C C Dynamical A L T E C S H Design of a Multi-Moon Orbiter Shane D. Ross Control and Dynamical Systems and JPL, Caltech W.S. Koon, M.W. Lo, J.E. Marsden AAS/AIAA Space Flight Mechanics Meeting Ponce, Puerto Rico February 9-13, 2003
Interplanetary Mission Design Use natural dynamics for fuel efficiency Dynamical channels connect planets and moons. Trajectory generation using invariant manifolds in the 3-body problem suggests new numerical algorithms for interplanetary missions Current research importance Design fuel efficient interplanetary trajectories (1) Multi-Moon Orbiter to multiple Jovian moons (2) Earth orbit to lunar orbit 2
Mission to Europa Motivation: Oceans and life on Europa? There is international interest in sending a scientific spacecraft to orbit and study Europa. 3
Multi-Moon Orbiter Orbit each moon in a single mission Other Jovian moons are also worthy of study All may have oceans, evidence from Galileo suggests 5
Multi-Moon Orbiter We propose a trajectory design procedure which uses little fuel and allows a single spacecraft to orbit multiple moons Orbit each moon for much longer than the quick flybys of previous missions Using a standard patched-conics approach, the V necessary would be prohibitively high By decomposing the N-body problem into 3-body problems and using the natural dynamics of the 3-body problem, the V can be lowered significantly 6
Multi-Moon Orbiter First attempt: A Ganymede-Europa Orbiter was constructed V of 1400 m/s was half the Hohmann transfer Gómez, Koon, Lo, Marsden, Masdemont, and Ross [2001] Maneuver performed Jupiter Ganymede's orbit Europa's orbit 0.02 z 1.5 1 0.5 Close approach to Ganymede y 0-0. 5-1 -1. 5-1. 5-1 (a) -0. 5 0 x 0.5 1 1.5 Injection into high inclination orbit around Europa 0.01 0.015 0.01 0.005 0.005 z 0-0.005 z 0-0.01-0.015-0.005-0.02 0.02 0.01 y 0-0.01-0.02 0.98 0.99 1 x 1.01 1.02-0.01 0.99 0.995 x 1 1.005 1.01-0.01-0.005 0 y 0.005 0.01 (b) (c) 7
Multi-Moon Orbiter Refinement Preceding V of 1400 m/s for the Ganymede-Europa orbiter was half the Hohmann transfer (that is, using patched conics, as in manned moon missions) Desirable to decrease V further one now does not directly tube-hop, but rather makes more refined use of the phase space structure New things: resonant gravity assists with the moons Interesting: still fits well with the tube-hopping method 32
Multi-Moon Orbiter Second attempt: Desirable to decrease V further One can consider using resonant gravity assists with the moons, leading to ballistic captures Consider the following tour of Jupiter s moons Begin in an eccentric orbit with perijove at Callisto s orbit, achievable using a patched-conics trajectory from the Earth to Jupiter Orbit Callisto, Ganymede, and Europa 8
Multi-Moon Orbiter V = 22 m/s, but flight time is a few years Low Energy Tour of Jupiter s Moons Seen in Jovicentric Inertial Frame Callisto Ganymede Europa Jupiter 9
Multi-Moon Orbiter Results are promising This result is preliminary Model is a restricted bicircular 5-body problem A user-assisted algorithm was necessary to produce it An automated algorithm is a future goal Future challenges The flight time is too long; should be reduced below 18 months Evidence to be presented later in this talk suggests that a significant decrease in flight time can be gained for a modest increase in V Radiation dose is not accounted for; will be included in future models 10
Construction Procedure Building blocks Patched three-body model: linking two adjacent three-body systems Inter-moon transfer: decreasing Jovian energy via resonant gravity assists Orbiting each moon: ballistic capture and escape Small impulsive manuevers: to steer spacecraft in sensitive phase space We will give some background on these issues 11
Inter-Moon Transfer Spacecraft gets a gravity assist from outer moon M 1 when it passes through apoapse if near a resonance When periapse close to inner moon M 2 s orbit is reached, it takes control ; this occurs for ellipse E Leaving moon M 1 Approaching moon M 2 Apoapse A fixed Periapse P fixed 15
Inter-Moon Transfer Small impulsive maneuvers are performed at opposition Exterior Realm Forbidden Realm Semimajor Axis vs. Time a+ S/C J L 1 M Interior Realm L 2 Spacecraft Maneuvers Performed When Opposite Moon a a a+ Moon Realm 16
