The structure of the co-orbital stable regions as a function of the mass ratio

TL;DR

This study investigates the structure of co-orbital stable regions across various mass ratios using numerical simulations and develops empirical equations to predict their size and location, aiding future searches for co-orbital bodies.

Contribution

It provides the first comprehensive empirical model linking co-orbital stable region characteristics to system mass ratios over a wide range.

Findings

Horseshoe region upper limit between 9.539e-4 and 1.192e-3 in mass ratio

Stability limit at mass ratio approximately 2.3313e-2, lower than linear theory predicts

Polynomial equations accurately estimate co-orbital stable region parameters

Abstract

Although the search for extra-solar co-orbital bodies has not had success so far, it is believed that they must be as common as they are in the Solar System. Co-orbital systems have been widely studied, and there are several works on stability and even on formation. However, for the size and location of the stable regions, authors usually describe their results but do not provide a way to find them without numerical simulations, and, in most cases, the mass ratio value range is small. In the current work, we study the structure of co-orbital stable regions for a wide range of mass ratio systems and built empirical equations to describe them. It allows estimating the size and location of co-orbital stable regions from a few system's parameters. Thousands of massless particles were distributed in the co-orbital region of a massive secondary body and numerically simulated for a wide range of mass ratios ($\mu$) adopting the planar circular restricted three-body problem. The results show that the horseshoe regions upper limit is between $9.539 \times 10^{-4} < \mu < 1.192 \times 10^{-3}$, which correspond to a minimum angular distance from the secondary to the separatrix between $27.239^{o} $ and $27.802^{o} $. We also found that the limit to exist stability in the co-orbital region is about $\mu = 2.3313 \times 10^{-2}$, much smaller than the value predicted by the linear theory. Polynomial functions to describe the stable region parameters were found, and they represent estimates of the angular and radial widths of the co-orbital stable regions for any system with $9.547 \times 10^{-5} \leq \mu \leq 2.331 \times 10^{-2}$.