Design of Low Energy Space Missions using Dynamical Systems Theory

Transcription

1 Design of Low Energy Space Missions using Dynamical Systems Theory Koon, Lo, Marsden, and Ross W.S. Koon (Caltech) and S.D. Ross (USC) CIMMS Workshop, October 7, 24

2 Acknowledgements H. Poincaré, J. Moser C. Conley, R. McGehee, D. Appleyard C. Simó, J. Llibre, R. Martinez B. Farquhar, D. Dunham E. Belbruno, B. Marsden, J. Miller K. Howell, B. Barden, R. Wilson S. Wiggins, V. Rom-Kedar, F. Lekien

3 Outline Main Theme how to use dynamical systems theory of 3-body problem in low energy trajectory design. Background and Motivation: NASA s Genesis Discovery Mission. A Low Energy Tour of Jupiter s Moons. Restricted 3-Body Problem. Main Results. Ongoing Work. Low Thrust Trajectories in a Multi-Body Environment. Parking a Satellite near an Asteroid Pair.

4 Motivation: Genesis Discovery Mission Genesis spacecraft collected solar wind sample from a L 1 halo orbit, returned them to Earth. Halo orbit, transfer/ return trajectories in rotating frame. 1E+6 5 y (km) L 1 L E+6-1E+6 1E+6 x (km)

5 Motivation: Genesis Discovery Mission Designed using dynamical systems theory (Barden, Howell, and Lo). Followed natural dynamics, little propulsion after launch. Return-to-Earth portion utilized heteroclinic dynamics. 1E+6 5 y (km) L 1 L E+6-1E+6 1E+6 x (km)

6 Motivation: Petit Grand Tour of Jupiter s Moons Construct a low energy trajectory to visit several moons in one mission. Instead of flybys, can orbit each moon for any duration. NASA is considering a Jupiter Icy Moon Orbiter (JIMO).

7 Ganymede Ganymede Design Strategy: Patched 3-Body Solutions Jupiter-Ganymede-Europa-SC 4-body system approximated as 2 coupled 3-body systems 3-body solutions of each 3-body systems are linked in right order to construct orbit with desired itinerary. Try to minimize V at each transfer patch point. Initial solution refined in 4-body model. 3-body solutions offer a large class of low energy trajectories. Spacecraft transfer trajectory V at transfer patch point Jupiter Europa Jupiter Europa

8 Planar Circular Restricted 3-Body Problem 2 main bodies Total mass normalized to 1: m J = µ, m S = 1 µ. Rotate about center of mass, angular velocity normalized to 1. Choose rotating coordinate system with origin at center of mass, 2 main bodies fixed at ( µ, ) and (1 µ, ). y (nondimensional units, rotating frame) L 3 S L 1 J m S = 1 - µ m J = µ L 4 L 5 comet L 2 Jupiter's orbit x (nondimensional units, rotating frame)

9 Equilibrium Points Equations of motion for SC are ẍ 2ẏ = U x, where U(x, y) = x2 +y µ r s µ r j. Five equilibrium points: ÿ + 2ẋ = U y, 3 unstable collinear equilbrium points, L 1, L 2, L 3. 2 equilateral equilibrium points, L 4, L 5. y (nondimensional units, rotating frame) L 3 S L 1 J m S = 1 - µ m J = µ L 4 L 5 comet L 2 Jupiter's orbit x (nondimensional units, rotating frame)

10 Hill s Realm Energy integral: E(x, y, ẋ, ẏ) = (ẋ 2 + ẏ 2 )/2 + U(x, y). E can be used to determine (Hill s ) realm in position space where SC is energetically permitted to move. Effective potential: U(x, y) = x2 +y µ r s µ r j.

