Combining magneto-hydrostatic constraints with Stokes profiles inversions

TL;DR

This paper introduces a method to improve the accuracy of converting optical depth to geometrical height in solar atmosphere models by applying magneto-hydrostatic equilibrium, accounting for magnetic forces neglected in traditional hydrostatic assumptions.

Contribution

The authors develop a 3D magneto-hydrostatic equilibrium method that enhances the conversion of optical depth to height, outperforming hydrostatic equilibrium in accuracy and applicability.

Findings

MHS equilibrium reduces height conversion errors to 30-70 km.

HE retrieves pressure and density accurately in 47% of the domain.

MHS equilibrium achieves this in 84% of the domain.

Abstract

Inversion codes for the polarized radiative transfer equation can be used to infer the temperature $T$, line-of-sight velocity $v_{\rm los}$, and magnetic field $\rm{\bf B}$ as a function of the continuum optical-depth $\tau_{\rm c}$. However, they do not directly provide the gas pressure $P_{\rm g}$ or density $\rho$. In order to obtain these latter parameters, inversion codes rely instead on the assumption of hydrostatic equilibrium (HE) in addition to the equation of state (EOS). Unfortunately, the assumption of HE is rather unrealistic across magnetic field lines. This is because the role of the Lorentz force, among other factors, is neglected. This translates into an inaccurate conversion from optical depth $\tau_{\rm c}$ to geometrical height $z$. We aim at improving this conversion via the application of magneto-hydrostatic (MHS) equilibrium instead of HE. We develop a method to solve the momentum equation under MHS equilibrium (i.e., taking the Lorentz force into account) in three dimensions. The method is based on the solution of a Poisson-like equation. Considering the gas pressure $P_{\rm g}$ and density $\rho$ from three-dimensional magneto-hydrodynamic (MHD) simulations of sunspots as a benchmark, we compare the results from the application of HE and MHS equilibrium. We find that HE retrieves the gas pressure and density within an order of magnitude of the MHD values in only about 47 \% of the domain. This translates into an error of about $160-200$ km in the determination of the $z-\tau_{\rm c}$ conversion. On the other hand, the application of MHS equilibrium allows determination of $P_{\rm g}$ and $\rho$ within an order of magnitude in 84 \% of the domain. In this latter case, the $z-\tau_{\rm c}$ conversion is obtained with an accuracy of $30-70$ km.