P-modes in rapidly rotating stars -- looking for regular patterns in synthetic asymptotic spectra

According to a recent ray-based asymptotic theory, the high-frequency p-mode spectrum of rapidly rotating stars is a superposition of frequency subsets associated with dynamically independent regions of the ray-dynamics phase space. At high rotation rates corresponding to typical $\delta$ Scuti stars, two frequency subsets are expected to be visible : a regular frequency subset described by a Tassoul like formula and an irregular frequency subset with specific statistical properties. In this paper, we investigate whether the regular patterns can be detected in the resulting spectrum. We compute the autocorrelation function of synthetic spectra where the frequencies follow the asymptotic theory, the relative amplitudes are simply given by the modes' disk-averaging factors, and the frequency resolution is that of a CoRoT long run. Our first results are that (i) the detection of regular patterns strongly depends on the ratio of regular over irregular modes, (ii) low inclination angle configurations are more favorable than near equator-on configurations, (iii) in the absence of differential rotation, the $2 \Omega$ rotational splitting between $m=1$ and $m=-1$ modes is an easy feature to detect.Comment: 4 pages, 4 figures, proceedings of the HELAS-IV International Conference, accepted for publication in Astronomische Nachrichte

Shear mixing in stellar radiative zones I. Effect of thermal diffusion and chemical stratification

Turbulent transport of chemical elements in radiative zones of stars is considered in current stellar evolution codes thanks to phenomenologically derived diffusion coefficients. Recent local numerical simulations (Prat & Ligni\`eres 2013, A&A, 551, L3) suggest that the coefficient for radial turbulent diffusion due to radial differential rotation satisfies $D_{\rm t}\simeq0.058\kappa/Ri$, in qualitative agreement with Zahn's model. However, this model does not apply when differential rotation is strong with respect to stable thermal stratification or when chemical stratification has a significant dynamical effect, a situation encountered at the outer boundary of nuclear-burning convective cores. We extend our numerical study to consider the effects of chemical stratification and of strong shear, and compare the results with prescriptions used in stellar evolution codes. We performed local, direct numerical simulations of stably stratified, homogeneous, sheared turbulence in the Boussinesq approximation. The regime of high thermal diffusivities, typical of stellar radiative zones, is reached thanks to the so-called small-P\'eclet-number approximation, which is an asymptotic development of the Boussinesq equations in this regime. The dependence of the diffusion coefficient on chemical stratification was explored in this approximation. Maeder's extension of Zahn's model in the strong-shear regime is not supported by our results, which are better described by a model found in the geophysical literature. As regards the effect of chemical stratification, our quantitative estimate of the diffusion coefficient as a function of the mean gradient of mean molecular weight leads to the formula $D_{\rm t}\simeq 0.45\kappa(0.12-Ri_\mu)/Ri$, which is compatible in the weak-shear regime with the model of Maeder & Meynet (1996, A&A, 313, 140).Comment: 10 pages, 9 figures, accepted in A&

Regular Oscillation Sub-spectrum of Rapidly Rotating Stars

We present an asymptotic theory that describes regular frequency spacings of pressure modes in rapidly rotating stars. We use an asymptotic method based on an approximate solution of the pressure wave equation constructed from a stable periodic solution of the ray limit. The approximate solution has a Gaussian envelope around the stable ray, and its quantization yields the frequency spectrum. We construct semi-analytical formulas for regular frequency spacings and mode spatial distributions of a subclass of pressure modes in rapidly rotating stars. The results of these formulas are in good agreement with numerical data for oscillations in polytropic stellar models. The regular frequency spacings depend explicitly on internal properties of the star, and their computation for different rotation rates gives new insights on the evolution of mode frequencies with rotation.Comment: 14 pages, 10 figure

