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

TL;DR

The paper addresses how near-degeneracy effects (NDE) in rotating red-giant stars bias linear rotational inversions used to infer the internal rotation profile. It develops a framework to compute oscillation frequencies including lowest-order rotational perturbations, solving a quadratic eigenvalue problem to capture off-diagonal mode coupling, and derives perturbed frequencies via $\delta\omega_{ij}= m \int_0^R \mathcal{K}_{ij}(r)\Omega(r)\,dr$. Using $1\,M_\odot$ red-giant models with $\Delta\nu$ in $[9,16]~\mu\mathrm{Hz}$ and core rotation rates $\Omega_{\rm core}$ in $[500,1500]~\mathrm{nHz}$, the authors generate synthetic splittings including NDE and invert them with eMOLA (and related OLA methods) to quantify systematic errors. They find that NDE-induced errors in the inferred envelope rotation rates can exceed the observational uncertainties for more evolved or faster-rotating stars (e.g., for $\Omega_{\rm core}=1000$ nHz at $\Delta\nu \lesssim 13~\mu$Hz), while core rotation estimates remain more robust; the results delineate RGB regions where linear inversions remain valid and highlight the potential to exploit NDE for improved rotation constraints.

Abstract

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.

Figures (4)

• Figure 1: Rotational splittings and their asymmetries as a function of mode frequency. Left panel: Rotational splittings of the evolved model using a core rotation rate of $\Omega_{\rm core}=750$ nHz and an envelope rotation rate of $\Omega_{\rm env}=50$ nHz. Rotational splittings that have the same $n_{pg}$ are connected with a grey line. We note that we select only a subset of the rotational splittings centred around $\nu_{\rm max}\approx100~\mu$Hz for the rotational inversions (see text for details). The dotted, blue line connects the symmetric splittings for the sake of readability. The rotational splittings $\delta\omega_{n\ell m}$ were obtained by computing the frequency differences according to Eq. \ref{['eqsplittingsasym1']} and \ref{['eqsplittingsasym2']} using the frequencies obtained from solving the full QEP posed by Eq. \ref{['eqoscmat']} as described in Appendix \ref{['secqep']}, while the symmetric rotational splittings $\delta\omega_{n\ell m,\, {\rm symm}}$ where obtained from Eq. \ref{['eqsplitting']}. Right panel: Asymmetries of the rotational splittings ($\Delta\omega_{\rm asym}$) of the evolved model over a range of core rotation rates $\Omega_{\rm core}$ and for $\Omega_{\rm env}=50$ nHz. The lines are colour-coded by the core rotation rate.
• Figure 2: Error on the estimated core and envelope rotation rates introduced by neglecting the NDE. The errors are shown in terms of the individual uncertainties $\sigma_\Omega$ as a function of the large frequency separation $\Delta\nu$. We generated synthetic splittings for three different core rotation rates of $\Omega_{\rm core}=500, 750, 1000$ nHz and a envelope rotation rate of $\Omega_{\rm env}=50$ nHz indicated with crosses, circles and squares, respectively. Results for the core and envelope are shown in different shades of blue and red, respectively. Note that the models evolve from left to right in this figure.
• Figure 3: Absolute error of the estimated envelope and core rotation rates for the evolved model. Left panel: The absolute error of the estimated envelope rotation rate as a function of input core and envelope rotation rate is shown. The input core and envelope rotation rates are given on the $x$- and $y$-axis, respectively. The error of the estimated envelope rotation rate in terms of nHz is colour coded and cut at a maximum value of 100 nHz. The grey dots indicate the underlying grid of input core and envelope rotation rates. The contour lines have a separation of 10 nHz. Right panel: Same as the left panel now for the absolute error of the estimated core rotation rate. Note that the colour scale is inverted, as the errors on the core rotation rate are negative.
• Figure 4: Comparison of rotational splitting asymmetries $\Delta\omega_{\rm asym}$ computed from the NDE, generalised dziembowski1992 and $\pi$-$\gamma$ formalisms. Results for the $\pi$-$\gamma$ formalisms and the generalised dziembowski1992 are shown in the left and right panel, respectively. Here, the evolved model was used to compute the synthetic data. The shaded band indicates two times the minimum uncertainty of the averaged splittings obtained from Eq. \ref{['eqsplittingsymm']}. The multiplication by two is necessary because $\Delta\omega_{\rm asym}$ is twice the frequency difference between the averaged and actual rotational splittings. As a minimum uncertainty of the observed frequencies we assume the frequency resolution. Then the minimum uncertainty of the averaged splittings is computed from error propagation as $\sigma_{\delta\omega, {\rm min}}/(2\pi)=1/\sqrt{2}\,\delta\nu_{\rm res}=1/\sqrt{2}\times0.0078~\mu$Hz, given the four years of Kepler observations.