Multistability and Gibbs entropy in the planar dissipative spin-orbit problem

TL;DR

This study explores the complex dynamical behavior of the dissipative spin-orbit problem, revealing multiple attractors and basin structures, and introduces Gibbs entropy as a measure of basin dominance, with applications to celestial bodies like Hyperion, the Moon, and Mercury.

Contribution

The paper provides a numerical analysis of multistability and basin complexity in the dissipative spin-orbit problem, applying Gibbs entropy to quantify basin dominance, and extends the methodology to real celestial systems.

Findings

Basins of attraction have intricate structures that vary with orbital eccentricity.

Gibbs entropy effectively measures the dominance of specific basins in phase space.

The methodology is applicable to celestial bodies such as Hyperion, the Moon, and Mercury.

Abstract

In this work, we numerically investigate and visually illustrate the dynamical properties of the dissipative spin-orbit problem such as the co-existence of multiple periodic and quasi-periodic attractors, and the complexity of the corresponding basins of attraction. Our model is composed by a triaxial satellite (planet) orbiting a planet (star) in a fixed Keplerian orbit with zero obliquity. A dissipative tidal torque that is proportional to its rotational angular velocity is assumed to be acting on the satellite. We use Hyperion as a toy model to characterize the methodology used, since this system has a very rich conservative dynamical scenario, and we later apply our methodology to the Moon and Mercury. Our results show that the basins of attraction may possess an intricate structure in all cases which changes with the orbital eccentricity, and that the Gibbs entropy is a good measure on how dominant one basin is over the others in the phase space.