11 Hill s Realm To fix energy value E is to fix height of plot of U(x, y). Contour plots give 5 cases of Hill s realm. Case 1 : C>C 1 Case 2 : C 1 >C>C S J S J Case 3 : C 2 >C>C 3 Case 4 : C 3 >C>C 4 =C S J S J

12 The Flow near L 1 and L 2 For energy value just above that of L 2, Hill s realm contains a neck about L 1 & L 2. SC can make transition through these equilibrium realms. 4 types of orbits: periodic, asymptotic, transit & nontransit. Forbidden Region Exterior Region y (rotating frame) Interior Region J L 1 M L 2 L 2 y (rotating frame) x (rotating frame) Moon Region x (rotating frame)

13 Invariant Manifold as Separatrix Asymptotic orbits form 2D invariant manifold tubes in 3D energy surface. They separate transit and non-transit orbits: Transit orbits are those inside the tubes. Non-transit orbits are those outside the tubes..6.4 Unstable Manifold Forbidden Region Stable Manifold Jupiter y (rotating frame) L 1 Periodic Orbit Stable Manifold x (rotating frame) Forbidden Region Unstable Manifold Moon

14 Invariant Manifold as Separatrix Invariant Manifold Tubes associated with periodic orbits about L 1, L 2 control ballistic capture and escape. P Moon Earth L 2 orbit Moon Ballistic Capture Into Elliptical Orbit L 2

15 Heteroclinic Connection Found heteroclinic connection between pair of periodic orbits. Found a large class of orbits near this (homo/heteroclinic) chain. SC can follow these channels in rapid transition.! '! "$#% &.8 y (AU, Sun-Jupiter rotating frame) !! "$#% & L1 L 2 L 1 L 2 (&)( #!! * + +(,&-! y (AU, Sun-Jupiter rotating frame) x (AU, Sun-Jupiter rotating frame) x (AU, Sun-Jupiter rotating frame)

16 Existence of Transitional Orbits Main Theorem: For any admissible itinerary, e.g., (..., X, J; S, J, X,...), there exists an orbit whose whereabouts matches this itinerary. Can even specify number of revolutions the comet makes around Sun & Jupiter (plus L 1 & L 2 ). 3-Body trajectories much richer than 2-body trajectories.

17 Numerical Construction of Orbits Developed procedure to construct orbit with prescribed itinerary. Example: An orbit with itinerary (X, J; S, J, X). O PRQ S7Q TUQ S7Q PWV X :@?76 8 ;=<>:@?76 ABAC9D EF9GB6<>D / + *) (! $ %# &' # $ "! IBJ L 1 L : -. / ;=<>:@?76 ABAC9DEF9GB6<HD, -,., /, K>L CLMN / + *) (! $ %# &' # $ "!

18 Construction of (M, X; M, I, M) Orbits Invariant mfd. tubes (S I) separate transit/nontransit orbits. Red curve (S 1 ) (Poincaré cut of L 2 unstable manifold). Green curve (S 1 ) (cut of L 1 stable manifold). Any point inside the intersection region M is a (X; M, I) orbit. y (Jupiter-Moon rotating frame) Forbidden Region L 1 M Stable Unstable L 2 Manifold Manifold Forbidden Region x (Jupiter-Moon rotating frame) otating frame) y ( (;M,I) (X;M) Intersection Region M = (X;M,I) Stable Manifold Cut Unstable Manifold Cut y ( otating frame)

19 Construction of (M,X;M,I,M) Orbits The desired orbit can be constructed by Choosing appropriate Poincaré sections and linking invariant manifold tubes in right order. X U 3 U 4 U 1 I U 3 M U 2 L 1 L 2 M U 2

20 Petit Grand Tour of Jupiter s Moon Petit Grand Tour can be constructed similarly Approximate 4-body system as 2 nested 3-body systems. Choose appropriate Poinaré section. Link invariant manifold tubes in right order. Refine initial solution in 4-body model. Spacecraft transfer trajectory V at transfer patch point Jupiter Europa opa rotating frame) Gan γ1 zż Eur γ2 zż Transfer patch point. x Ganymede x opa rotating frame)

21 Petit Grand Tour of Jupiter s Moons (Planar Model) Used invariant manifolds to construct trajectories with interesting characteristics: Petit Grand Tour of Jupiter s moons. 1 orbit around Ganymede. 4 orbits around Europa. A V nudges the SC from Jupiter-Ganymede system to Jupiter-Europa system. Instead of flybys, can orbit several moons for any duration. Ganymede Transfer orbit V Jupiter Ganymede s orbit Europa Europa s orbit Jupiter Jupiter

