Scattering Matrix for the Interaction between Solar Acoustic Waves and Sunspots. II. Theory

Abstract

Abstract The interaction between solar acoustic waves and an isolated sunspot is a scattering problem. A common tool to study scattering problems is the scattering matrix, which is the amplitude for the transition from one mode to another after the interaction. In the previous study (Part I), the scattering matrix elements were determined with the measured scattered wavefunction of the sunspot. In this study (Part II), we obtain an analytical relation between the scattering matrix elements and the perturbed quantities of the background medium of the sunspot region. The sunspot is considered a perturbed region relative to the quiet Sun. The perturbation of the background medium includes the magnetic field, the flow velocity, and perturbed thermodynamics quantities, such as the density and pressure perturbations. Inferring these perturbed quantities from measured quantities is one of the goals of helioseismology. Here, with the help of Green’s functions, the scattering matrix elements are expressed as a spatial integral, which contains these unknown perturbed quantities. This integral equation, together with the measured scattering matrix elements, could be used to infer the perturbed quantities with the forward and inversion methods. Besides the typical approximations for solar acoustic waves, two additional assumptions are made here: one is the Born approximation, and the other is that the background medium of the sunspot region does not change with time.