The Coupled Three-Body Problem and Ballistic Lunar Capture

Transcription

1 The Coupled Three-Bod Problem and Ballistic Lunar Capture Shane Ross Martin Lo (JPL), Wang Sang Koon and Jerrold Marsden (Caltech) Control and Dnamical Sstems California Institute of Technolog Three Bod Fest, Ma 24, 2000 Control and Dnamical Sstems

2 2 Efficient Lunar Transfer: How to Shoot the Moon Our goal: Design an efficient trajector to the Moon using less fuel than the traditional Hohmann transfer (e.g., the Apollo missions). Find and classif sets of solutions to the Sun--Moon-spacecraft 4-bod problem; in particular, find solutions starting from and ending in ballistic capture b the Moon. Approimate 4-bod sstem as two coupled 3-bod sstems. Review results on L 1 and dnamical channels in the planar circular restricted 3-bod problem. Find intersections between dnamical channels between the two sstems to construct complete -to-moon trajector, which utilizes perturbation b Sun.

3 Lunar Capture: How to get to the Moon Cheapl In 1991, the failed Japanese mission, Muses-A (Uesugi [1986]), was given new life with a radical new mission concept and renamed as the Hiten Mission (Tanabe et al. [1982], Belbruno [1987], Belbruno and Miller [1993]). 3 (from Belbruno and Miller [1993])

4 We present an approach to the problem of the orbital dnamics of this interesting trajector b implementing in a sstematic wa the view that the Sun--Moon-spacecraft 4-bod sstem can be modeled as two coupled 3-bod sstems. Inertial Frame V Sun- Rotating Frame 4 V Moon's Orbit Ballistic Capture L 1 Sun Moon's Orbit Ballistic Capture

5 Below is a schematic of the Shoot the Moon trajector, showing the two legs of the trajector in the Sun- rotating frame: backward targeting portion Lunar capture portion 5 Maneuver ( V) at Patch Point Sun--S/C Sstem -Moon-S/C Sstem Sun Moon

6 Within each 3-bod sstem, using our understanding of the invariant manifold structures associated with the Lagrange points L 1 and, we transfer from a 200 km altitude orbit into the region where the invariant manifold structure of the Sun- Lagrange points interact with the invariant manifold structure of the -Moon Lagrange points. 6 Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's Orbit orbit

7 7 Schematic with Lagrange point invariant manifold structures: backward targeting portion Lunar capture portion Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's Orbit orbit

8 One utilizes the sensitivit of the twisting near the invariant manifold tubes to target back to a suitable parking orbit. Targeting Portion of Trajector Using "Twisting" Near Tubes 8 Sun orbit

9 This interaction permits a fuel efficient transfer from the Sun- sstem to the -Moon sstem. The invariant manifold tubes of the -Moon sstem provide the dnamical channels in phase space that enable ballistic captures of the spacecraft b the Moon. Tube Containing Lunar Capture Orbits 9 Lunar Capture Portion of Trajector Sun Moon

10 The results are then checked b integration in the bicircular 4-bod problem. It works! Inertial Frame V Sun- Rotating Frame 10 V Moon's Orbit Ballistic Capture L 1 Sun Moon's Orbit Ballistic Capture

11 This technique is cheaper (about 20% less V ) than the usual Hohmann transfer (jumping onto an ellipse that reaches the Moon, then accelerating to catch it, then circularizing). However, it also takes longer (4 to 6 months compared to 3 das). 11 V2 V1 Moon's Orbit Transfer Trajector Ballistic Capture (from Brown [1992])

12 Planar Circular Restricted Three Bod Problem PCR3BP General Comments Describes the motion of a bod moving in the gravitational field of two main bodies (the primaries) that are moving in circles. The two main bodies could be the Sun and,orthe and Moon, etc. The total mass is normalized to 1; the are denoted m S =1 µand m E = µ, so0<µ< 1 2. Let µ be the ratio between the mass of the and the mass of the Sun- sstem, m µ = E, m E + m S For the Sun- sstem, µ = For the -Moon sstem, µ =

