Gauss's method for secular dynamics, softened

TL;DR

This paper extends Gauss's method for secular gravitational dynamics to include softened interactions, providing a versatile and efficient tool for studying the long-term evolution of nearly Keplerian systems like stellar clusters and planetary disks.

Contribution

The authors adapt Gauss's classical algorithm to softened gravity, enabling accurate and efficient analysis of secular dynamics in various astrophysical systems.

Findings

Validated the method's accuracy across different configurations

Demonstrated the instability and mode saturation in counter-rotating disks

Showcased the method's efficiency for long-term evolution studies

Abstract

We show that the algorithm proposed by Gauss to compute the secular evolution of gravitationally interacting Keplerian rings extends naturally to softened gravitational interactions. The resulting tool is ideal for the study of the secular dynamical evolution of nearly Keplerian systems such as stellar clusters surrounding black holes in galactic nuclei, cometary clouds, or planetesimal discs. We illustrate its accuracy, efficiency and versatility on a variety of configurations. In particular, we examine a secularly unstable unstable system of counter-rotating disks, and follow the unfolding and saturation of the instability into a global, uniformly precessing, lopsided (m=1) mode.