Magnetoseismology of the Sun: Waves, exact solutions, and dispersion relations
The study of waves inside the Sun and stars unveil the structure of stellar interior, which otherwise is less accessible to direct observations. This kind of study called 'asteroseismology' can, in principle, be used to infer the emergence of the subsurface solar magnetic field before they actually emerge on the surface. We look for the signatures of these emerging magnetic fields in the dispersion relation of acoustic waves, trapped near the solar surface. We first begin with an isothermal, stratified atmosphere, permeated by a uniform (and a non-uniform) horizontal magnetic field. After solving for waves and its dispersion relation in such an idealized case, we consider a more realistic polytropic atmosphere. We formulate a linear eigenvalue problem with a second-order differential operator in two different cases: (i) without a magnetic field, where we get symmetric rings of constant frequencies projected over the horizontal wavenumber plane, and (ii) with a horizontal magnetic field, which gives rise to asymmetry in the rings of constant frequencies. We suggest that these signatures may have the potential to reveal the presence and strength of subsurface magnetic field observationally. We also present exact analytical solutions to waves in terms of hypergeometric functions and confluent Heun function in separate cases.