13 The two main bodies rotate in the plane in circles counterclockwise about their common center of mass and with angular velocit ω (also normalized to 1). The third bod, the spacecraft, has mass zero and is free to move in the plane. The planar restricted three bod problem is used for simplicit. Generalization to the three dimensional problem is of course important, but man of the effects can be described well with the planar model. 13

14 Equations of Motion Notation: Choose a rotating coordinate sstem so that the origin is at the center of mass the Sun and are on the -aisatthepoints( µ, 0) and (1 µ, 0) respectivel i.e., the distance from the Sun to is normalized to be 1. Let (, ) be the position of the comet in the plane relative to the positions of the Sun and distances to the Sun and : r 1 = ( + µ) and r 2 = ( 1+µ)

15 15 spacecraft 1 (rotating frame) = - µ m S = 1 - µ m E = µ Sun r 2 r 1 = 1 - µ -1 s orbit (rotating frame)

16 Equations of motion: where ẍ 2ẏ = Ω, Ω= (2 + 2 ) 2 ÿ +2ẋ= Ω + 1 µ r 1 + µ r 2 + µ(1 µ). 2 The have a first integral, the Jacobi constant, givenb C(,, ẋ, ẏ) = (ẋ 2 +ẏ 2 ) + 2Ω(, ), which is related to the Hamiltonian energ b C = 2H. Energ manifolds are 3-dimensional surfaces foliating the 4-dimensional phase space. For fied energ, Poincaré sections are then 2-dimensional, making visualization of intersections between sets in the phase space particularl simple. 16

17 17 Five Equilibrium Points Three collinear (Euler, 1750) on the -ais L 1,,L 3 Two equilateral points (Lagrange, 1760) L 4,L 5. 1 L 4 spacecraft 0.5 (rotating frame) L 3 S L 1 E m S = 1 --µ m E = µ -1 L 5 s orbit (rotating frame)

18 18 Stabilit of Equilibria Eigenvalues of the linearized equations at L 1 and have one real and one imaginar pair. The stable and unstable manifolds of these equilibria pla an important role. Associated periodic orbits are called the Lapunov orbits. Their stable and unstable manifolds are also important. For space mission design, the most interesting equilibria are the unstable ones, not the stable ones! Consider the dnamics plus the control! Control often makes unstable objects, not attractors, of interest! 1 Under proper control management the are incredibl energ efficient. 1 This is related to control of chaos; see Bloch, A.M. and J.E. Marsden [1989] Controlling homoclinic orbits, Theor. & Comp. Fluid Mech. 1,

19 Hill s Regions Our main concern is the behavior of orbits whose energ is just above that of. Roughl, we refer to this small energ range as the temporar capture energ range. Fortunatel, the temporar capture energ surfaces for the Sun- and -Moon sstems intersect in phase space, making a fuel efficient transfer possible. The Hill s region is the projection of this energ region onto position space. The region not accessible for these energies is called the forbidden region. For this case, the Hill s region contains a neck about L 1 and. This equilibrium neck region and its relation to the global orbit structure is critical: it was studied in detail b Conle, McGehee and the Barcelona group. 19

20 Orbits with energ just above that of can be transit orbits, passing through the neck region between the eterior region (outside s orbit) and the temporar capture region (bubble surrounding ). The can also be nontransit orbits or asmptotic orbits Forbidden Region Eterior Region (rotating frame) Interior Region S L 1 E (rotating frame) (rotating frame) Capture Region (rotating frame)

