Dynamical Systems and Space Mission Design

Transcription

1 Dynamical Systems and Space Mission Design Jerrold Marsden, Wang Koon and Martin Lo Wang Sang Koon Control and Dynamical Systems, Caltech

2 The Flow near L and L 2 : Outline Outline of Lecture 2B: Equilibrium Regions near L and L 2 Four Types of Orbits. Invariant Manifold as Separatrix. Flow Mappings in Equilibrium Region. y (nondimensional units, rotating frame) forbidden interior S Zero velocity curve bounding forbidden L exterior J L 2 Jupiter L y (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)

3 Jupiter Comets Rapid transition from outside to inside Jupiter s orbit. Captured temporarily by Jupiter during transition. Exterior (2:3 resonance). Interior (3:2 resonance). 8 Oterma s orbit 2:3 resonance Second encounter y (AU, Sun-centered inertial frame) START Jupiter Sun 3:2 resonance END First encounter -8 Jupiter s orbit x (AU, Sun-centered inertial frame)

4 The Flow near L and L 2 : Overview Lo and Ross used PCR3BP as model and noticed that the comets follow closely the invariant manifolds of L and L 2. Near L and L 2 are where Jupiter comets make resonance transition. y (AU, Sun-Jupiter rotating frame) L 3 S L 4 L L L 2 x (AU, Sun-Jupiter rotating frame) J Jupiter s orbit y (AU, Sun-Jupiter rotating frame) END START Oterma s orbit S J x (AU, Sun-Jupiter rotating frame)

5 The Flow near L and L 2 : Overview For energy value just above that of L 2, Hill s contains a neck aboutl and L 2. Comets are energetically permitted to make transition through these equilibrium s. [Conley] Theflow has4 typesof orbits: periodic, asymptotic, transit and non-transit orbits. y (nondimensional units, rotating frame) forbidden interior S Zero velocity curve bounding forbidden L exterior J L 2 Jupiter L y (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)

6 The Flow near L and L 2 : Linearization [Moser] All the qualitative results of the linearized equations carry over to the full nonlinear equations. Recall equations of PCR3BP: After linearization, ẋ = v x, v x =2v y Ω x, ẏ = v y, v y =2v x Ω y. () ẋ = v x, v x =2v y ax, ẏ = v y, v y = 2v x by. (2) Eigenvalues have the form ±λ and ±iν. Corresponding eigenvectors are u =(,σ, λ, λσ), u 2 =(,σ,λ, λσ), w =(,iτ, iν, ντ), w 2 =(,iτ,iν, ντ).

7 The Flow near L and L 2 : Linearization After linearization & making eigenvectors as new coordinate axes, equations assume simple form ξ = λξ, η = λη, ζ = νζ 2, ζ2 = νζ, with energy function E l = ληξ ν 2 (ζ2 ζ2 2 ). The flow near L,L 2 have the form of saddle center. ξ ηξ= ζ 2 = η ηξ=c ηξ=c ηξ= ζ 2 =ρ ζ 2 = ζ 2 =ρ

8 Flow near L and L 2 : Equilibrium Region R S 2 I Recall that energy function E l = ληξ ν 2 (ζ2 ζ2 2 ). Equilibrium R on 3D energy manifold is homeomorphic to S 2 I. Because for each fixed value of η ξ, E l = E is -sphere λ 4 (η ξ)2 ν 2 (ζ2 ζ2 2 )=Eλ 4 (ηξ)2. ξ ηξ= ζ 2 = η ηξ=c ηξ=c ηξ= ζ 2 =ρ ζ 2 = ζ 2 =ρ

9 Flow near L and L 2 : Equilibrium Region R S 2 I Recall that energy function E l = ληξ ν 2 (ζ2 ζ2 2 ). Each point in (η, ξ)-plane corr. to a circle S in R with radius ζ = 2ν (Eληξ), with point on the bounding hyperbola corr. to point-circle. Thus, line segment (η ξ = ±c) corr. to S 2 in R because each corr. to S I with 2 ends pinched to a point. ξ η ηξ= ζ 2 = d a ηξ=c ηξ=c ηξ= r Liapunov Orbit, l d 2 r 2 r 2 a 2 n b a ζ 2 =ρ ζ 2 = ζ 2 =ρ r d

