Analytical calculation of Stokes profiles of rotating stellar magnetic dipole

TL;DR

This paper develops an analytical method to calculate Stokes profiles of rotating stars with magnetic dipoles, incorporating the effects of rotation, magnetic fields, and polarization, extending the weak-field approximation.

Contribution

It generalizes the weak-field approximation for polarized line profiles to rotating stars, including Doppler effects and surface integration, with explicit formulas for magnetic dipoles.

Findings

Profiles expressed as Hermite polynomial series expansions.

Dependence of polarization on intensity derivatives and profiles.

Numerical evaluation limited to moderate rotation velocities.

Abstract

The observation of the polarization emerging from a rotating star at different phases opens up the possibility to map the magnetic field in the stellar surface thanks to the well-known Zeeman Doppler Imaging. When the magnetic field is sufficiently weak, the circular and linear polarization profiles locally in each point of the star are proportional to the first and second derivatives of the unperturbed intensity profile, respectively. We show that the weak-field approximation (for weak lines in the case of linear polarization) can be generalized to the case of a rotating star including the Doppler effect and taking into account the integration on the stellar surface. The Stokes profiles are written as a linear combination of wavelength-dependent terms expressed as series expansions in terms of Hermite polynomials. These terms contain the surface integrated magnetic field and velocity components. The direct numerical evaluation of these quantities is limited to rotation velocities not larger than 8 times the Doppler width of the local absorption profiles. Additionally, we demonstrate that, in a rotating star, the circular polarization flux depends on the derivative of the intensity flux with respect to the wavelength and also on the profile itself. Likewise, the linear polarization depends on the profile and on its first and second derivative with respect to the wavelength. We particularize the general expressions to a rotating dipole.