Dynamical Systems and Space Mission Design

Transcription

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

2 Constructing Orbits of Prescribed Itineraries: Outline Outline of Lecture 4A: Using the proof of the Main Theorem as the guide, we develop a procedure to construct orbits with prescribed itineraries. Example: An orbit with itinerary (X, J, S, J, X). Application: Petit Grand Tour of Jovian moons. 1.5 (X,J,S,J,X) Orbit Forbidden Region y (rotating frame) Sun L 1 L y (rotating frame) Forbidden Region x (rotating frame) x (rotating frame)

3 The Petit Grand Tour of the Jovian Moons x (nondimensional, -Europa rotating frame) Ganymede's L 1 periodic orbit unstable manifold Heteroclinic intersection Europa's L 2 periodic orbit stable manifold x (nondimensional, -Europa rotating frame) y (-Ganymede rotating frame) L 1 Ganymede L 2 y (-centered inertial frame) Transfer orbit V Europa s orbit Ganymede s orbit y (-Europa rotating frame) Europa L 2 x (-Ganymede rotating frame) x (-centered inertial frame) x (-Europa rotating frame)

4 Global Orbit Structure: Review Symbolic sequence used to label itinerary of each orbit. Main Theorem: For any admissible itinerary, e.g., (...,X,J,S,J,X,...) in the informal notation, there exists an orbit whose whereabouts matches this itinerary. Can even specify number of revolutions the comet (or spacecraft) makes around L 1 & L 2 as well as Sun &. L1 L2 S J X

5 Construction of Orbits with Prescribed Itineraries Recall from Lecture 3B, that the Main Theorem concerning biinfinite itineraries (e.g., (...,X,J,S,J,X,...)) relied on the construction of the invariant set, a cloud of points. To generate usable trajectories, truncate construction of invariant set at a finite number of iterations of the Poincaré mapp. U 3 E 1 H n 31 C 1 D 1 A 1 B 1 A 2 B 2 A 1 B 1 F 1 V m 34 H n 32 V m 32 H 2 ' C 1 V m 13 H n 11 V m 11 A 2 B 2 E 2 V m 44 H n 43 V m 42 D 1 H n 12 F 2 H n 44 H 2 ' C 2 D 2 3;12 Q m;n U 1 C 2 V m 21 H n 23 V m 23 G 2 H 2 U 4 D 2 H n 24 H 2 ' H 1 G 1 U 2

6 Construction of Orbits with Prescribed Itineraries The sets of orbits with different itineraries are easily visualizable on our chosen Poincaré section as areas in which all the orbits have the same finite itinerary. We will also no longer be limited to a small neighborhood of a homoclinic-heteroclinic chain, but can obtain more global results. Example: An orbit with itinerary (J, X, J). Poincare Section Stable Manifold Unstable Manifold Cut (;X,J) S J L 2 L 1 X = (J;X,J) (J;X) Unstable Manifold Stable Manifold Cut

7 Construction of Orbits with Prescribed Itineraries Example: An orbit with itinerary (X, J, S), passing from the exterior region to the interior region via the region. Look at Poincaré section where L 2 periodic orbit unstable manifold intersects L 1 periodic orbit stable manifold. Intersection region contains orbits with itinerary (X, J, S). y (rotating frame) Forbidden Region Forbidden Region Poincare section J L 1 L Stable Unstable 2 Manifold Manifold y (;J,S) Stable Manifold Cut (X;J) Unstable Manifold Cut y (;J,S) (X;J) Intersection Region J = (X;J,S) Stable Manifold Cut Unstable Manifold Cut x (rotating frame) y y

8 The Flow near L 1 and L 2 :Review Recall from Lecture 2B, for energy values just above that of L 2, Hill s region contains a neck about L 1 and L 2. [Conley] The flow in the equilibrium region R has 4types of orbits: periodic, asymptotic, transit and non-transit orbits. y (nondimensional units, rotating frame) forbidden region interior region S Zero velocity curve bounding forbidden region L 1 exterior region J L 2 region L y (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)

