Abstract
A modification of the Hill problem when the larger primary is a source of radiation is considered and asymptotic motions around the collinear equilibrium points are studied. Our work focuses on the computation of homoclinic orbits to the collinear equilibrium points themselves or to the Lyapunov orbits emanating from each equilibrium point. These orbits depart asymptotically from an equilibrium point (or a Lyapunov orbit) and return to the same point (or orbit) asymptotically. In both cases, semi-analytical solutions have been obtained in order to determine appropriate initial conditions which have been used as suitable seed for the numerical computation of the asymptotic orbits with a predetermined accuracy. In addition, for homoclinic orbits to the Lyapunov periodic orbits, transversality is achieved by the construction of appropriate surface of section portraits of the unstable manifolds.
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C.C. Conley, Low energy transit orbits in the restricted three-body problem. SIAM J. Appl. Math. 16, 732–746 (1968)
M.K. Das, P. Narang, S. Mahajan, M. Yuasa, Effect of radiation on the stability of equilibrium points in the binary stellar systems: RW–Monocerotis, Kr\(\ddot u\)ger 60. Astrophys. Space Sci. 314, 261–274 (2008)
K.E. Davis, R.L. Anderson, D.J. Scheeres, G.H. Born, The use of invariant manifolds for transfers between unstable periodic orbits of different energies. Celest. Mech. Dyn. Astron. 107, 471–485 (2010)
A. Deprit, J. Henrard, Symmetric doubly asymptotic orbits in the restricted three-body problem. Astron. J. 70, 271–274 (1965)
A. Deprit, J. Henrard, Construction of orbits asymptotic to a periodic orbit. Astron. J. 74, 308–316 (1969)
G. Gómez, J.M. Mondelo, The dynamics around the collinear equilibrium points of the RTBP. Phys. D 157(4), 283–321 (2001)
G. Gómez, M. Marcote, J.M. Mondelo, The invariant manifold structure of the spatial Hill’s problem. Dyn. Syst. Int. J. 20, 115–147 (2005)
M. Hénon, Exploration Numérique du Problème Restreint II – Masses Égales, Stabilité des Orbites Périodiques. Ann. Astrophys. 28, 992–1007 (1965)
K.C. Howell, D.C. Davis, A.F. Haapala, Application of periapse maps for the design of trajectories near the smaller primary in multi–body regimes. Math. Probl. Eng. 2012, Article ID 351759, 22 pp. (2012)
V.S. Kalantonis, C.N. Douskos, E.A.Perdios, Numerical determination of homoclinic and heteroclinic orbits at collinear equilibria in the restricted three-body problem with oblateness. Celest. Mech. Dyn. Astron. 94, 135–153 (2006)
W.S. Koon, M.W. Lo, J.E. Marsden, S.D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics. Chaos 10(2), 427–469 (2000)
A.L. Kunitsyn, A.T. Tureshbaev, On the collinear libration points in the photo-gravitational three-body problem. Celest. Mech. 35, 105–112 (1985)
A.L. Kunitsyn, E.N. Polyakhova, The restricted photogravitational three-body problem: a modern state. Astron. Astrophys. Trans. 6, 283–293 (1995)
J. Llibre, R. Martínez, C. Simó, Transversality of the invariant manifolds associated to the Lyapunov family of the periodic orbits near L2 in the restricted three-body problem. J. Differ. Equ. 58, 104–156 (1985)
V.V. Markellos, A.E. Roy, M.J. Velgakis, S.S. Kanavos, A photogravitational Hill problem and radiation effects on Hill stability of orbits. Astrophys. Space Sci. 271, 293–301 (2000)
Z. Niedzielska, Nonlinear stability of the libration points in the photogravitational restricted three body problem. Celest. Mech. Dyn. Astron. 58, 203–213 (1994)
K.E. Papadakis, Homoclinic and heteroclinic orbits in the photogravitational restricted three–body problem. Astrophys. Space Sci. 302, 67–82 (2006)
N. Pathak, V.O. Thomas, Evolution of the f family orbits in the photo gravitational Sun-Saturn system with oblateness. Int. J. Astron. Astrophys. 6, 254–271 (2016)
E.A. Perdios, V.V. Markellos, Symmetric doubly-asymptotic periodic orbits at collinear equilibria. Astrophys. Space Sci. 166, 129–149 (1990)
D.W. Schuerman, Roche potentials including radiation effects. Astrophys. Space Sci. 19, 351–358 (1972)
D.W. Schuerman, The restricted three–body problem including radiation pressure. Astrophys. J. 238, 337–342 (1980)
C. Simó, T.J. Stuchi, Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Phys. D 140, 1–32 (2000)
J.F.L. Simmons, A.J.C. McDonald, J.C. Brown, The restricted 3-body problem with radiation pressure. Celest. Mech. 35, 145–187 (1985)
P. Verrier, T. Waters, J.Sieber, Evolution of the L1 halo family in the radial solar sail circular restricted three–body problem. Celest. Mech. Dyn. Astron. 120, 373–400 (2014)
B.F. Villac, D.J. Scheeres, Escaping trajectories in the Hill three–body problem and applications. J. Guid. Control Dyn. 26, 224–232 (2003)
C. Zagouras, V.V. Markellos, Three-dimensional periodic solutions around equilibrium points in Hill’s problem. Celest. Mech. 35, 257–267 (1985)
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The authors would like to thank Prof. V.V. Markellos for valuable discussions during this work.
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Kalantonis, V.S., Perdiou, A.E., Douskos, C.N. (2018). Asymptotic Orbits in Hill’s Problem When the Larger Primary is a Source of Radiation. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_18
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