Damped perturbations in stellar systems: Genuine modes and Landau-damped waves

TL;DR

This paper investigates the nature of damped perturbations in stellar systems, revealing the existence of discrete damped modes and clarifying the distinction between genuine modes and Landau-damped waves in a simplified stellar medium model.

Contribution

It introduces a detailed analysis of damped modes in stellar systems, highlighting the superposition of van Kampen modes and the nature of Landau damping, with implications for stability studies.

Findings

Landau-damped waves can be represented as superpositions of van Kampen modes plus a discrete damped mode.

Genuine modes depend on time as exp(-iωt), unlike Landau-damped waves which lack eigenfunctions on the real velocity axis.

Deviations from Landau damping can occur due to singularities or velocity cut-offs in initial perturbations.

Abstract

This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes {\it plus} a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as $\exp (-{\rm i} \omega t)$, while the waves do not have eigenfunctions on the real $v$-axis at all. However, `eigenfunctions' on the complex $v$-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space.