The asteroseismology of rapidly rotating pulsating stars is hindered by our poor knowledge of the effect of the rotation on the oscillation properties. Here we present an asymptotic analysis of high-frequency acoustic modes in rapidly rotating stars. We study the Hamiltonian dynamics of acoustic rays in uniformly rotating polytropic stars and show that the phase space structure has a mixed character, regions of chaotic trajectories coexisting with stable structures like island chains or invariant tori. In order to interpret the ray dynamics in terms of acoustic mode properties, we then use tools and concepts developed in the context of quantum physics. Accordingly, the high-frequency acoustic spectrum is a superposition of frequency subsets associated with dynamically independent phase space regions. The sub-spectra associated with stable structures are regular and can be modelled through EBK quantization methods while those associated with chaotic regions are irregular but with generic statistical properties. The results of this asymptotic analysis are successfully confronted with the properties of numerically computed high-frequency acoustic modes. The implications for the asteroseismology of rapidly rotating stars are discussed.
Interpreting the oscillation spectra of rapidly rotating stars is a long standing unsolved problem of asteroseismology. It so far prevented to directly probe the interior of stars in large fractions of the HR diagram, mostly in the range of intermediate and massive stars. Important progresses are nevertheless expected from the current spatial seismology missions (in particular COROT and KEPLER) as well as from recent modelling efforts on the effect of rotation on stellar oscillations. On the modelling side, new numerical codes have been able to accurately compute oscillation frequencies in significantly centrifugally distorted polytropic models of stars (Lignières et al., 2006;Reese et al., 2006) and are being extended to more realistic models (Reese et al., 2009). In particular the previous calculations based on perturbative methods were shown to be inadequate for these stars (Reese et al., 2006;Lovekin & Deupree, 2008). Nevertheless, interpreting the data requires an understanding of the mode properties that goes far beyond the accurate computation of frequencies. Indeed, the necessary identification of the observed frequencies with theoretical frequencies is a largely underconstrained problem which requires a priori information on the spectrum to be successful. The knowledge of the structure of the frequency spectrum is of primary importance in this context. For slowly rotating pulsating stars, this structure is characterized by regular frequency patterns which can be analytically derived from an asymptotic theory of the high-frequency acoustic modes.
Send offprint requests to: F. Lignières Until recently, the asymptotic structure of the frequency spectrum of rapidly rotating stars was completely unknown. Our first calculations of low degree (ℓ = 0 -7) and low order (n = 1 -10) acoustic axisymmetric modes in centrifugally distorted polytropic stars (Lignières et al., 2006) have revealed the presence of regular frequency patterns similar albeit different from those of spherically symmetric stars. This was confirmed using more realistic calculations including the Coriolis force and was also extended to non-axisymmetric and higher-frequency modes (Reese et al., 2008). The analogy with the non-rotating case suggests an asymptotic analysis could model these empirical regular patterns.
The asymptotic analysis presented in this paper is based on the acoustic ray dynamics. This approach can be viewed as a natural extension of the asymptotic analysis developed for non-rotating stars (Vandakurov, 1967;Tassoul, 1980Tassoul, , 1990;;Deubner & Gough, 1984;Roxburgh & Vorontsov, 2000). In this case, spherical symmetry enables to reduces the initial 3D boundary value problem to a 1D boundary value problem in the radial direction. Asymptotic solutions of this 1D boundary value problem can then be obtained using a short-wavelength approximation which consists in looking for wave-like solutions under the assumption that their wavelength is much smaller than the typical lengthscale of the background medium. As rotation breaks the spherical symmetry, the eigenmodes are no longer separable in the latitudinal and radial directions and the 3D boundary value problem of acoustic modes in rapidly rotating stars can not be reduced to a 1D boundary value problem. Thus, the short-wavelength approximation is directly applied to the 3D equations governing linear adiabatic stellar perturbations. This leads to an acoustic ray model which describes the propagation of locally plane waves. Since we are concerned by oscillation modes, the main issue of an asymptotic analysis based on ray dynamics is to construct standing waves solutions from the shortwavelength propagating waves described by the acoustic rays.
The short-wavelength approximation of wave equations is standard in physics, best known examples being the geometrical optics limit of electromagnetic waves or the classical limit of quantum mechanics. We shall see that similarly to these cases the acoustic rays in stars can be described as trajectories of a particle in the framework of classical Hamiltonian mechanics. As is well known in quantum physics (Gutzwiller, 1990;Ott, 1993), the issue of construction modes from the ray dynamics depends crucially on the nature of this Hamiltonian motion.
Indeed, Hamiltonian systems can have one of two very different behaviors. If there are enough constants of motion, the dynamics is integrable, and trajectories organize themselves in families on well-defined phase space structures. In contrast, chaotic systems display exponential divergence of nearby trajectories, and a typical orbit is ergodic in phase space. The modes constructed from these different dynamics are markedly different. For integrable systems, the eigenfrequency spectrum can be described by a function of N integers, N being the number of degrees of freedom of the system. In contrast, the spectrum of chaotic systems shows no such regularities. However, the spectrum possesses g
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