# Vector Resonant Relaxation of Stars around a Massive Black Hole

## TL;DR

This paper develops an analytical framework for vector resonant relaxation of stars near a massive black hole, comparing it with simulations and providing a stochastic model for orbital reorientation.

## Contribution

It introduces a first-principles derivation of potential fluctuation correlations and a stochastic scheme for orbital orientation evolution, advancing understanding of stellar dynamics near black holes.

## Key findings

- Analytical correlation functions for potential fluctuations
- Agreement between theory and N-body simulations
- Quantitative estimates for Milky Way-like clusters

## Abstract

In the vicinity of a massive black hole, stars move on precessing Keplerian orbits. The mutual stochastic gravitational torques between the stellar orbits drive a rapid reorientation of their orbital planes, through a process called vector resonant relaxation. We derive, from first principles, the correlation of the potential fluctuations in such a system, and the statistical properties of random walks undergone by the stellar orbital orientations. We compare this new analytical approach with effective $N$-body simulations. We also provide a simple scheme to generate the random walk of a test star's orbital orientation using a stochastic equation of motion. We finally present quantitative estimations of this process for a nuclear stellar cluster such as the one of the Milky Way.