An orbit-averaged generalized-Landau kinetic equation for the relaxation evolution of finite weakly-coupled star clusters: `Discreteness' stochastic acceleration and anti-normalization

TL;DR

This paper develops an orbit-averaged generalized-Landau kinetic equation to model the relaxation and stochastic acceleration in finite star clusters, emphasizing the importance of discreteness effects and their role in cluster evolution.

Contribution

It introduces a detailed kinetic formulation for stochastic acceleration in star clusters, highlighting the unique properties of the generalized-Landau equation and its physical consistency.

Findings

The generalized-Landau equation satisfies the anti-normalization condition.

Stochastic acceleration can isotropize stellar distributions independently of two-body relaxation.

The equation conserves energy and star number, obeying the H-theorem.

Abstract

In the relaxation evolution of finite weakly-coupled star clusters, stars undergo stochastic acceleration due to the `discreteness' of the clusters (the finiteness of the total stellar number), in addition to fundamental two-body relaxation processes. The acceleration is the essential non-collective many-body relaxation process. However, existing works have never detailed the `discreteness' stochastic acceleration and the corresponding mathematical model, i.e., the generalized-Landau (g-Landau) kinetic equation for the stellar distribution function. The present paper shows the kinetic formulation of an orbit-averaged g-Landau equation in action-angle coordinates, beginning with Bogoliubox-Born-Green-Kirkwood-Yvon hierarchy. We show that only the g-Landau equation can satisfy the anti-normalization condition among existing approximated star-cluster kinetic equations if gravitational polarization is neglected. Accordingly, only the equation can correctly define the total energy and the total number of stars in phase space. It also holds the conservation laws and the H-theorem. We further show that stars undergoing the discreteness stochastic acceleration can hold these physical properties independently of stars experiencing the two-body relaxation. Lastly, we suggest that the stochastic acceleration tends to isotropize the stellar DF and to drive star clusters into a stationary state in the early relaxation-evolution stage before the two-body relaxation thermalizes stars.