Invariant Manifolds, Spatial 3-Body Problem and Space Mission Design

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1 Invariant Manifolds, Spatial 3-Bod Problem and Space Mission Design Góme, Koon, Lo, Marsden, Masdemont and Ross Wang Sang Koon Control and Dnamical Sstems, Caltech

2 Acknowledgements H Poincaré, J Moser C Conle, R McGehee, D Appleard B Farquhar, D Dunham C Simó, J Llibre, R Martine E Belbruno, B Marsden, J Miller K Howell, B Barden S Wiggins, L Wiesenfeld, C Jaffé, T Uer, L Vela-Arevalo

3 Petit Grand Tour of s Moons (Planar Model) Used invariant manifolds to construct trajectories with interesting characteristics: Petit Grand Tour of s moons orbit around Ganmede 4 orbits around Europa A V nudges the SC from -Ganmede sstem to -Europa sstem Instead of flbs, can orbit several moons for an duration L Ganmede L 2 Transfer orbit V Europa s orbit Ganmede s orbit Europa L 2

4 Etend from Planar Model to Spatial Model Previous work based on planar 3-bod problem Future missions will require 3D capabilities Europa Orbiter mission needs a capture into a high inclination orbit around Europa Current stud has etended from planar to spatial model Close approach to Ganmede Injection into high inclination orbit around Europa

5 Etend from Planar Model to Spatial Model Ganmede's orbit Europa's orbit Close approach to Ganmede (a) Injection into high inclination orbit around Europa (b) (c)

6 Petit Grand Tour of s Moons -Ganmede-Europa-SC 4-bod sstem approimated as 2 coupled 3-bod sstems Invariant manifold tubes of spatial 3-bod sstems are linked in right order to construct orbit with desired itinerar Initial solution refined in 4-bod model Spacecraft transfer trajector V at transfer patch point Europa Europa Ganmede Ganmede

7 Planar Restricted 3-Bod Problem Recall: for energ value just above that of L 2, Hill s region contains a neck about L & L 2 Dnamics in each equilibrium region: saddle center 4 tpes of orbits: periodic, asmptotic, transit & nontransit Forbidden Region Eterior Region (rotating frame) Interior Region J L M L 2 L 2 (rotating frame) (rotating frame) Moon Region (rotating frame)

8 Planar: Invariant Manifold as Separatri Asmptotic orbits form 2D invariant manifold tubes in 3D energ surface The 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 (rotating frame) L Periodic Orbit Stable Manifold (rotating frame) Forbidden Region Unstable Manifold Moon

9 Spatial Restricted 3-Bod Problem (CR3BP) Dnamics near equilibrium point: saddle center center bounded orbits (periodic/quasi-periodic): S 3 (3-sphere) asmptotic orbits to 3-sphere: S 3 I ( tubes ) transit and nontransit orbits Forbidden Region Eterior Region (rotating frame) Interior Region J L M L 2 L 2 (rotating frame) (rotating frame) Moon Region (rotating frame)

10 Spatial: Invariant Manifold as Separatri Asmptotic orbits form 4D invariant manifold tubes (S 3 I) in 5D energ surface The 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 (rotating frame) L Periodic Orbit Stable Manifold (rotating frame) Forbidden Region Unstable Manifold Moon

11 Construct Orbit with Desired Itinerar How to link invariant manifold tubes to construct orbit with desired itinerar Construction of (X; M,I) orbit X U 3 U 4 U I U 3 M U 2 L L 2 M U 2

12 Planer: Construction of (X; M, I) Orbits Invariant mfd tubes (S I) separate transit/nontransit orbits Red curve (S ) (Poincaré cutofl 2 unstable manifold) Green curve (S ) (cut of L stable manifold) An point inside the intersection region M is a (X; M,I) orbit (-Moon rotating frame) Forbidden Region L M Stable Unstable L 2 Manifold Manifold Forbidden Region (-Moon rotating frame) (-Moon rotating frame) (;M,I) (X;M) Intersection Region M = (X;M,I) Stable Manifold Cut Unstable Manifold Cut (-Moon rotating frame)

13 Spatial: Construction of (X; M,I) orbits Invariant manifold tubes: (S 3 I) Poincaré cut 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 If =c, ż =,itsprojection on (, ẏ) plane is a curve An point inside this curve is a (X; M) orbit γ 4 2 (, )

14 Spatial: Construction of (X; M, I) orbits Similarl, while cut of stable manifold tube is a S 3, its projection on (, ẏ) plane is a curve for = c, ż = An point inside this curve is a (M,I) orbit Hence, an point inside the intersection region M is a (X; M,I) orbit C +u C +s C +s2 C +u γ2 '' γ '' Initial condition corresponding to itinerar (X;M,I)

15 Spatial: Construction of (X; M, I) orbits

16 Petit Grand Tour of s Moon Petit Grand Tour can be constructed similarl Approimate 4-bod sstem as 2 nested 3-bod sstems Choose appropriate Poinaré section Link invariant manifold tubes in right order Refine initial solution in 4-bod model Spacecraft transfer trajector V at transfer patch point Europa (-Europa rotating frame) Gan γ ż Eur γ2 ż Transfer patch point Ganmede (-Europa rotating frame)

17 Conclusion and Future Work In our stud of spatial CR3BP: Invariant manifolds still act as separtrices Construct orbit with prescribed itinerar Construct Petit Grand Tour of s moons that ends in a high inclination orbit around Europa To construct useful trajectories in Sun-Earth-Moon sstem Close approach to Ganmede Injection into high inclination orbit around Europa