9 McGehee Representation: Equilbrium Region R 4typesof orbits in equilibrium region 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, separating spherical caps, which contain transit orbits, from spherical zones, which contain non-transit orbits. ω d 1 n 1 d 1 r 1 a 1 d 2 + a 2 + r 2 + l b 1 b 2 r 2 a 1 n 2 r 1 Liapunov Orbit, l r 2 + r 2 d 2 + n 2 a 2 + b 2 n 1 b 1 r 1 + a1 + a 2 d 2 r 1 + d 2 a 2 a 1 + d 1 + d 1 + (a) (b)

10 Tubes Partition the Energy Surface Cylinders of asymptotic orbits form tubes in the energy surface. Periodic orbit stable and unstable manifold tubes. To transit from one region to another, must be inside the tubes. e.g., transit from to Sun region via L 1 periodic orbit tubes..6.4 Unstable Manifold Forbidden Region Stable Manifold y (rotating frame) Sun L 1 Periodic Orbit Stable Manifold x (rotating frame) Forbidden Region Unstable Manifold

11 Spherical Caps Containing Transit Orbits We will be focusing on the spherical caps of transit orbits. In particular, we will look at their images and pre-images on a suitable Poincaré section, since these spherical caps will be building blocks from which we construct orbits of prescribed itineraries. The images and pre-images of the spherical caps form the tubes that partition the energy surface. Poincare Section U 3 (x = 1 µ, y > ) d 1,2 + d 2,1 J Stable Manifold Unstable Manifold Right bounding sphere n 1,2 of L 1 equilibrium region (R 1 ) region (J ) Left bounding sphere n 2,1 of L 2 equilibrium region (R 2 )

12 Poincaré Section: Images of Bounding Sphere Caps For instance, on a Poincaré section between L 1 and L 2, We look at the image of the cap on the left bounding sphere of the L 2 equilibrium region R 2 containing orbits leaving R 2. Wealsolookatthepre-image of the cap on the right bounding sphere of R 1 containing orbits entering R 1. Poincare Section U 3 (x = 1 µ, y > ) d 1,2 + d 2,1 J Stable Manifold Unstable Manifold Right bounding sphere n 1,2 of L 1 equilibrium region (R 1 ) region (J ) Left bounding sphere n 2,1 of L 2 equilibrium region (R 2 )

13 Poincaré Section: Images of Bounding Sphere Caps Pre-Image of Spherical Cap d 1,2 + (;J,S) y (X;J) Image of Spherical Cap d 2, y Poincare Section U 3 (x = 1 µ, y > ) d 1,2 + d 2,1 J Stable Manifold Unstable Manifold Right bounding sphere n 1,2 of L 1 equilibrium region (R 1 ) region (J ) Left bounding sphere n 2,1 of L 2 equilibrium region (R 2 )

14 Construction of Orbits with Prescribed Itineraries The Poincaré cut of the unstable manifold of the L 2 periodic orbit forms the boundary of the image of the cap containing transit orbits leaving R 2. All of these orbits came from the exterior region and are now in the region, so we label this region (X; J). y (rotating frame) Forbidden Region Forbidden Region Poincare section J L 1 L Stable Unstable 2 Manifold Manifold y (;J,S) Stable Manifold Cut (X;J) Unstable Manifold Cut y (;J,S) (X;J) Intersection Region J = (X;J,S) Stable Manifold Cut Unstable Manifold Cut x (rotating frame) y y

15 Construction of Orbits with Prescribed Itineraries Similarly, the Poincaré cut of the stable manifold of the L 1 periodic orbit forms the boundary of the pre-image of the cap of transit orbits entering R 1. All of these orbits are now in the region and are headed for the interior (Sun) region, so we label this region (; J, S). y (rotating frame) Forbidden Region Forbidden Region Poincare section J L 1 L Stable Unstable 2 Manifold Manifold y (;J,S) Stable Manifold Cut (X;J) Unstable Manifold Cut y (;J,S) (X;J) Intersection Region J = (X;J,S) Stable Manifold Cut Unstable Manifold Cut x (rotating frame) y y