22 Extend from Planar Model to Spatial Model Jupiter Ganymede's orbit Europa's orbit z Close approach to Ganymede y (a) -. 5 x Injection into high inclination orbit around Europa z -.5 z y x x y.5.1 (b) (c)

23 Look for Natural Pathways to Bridge the Gap Tubes of two 3-body systems may not intersect for awhile. May need large V to jump from one tube to another. Look for natural pathways to bridge the gap between z where tube of one system (Ganymede) enters and z 2 where tube of another system exits (into Europa realm) by hopping through phase space (z 1 ). Spacecraft transfer trajectory V at transfer patch point f 2 z 5 U 2 z f 2 4 Entrance z 3 Jupiter Europa f 12 U 1 f 1 f 1 z (a) Ganymede (b) z 2 Exit z 1

24 Transport in Phase Space via Tube & Lobe Dynamics By using tubes of rapid transition that connect realms lobe dynamics to hop through phase space, New tour only needs V = 2m/s (5 times less). Low Energy Tour of Jupiter s Moons Seen in Jovicentric Inertial Frame z 5 U 2 f 2 z 4 Entrance f 2 z 3 f 12 U 1 (a) z 2 Exit f 1 f 1 z z 1 Callisto Ganymede Europa (b) Jupiter

25 Lobe Dynamics: Mixed Phase Space Poincaré section reveals mixed phase space: resonance regions and chaotic sea. Identify Poincare section spacecraft Forbidden Realm (at a particular energy level) Interior (Jupiter) Realm Exterior Realm L 1 L 2 Semimajor Axis (Neptune = 1) (a) Moon Realm Argument of Periapse (radians) (b)

26 Transport between Regions via Lobe Dynamics Invariant manifolds divide phase space into resonance regions. Transport between regions can be studied via lobe dynamics. Identify Semimajor Axis Argument of Periapse L 2,1 (1) R 2 q f (L 1,2 (1)) f -1 (q ) q 1 p i L 1,2 (1) f (L 2,1 (1)) p j R 1

27 Transport between Regions via Lobe Dynamics Segments of unstable and stable manifolds form partial barriers between regions R 1 and R 2. L 1,2 (1), L 2,1 (1) are lobes; they form a turnstile. In one iteration, only points from R 1 to R 2 are in L 1,2 (1) only points from R 2 to R 1 are in L 2,1 (1). By studying pre-images of L 1,2 (1), one can find efficient way from R 1 to R 2. L 2,1 (1) R 2 q f (L 1,2 (1)) f -1 (q ) q 1 p i L 1,2 (1) f (L 2,1 (1)) p j R 1

28 Hopping through Resonaces in Low Energy Tour Guided by lobe dynamics, hopping through resonances (essential for low energy tour) can be performed. To get SC captured by secondary (m 2 ), need to decrease semi-major axis passing through resonances..8 P.75.7 m 1 m Surface-of-section Large orbit changes

29 Tube/Lobe Dynamics: Transport in Solar System To use tube dynamics/lobe dynamics of spatial 3-body problem to systematically design low-fuel trajectory. Part of our program to study transport in solar system using tube and lobe dynamics. Low Energy Tour of Jupiter s Moons Seen in Jovicentric Inertial Frame z 5 U 2 f 2 z 4 Entrance f 2 z 3 f 12 U 1 (a) z 2 Exit f 1 f 1 z z 1 Callisto Ganymede Europa (b) Jupiter

30 Low Thrust Trajectories in a Multi-Body Environment Incorporation of low thrust. Design to take best advantage of natural dynamics. See Shane D. Ross. Spiral out from Europa Europa to Io transfer

31 Parking a Satellite near an Asteroid Pair Find stable periodic/quasi-periodic orbits for SC to observe binary as it orbits the Sun. Model for asteroid pair: sphere and rigid body (3 connected masses). Model for SC motion: binary in relative equilibrium. See Gabern, Koon, and Marsden [24]