Continuation of periodic orbits in two-planet resonant systems

TL;DR

This paper systematically studies the continuation of resonant periodic orbits from simplified to full two-planet systems, revealing bifurcations and including asymmetric orbits across all planetary mass ratios.

Contribution

It provides a comprehensive method to trace resonant periodic orbit families from restricted to general three-body problems, including asymmetric orbits and a full range of mass ratios.

Findings

Continuation follows Bozis and Hadjidemetriou's scheme for symmetric orbits.

Includes asymmetric periodic orbits in external resonances.

Analyzes bifurcations caused by family collisions and bifurcation point vanishing.

Abstract

The continuation of resonant periodic orbits from the restricted to the general three body problem is studied in a systematic way. Starting from the Keplerian unperturbed system we obtain the resonant families of the circular restricted problem. Then we find all the families of the resonant elliptic restricted three body problem which bifurcate from the circular model. All these families are continued to the general three body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. We consider planar motion only. We show that the continuation follows a scheme proposed by Bozis and Hadjidemetriou (1976) for symmetric orbits. Our study includes also asymmetric periodic orbits, which exist in cases of external resonances. The families formed by passing from the restricted to the general problem are continued within the framework of the general problem by varying the planetary mass ratio $\rho$. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values ($\rho \in (0,\infty)$) and, therefore we include the passage from external to internal resonances. In the present work, our numerical study includes the case of the 2/1 and 1/2 resonance. The same method can be used to study all other planetary resonances.