16 Construction of Orbits with Prescribed Itineraries Note that the image of the R 2 cap and the pre-image of the R 1 cap intersect. We label this intersection region J. This is the set of orbits which came from the exterior region, are now in the region, and are headed for the interior (Sun) region, so we label this region (X; J, S). Integrating an initial condition in this set forwards and backwards would give us an orbit with the desired itinerary (X, J, S). y (rotating frame) Forbidden Region Forbidden Region Poincare section J L 1 L Stable Unstable 2 Manifold Manifold y (;J,S) Stable Manifold Cut (X;J) Unstable Manifold Cut y (;J,S) (X;J) Intersection Region J = (X;J,S) Stable Manifold Cut Unstable Manifold Cut x (rotating frame) y y

17 .. Construction of Orbits with Prescribed Itineraries We can partition the intersection region even further using this technique. Smaller regions of orbits with longer itineraries emerge. (J,X,J,S,J) Orbit y (nondimensional units, Sun- rotating frame) Forbidden Region Poincare Forbidden Region section J L 1 Stable L 2 Manifold Manifold Unstable x (nondimensional units, Sun- rotating frame) y (nondimensional units, Sun- rotating frame) (;J,S) (X;J) Intersection Region J = (X;J,S) Stable Manifold Cut Unstable Manifold Cut y (nondimensional units, Sun- rotating frame) Iterate Poincare Return Map y (nondimensional units, Sun- rotating frame) Sun y (nondimensional units, Sun- rotating frame) (J,X;J) (;J,S,J) (J,X;J,S,J) x (nondimensional units, Sun- rotating frame) y (nondimensional units, Sun- rotating frame) Integrate Initial Condition in (J,X;J,S,J) Square

18 Example Itinerary: (X, J, S, J, X) We wish to illustrate these techniques by constructing an orbit with an itinerary (X, J, S, J, X). This is the itinerary for the comet Oterma in the Sun- system during the time interval of interest. y (AU, Sun- rotating frame) Sun 191 Oterma s Trajectory Oterma s Trajectory Forbidden Region L 1 L y (AU, Sun- rotating frame) -8 Forbidden Region x (AU, Sun- rotating frame) x (AU, Sun- rotating frame)

19 Example Itinerary: (X, J, S, J, X) Before beginning, we review the notation regarding the location of the Poincaré sections used for the construction. U 1 in interior (Sun) region, U 4 in exterior region U 2 and U 3 in region X U 3 U 4 U 1 S J U 2

20 Example Itinerary: (X, J, S, J, X) Note also the locations of the stable and unstable manifolds of the L 1 and L 2 periodic orbits. Note the mirror symmetry about the Sun- line. X U 3 U 4 U 1 S J U 2

21 Example Itinerary: (X, J, S, J, X) We compute the first few transversal cuts of the L 1 stable and L 2 unstable periodic orbit manifolds on the U 3 Poincaré section in the region. Notice the intersection region J, in which all orbits have the central block itinerary (X; J, S). y (nondimensional units, rotating frame) Γ s,j L 1,1 Γ u,j L 2,1 U 3 J =(X;J,S) Γ u,j L 2,3 Γ u,j L 2, y (nondimensional units, rotating frame)

22 Example Itinerary: (X, J, S, J, X) All orbits entering the Sun region must pass through the first cut of the L 1 p.o. stable manifold. Notice that the L 2 p.o. unstable manifold swings around several times before intersecting the first cut of the L 1 p.o. stable manifold. The intersection, containing (X; J, S) orbits, is shown shaded. y (nondimensional units, rotating frame) Γ s,j L 1,1 Γ u,j L 2,1 U 3 J =(X;J,S) Γ u,j L 2,3 Γ u,j L 2, y (nondimensional units, rotating frame)

23 Example Itinerary: (X, J, S, J, X) Follow the shaded J =(X;J,S) region forward under the flow until it intersects the U 1 Poincaré section in the Sun region. The image (X, J; S) is contained entirely inside the region bounded by the L 1 p.o. unstable manifold, as expected. Pieces that also intersect the L 1 p.o. stable manifold will return to the region and are labelled S =(X,J;S,J). x (nondimensional units, rotating frame) (X,J;S,J) U 1 (X,J;S) Γ u,s L 1,1 Γ s,s L 1,1 s,s L 1,1 u,s L 1,1 S = (X,J;S,J) x (nondimensional units, rotating frame) x (nondimensional units, rotating frame) (a) x (nondimensional units, rotating frame) (b)

