# Scalar Resonant Relaxation of Stars Around a Massive Black Hole

## TL;DR

This paper develops a first-principles, parameter-free model for scalar resonant relaxation in nuclear star clusters around massive black holes, describing how stellar angular momentum evolves due to potential fluctuations.

## Contribution

It provides the first complete calculation of diffusion coefficients for scalar resonant relaxation in spherical systems from fundamental principles, with an open source implementation.

## Key findings

- Diffusion coefficients depend explicitly on the mean field stellar distribution.
- The model describes potential fluctuations as correlated noise based on stellar orbits.
- The approach enables parameter-free, first-principles predictions of angular momentum evolution.

## Abstract

In nuclear star clusters, the potential is governed by the central massive black hole, so that stars move on nearly Keplerian orbits and the total potential is almost stationary in time. Yet, the deviations of the potential from the Keplerian one, due to the enclosed stellar mass and general relativity, will cause the stellar orbits to precess. Moreover, as a result of the finite number of stars, small deviations of the potential from spherical symmetry induce residual torques that can change the stars' angular momentum faster than the standard two-body relaxation. The combination of these two effects drives a stochastic evolution of orbital angular momentum, a process named "resonant relaxation". Owing to recent developments in the description of the relaxation of self-gravitating systems, we can now fully describe scalar resonant relaxation (relaxation of the magnitude of the angular momentum) as a diffusion process. In this framework, the potential fluctuations due to the complex orbital motion of the stars are described by a random correlated noise, whose statistical properties are fully characterized by the stars' mean field motion. On long timescales, the cluster can be regarded as a diffusive system, whose diffusion coefficients depend explicitly on the mean field stellar distribution through the properties of the noise. We show here, for the first time, how the diffusion coefficients of scalar resonant relaxation, for a spherically symmetric system, can be fully calculated from first principles, without any free parameters. We also provide an open source code that evaluates these diffusion coefficients numerically.