# An Approximate Analytic Model of a Star Cluster with Potential Escapers

## TL;DR

This paper develops an approximate analytic model of star clusters that includes potential escapers—stars with energies above escape energy but still within the cluster—using perturbation theory and orbital analysis.

## Contribution

It introduces a new phase space model incorporating potential escapers, extending traditional models that exclude unbound stars.

## Key findings

- Potential escapers can be characterized by the Jacobi integral and additional approximate integrals.
- The model successfully describes the population of stars above the escape energy within the cluster.
- Provides a framework for including unbound stars in star cluster models.

## Abstract

In the context of a star cluster moving on a circular galactic orbit, a "potential escaper" is a cluster star that has orbital energy greater than the escape energy, and yet is confined within the Jacobi radius of the stellar system. On the other hand analytic models of stellar clusters typically have a truncation energy equal to the cluster escape energy, and therefore explicitly exclude these energetically unbound stars. Starting from the landmark analysis performed by Henon of periodic orbits of the circular Hill equations, we present a numerical exploration of the population of "non-escapers", defined here as those stars which remain within two Jacobi radii for several galactic periods, with energy above the escape energy. We show that they can be characterised by the Jacobi integral and two further approximate integrals, which are based on perturbation theory and ideas drawn from Lidov-Kozai theory. Finally we use these results to construct an approximate analytic model that includes a phase space description of a population resembling that of potential escapers, in addition to the usual bound population.