Exact Analytical Solutions in Inhomogeneous Magnetic Fields for Linear Asteroseismic Waves

Abstract

We solve for waves in an isothermal, stratified medium with a magnetic field that points along a direction perpendicular to that of gravity and varies exponentially in the direction of gravity. We find exact analytical solutions for two different cases: (a) waves propagating along the direction of the magnetic field and (b) waves propagating along the direction of the gravity. In each case, we find solutions in terms of either the hypergeometric functions or their confluent cousins. We solve the resultant transcendental dispersion relation numerically. The eigenfrequencies decrease with increasing degree of the spatial inhomogeneity of the magnetic field. Further, the nodes of the eigenfunctions leak toward regions of lower Alfvén wave speed due to softened wave-reflection in such regions. Such changes in the dispersion relation and the mode structures may allow the detection of magnetic fields buried in the stellar interior.