21 Equilibrium Region near a Lagrange Point 4tpesof orbits in the equilibrium region. Black circle is the unstable periodic Lapunov orbit. 4 clinders of asmptotic orbits form pieces of stable and unstable manifolds. The intersect the bounding spheres at asmptotic circles, separating spherical caps, which contain transit orbits, from spherical zones, which contain nontransit orbits. 21 Transit Orbits Asmptotic Orbits ω Spherical Cap of Transit Orbits Asmptotic Circle Nontransit Orbits l Spherical Zone of Nontransit Orbits Lapunov Orbit Cross-section of Equilibrium Region Equilibrium Region

22 22 Roughl speaking, the equilibrium region has the dnamics of a saddle harmonic oscillator. Asmptotic Orbits Transit Orbits ω Spherical Cap of Transit Orbits Asmptotic Circle Nontransit Orbits l Spherical Zone of Nontransit Orbits Lapunov Orbit Cross-section of Equilibrium Region Equilibrium Region

23 Tubes Partition the Energ Surface Stable and unstable manifold tubes act as separatrices for the flow in the equilibrium region. Those inside the tubes are transit orbits. Those outside the tubes are nontransit orbits. e.g., transit from outside s orbit to capture region possible onl through periodic orbit stable tube. 23 Forbidden Region Unstable Manifold Stable Manifold Sun (rotating frame) Stable Manifold Forbidden Region Periodic Orbit Unstable Manifold (rotating frame)

24 Stable and unstable manifold tubes effect the transport of material to and from the capture region. 24 Forbidden Region Stable Manifold Unstable Manifold Sun (rotating frame) Stable Manifold Forbidden Region Periodic Orbit Unstable Manifold (rotating frame)

25 25 Tubes of transit orbits can be utilized for ballistic capture. Trajector Begins Inside Tube Sun (rotating frame) Ends in Ballistic Capture Passes through Equilibrium Region (rotating frame)

26 Invariant manifold tubes are global objects etend far beond vicinit of Lagrange points. 26 Ballistic Capture Orbit Tube of Transit Orbits (rotating frame) Sun orbit Forbidden Region (rotating frame)

27 Twisting of Orbits in the Equilibrium Region Recall that the equilibrium region roughl has the dnamics of a saddle harmonic oscillator. Orbits twist in the equilibrium region, roughl following the Lapunov orbit. The closer the approach to the Lapunov orbit, the more the orbit twists. Thus, the closer an orbit begins to the tube on its approach to the equilibrium region, the more it will be twisted when it eits the equilibrium region. This twisting near the tubes will be used in the backward targeting portion of the -to-moon trajector. 27

28 Twisting in the equilibrium region can be understood in terms of mappings between the bounding spheres. 28 Twisting of Orbits in Equilibrium Region ψ 2 (γ 2 ) δ 2 γ 2 Q 3 P 4 Q 2 P 2 ψ 3 (γ 3 ) δ 3 γ 4 γ 3 ψ 4 (γ 4 ) δ 4 P 1 P 3 Q 1 Q 4 γ 1 ψ 1 (γ 1 ) δ 1 Left Bounding Sphere Right Bounding Sphere

29 We select a line of constant -position passing through the in the capture region. Pick a few points near the unstable tube, with the same position but slightl different velocities. Targeting Using "Twisting" 29 Sun orbit

30 With a slight change in the velocit, we can target an position on the lower line, where the mirror image stable tube intersects. Targeting Using "Twisting" 30 Sun orbit

31 Look at Poincaré section along this line of constant -position. With infinitesimal changes in velocit, an point near lower tube cross-section can be targeted. 31 Poincare Section Targeting Using "Twisting" P -1 (q 2 ) P -1 (q 1 ) q 1 q 2 -velocit Pre-Image of Strip S Stable Manifold Strip S Unstable Manifold -position Sun orbit -position -position

32 Two Coupled 3-Bod Sstems We obtain the fuel efficient tranfer b taking full advantage of the dnamics of the 4-bod sstem (, Moon, Sun, and spacecraft) b initiall modeling it as two coupled planar circular restricted 3-bod sstems. In this approach, we utilize the Lagrange point dnamics of both the -Moon-spacecraft and Sun--spacecraft sstems. 32 Maneuver ( V) at Patch Point Sun--S/C Sstem -Moon-S/C Sstem Sun Moon

