The spin-spin model and the capture into the double synchronous resonance

TL;DR

This paper introduces a planar spin-spin model for two extended bodies orbiting each other, analyzes the existence and stability of double synchronous resonances, and explores their capture mechanisms under dissipative tidal effects, with applications to celestial systems.

Contribution

It extends the classical spin-orbit problem to a coupled spin-spin model, providing theoretical conditions for resonance existence and stability, including in dissipative regimes.

Findings

Existence of a unique, linearly stable double synchronous resonance in the conservative model.

Continuation of the resonance solution into the dissipative regime showing asymptotic stability.

Application of the model to real celestial systems like Pluto-Charon and 617 Patroclus.

Abstract

The aim of this article is to propose a model, that is a planar version of the Full Two-Body Problem, and discuss the existence and stability of a relevant periodic solution. Consider two homogeneous ellipsoids orbiting around each other in fixed coplanar Keplerian orbits. Moreover, their respective spin axes are assumed to be perpendicular to the orbital plane, that is also a common equatorial plane. The spin-spin model deals with the coupled rotational dynamics of both ellipsoids. For a non-zero orbital eccentricity, it has the structure of a non-autonomous system of coupled pendula. This model is a natural extension of the classical spin-orbit problem for two extended bodies. In addition, we consider dissipative tidal torques, that can trigger the capture of the system into spin-orbit and spin-spin resonances. In this paper we give some theoretical results for both the conservative model and the dissipative one. The conservative model has a Hamiltonian structure. We use properties of Hamiltonian systems to give some sufficient conditions in the space of parameters of the model, that guarantee existence, uniqueness and linear stability of an odd periodic solution. This solution represents a double synchronous resonance in the conservative regime. Such solution can be continued to the dissipative regime, where it becomes asymptotically stable. We see asymptotic stability as a dynamical mechanism for the capture into the double synchronous resonance. Finally we apply our results to several cases including the Pluto-Charon binary system and the Trojan binary asteroid 617 Patroclus, target of the LUCY mission.