The linear eigenvalue problem for barotropic selfgravitating discs

TL;DR

This paper introduces a new method for analyzing spiral patterns in gaseous galactic discs by transforming hydrodynamical equations into a linear eigenvalue problem, facilitating more accurate stability and pattern formation studies.

Contribution

It presents a novel effective approach that converts the complex hydrodynamical equations into a linear algebraic eigenvalue problem without approximations, improving analysis of spiral structures.

Findings

Method successfully applied to an exactly solvable model.

Method provides insights into gravitational instability in galactic discs.

Potential to improve understanding of spiral pattern formation.

Abstract

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured, presumably due to a difficult outer boundary condition. The paper suggests a new effective method for finding the spiral patterns based on an expansion of small amplitude perturbations over finite radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to an exactly solvable model with finite outer boundary and to a galactic disc model.