10 Flow near L and L 2 : 4 Types of Orbits The flow (η(t),ξ(t),ζ (t),ζ 2 (t)) is given by η(t) =η e λt, ξ(t)=ξ e λt, ζ(t) =ζ (t)iζ 2 (t) =ζ e iνt, with 2 additional integrals ηξ(= η ξ ), ζ 2 = ζ 2 ζ2 2 (= ζ 2 ). 9 classes of orbits grouped into 4types: black point η = ξ = corr. to a periodic orbit. 4 half green segments of axis ηξ = corr. to 4 cylinders of orbits asymptotic to this periodic orbit. ξ ηξ= ζ 2 = η ηξ=c ηξ=c ηξ= ζ 2 =ρ ζ 2 = ζ 2 =ρ

11 Flow near L and L 2 : 4 Types of Orbits 9 classes of orbits grouped into 4types: Shown are 2 red hyperbolic segments ηξ = constant > corr. to 2 cylinders of transit orbits which transit from bounding sphere to the other. 2 blue hyperbolic segments ηξ = constant < corr. to 2 cylinders of non-transit orbits each of which runs from hemisphere to the other on same bounding sphere. ξ ηξ= ζ 2 = η ηξ=c ηξ=c ηξ= ζ 2 =ρ ζ 2 = ζ 2 =ρ

12 McGehee Representation: Equilbrium Region R [McGehee] To visualize R S 2 I. ξ η ηξ= ζ 2 = ηξ=c ηξ=c ηξ= ζ 2 =ρ ζ 2 = ζ 2 =ρ ω a r d l b n a r Liapunov Orbit, l d n b r a r a d d (a) (b)

13 McGehee Representation: Equilbrium Region R 4typesof orbits in equilibrium R: Black circle l is the unstable periodic orbit. 4 cylinders of asymptotic orbits form pieces of stable and unstable manifolds. They intersect the bounding spheres at asymptotic circles. ω d n d r a l b a r Liapunov Orbit, l n b r a r a d d (a) (b)

14 McGehee Representation: Equilibrium Region R Asymptotic circles divide bounding sphere into spherical caps and spherical zones. 4typesof orbits in equilibrium R: Transit orbits entering R through a cap on a bounding sphere will leave through a cap on another bounding sphere. Non-transit orbits entering R through a spherical zone will leave through another zone of the same bounding sphere. ω d n d a r l b a r Liapunov Orbit, l n b r a r a d d (a) (b)

15 McGehee Representation: Separatrix Invariant manifold tubes act as separatrices for the flow in R: Those inside the tubes are transit orbits. Those outside of the tubes are non-transit orbits. ω d n d r a l b a r Liapunov Orbit, l n b r a r a d d (a) (b)

16 McGehee Representation: Separatrix Stable and unstable manifold tubes act as separatrices for the flow in R: Those inside the tubes are transit orbits. Those outside of the tubes are non-transit orbits..6 Unstable Manifold Forbidden Region.4 Stable Manifold y (rotating frame) Sun L Periodic Orbit Stable Manifold x (rotating frame) Forbidden Region Unstable Manifold

17 Applications: Jupiter Comets By linking invariant manifold tubes, we found dynamical channels for a fast transport mechanism between exterior and interior Hill s s. Jupiter comets can follow these channels in rapid transition between exterior and interior passing through Jupiter..8.6 Forbidden Region.6.4 (;J,S) Stable Manifold Cut.4 y (rotating frame) J L L Stable Unstable 2 Manifold Manifold. y (rotating frame) Intersection Region J = (X;J,S) Forbidden Region x (rotating frame) (X;J) Unstable Manifold Cut y (rotating frame)

