Families of Symmetric Exchange Orbits in the Planar (1+2n)-Body Problem

• Bengochea, Abimael ^[2] ; Galán-Vioque, Jorge ^[1] ; Pérez-Chavela, Ernesto ^[2]
  1. [1] Universidad de Sevilla

     Universidad de Sevilla

     Sevilla, España

     
  2. [2] Department of Mathematics, ITAM
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00473-x
• Enlaces
  • Texto completo
• Resumen

  • We study a particular (1+2n)-body problem, conformed by a massive body and 2n equal small masses, since this problem is related with Maxwell’s ring solutions, we call planet to the massive body, and satellites to the other 2n masses. Our goal is to obtain doubly-symmetric orbits in this problem. By means of studying the reversing symmetries of the equations of motion, we reduce the set of possible initial conditions that leads to such orbits, and compute the 1-parameter families of time-reversible invariant tori. The initial conditions of the orbits were determined as solutions of a boundary value problem with one free parameter, in this way we find analytically and explicitly a new involution, until we know this is a new and innovative result. The numerical solutions of the boundary value problem were obtained using pseudo arclength continuation. For the numerical analysis we have used the value of 3.5×10−4 as mass ratio of some satellite and the planet, and it was done for n=2,3,4,5,6. We show numerically that the succession of families that we have obtained approach the Maxwell solutions as n increases, and we establish a simple proof why this should happen in the configuration.

• Referencias bibliográficas
  • Bengochea, A., Piña, E.: The Saturn, Janus and Epimetheus dynamics as a gravitational three body problem in the plane. Rev. Mexicana Fís....
  • Bengochea, A., Falconi, M., Pérez-Chavela, E.: Symmetric horseshoe periodic orbits in the general planar three-body problem. Astrophys. Space...
  • Bengochea, A., Falconi, M., Pérez-Chavela, E.: Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete...
  • Bengochea, A., Galán, J., Pérez-Chavela, E.: Doubly-symmetric horseshoe orbits in the general planar three-body problem. Astrophys. Space...
  • Bengochea, A., Galán, J., Pérez-Chavela, E.: Exchange orbits in the planar (1 + 4)-body problem. Physica D 301–302, 21–35 (2015). https://doi.org/10.1016/j.physd.2015.03.006
  • Bruno, A.D., Varin, V.P.: Periodic solutions of the restricted three-body problem for a small mass ratio. J. Appl. Math. Mech. 71, 933–960...
  • Cors, J.M., Hall, G.R.: Coorbital periodic orbits in the three body problem. SIAM J. Appl. Dyn. Syst. 2, 219–237 (2003). https://doi.org/10.1137/S1111111102411304
  • 8. Cors, J.M., Llibre, J., Ollé, M.: Central configurations of the planar coorbital satellite problem. Celest. Mech. Dynam. Astron. 89, 319–342...
  • Cors, J.M., Palacián, J.F., Yanguas, P.: On co-orbital quasi-periodic motion in the three-body problem. SIAM J. Appl. Dyn. Syst. 18, 334–353...
  • Doedel, E.J.: AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr, Numer. 30, 265–284 (1981)
  • Doedel, E.J., Paffenroth, R.C., Keller, H.B., Dichmann, D.J., Galán-Vioque, J., Vanderbauwhede, A.: Computation of Periodic Solutions of Conservative...
  • Dermott, S.F., Murray, C.D.: The dynamics of tadpole and horseshoe orbits: I. Theory. Icarus 48, 1–11 (1981). https://doi.org/10.1016/0019-1035(81)90147-0
  • Dermott, S.F., Murray, C.D.: The dynamics of tadpole and horseshoe orbits: II. The coorbital satellites of saturn. Icarus 48, 12–22 (1981)....
  • Funk, B., Dvorak, R., Schwarz, R.: Exchange orbits: an interesting case of co-orbital motion. Celest. Mech. Dynam. Astron. 117, 41–58 (2013)....
  • Galán, J., Muñoz-Almaraz, F. J., Freire, E., Doedel, E., Vanderbauwhede, A.: Stability and Bifurcations of the Figure-8 Solution of the Three-Body...
  • Lamb, J.S.W., Roberts, J.A.G.: Time-reversal symmetry in dynamical systems: A survey. Phys. D 112, 1–39 (1998). https://doi.org/10.1016/S0167-2789(97)00199-1
  • Laughlin, G., Chambers, J.E.: Extrasolar trojans: the viability and detectability of planets in the 1:1 resonance. Astrophys. J. 124, 592–600...
  • Llibre, J., Ollé, M.: The motion of Saturn coorbital satellites in the Restricted Three-Body Problem. Astron. Astrophys. 378, 1087–1099 (2001)....
  • Maxwell, C.J.: On the stability of the motion of Saturn’s Rings. In: Niven, W.D. (ed.) The Scientific Papers of James Clerk Maxwell, pp. 288–376....
  • Muñoz-Almaraz, F.J., Freire, E., Galán, J., Doedel, E., Vanderbauwhede, A.: Continuation of periodic orbits in conservative and hamiltonian...
  • Muñoz-Almaraz, F.J., Freire, E., Galán, J., Vanderbauwhede, A.: Continuation of normal doubly symmetric orbits in conservative reversible...
  • Niederman, L., Pousse, A., Robutel, P.: On the co-orbital motion in the three-body problem: existence of quasi-periodic horseshoe-shaped orbits....
  • Renner, S., Sicardy, B.: Stationary configurations for co-orbital satellites with small arbitrary masses. Celest. Mech. Dynam. Astron. 88,...
  • Vanderbauwhede, A.: Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete Contin. Dyn. Syst....
  • Verrier, P.E., McInnes, C.: Periodic orbits for three and four co-orbital bodies. Mon. Not. R. Astron. Soc. 442, 3179–3191 (2014). https://doi.org/10.1093/mnras/stu1056
  • Salo, H., Yoder, C. F.: Dynamics of coorbital satellite rings. In: Valtonen, M. J. (ed.) The few body problem, Astrophys. Space Sci. Libr.,...