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Comet and moon solutions in the time-dependent restricted $(n+1)$-body problem

Carlos Barrera, Abimael Bengochea, Carlos García-Azpeitia

Abstract

The time-dependent restricted $(n+1)$-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by $n$ primary bodies following a periodic solution of the $n$-body problem. We prove that the satellite has periodic solutions close to the large-amplitude circular orbits of the Kepler problem (comet solutions), and in the case that the primaries are in a relative equilibrium, close to small-amplitude circular orbits near a primary body (moon solutions). The comet and moon solutions are constructed with the application of a Lyapunov-Schmidt reduction to the action functional. In addition, using reversibility technics, we compute numerically the comet and moon solutions for the case of four primaries following the super-eight choreography.

Comet and moon solutions in the time-dependent restricted $(n+1)$-body problem

Abstract

The time-dependent restricted -body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by primary bodies following a periodic solution of the -body problem. We prove that the satellite has periodic solutions close to the large-amplitude circular orbits of the Kepler problem (comet solutions), and in the case that the primaries are in a relative equilibrium, close to small-amplitude circular orbits near a primary body (moon solutions). The comet and moon solutions are constructed with the application of a Lyapunov-Schmidt reduction to the action functional. In addition, using reversibility technics, we compute numerically the comet and moon solutions for the case of four primaries following the super-eight choreography.

Paper Structure

This paper contains 16 sections, 15 theorems, 101 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting up the problem
  3. The comet problem
  4. The moon problem
  5. Lyapunov-Schmidt reduction
  6. Non-degeneracy condition
  7. Lyapunov-Schmidt reduction
  8. Comet and moon solutions
  9. Main theorems for $\alpha\geq1$ ($\alpha\neq2$)
  10. Main theorems for $\alpha=2$
  11. Numerical study of periodic orbits
  12. Super-eight choreography of the 4-body problem
  13. The restricted 5-body problem for the super-eight choreography
  14. Reversing symmetries of the restricted five-body problem
  15. Comet orbits
  16. ...and 1 more sections

Key Result

Proposition 1

Set $\omega^{2}=\varepsilon^{(\alpha+1)}$. For $\alpha\geq1$, the solutions of restricted problem body in the coordinate $x(\tau)$ (given by cambio_q_c) are critical points of the action where $\mathcal{H}(x)=\int_{0}^{2\pi}h(x(\tau),\tau)d\tau$ with and $x_{j}(\tau)$ are given by cambio_qj_c.

Figures (2)

  • Figure 1: Left: Multiple symmetric periodic orbits of comet type in the restricted 5-body problem. Right: A symmetric periodic orbit of moon type in the restricted 5-body problem.
  • Figure 2: Gerver's super eight choreography. At left an isosceles reversible configuration, at time $t=0$. At right an orthogonal reversible configuration, at time $t = \overline{T}$.

Theorems & Definitions (19)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Remark 8
  • Proposition 9
  • Theorem 10
  • ...and 9 more