marsden, J. Masdemont Special Session on Celestial Mechanics, January 7, 24
Introduction Goal Use dynamical systems techniques to identify key transport mechanisms and useful orbits for space missions. Outline Circular restricted three-body problem Equilibrium points and invariant manifold structures Construction of trajectories with prescribed itineraries Connecting orbits, e.g., heteroclinic connections Tours of Jupiter s moons 2
Introduction Current research importance Development of some NASA mission trajectories, such as the recently launched Genesis Discovery Mission, and the upcoming Jupiter Icy Moons Orbiter Of current astrophysical interest for understanding the transport of solar system material (eg, how ejecta from Mars gets to Earth, etc.) 3
Three-Body Problem Circular restricted 3-body problem the two primary bodies move in circles; the much smaller third body moves in the gravitational field of the primaries, without affecting them the two primaries could be Jupiter and a moon the smaller body could be a spacecraft or asteroid we consider the planar and spatial problems there are five equilibrium points in the rotating frame, places of balance which generate interesting dynamics 4
Three-Body Problem Circular restricted 3-body problem Consider the two unstable points on line joining the two main bodies L 1, L 2 z S/C M L 2 r L 1 1 r 2 x y J 1 µ µ Equilibrium points L 1, L 2 5
Three-Body Problem orbits exist around L 1 and L 2 ; both periodic and quasiperiodic Lyapunov, halo and Lissajous orbits one can draw the invariant manifolds assoicated to L 1 (and L 2 ) and the orbits surrounding them these invariant manifolds play a key role in what follows 6
Three-Body Problem Equations of motion (planar): where ẍ 2ẏ = Ūx, ÿ + 2ẋ = Ūy Ū = (x2 + y 2 ) 1 µ µ. 2 r 1 r 2 Have a first integral, the Hamiltonian energy, given by E(x, y, ẋ, ẏ) = 1 2 (ẋ2 + ẏ 2 ) + Ū(x, y). Energy manifolds are codimension 1 in the phase space. 7
Realms of Possible Motion Effective potential In a rotating frame, the equations of motion describe a particle moving in an effective potential plus a magnetic field (goes back to work of Jacobi, Hill, etc). 8
Realms of Possible Motion _ U(x,y) L 3 L 4 Forbidden Realm Interior Realm Exterior Realm L 5 L 1 J L 1 M L 2 L 2 Effective potential Moon Realm Level set shows accessible realms 9
Motion Near Equilibria For saddles of rank 1 Near equilibrium point, suppose linearized Hamiltonian vector field has eigenvalues ±iω j, j = 1,..., N 1, and ±λ. Assume the complexification is diagonalizable. Hamiltonian normal form theory tranforms Hamiltonian into a lowest order form: H(q, p) = N 1 i=1 ω i 2 ( p 2 i + q 2 i ) + λqn p N. Equilibrium point is of type center center saddle (N 1 centers). 1
Motion Near Equilibria Multidimensional saddle point For fixed energy H = h, energy surface S 2N 2 R. Constants of motion: I j = q 2 j + p2 j, j = 1,..., N 1, and I N = q N p N. p p N-1 1 q 1 q N-1 X X X p N q N The N Canonical Planes 11
Motion Near Equilibria Normally hyperbolic invariant manifold at q N = p N =, M h = n 1 i=1 ω i 2 ( p 2 i + q 2 i ) = h >. Note that M h S 2N 3, not a single trajectory. Four cylinders of asymptotic orbits: the stable and unstable manifolds W s ±(M h ), W u ±(M h ), which have the structure S 2N 3 R. 12
Flow Near Equilibria Dynamics near L 1 & L 2 in spatial problem: saddle center center. Hamiltonian for linearized equations has eigenvalues ±λ, ±iν, and ±iω, where ν ω, Change of coordinates yields H 2 = λq 1 p 1 + ν 2 (q2 2 + p 2 2) + ω 2 (q2 3 + p 2 3). For fixed energy H = h, energy surface S 4 R. Constants of motion: q 1 p 1, q 2 2 + p 2 2 and q 2 3 + p 2 3. 13
Flow Near Equilibria Normally hyperbolic invariant manifold at q 1 = p 1 =, M h = ν 2 (q2 2 + p 2 2) + ω 2 (q2 3 + p 2 3) = h >. Note that M h S 3, not a single trajectory. Four cylinders of asymptotic orbits: the stable and unstable manifolds W s ±(M h ), W u ±(M h ), which have the structure S 3 R. 14
Flow Near Equilibria B : bounded orbits (periodic/quasi-periodic): S 3 (3-sphere) A : asymptotic orbits to 3-sphere: S 3 I ( tubes ) T : transit and NT : non-transit orbits. q 1 A NT p 1 q 1 = c p 1 q 1 = T q 1 p 1 =h/λ p 1 q 1 =+c p 1 A NT p 1 +q 1 = p 2 q 2 p 3 q 3 n 1 n T 2 A q 1 p 1 =h/λ A planar oscillations projection vertical oscillations projection saddle projection The flow in the equilibrium region. 15
Flow Near Equilibria B : bounded orbits (periodic/quasi-periodic): S 3 (3-sphere) A : asymptotic orbits to 3-sphere: S 3 I ( tubes ) T : transit and NT : non-transit orbits. Forbidden Realm Interior Realm Exterior Realm T y J L 1 M L 2 NT A L 2 B NT T x Moon Realm Projection to configuration space. 16
Transport Between Realms Asymptotic orbits form 4D invariant manifold tubes (S 3 I) in 5D energy surface. red = unstable, green = stable z.1.1.1.82.84.86.88.9 x.92.94.96.98 1 1.2.2.4.2.8.6 y 17