33 In this simplified model, the Moon is on a circular orbit about the, the (or rather the -Moon center of mass) is on a circular orbit about the Sun, and the sstems are coplanar. In the actual solar sstem: Moon s eccentricit is s eccentricit is Moon s orbit is inclined to s orbit b 5 These values are low, so the coupled planar circular 3-bod problem is considered a good starting model. An orbit which becomes a real mission is tpicall obtained first in such an approimate sstem and then later refined through more precise models which inlude effects such as out-of-plane motion, eccentricit, the other planets, solar wind, etc. However, tremendous insight is gained b considering a simpler model which reveals the essence of the transfer dnamics. 33

34 Notice that in the Sun- rotating frame, patterns such as the Sun- portion of the trajector are made plain which are not discernable in the inertial frame. Inertial Frame 34 Sun- Rotating Frame Targeting Portion Using "Twisting" V Lunar Capture Portion V Moon's Orbit Ballistic Capture Sun orbit

35 The construction is done mainl in the Sun- rotating frame using the Poincaré section (that passes through the -position of the ) which help to glue the Sun- Lagrange point orbit portion of the trajector with the lunar ballistic capture portion. 35 Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon's Orbit orbit Poincare Section Taken Along This Line

36 Construction of -to-moon Transfer Strateg: Find an initial condition (position and velocit) for a spacecraft on the Poincaré section such that: when integrating forward, spacecraft will be guided b - Moon manifold and get ballisticall captured b the Moon; when integrating backward, spacecraft will hug Sun- manifolds and return to. We utilize two important properties of Lagrange point dnamics: Tube of transit orbits is ke in finding capture orbit for - Moon portion of the design; Twisting of orbits in equilibrium region is ke in finding a fuel efficient transfer for Sun- portion of the design. 36

37 Lunar Ballistic Capture Portion Recall that b targeting the region enclosed b the stable manifold tube of the -Moon-S/C sstem s Lapunov orbit, we can construct an orbit which will get ballisticall captured b the Moon Transit Orbits Ballistic Capture Orbit Tube of Transit Orbits Ballistic Capture Initial Condition Poincare Section. r -0.4 Poincare Cut of Stable Manifold orbit r Forbidden Region Moon

38 When we transform the Poincaré section of the stable manifold of the Lapunov orbit about the -Moon point into the Sun- rotating frame, we obtain a curve. A point interior to this curve, with the correct phasing of the Moon, will approach the Moon when integrated forward. The phasing of the Moon is determined b the orientation of the Poincaré section in the - Moon sstem Transit Orbits Ballistic Capture Orbit Tube of Transit Orbits Ballistic Capture Initial Condition Poincare Section. r -0.4 Poincare Cut of Stable Manifold orbit r Forbidden Region Moon

39 Assuming the Sun is a negligible perturbation to the -Moon- S/C 3-bod dnamics, an spacecraft with initial conditions within this closed loop will be ballisticall captured b the Moon. 39 Tube Containing Lunar Capture Orbits Sun- Rotating Frame (Sun- rotating frame) Moon's Orbit Initial Condition -Moon Orbit Stable Manifold Cut Ballistic Capture Orbit Sun Moon Initial Condition (Sun- rotating frame) Poincare Section

40 Ballistic capture b the Moon means an orbit which under natural dnamics gets within the sphere of influence of the Moon (20,000 km) and performs at least one revolution around the Moon. In such a state, a slight V will result in a stable capture (closing off the necks at L 1 and ). Trajector Begins Inside Tube 40 (rotating frame) Moon Ends in Ballistic Capture Passes through Equilibrium Region (rotating frame)

