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 koon@cdscaltechedu
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
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
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 2 5 5 5-5 - -5-5 -2 2 - -2 98 99 2-99 995 5 - -5 5
Etend from Planar Model to Spatial Model Ganmede's orbit Europa's orbit 2 5 5 5 Close approach to Ganmede - 5 - - 5-5 - (a) - 5 5 5 Injection into high inclination orbit around Europa 5 5-5 - -5-5 -2 2 - -2 98 99 2-99 995 5 - -5 5 (b) (c)
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
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)
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) 2-2 -4-6 L Periodic Orbit Stable Manifold 88 9 92 94 96 98 (rotating frame) Forbidden Region Unstable Manifold Moon
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)
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) 2-2 -4-6 L Periodic Orbit Stable Manifold 88 9 92 94 96 98 (rotating frame) Forbidden Region Unstable Manifold Moon
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
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) 8 6 4 2-2 -4-6 -8 Forbidden Region L M Stable Unstable L 2 Manifold Manifold Forbidden Region 92 94 96 98 2 4 6 8 (-Moon rotating frame) (-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 5 2 25 3 35 4 45 5 (-Moon rotating frame)
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 6 6 4 2 γ 4 2 (, ) -2-2 -4-4 -6 5 5-6 -5 5
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 6 4 2-2 -4-6 C +u C +s2 5 5 6 4 2-2 -4-6 C +s2 C +u -5 5 5 5 γ2 '' γ '' Initial condition corresponding to itinerar (X;M,I) 4 6 8 2 4
Spatial: Construction of (X; M, I) orbits 4 4 2 2-2 -4 4 2-2 -4 4 98 2 96-2 -4 96 98 2 4 4 5 2-5 -2 - -4 96 98 2 4-5 5 5 2
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) 2 8 6 4 2-2 -4-6 -8 Gan γ ż Eur γ2 ż Transfer patch point Ganmede - 5-4 - 3-2 - - (-Europa rotating frame)
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 2 5 5 5-5 - -5-5 -2 2 - -2 98 99 2-99 995 5 - -5 5