Unstable $m=1$ modes of counter-rotating Keplerian discs

TL;DR

This paper investigates the linear $m=1$ counter-rotating instability in nearly Keplerian discs around black holes, revealing how the growth rates and pattern speeds depend on the counter-rotating mass fraction and eigenfunction properties.

Contribution

It derives and solves coupled integral eigenvalue equations for two-component softened gravity discs, providing new insights into the stability and mode characteristics of counter-rotating galactic nuclei discs.

Findings

Eigenvalues are generally complex, real without counter-rotation, imaginary with identical surface density profiles.

Pattern speed is non-negative and correlates with growth or damping rates.

Growth rate increases with the mass in the counter-rotating component.

Abstract

We study the linear $m=1$ counter-rotating instability in a two-component, nearly Keplerian disc. Our goal is to understand these \emph{slow} modes in discs orbiting massive black holes in galactic nuclei. They are of interest not only because they are of large spatial scale--and can hence dominate observations--but also because they can be growing modes that are readily excited by accretion events. Self-gravity being nonlocal, the eigenvalue problem results in a pair of coupled integral equations, which we derive for a two-component softened gravity disc. We solve this integral eigenvalue problem numerically for various values of mass fraction in the counter-rotating component. The eigenvalues are in general complex, being real only in the absence of the counter-rotating component, or imaginary when both components have identical surface density profiles. Our main results are as follows: (i) the pattern speed appears to be non negative, with the growth (or damping) rate being larger for larger values of the pattern speed; (ii) for a given value of the pattern speed, the growth (or damping) rate increases as the mass in the counter-rotating component increases; (iii) the number of nodes of the eigenfunctions decreases with increasing pattern speed and growth rate. Observations of lopsided brightness distributions would then be dominated by modes with the least number of nodes, which also possess the largest pattern speeds and growth rates.