Vector Resonant Relaxation and Statistical Closure Theory. I. Direct Interaction Approximation

TL;DR

This paper applies statistical closure theory, specifically the Direct Interaction Approximation, to model vector resonant relaxation of stars near supermassive black holes, comparing theoretical predictions with numerical simulations.

Contribution

It introduces a novel application of the Martin-Siggia-Rose formalism to astrophysical stellar dynamics, providing explicit correlation function predictions for vector resonant relaxation.

Findings

The D.I.A. approximation captures key features of the correlation functions.

Theoretical predictions agree with simulations within certain limits.

Limitations of the approach are discussed and future directions are proposed.

Abstract

Stars orbiting a supermassive black hole in the center of galaxies undergo very efficient diffusion in their orbital orientations: this is "Vector Resonant Relaxation". Such a dynamics is intrinsically non-linear, stochastic, and correlated, hence bearing deep similarities with turbulence in fluid mechanics or plasma physics. In that context, we show how generic methods stemming from statistical closure theory, namely the celebrated "Martin-Siggia-Rose formalism", can be used to characterize the correlations describing the redistribution of orbital orientations. In particular, limiting ourselves to the leading order truncation in this closure scheme, the so-called "Direct Interaction Approximation", and placing ourselves in the limit of an isotropic distribution of orientations, we explicitly compare the associated prediction for the two-point correlation function with measures from numerical simulations. We discuss the successes and limitations of this approach and present possible future venues.