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

Context.Accurate estimates of internal red giant rotation rates are crucial for constraining and improving current models of stellar rotation. Asteroseismic rotational inversions provide a means of estimating these internal rotation rates. Aims. In this work, we focus on the observed differences in the rotationally induced frequency shifts between prograde and retrograde modes. These effects have been overlooked in previous studies 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 of the underlying rotation rates. Methods. We determine oscillation mode frequencies in the presence of rotation using the lowest-order perturbative approach and describe the differences between prograde and retrograde modes arising from the coupling of multiple mixed modes, also known as near-degeneracy effects. We computed synthetic rotational splittings, taking these near-degeneracy effects into account. We used red giant models with one solar mass, a large frequency separation between 16 and 9 μHz, and core rotation rates between 500 and 1500 nHz, covering the regime of observed parameters of Kepler red giant stars. Finally, we used these synthetic data to quantify the systematic errors in the internal rotation rates estimated by means of rotational inversions in the presence of near-degeneracy effects. Results. We show that the systematic errors in the estimated rotation rates introduced by near-degeneracy effects surpass the observational uncertainties for more evolved and faster-rotating stars. For a core rotation rate of 500 nHz, linear inversions remain applicable over the range of models considered here, while for a core rotation rate of 1000 nHz, systematic errors become significant below a large frequency separation of 13 μHz. Conclusions. The estimated rotation rates of some previously analysed red giants suffer from significant systematic errors that have not yet been accounted for. Nonetheless, reliable analyses with existing inversion methods are feasible for a number of red giants, and we expect there to be unexplored targets within the parameter ranges determined here. Finally, exploiting the observational potential of near-degeneracy effects is an important step towards obtaining more accurate estimates of internal red-giant rotation rates.