Satellites of the Broucke-Hadjidemetriou-H\'{e}non family of periodic unequal-mass three-body orbits

TL;DR

This paper reports an extensive discovery of nearly 600,000 new periodic three-body orbits and satellites with unequal masses, revealing new stability properties and differences in angular momentum-period relationships compared to equal-mass cases.

Contribution

It provides the largest set of newly identified unequal-mass three-body orbits and satellites, including stability analysis and new insights into their dynamical relationships.

Findings

Over 598,000 new orbits and satellites identified.

Approximately 33.5% of these orbits are linearly stable.

Discovered differences in angular momentum-period relationships for unequal masses.

Abstract

The Broucke-Hadjidemetriou-H\'{e}non's (BHH) orbits are a family of periodic orbits of the three-body system with the simplest topological free group word $a$, while the BHH satellites have free group words $a^k$ ($k>1$), where $k$ is the topological exponent. Jankovi\'{c} and Dmitra\v{s}inovi\'{c} [Phy. Rev. Lett. 116, 064301 (2016)] reported 57 new BHH satellites with equal mass and found that at a fixed energy the relationship between the angular momentum ($L$) and the topologically rescaled period ($T/k$) is the same for both of the BHH orbits ($k=1$) and the BHH satellites ($k>1$). In this letter, we report 419,743 new BHH orbits ($k=1$) and 179,253 new BHH satellites ($k>1$) of the three-body system with unequal mass, which have never been reported, to the best of our knowledge. Among these newly-found 598,996 BBH orbits and satellites, about 33.5 % (i.e., 200,686) are linearly stable and thus many among them might be observed in practice. Besides, we discover that, for the three-body system with unequal mass at a fixed energy, relationship between the angular momentum ($L$) and topologically rescaled period ($T/k$) of the BHH satellites ($k>1$) is different from that of the BHH orbits ($k=1$).