24 Example Itinerary: (X, J, S, J, X) Follow the larger strip of S =(X,J;S,J)forward until it returns to the region, intersecting the U 2 Poincaré section. The piece contained inside the region bounded by the L 2 p.o. stable manifold will return to the exterior region. Region = (X, J, S; J, X) contains orbits with desired itinerary. y (nondimensional units, rotating frame) Γ s,j L 2,5 U 2 Γ s,j L 2,3 Γ u,j L 1,1 Γ s,j L 2,1 s,j L 2,5 u,j L 1,1 (X,J,S;J) = (X,J,S;J,X) y (nondimensional units, rotating frame) y (nondimensional units, rotating frame) (a) y (nondimensional units, rotating frame) (b)

25 Example Itinerary: (X, J, S, J, X) Pick any initial condition in the region = (X, J, S; J, X) and integrate forward and backward. We obtain orbit with the desired prescribed itinerary (X, J, S, J, X). 1.5 (X,J,S,J,X) Orbit Forbidden Region y (rotating frame) Sun L 1 L y (rotating frame) Forbidden Region x (rotating frame) x (rotating frame)

26 Application: Petit Grand Tour of Jovian Moons Recently, there has been interest in sending a scientific spacecraft to orbit and study Europa. The Jovian moon may have subterranean oceans and life.

27 Application: Petit Grand Tour of Jovian Moons The trajectory design involved in effecting an orbital capture by Europa presents formidable challenges to the traditional twobody patched conic approach. Regimes of motion involved depend heavily on three-body dynamics. New three-body perspective is necessary to design a feasible trajectory from Earth to Europa using a very limited fuel budget. Fortunately, s vicinity and the Galilean moons provide enough interesting orbital dynamics that the construction of a relatively low fuel mission is possible.

28 Spacecraft Trajectory Petit Grand Tour: Ganymede and Europa For this initial exploration, we study the final leg of the trip, once the spacecraft is in a large orbit around. The goal is to perform a tour of two of the Jovian moons, Ganymede and Europa. Ending with a capture orbit around Europa. Europa s Orbit Capture at Europa Ganymede s Orbit

29 Spacecraft Trajectory Petit Grand Tour: Ganymede and Europa Itinerary: Starting from outside Ganymede s orbit, get captured for one orbit around Ganymede. Leaving Ganymede, transfer to Europa. Get captured by Europa for several orbits. Europa s Orbit Capture at Europa Ganymede s Orbit

30 Starting Model: Coupled Three-Body Problem Solution space of full 4-body problem is little known. Look at coupled 3-body problem: Two nested 3-body systems, -Ganymede-spacecraft and -Europa-spacecraft, approximate the dynamics well. Europa s Orbit Spacecraft Ganymede s Orbit

31 Ganymede s orbit Petit Grand Tour: Ganymede and Europa The transfer trajectory from Ganymede to Europa is the most difficult part of the tour to construct. That is where we begin. We wish to go from the capture region of Ganymede to the capture region of Europa. Use the intersection of unstable and stable manifold tubes. Europa s L 2 p.o. stable manifold Europa s orbit Ganymede s L 1 p.o. unstable manifold

32 Ganymede s orbit Intersection of Tubes Find intersection of Ganymede s L 1 periodic orbit unstable manifold and Europa s L 2 periodic orbit stable manifold. The two manifolds are at different energies, so a rocket burn ( V ) will be necessary. Europa s L 2 p.o. stable manifold Europa s orbit Ganymede s L 1 p.o. unstable manifold

33 Intersection of Tubes: Poincaré Section On a suitably chosen Poincaré section Look for an intersection between the manifolds. Compute magnitude of rocket burn ( V ). Intersection point Europa s L 2 p.o. stable manifold Poincare section Europa s orbit Ganymede s L 1 p.o. unstable manifold Ganymede s orbit

34 Intersection of Tubes: Poincaré Section Poincaré section: Vary configuration of the moons until, (x, ẋ)-plane: Intersection found! (x, ẏ)-plane: Velocity discontinuity since energies are different. A rocket burn V of this magnitude will make transfer trajectory jump from one tube to the other. x (nondimensional, -Europa rotating frame) Ganymede s L 1 periodic orbit unstable manifold Intersection point Europa s L 2 periodic orbit stable manifold x (nondimensional, -Europa rotating frame) y (nondimensional, -Europa rotating frame) Ganymede s L 1 periodic orbit unstable manifold Intersection point (and V) Europa s L 2 periodic orbit stable manifold x (nondimensional, -Europa rotating frame)

