Analytical maximum likelihood estimation of stellar magnetic fields

TL;DR

This paper derives explicit maximum likelihood formulas to estimate stellar magnetic field vectors from polarised spectra, applicable to both solar and stellar observations, including LSD profiles, under weak field assumptions.

Contribution

It provides novel explicit formulas for magnetic field estimation from polarised spectra using maximum likelihood, accounting for error and bias analysis, especially for slow rotators with dipolar fields.

Findings

Explicit formulas for magnetic field vector estimation from Stokes parameters.

Error and bias analysis of the estimators.

Application to LSD profiles without mean spectral line assumptions.

Abstract

The polarised spectrum of stellar radiation encodes valuable information on the conditions of stellar atmospheres and the magnetic fields that permeate them. In this paper, we give explicit expressions to estimate the magnetic field vector and its associated error from the observed Stokes parameters. We study the solar case where specific intensities are observed and then the stellar case, where we receive the polarised flux. In this second case, we concentrate on the explicit expression for the case of a slow rotator with a dipolar magnetic field geometry. Moreover, we also give explicit formulae to retrieve the magnetic field vector from the LSD profiles without assuming mean values for the LSD artificial spectral line. The formulae have been obtained assuming that the spectral lines can be described in the weak field regime and using a maximum likelihood approach. The errors are recovered by means of the hermitian matrix. The bias of the estimators are analysed in depth.