An MHD spectral theory approach to Jeans' magnetized gravitational instability

TL;DR

This paper develops a comprehensive mathematical framework for analyzing magnetohydrodynamic waves and gravitational instabilities in magnetized, self-gravitating slabs, extending classical theories to include non-uniform magnetic fields and self-gravity effects.

Contribution

It introduces new reformulations of the MHD wave equations, including a Hamiltonian form, and derives analytical eigenfunctions and dispersion relations for magnetized, self-gravitating slabs.

Findings

Derived exact dispersion relations for Jeans-unstable modes.

Showed how self-gravity modifies MHD wave spectra.

Provided analytical eigenfunctions for magnetized slabs.

Abstract

In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfv\'en and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans' instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans' swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.