An efficient integrator for stellar dynamics in effective gravity fields based on the isochrone potential

LTE, Observatoire de Paris, Université PSL, Sorbonne Université, Université de Lille, LNE, CNRS, 61 Avenue de l’Observatoire, 75014 Paris, France

^★ Corresponding author: alexandre.bougakov@obspm.fr

Received: 24 January 2025
Accepted: 19 March 2025

Abstract

Context. Integrating the motion of stars immersed in some smoothed potential is necessary in many stellar and galactic studies. Previous works have generally used numerical integrators that alternate between linear drifts and velocity kicks (such as the standard Leapfrog scheme). The low efficiency of this approach contrasts with the sophisticated methods developed in other fields such as planetary dynamics, for which integrators alternate between Keplerian drifts and velocity kicks.

Aims. Inspired by the splitting methods used in planetary dynamics, we aim to build an efficient integration scheme dedicated to stellar and galactic dynamics.

Methods. We took advantage of the peculiar properties of Hénon’s isochrone potential to design a new symplectic splitting scheme that can be used to integrate the motion of stars in any gravitational potential. This scheme alternates between isochrone drifts and velocity kicks. As a first application, we considered the motion of a star in a Plummer potential – an essential constituent of galactic potentials – and determined the set of integration parameters that provide the best integration efficiency (i.e. the best conservation of total energy for the lowest computational cost).

Results. We derived the analytical solution for all possible kinds of orbits in Hénon’s isochrone potential (bound and unbound trajectories) as needed in our integration scheme. Our numerical experiments for stars in a Plummer potential show excellent performances in regions in the inner and outer parts of the gravity field, that is, where the motion of stars is well approximated by isochrone trajectories (with perturbations of order 10⁻³ or less). For highly elongated orbits that cross the characteristic length of the Plummer potential, the performance is equivalent to that of previous methods.

Conclusions. The splitting scheme presented here is a good alternative to previous methods: it performs at least as well, and up to orders of magnitude better, depending on the dynamical regime of the star.