Context. Inversion codes for the polarized radiative transfer equation, when applied to spectropolarimetric observations (i.e., Stokes vector) in spectral lines, can be used to infer the temperature T, line-of-sight velocity vlos, and magnetic field B as a function of the continuum optical-depth τc. However, they do not directly provide the gas pressure Pg or density ρ. 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, causing estimations of Pg and ρ to be unreliable. This is because the role of the Lorentz force, among other factors, is neglected. Unreliable gas pressure and density also translate into an inaccurate conversion from optical depth τc to geometrical height z.
Aims: We aim at improving the determination of the gas pressure and density via the application of magnetohydrostatic (MHS) equilibrium instead of HE.
Methods: 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 iterative solution of a Poisson-like equation. Considering the gas pressure Pg and density ρ from three-dimensional magnetohydrodynamic (MHD) simulations of sunspots as a benchmark, we compare the results from the application of HE and MHS equilibrium using boundary conditions with different degrees of realism. Employing boundary conditions that can be applied to actual observations, we find that HE retrieves the gas pressure and density with an error smaller than one order of magnitude (compared to the MHD values) in only about 47% of the grid points in the three-dimensional domain. Moreover, the inferred values are within a factor of two of the MHD values in only about 23% of the domain. This translates into an error of about 160 - 200 km in the determination of the z - τc conversion (i.e., Wilson depression). On the other hand, the application of MHS equilibrium with similar boundary conditions allows determination of Pg and ρ with an error smaller than an order of magnitude in 84% of the domain. The inferred values are within a factor of two in more than 55% of the domain. In this latter case, the z - τc conversion is obtained with an accuracy of 30 - 70 km. Inaccuracies are due in equal part to deviations from MHS equilibrium and to inaccuracies in the boundary conditions.
Results: Compared to HE, our new method, based on MHS equilibrium, significantly improves the reliability in the determination of the density, gas pressure, and conversion between geometrical height z and continuum optical depth τc. This method could be used in conjunction with the inversion of the radiative transfer equation for polarized light in order to determine the thermodynamic, kinematic, and magnetic parameters of the solar atmosphere.