Inter-Moon Transfer The transfer between three-body systems occurs when energy surfaces intersect; can be seen on semimajor axis vs. eccentricity diagram (similar to Tisserand curves of Longuski et al.) Spacecraft jumping between resonances on the way to Europa 0.3 Spacecraft path 0.25 eccentricity 0.2 0.15 0.1 0.05 Curves of Constant 3 Body Energy within each system 0 E G C 1 1.5 2 2.5 3 3.5 4 semimajor axis (a Europa = 1) 17
Ballistic Capture An L 2 orbit manifold tube leading to ballistic capture around a moon is shown schematically Escape is the time reverse of ballistic capture 18
Why Does It Work? Recall the planar circular restricted three-body problem: motion of a spacecraft in the gravitational field of two larger bodies in circular motion. View in rotating frame = constant energy E Forbidden Realm (at a particular energy level) Exterior Realm Poincare section spacecraft Interior (Jupiter) Realm L 1 L 2 Moon Realm Rotating frame: different realms of motion at energy E. 19
Poincaré Surface of Section Study Poincaré surface of section at fixed energy E, reducing system to a 2-dimensional area preserving map. z P(z) Poincaré surface of section 20
Poincaré Surface of Section Poincaré section reveals mixed phase space structure: KAM tori and a chaotic sea are visible. Identify Semimajor Axis (Neptune = 1) Argument of Periapse (radians) Poincaré surface of section 21
Transport in Poincaré Section Phase space divided into regions R i, i = 1,..., N R bounded by segments of stable and unstable manifolds of unstable fixed points. R 2 q 1 q 4 q 2 R 1 B 12 = U [p 2,q 2 ] U S [p 1,q 2 ] q 5 p 2 p 3 p 1 q 3 R 3 R 4 q 6 R 5 22
Lobe Dynamics Transport btwn regions computed via lobe dynamics. L 2,1 (1) R 2 q 0 f (L 1,2 (1)) f -1 (q 0 ) q 1 p i L 1,2 (1) f (L 2,1 (1)) p j R 1 23
Movement btwn Resonances We can compute manifolds which naturally divide the phase space into resonance regions. Identify Semimajor Axis Argument of Periapse Unstable and stable manifolds in red and green, resp. 24
Movement btwn Resonances Transport and mixing between regions can be computed. Identify R 3 Semimajor Axis R 2 R 1 Argument of Periapse Four sequences of color coded lobes are shown. 25
Movement btwn Resonances Navigation from one resonance to another, essential for the Multi-Moon Orbiter, can be performed. Identify Semimajor Axis Resonance Region B } R 3 }R 2 Resonance Region A }R 1 Argument of Periapse 26
Resonances and Tubes Resonances and tubes are linked It has been observed that the tubes of capture (resp., escape) orbits are coming from (resp., going to) certain resonances. Resonances are a function of energy E and the mass parameter µ Koon, Lo, Marsden, Ross [2001] 27
Earth to Moon Trajectories Similar methods can be applied to near-earth space to study the V verses time trade-off 68
Earth to Moon Trajectories Results: much shorter transfer times than previous authors for only slightly more V Trajectories from a circular Earth orbit (r=59669 km) to a stable lunar orbit 1300 1200 Hohmann transfer V = 1220 m/s TOF = 6.6 days 1100 V (m/s) 1000 900 Present Work 800 Bollt & Meiss [1995] Schroer & Ott [1997] 700 0 100 200 300 400 500 600 700 800 Time of Flight (days) 39
Earth to Moon Trajectories Compare with Bollt and Meiss [1995] A tenth of the time for only 100 m/s more Current Result Bollt and Meiss [1995] 65 days, V = 860 m/s 748 days, V = 750 m/s TOF = 65 days V = 860 m/s 1 Day Tick Marks Earth Moon 40
e.g., GEO to Lunar Orbit GEO to Moon Orbit Transfer Seen in Geocentric Inertial Frame TOF = 63 days V = 1211 m/s 1 Day Tick Marks Moon s Orbit Earth 41
References Trade-Off Between Fuel and Time Optimization, in preparation. Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2002] Constructing a low energy transfer between Jovian moons. Contemporary Mathematics 292, 129 145. Gómez, G., W.S. Koon, M.W. Lo, J.E. Marsden, J. Masdemont and S.D. Ross [2001] Invariant manifolds, the spatial three-body problem and space mission design. AAS/AIAA Astrodynamics Specialist Conference. Koon, W.S., M.W. Lo, J.E. Marsden & S.D. Ross [2001] Resonance and capture of Jupiter comets. Celestial Mechanics & Dynamical Astronomy 81(1-2), 27 38. 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. Chaos 10(2), 427 469. For papers, movies, etc., visit the websites: http://www.cds.caltech.edu/ marsden http://www.cds.caltech.edu/ shane 27