Schr\"odinger Evolution of Self-Gravitating Disks

TL;DR

This paper models the long-term evolution of self-gravitating disks using a Schr"odinger equation analogy, revealing insights into their inclination dynamics and response to external perturbations.

Contribution

It introduces a novel formalism linking disk inclination dynamics to quantum mechanics, providing an analytic criterion for gravitational rigidity.

Findings

Inclination dynamics described by the Schr"odinger equation.

Eigenmodes correspond to nodal bending waves.

External perturbations resemble quantum scattering.

Abstract

An understanding of the long-term evolution of self-gravitating disks ranks among the classic outstanding problems of astrophysics. In this work, we show that the secular inclination dynamics of a geometrically thin quasi-Keplerian disk, with a surface density profile that scales as the inverse square-root of the orbital radius, are described by the time-dependent Schr\"odinger equation. Within the context of this formalism, nodal bending waves correspond to the eigenmodes of a quasiparticle's wavefunction, confined in an infinite square well with boundaries given by the radial extent of the disk. We further show that external secular perturbations upon self-gravitating disks exhibit a mathematical similarity to quantum scattering theory. Employing this framework, we derive an analytic criterion for the gravitational rigidity of a nearly-Keplerian disk under external perturbations. Applications of the theory to circumstellar disks and Galactic nuclei are discussed.