Eigenstates of Quasi-Keplerian Self-Gravitating Particle Discs

TL;DR

This paper models the secular small-inclination dynamics of quasi-Keplerian self-gravitating particle discs using a quantum mechanics analogy, revealing that local models accurately predict mode shapes but not eigenfrequencies.

Contribution

It introduces a quantum-mechanical framework for analyzing the secular evolution of self-gravitating discs and compares local and non-local coupling models.

Findings

Local coupling models accurately predict mode shapes in narrow annuli.

Eigenfrequency predictions are less accurate than mode shape predictions.

The quantum analogy provides new insights into disc dynamics.

Abstract

Although quasi-Keplerian discs are among the most common astrophysical structures, computation of secular angular momentum transport within them routinely presents a considerable practical challenge. In this work, we investigate the secular small-inclination dynamics of a razor-thin particle disc as the continuum limit of a discrete Lagrange-Laplace secular perturbative theory and explore the analogy between the ensuing secular evolution -- including non-local couplings of self-gravitating discs -- and quantum mechanics. We find the 'quantum' Hamiltonian that describes the time evolution of the system and demonstrate the existence of a conserved inner product. The lowest-frequency normal modes are numerically approximated by performing a Wick rotation on the equations of motion. These modes are used to quantify the accuracy of a much simpler local-coupling model, revealing that it predicts the shape of the normal modes to a high degree of accuracy, especially in narrow annuli, even though it fails to predict their eigenfrequencies.