Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero Collisions

TL;DR

This paper demonstrates that Landau modes become true eigenmodes in stellar systems as collisions approach zero, clarifying their role in the spectrum and implications for stellar dynamics and fluctuations.

Contribution

It analytically and numerically shows Landau modes are true eigenmodes in the zero-collision limit, resolving their previous classification as non-eigenmodes in collisionless systems.

Findings

Landau modes become true eigenmodes as collisions vanish.

The continuous spectrum of collisionless systems is eliminated in this limit.

Deviations from Maxwellian distribution significantly alter the Landau mode spectrum.

Abstract

We consider the spectrum of eigenmodes in a stellar system dominated by gravitational forces in the limit of zero collisions. We show analytically and numerically using the Lenard-Bernstein collision operator that the Landau modes, which are not true eigenmodes in a strictly collisionless system (except for the Jeans unstable mode), become part of the true eigenmode spectrum in the limit of zero collisions. Under these conditions, the continuous spectrum of true eigenmodes in the collisionless system, also known as the Case-van Kampen modes, is eliminated. Furthermore, since the background distribution function in a weakly collisional system can exhibit significant deviations from a Maxwellian distribution function over long times, we show that the spectrum of Landau modes can change drastically even in the presence of slight deviations from a Maxwellian, primarily through the appearance of weakly damped modes that may be otherwise heavily damped for a Maxwellian distribution. Our results provide important insights for developing statistical theories to describe thermal fluctuations in a stellar system, which are currently a subject of great interest for N-body simulations as well as observations of gravitational systems.