Transport Between Realms These manifold tubes play an important role in governing what orbits approach or depart from a moon (transit orbits) and orbits which do not (non-transit orbits) transit possible for objects inside the tube, otherwise no transit this is important for transport issues 18
Transport Between Realms Jupiter y (rotating frame) Europa L 2 Spacecraft Begins Inside Tube Passes through L2 Equilibrium Region Ends in Capture Around Moon x (rotating frame) 19
Transport Between Realms Transit orbits can be found using a Poincaré section transversal to a tube. Poincare Section Tube 2
Construction of Trajectories One can systematically construct new trajectories, which use little fuel. by linking stable and unstable manifold tubes in the right order and using Poincaré sections to find trajectories inside the tubes One can construct trajectories involving multiple 3-body systems. 21
Construction of Trajectories For a single 3-body system, we wish to link invariant manifold tubes to construct an orbit with a desired itinerary Construction of (X; M, I) orbit. X U 3 U 4 U 1 I U 3 M U 2 L 1 L 2 M U 2 The tubes connecting the X,M, and I regions. 22
Construction of Trajectories First, integrate two tubes until they pierce a common Poincaré section transversal to both tubes. Second, pick a point in the region of intersection and integrate it forward and backward. 23
Construction of Trajectories Integrate two tubes Integrate a point in the region of intersection Poincare Section Tube B Tube A 24
Construction of Trajectories Planar: tubes (S I) separate transit/non-transit orbits. Red curve (S 1 ) : slice of L 2 unstable manifold Green curve (S 1 ) : slice of L 1 stable manifold Any point inside the intersection region M is a (X; M, I) orbit. y (Jupiter-Moon rotating frame).8.6.4.2 -.2 -.4 -.6 -.8 Forbidden Region L 1 M Stable Unstable L 2 Manifold Manifold Forbidden Region.92.94.96.98 1 1.2 1.4 1.6 1.8 x (Jupiter-Moon rotating frame) Tubes intersect in position y (Jupiter-Moon rotating frame)..6.4.2 -.2 -.4 -.6 (;M,I) (X;M) Intersection Region M = (X;M,I) Stable Manifold Cut Unstable Manifold Cut.1.15.2.25.3.35.4.45.5 y (Jupiter-Moon rotating frame) Poincaré section of intersection 25
Construction of Trajectories Spatial: Invariant manifold tubes (S 3 I) Poincaré slice is a topological 3-sphere S 3 in R 4. S 3 looks like disk disk: ξ 2 + ξ 2 + η 2 + η 2 = r 2 = r 2 ξ + r2 η Find (X; M) orbit..6.6. y.4.2 γ z z.. z.4.2. (z,z ) -.2 -.2 -.4 -.4 -.6.5.1 y.15 -.6 -.5.5 z (y, ẏ) Plane (z, ż) Plane 26
Construction of Trajectories Similarly, while the cut of the stable manifold tube is S 3, its projection on (y, ẏ) plane is a curve for z = c, ż =. Any point inside this curve is a (M, I) orbit. Hence, any point inside the intersection region M (X; M, I) orbit. is a 27
Construction of Trajectories.6.4.2 C +u1 1.6.4.2 C +s2 1 C +u1 1 y. -.2 -.4 C +s2 1 z. -.2 -.4 -.6.5.1 y.15 -.6 -.5.5 z. y.1.5 γ2 z'z'. Initial condition corresponding to itinerary (X;M,I).5.1 γ1 z'z'..4.6.8.1.12.14 y Intersection Region 28
Construction of Trajectories.4.4.2.2 z -.2 y -.4.4.2 y -.2 -.4 1.4.98 1 1.2.96 x -.2 -.4.96.98 1 1.2 1.4 x.4.5.2 z z -.5 -.2 -.1 -.4.96.98 1 1.2 1.4 x -.5.5.1.15.2 y Construction of an (X, M, I) orbit 29
Connecting Orbits.6.4 y.2.2.4.99.995 1 1.5 1.1 x.4.4.2.2 z z.2.2.4.4.99.995 1 1.5 1.1.4.2.2.4 x y An L 1 -L 2 heteroclinic connection.6 3
Tours of Jupiter s Moons Tours of planetary satellite systems. Example 1: Europa Io Jupiter 31
Tours of Jupiter s Moons Example 2: Ganymede Europa injection into Europa orbit Maneuver performed Jupiter Ganymede's orbit Europa's orbit.2.1 z 1.5 1.5 Close approach to Ganymede y -. 5-1 -1. 5-1. 5-1 (a) -. 5 x.5 1 1.5 Injection into high inclination orbit around Europa.15.1.5.5 z -.5 z -.1 -.15 -.5 -.2.2.1 y -.1 -.2.98.99 1 x 1.1 1.2 -.1.99.995 x 1 1.5 1.1 -.1 -.5 y.5.1 (b) (c) 32
Tours of Jupiter s Moons The Petit Grand Tour can be constructed as follows: Approximate 4-body system as 2 nested 3-body systems. Choose an appropriate Poinaré section. Link the invariant manifold tubes in the proper order. Integrate initial condition (patch point) in the 4-body model. Spacecraft transfer trajectory V at transfer patch point Jupiter Europa opa rotating frame).12.1.8.6.4.2 -.2 -.4 Gan γ1 zż Eur γ2 zż Transfer patch point. x -.6 -.8 Ganymede Look for intersection of tubes -1. 5-1. 4-1. 3-1. 2-1. 1-1 x opa rotating frame) Poincaré section at intersection 33
Some References Gómez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [21] Connecting orbits and invariant manifolds in the spatial three-body problem. submitted to Nonlinearity. Gómez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [21] Invariant manifolds, the spatial three-body problem and space mission design. AAS/AIAA Astrodynamics Specialist Conference. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2] Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 1(2), 427 469. The End 34