Near-degeneracy effects in Quadrupolar Mixed Modes. From an Asymptotic Description to Data Fitting

TL;DR

This work addresses the challenge of extracting internal rotation information from quadrupolar ($ abla$,$\ell$=2) mixed modes in evolved solar-like stars, where near-degeneracy distorts rotational multiplets. It develops a comprehensive asymptotic framework that incorporates non-diagonal near-degeneracy terms into the mixed-mode basis and embeds it in a global Bayesian PSD-fitting pipeline to simultaneously fit $\ell$=0,1,2 modes. The authors derive a closed-form asymptotic expression for off-diagonal couplings $\gamma_{c,ij}$, validate it against numerical models for KIC 7341231, and apply it to Kepler targets KIC 7341231 and KIC 8179973 to obtain core and envelope rotation rates with improved precision, as well as first observational constraints on asymptotic $\ell$=2 parameters $\Delta\Pi_2$, $\varepsilon_{g,2}$, and $q_2$. This approach enables model-independent, high-precision rotation inversions that leverage the richer information content of $\ell$=2 mixed modes, paving the way for asymptotic seismology beyond $\ell$=1 and contributing to the understanding of angular momentum transport in evolved stars.

Abstract

Dipolar (l=1) mixed modes revealed surprisingly weak differential rotation between the core and the envelope of evolved solar-like stars. Quadrupolar (l=2) mixed modes also contain information on the internal dynamics, but are very rarely characterised due to their low amplitude and the challenging identification of adjacent or overlapping rotationally split multiplets affected by near-degeneracy effects. We aim to extend broadly used asymptotic seismic diagnostics beyond l=1 mixed modes by developing an analogue asymptotic description of l=2 mixed modes, explicitly accounting for near-degeneracy effects that distort their rotational multiplets. We derive a new asymptotic formulation of near-degenerate mixed l=2 modes that describes off-diagonal terms representing the interaction between modes of adjacent radial orders. We implement the formalism within a global Bayesian mode-fitting framework, for a direct fit of all l=0,1,2 modes in the power spectrum density. We are able to asymptotically model the asymmetric rotational splitting present in various radial orders of l=2 modes observed in young red giant stars without the need for any numerical stellar modelling. Applied to the Kepler target KIC 7341231, our formalism yields core and envelope rotation rates consistent with previous numerical modelling, while providing improved constraints from the global and model-independent approach. We also characterise the new target KIC 8179973, measuring its rotation rate and mixed-mode parameters for the first time. The global fit allows for much better precision than standard methods, yielding better constraints for rotation inversions. We place the first observational constraints on the asymptotic l=2 mixed mode parameters (DPi_2,q_2,eps_g2), paving the way towards the use of asymptotic seismology beyond l=1 mixed modes.

Figures (11)

• Figure 1: Illustration of the effect of near degeneracy on two coupled $\ell=2$ multiplets as the core rotation rate $\nu_\mathrm{c}=\Omega_\mathrm{core}/2\pi$ increases, for a fixed inclination angle of $i=42^\circ$. Each horizontal modelled PSD shows the combined visibility profile of the ten components (two sets of $m=-2,-1, 0, +1,+2$ modes) at a given $\nu_\mathrm{c}$. The greyscale background indicates how the multiplet structure evolves continuously with rotation. All the frequencies were computed by solving Eq. (\ref{['eq:mat_eigval']}) with off-diagonal terms provided by the asymptotic formulation of Eq. (\ref{['off-diag-factor']}) and diagonal terms given by Eq. (\ref{['split_first_order']}). This example highlights how near-degenerate interactions modify the usual pattern of rotationally split multiplets and generate asymmetries in the frequency pattern.
• Figure 2: Asymptotic fit of KIC 7341231. In both panels, the Kepler data is in grey and our asymptotic fit in red. Radial, $\ell=1$, and $\ell=2$ modes are highlighted by the shading in green, blue, and yellow, respectively. (a) Two orders showing mixed $\ell=2$ modes with asymmetric splittings; (b) Zoom near 411 µHz on split $\ell=2$ modes; $m=0$, $\pm1$, and $\pm2$ components are marked in yellow, orange, and red. (c) Zoom near 381 µHz on split $\ell=2$ modes; model in thick red, data in grey; $m=0$, $\pm1$, and $\pm2$ components of the $\ell=2$ mode are marked in yellow, orange, and red and $m=\pm1$ components of the $\ell=1$ mode are marked in dark blue.
• Figure 3: Asymptotic fit of KIC 8179973. In both panels, the Kepler data is in grey and our asymptotic fit in red. Radial, $\ell=1$, and $\ell=2$ modes are highlighted by the shading in green, blue, and yellow, respectively. (a) Two orders showing mixed $\ell=2$ modes with asymmetric splittings; (b) Zoom near 352 µHz on split near-degenerate $\ell=2$ modes. The quadrupolar mixed $m=0$, $\pm1$, and $\pm2$ components are marked in yellow, orange, and red, respectively. The full fit can be found in Fig. \ref{['fig:echelle_Alice']}
• Figure 4: Histograms showing the result of the Monte-Carlo analysis with the asymptotic formulation
• Figure 5: Echelle diagram comparing the oscillation frequencies derived from the Bayesian asymptotic model (circles, this work) with the peakbagged frequencies of KIC 7341231 (squares, 2012ApJ...756...19D2017AA...605A..75D, referred to as Dh12 in the legend of the figure). The markers for the model frequencies are rendered semi-transparent when the separation from a peakbagged frequency exceeds 1 µHz.