41 Backward Targeting Portion Pick an energ in the temporar capture range of the Sun- sstem which has orbit manifolds that come near a 200 km altitude parking orbit. Targeting Portion of Trajector Using "Twisting" Near Tubes 41 Sun Moon's Orbit orbit

42 Compute Poincaré section along line of constant -position passing through : The red curve is the Poincaré cut of the unstable manifold of the Lapunov orbit around the Sun-. 42 Poincare Section Backward Targeting Portion Using "Twisting" Near Tubes (Sun- rotating frame) Moon's Orbit Sun- Orbit Unstable Manifold Cut Initial Condition Sun Moon's Orbit Orbit (Sun- rotating frame)

43 Picking an initial condition just outside this curve, we can backward integrate to produce a trajector coming back to the parking orbit. 43 Poincare Section Backward Targeting Portion Using "Twisting" Near Tubes (Sun- rotating frame) Moon's Orbit Sun- Orbit Unstable Manifold Cut Initial Condition Sun Moon's Orbit Orbit (Sun- rotating frame)

44 Connecting the Two Portions Var the phase of the Moon until -Moon manifold curve intersects Sun- manifold curve. 44 (Sun- rotating frame) Poincare Section in Sun- Rotating Frame -Moon Orbit Stable Manifold Cut with Moon at Different Phases B Sun- Orbit Unstable Manifold Cut (Sun- rotating frame) A Stable Manifold Tube in -Moon Rotating Frame Stable Manifold Tube A B orbit Moon

45 Intersection is found! In the region which is in the interior of the green curve but in the eterior of the red curves, 45 Sun- Orbit Unstable Manifold Cut Initial Condition Backward Targeting Portion Using "Twisting" (Sun- rotating frame) Initial Condition -Moon Orbit Stable Manifold Cut Lunar Capture Portion Sun Moon's Orbit orbit (Sun- rotating frame) Poincare Section

46 an orbit will get ballisticall captured b the Moon when integrated forward; when integrated backward, orbit will hug the unstable manifold back to the Sun- equilibrium region with a twist, and then hug the stable manifold back towards. 46 Sun- Orbit Unstable Manifold Cut Initial Condition Backward Targeting Portion Using "Twisting" (Sun- rotating frame) Initial Condition -Moon Orbit Stable Manifold Cut Lunar Capture Portion Sun Moon's Orbit orbit (Sun- rotating frame) Poincare Section

47 With onl a slight modification, a midcourse V at the patch point (34 m/s), this procedure produces a genuine solution integrated in the bicircular 4-bod problem. Since capture at Moon is natural (zero V ), the amount of onboard fuel necessar is lowered (b about 20%). 47 Maneuver ( V) at Patch Point Targeting Portion Using "Twisting" Lunar Capture Portion Sun Moon s Orbit

48 Wh Does It Work? Heuristic arguments for using the coupled 3-bod model: When outside the Moon s small sphere of influence (20,000 km), which is most of pre-capture flight, we can consider the Moon s perturbation on the Sun--S/C 3-bod sstem to be negligible. Thus, can utilize Sun- Lagrange point invariant manifold structures. The midcourse V is performed at a point where the spacecraft is entering the s sphere of influence (900,000 km), where we can consider the Sun s perturbation on the -Moon-S/C 3-bod sstem to be negligible. Thus, -Moon Lagrange point structueres can be utilized for capture sequence. 48

49 What Net? Future Work Using differential correction and continuation, we epect this trajector can be used as an initial guess to generalize to the threedimensional 4-bod problem as well as the full solar sstem model. Optimize trajector (minimize total fuel consumption): Incorporate initial lunar swingb Appl optimal control (e.g., COOPT) Use continuous thrust (low-thrust) Develop sstematic procedure for coupling multiple 3-bod sstems. This will aid in the design of innovative space missions and help aid in understanding subtle non-keplerian transport throughout the solar sstem. 49

50 More Information and References Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, available from marsden/ 50 Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000] Shoot the Moon, available at request