The Direct Effect of Toroidal Magnetic Fields on Stellar Oscillations: An Analytical Expression for the General Matrix Element

TL;DR

This paper develops an analytical formalism to quantify how large-scale toroidal magnetic fields influence solar and stellar oscillation eigenfunctions and frequencies, aiding the understanding of subsurface magnetic structures.

Contribution

It provides the first analytical expression for the general matrix element describing the direct effect of toroidal magnetic fields on stellar oscillations.

Findings

Analytical expression for the matrix element derived.

Facilitates forward modeling of magnetic effects on oscillations.

Helps infer subsurface magnetic field properties from seismic data.

Abstract

Where is the solar dynamo located and what is its modus operandi? These are still open questions in solar physics. Helio- and asteroseismology can help answer them by enabling us to study solar and stellar internal structures through global oscillations. The properties of solar and stellar acoustic modes are changing with the level of magnetic activity. However, until now, the inference on subsurface magnetic fields with seismic measures has been very limited. The aim of this paper is to develop a formalism to calculate the effect of large-scale toroidal magnetic fields on solar and stellar global oscillation eigenfunctions and eigenfrequencies. If the Lorentz force is added to the equilibrium equation of motion, stellar eigenmodes can couple. In quasi-degenerate perturbation theory, this coupling, also known as the direct effect, can be quantified by the general matrix element. We present the analytical expression of the matrix element for a superposition of subsurface zonal toroidal magnetic field configurations. The matrix element is important for forward calculations of perturbed solar and stellar eigenfunctions and frequency perturbations. The results presented here will help to ascertain solar and stellar large-scale subsurface magnetic fields, and their geometric configuration, strength, and their change over the course of activity cycles.