Impact of near-degeneracy effects on linear rotational inversions for red-giant stars

Accurate estimates of internal red-giant rotation rates are a crucial ingredient for constraining and improving current models of stellar rotation. Asteroseismic rotational inversions are a method to estimate these internal rotation rates. In this work, we focus on the observed differences in the rotationally-induced frequency shifts between prograde and retrograde modes, which were ignored in previous works when estimating internal rotation rates of red giants using inversions. We systematically study the limits of applicability of linear rotational inversions as a function of the evolution on the red-giant branch and the underlying rotation rates. We solve for the oscillation mode frequencies in the presence of rotation in the lowest-order perturbative approach. This enables a description of the differences between prograde and retrograde modes through the coupling of multiple mixed modes. We compute synthetic rotational splittings taking these near-degeneracy effects into account. We use red-giant models with one solar mass, a large frequency separation between 16 and 9 microhertz and core rotation rates between 500 and 1500 nHz covering the regime of observed parameters of Kepler red-giant stars. Finally, we use these synthetic data to quantify the systematic errors of internal rotation rates estimated by means of rotational inversions in the presence of near-degeneracy effects. We show that the systematic errors in the estimated rotation rates introduced by near-degeneracy effects surpass observational uncertainties for more evolved and faster rotating stars. The estimated rotation rates of some of the previously analysed red giants suffer from significant systematic errors that have not been taken into account yet. Notwithstanding, reliable analyses with existing inversion methods are feasible for a number of red giants within the parameter ranges determined here.