Abstract
The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.
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References
Carletta, S., Pontani, M., Teofilatto, P.: Long-term capture orbits for low-energy space missions. Celest. Mech. Dyn. Astron. 130, 46 (2018). https://doi.org/10.1007/s10569-018-9843-7
Conley, C.C.: Low energy transit orbits in the restricted three-body problems. SIAM J. Appl. Math. 16, 732–746 (1968). https://doi.org/10.1137/0116060
Cox, A.D., Howell, K.C., Folta, D.C.: Dynamical structures in a low-thrust, multi-body model with applications to trajectory design. Celest. Mech. Dyn. Astron. 131, 12 (2019). https://doi.org/10.1007/s10569-019-9891-7
Giancotti, M., Pontani, M., Teofilatto, P.: Lunar capture trajectories and homoclinic connections through isomorphic mapping. Celest. Mech. Dyn. Astron. 114, 55–76 (2012). https://doi.org/10.1007/s10569-012-9435-x
Giancotti, M., Pontani, M., Teofilatto, P.: Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon missions. Celest. Mech. Dyn. Astron. 120, 249–268 (2014). https://doi.org/10.1007/s10569-014-9563-6
Hénon, M.: Numerical exploration of the restricted problem. V. Hill’s case: periodic orbits and their stability. A&A 1, 223–238 (1969)
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10, 427–469 (2000). https://doi.org/10.1063/1.166509
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Low energy transfer to the Moon. Celest. Mech. Dyn. Astron. 81, 63–73 (2001). https://doi.org/10.1023/A:1013359120468
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems, the Three-Body Problem and Space Mission Design. Marsden Books, Wellington (2011)
Lega, E., Guzzo, M., Froeschlé, C.: Detection of close encounters and resonances in three-body problems through Levi-Civita regularization. MNRAS 418, 107–113 (2011). https://doi.org/10.1111/j.1365-2966.2011.19467.x
Levi-Civita, T.: Sur la résolution qualitative du probleme restrient des trios corps. Acta Math. 30, 305–327 (1906). https://doi.org/10.1007/BF02418577
Moser, J.: On the generalization of a theorem of A. Liapounoff. Commun. Pure Appl. Math. 11, 257–271 (1958). https://doi.org/10.1002/cpa.3160110208
Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, New York (1999)
Oshima, K.: The role of vertical instability of Jupiter’s quasi-satellite orbits: making hazardous asteroids less detectable? MNRAS 482, 5441–5447 (2019a). https://doi.org/10.1093/mnras/sty3125
Oshima, K.: The use of vertical instability of \(L_1\) and \(L_2\) planar Lyapunov orbits for transfers from near rectilinear halo orbits to planar distant retrograde orbits in the Earth–Moon system. Celest. Mech. Dyn. Astron. 131, 14 (2019b). https://doi.org/10.1007/s10569-019-9892-6
Oshima, K., Yanao, T.: Jumping mechanisms of Trojan asteroids in the restricted three- and four-body problems. Celest. Mech. Dyn. Astron. 122, 53–74 (2015). https://doi.org/10.1007/s10569-015-9609-4
Oshima, K., Topputo, F., Campagnola, S., Yanao, T.: Analysis of medium-energy transfers to the Moon. Celest. Mech. Dyn. Astron. 127, 285–300 (2017). https://doi.org/10.1007/s10569-016-9727-7
Pontani, M., Teofilatto, P.: Low-energy Earth–Moon transfers involving manifolds through isomorphic mapping. Acta Astronaut. 91, 96–106 (2013). https://doi.org/10.1016/j.actaastro.2013.05.009
Pontani, M., Giancotti, M., Teofilatto, P.: Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies. Acta Astronaut. 105, 218–229 (2014). https://doi.org/10.1016/j.actaastro.2014.08.029
Ross, S.D., Scheeres, D.J.: Multiple gravity assists, capture, and escape in the restricted three-body problem. SIAM J. Appl. Dyn. Syst. 6, 576–596 (2007). https://doi.org/10.1137/060663374
Swenson, T., Lo, M., Anderson, B., Gorordo, T.: The topology of transport through planar Lyapunov orbits. Space Flight Mechanics Meeting, AIAA 2018-1692, Kissimmee, USA, 8–12 January (2018). https://doi.org/10.2514/6.2018-1692
Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York (1967)
Topputo, F., Vasile, M., Bernelli-Zazzera, F.: Low energy interplanetary transfers exploiting invariant manifolds of the restricted three-body problem. J. Astronaut. Sci. 53, 353–372 (2005)
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This study has been partially supported by Grant-in-Aid for JSPS Fellows No. 18J00678.
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Oshima, K. Linking low- to high-energy dynamics of invariant manifolds, transit orbits, and singular collision orbits in the planar circular restricted three-body problem. Celest Mech Dyn Astr 131, 53 (2019). https://doi.org/10.1007/s10569-019-9934-0
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DOI: https://doi.org/10.1007/s10569-019-9934-0