Gravity modes in rapidly rotating stars. Limits of perturbative methods

CoRoT and Kepler missions are now providing high-quality asteroseismic data for a large number of stars. Among intermediate-mass and massive stars, fast rotators are common objects. Taking the rotation effects into account is needed to correctly understand, identify, and interpret the observed oscillation frequencies of these stars. A classical approach is to consider the rotation as a perturbation. In this paper, we focus on gravity modes, such as those occurring in gamma Doradus, slowly pulsating B (SPB), or Be stars. We aim to define the suitability of perturbative methods. With the two-dimensional oscillation program (TOP), we performed complete computations of gravity modes -including the Coriolis force, the centrifugal distortion, and compressible effects- in 2-D distorted polytropic models of stars. We started with the modes l=1, n=1-14, and l=2-3, n=1-5,16-20 of a nonrotating star, and followed these modes by increasing the rotation rate up to 70% of the break-up rotation rate. We then derived perturbative coefficients and determined the domains of validity of the perturbative methods. Second-order perturbative methods are suited to computing low-order, low-degree mode frequencies up to rotation speeds ~100 km/s for typical gamma Dor stars or ~150 km/s for B stars. The domains of validity can be extended by a few tens of km/s thanks to the third-order terms. For higher order modes, the domains of validity are noticeably reduced. Moreover, perturbative methods are inefficient for modes with frequencies lower than the Coriolis frequency 2Omega. We interpret this failure as a consequence of a modification in the shape of the resonant cavity that is not taken into account in the perturbative approach.Comment: 8 pages, 6 figures, Astronomy & Astrophysics (in press

Pulsation modes in rapidly rotating stellar models based on the Self-Consistent Field method

Context: New observational means such as the space missions CoRoT and Kepler and ground-based networks are and will be collecting stellar pulsation data with unprecedented accuracy. A significant fraction of the stars in which pulsations are observed are rotating rapidly. Aims: Our aim is to characterise pulsation modes in rapidly rotating stellar models so as to be able to interpret asteroseismic data from such stars. Methods: The pulsation code developed in Ligni\`eres et al. (2006) and Reese et al. (2006) is applied to stellar models based on the self-consistent field (SCF) method (Jackson et al. 2004, 2005, MacGregor et al. 2007). Results: Pulsation modes in SCF models follow a similar behaviour to those in uniformly rotating polytropic models, provided that the rotation profile is not too differential. Pulsation modes fall into different categories, the three main ones being island, chaotic, and whispering gallery modes, which are rotating counterparts to modes with low, medium, and high l-|m| values, respectively. The frequencies of the island modes follow an asymptotic pattern quite similar to what was found for polytropic models. Extending this asymptotic formula to higher azimuthal orders reveals more subtle behaviour as a function of m and provides a first estimate of the average advection of pulsation modes by rotation. Further calculations based on a variational principle confirm this estimate and provide rotation kernels that could be used in inversion methods. When the rotation profile becomes highly differential, it becomes more and more difficult to find island and whispering gallery modes at low azimuthal orders. At high azimuthal orders, whispering gallery modes, and in some cases island modes, reappear.Comment: 16 pages, 11 figures, accepted for publication in A&

Mode identification in rapidly rotating stars

Context: Recent calculations of pulsation modes in rapidly rotating polytropic models and models based on the Self-Consistent Field method have shown that the frequency spectrum of low degree pulsation modes can be described by an empirical formula similar to Tassoul's asymptotic formula, provided that the underlying rotation profile is not too differential. Aims: Given the simplicity of this asymptotic formula, we investigate whether it can provide a means by which to identify pulsation modes in rapidly rotating stars. Methods: We develop a new mode identification scheme which consists in scanning a multidimensional parameter space for the formula coefficients which yield the best-fitting asymptotic spectra. This mode identification scheme is then tested on artificial spectra based on the asymptotic formula, on random frequencies and on spectra based on full numerical eigenmode calculations for which the mode identification is known beforehand. We also investigate the effects of adding random frequencies to mimic the effects of chaotic modes which are also expected to show up in such stars. Results: In the absence of chaotic modes, it is possible to accurately find a correct mode identification for most of the observed frequencies provided these frequencies are sufficiently close to their asymptotic values. The addition of random frequencies can very quickly become problematic and hinder correct mode identification. Modifying the mode identification scheme to reject the worst fitting modes can bring some improvement but the results still remain poorer than in the case without chaotic modes