18 Applications: Shoot the Moon Stable manifold tube provide a temporary capture mechanism by the second primary. Stable manifold tube of a periodic orbit around L 2 guides spacecraft towards a ballistic capture by the Moon. By saving fuel for this lunar ballistic capture leg, the design uses less fuel than a Earth-to-Moon Hohmann transfer y -.2 Moon Earth Sun Forbidden Region L 2 orbit x

19 Flow Mappings in R: 4 Flow Mappings The flow in R defines 4 mappings: 2 between pairs of spherical caps of different bounding spheres. 2 between pairs of spherical zones of same bounding sphere. They are Poincaré mapsinrand they will be used later to build the global Poincaré map. ω d n d r a l b a r Liapunov Orbit, l n b r a r a d d (a) (b)

20 Flow Mappings in R: Infinite Twisting of the Map To visualize the infinite twisiting of the maps. ω a r d l b n a r Liapunov Orbit, l d n b r a r a d d (a) (b) n ψ 2 (γ 2 ) δ 2 d a γ 2 Q 3 r P 4 Q 2 P 2 ψ 3 (γ 3 ) δ 3 b γ 4 r γ 3 a ψ 4 (γ 4 ) δ 4 P γ P 3 d Q ψ (γ ) δ Q 4

21 Flow Mappings in R: Infinite Twisting of the Map From ζ(t) =ζ e iνt,weobtaindt d (arg ζ) =ν. Amount of twisting a point will undergo is proportional to the time required for its corr. trajectory to go from domain to range. This time approaches infinity as the flow approaches an asymtotic circle. Hence, amount of twisiting a point will undergo depends very sensitively on its distance from an aymptotic circle. n ψ 2 (γ 2 ) δ 2 d a γ 2 Q 3 r P 4 Q 2 P 2 ψ 3 (γ 3 ) δ 3 b γ 4 r γ 3 a ψ 4 (γ 4 ) δ 4 P γ P 3 d Q ψ (γ ) δ Q 4

22 Flow Mappings in R: Infinite Twisting of the Map Images ψ 2 (γ 2 )ofabutting arc γ 4 spirals infinitely many times around and towards asymptotic circles a. n ψ 2 (γ 2 ) δ 2 d a γ 2 Q 3 r P 4 Q 2 P 2 ψ 3 (γ 3 ) δ 3 b γ 4 r γ 3 a ψ 4 (γ 4 ) δ 4 P γ P 3 d Q ψ (γ ) δ Q 4

23 Applications: Shoot the Moon Find position and velocity for a spacecraft such that when integrating forward, SC will be guided by Earth-Moon manifold and get ballistically captured at the Moon; when integrating backward, SC will hug Sun-Earth manifolds and return to the Earth. Infinite twisting of flow map is key in finding a low energy transfer in Sun-Earth leg of the design y -.2 Moon Earth Sun Forbidden Region L 2 orbit x

24 Ballistic Capture of a Spacecraft by the Moon from 2 km Altitude Earth Parking Orbit (Shown in an Inertial Frame) y (Earth-Moon distances) Transfer time is about 6 months (5 day tick marks) V 2 =34 m/s V L =32 m/s Earth Moon's Orbit Ballistic Capture x (Earth-Moon distances) Integrated in 4-body system: Sun-Earth-Moon-Spacecraft y (AU) y (Earth-Moon distances) Sun-Earth Rotating Frame L Sun Earth Moon's Orbit V 2 V L Ballistic Capture x (AU) L L 2 Earth Ballistic Capture by the Moon Moon x (Earth-Moon distances) L 2 Earth-Moon Rotating Frame

25 Appearance of Orbits in Position Space Tilted projection of R on (x, y)-plane. ω a r d l b n d a r Liapunov Orbit, l n b r a r a d d (a) (b) y (nondimensional units, rotating frame) forbidden interior S Zero velocity curve bounding forbidden L exterior J L 2 Jupiter L y (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)

26 Appearance of Orbits in Position Space 4typesof orbits: Shown are A periodic orbit. A typical aymptotic orbit. 2 transit orbits. 2 non-transit orbits. y (nondimensional units, rotating frame) forbidden interior S Zero velocity curve bounding forbidden L exterior J L 2 Jupiter L y (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)