Constructing semi-empirical sunspot models for helioseismology

TL;DR

This paper develops simplified, cylindrically-symmetric sunspot models incorporating magnetic and thermodynamic effects, validated through helioseismic simulations against observations to improve understanding of sunspot subsurface structures.

Contribution

It introduces a semi-empirical, magneto-thermodynamic sunspot model that captures near-surface effects and is validated with helioseismic wave simulations.

Findings

Model reproduces observed helioseismic signatures of sunspots.

Magnetic and thermodynamic effects are effectively integrated.

Model provides a basis for helioseismic inversion of sunspot structures.

Abstract

One goal of helioseismology is to determine the subsurface structure of sunspots. In order to do so, it is important to understand first the near-surface effects of sunspots on solar waves, which are dominant. Here we construct simplified, cylindrically-symmetric sunspot models, which are designed to capture the magnetic and thermodynamics effects coming from about 500 km below the quiet-Sun $\tau_{5000}=1$ level to the lower chromosphere. We use a combination of existing semi-empirical models of sunspot thermodynamic structure (density, temperature, pressure): the umbral model of Maltby et al. (1986) and the penumbral model of Ding and Fang (1989). The OPAL equation of state tables are used to derive the sound speed profile. We smoothly merge the near-surface properties to the quiet-Sun values about 1mm below the surface. The umbral and penumbral radii are free parameters. The magnetic field is added to the thermodynamic structure, without requiring magnetostatic equilibrium. The vertical component of the magnetic field is assumed to have a Gaussian horizontal profile, with a maximum surface field strength fixed by surface observations. The full magnetic field vector is solenoidal and determined by the on-axis vertical field, which, at the surface, is chosen such that the field inclination is 45$^\circ$ at the umbral-penumbral boundary. We construct a particular sunspot model based on SOHO/MDI observations of the sunspot in active region NOAA 9787. The helioseismic signature of the model sunspot is studied using numerical simulations of the propagation of f, p$_1$, and p$_2$ wave packets. These simulations are compared against cross-covariances of the observed wave field. We find that the sunspot model gives a helioseismic signature that is similar to the observations.