Inequalities on stellar rotational splittings derived from assumptions on the rotation profile

TL;DR

This paper develops inequalities to assess whether observed stellar rotational splittings are compatible with certain rotation profiles, aiding the interpretation of asteroseismic data from evolved stars.

Contribution

It introduces a set of inequalities based on the relation between splittings and rotation profiles, providing criteria to evaluate profile plausibility from observational data.

Findings

Inequalities can identify increasing rotation profiles or violations of Rayleigh's criterion.

Mismatches between models and stars can lead to incorrect inferences about rotation.

Near-surface effects can obscure frequency differences, affecting profile assessments.

Abstract

Context: A number of pulsating stars with rotational splittings have been observed thanks to the CoRoT and Kepler missions. This is particularly true of evolved (sub-giant and giant) stars, and has led various groups to investigate their rotation profiles via different methods. Aims: We would like to set up some criteria which will help us to know whether a decreasing rotation profile, or one which satisfies Rayleigh's stability criterion, is compatible with a set of observed rotational splittings for a given reference model. Methods: We derive inequalities on the rotational splittings using a reformulated version of the equation which relates the splittings to the rotation profile and kernels. Results: These inequalities are tested out on some simple examples. The first examples show how they are able to reveal when a rotation profile is increasing somewhere or inconsistent with Rayleigh's criterion in a main sequence star, depending on the profile and the $\ell$ values of the splittings. The next example illustrates how a slight mismatch between an observed evolved star and a reference model can lead to erroneous conclusions about the rotation profile. We also show how frequency differences between the star and the model, which should normally reveal this mismatch, can be masked by frequency corrections for near-surface effects.