35 Intersection of Tubes: Transfer Trajectory Using an initial condition in the intersection region, forward and backward integrate to generate a transfer trajectory. Integrate full 4-body problem: -Ganymede-Europa-S/C. Spacecraft transfer trajectory Europa s L 2 p.o. stable manifold V Europa s orbit Ganymede s L 1 p.o. unstable manifold Ganymede s orbit

36 Intersection of Tubes: Transfer Trajectory Rocket burn required for transfer trajectory: V = 12 m/s Compare with standard Hohmann transfer: V = 28 m/s Coupled 3-body approach uses only 4% of the fuel of the conventional coupled 2-body approach. Significant savings! Trajectory shown below in both inertial and rotating frames. Transfer trajectory takes only 25 days. y (-Ganymede rotating frame) L 1 Ganymede L 2 y (-centered inertial frame) Transfer orbit V Europa s orbit Ganymede s orbit y (-Europa rotating frame) Europa L 2 x (-Ganymede rotating frame) x (-centered inertial frame) x (-Europa rotating frame)

37 Summary: Using Tubes to Construct Trajectories The study of the planar circular restricted 3-body problem (PCR3BP) provides a systematic method for the numerical construction of a transfer trajectory for a space mission. We work in the Case 3 energy regime of the PCR3BP. 1 S J Case 3 : E 2 < E < E 3 or C 2 > C > C 3

38 Summary: Using Tubes to Construct Trajectories L 1 and L 2 periodic orbit stable and unstable manifold tubes partition the energy surface into transit and nontransit orbits. Transit orbits, contained inside the tubes, transit between the interior, capture, and exterior regions. X.6 Unstable Manifold Forbidden Region.4 Stable Manifold U 4 U 1 S U 3 J U 2 y (rotating frame) Sun L 1 Periodic Orbit Stable Manifold Forbidden Region Unstable Manifold x (rotating frame)

39 Summary: Using Tubes to Construct Trajectories Construct leap-frogging transfer trajectories between moons: Jump from inside of unstable manifold tube of one 3-body system to the inside of stable manifold tube of an adjacent 3-body system. Use rocket burn ( V ) at tube intersection point. Spacecraft transfer trajectory Europa s L 2 p.o. stable manifold V Europa s orbit Ganymede s L 1 p.o. unstable manifold Ganymede s orbit

40 Summary: Using Tubes to Construct Trajectories e.g., Petit Grand Tour of Jovian moons Ganymede and Europa. Uses less than half the fuel of standard 2-body approach. y (-Ganymede rotating frame) L 1 Ganymede L 2 y (-centered inertial frame) Transfer orbit V Europa s orbit Ganymede s orbit y (-Europa rotating frame) Europa L 2 x (-Ganymede rotating frame) x (-centered inertial frame) x (-Europa rotating frame)

41 Summary: Using Tubes to Construct Trajectories Construction of many other low-fuel transfers is possible. Earth to Moon transfer: Shoot the Moon in an upcoming lecture. y Moon Sun Earth Forbidden Region L 2 orbit x y (Sun-Earth rotating frame) Earth Moon's Orbit Sun-Earth L 2 Orbit Unstable Manifold Cut Transfer Patch Point ( V) Earth-Moon L 2 Orbit Stable Manifold Cut y (Sun-Earth rotating frame) y (Inertial frame, lunar distances) 2 V = 34 m/s Launch Earth Moon s Orbit Ballistic Capture x (Inertial frame, lunar distances) (a) (b) (c)

42 Looking Ahead: Natural Transfers in the Solar System Natural (zero-fuel) transfers may exist in the Solar System. Comet transitions between resonances of (e.g., Oterma). Transfer of comets between and Saturn systems. Transfer of material between Asteroid Belt and Kuiper Belt. 1.9 Orbital Eccentricity Asteroids Comets Pluto.2 Trojans.1 Saturn Uranus Neptune Kuiper Belt Objects Semi-major Axis (AU)