[Paper Review] Acoustic oscillations in rapidly rotating polytropic stars I. Effects of the centrifugal distortion

F. Lignières, M. Rieutord+1 more|CERN Bulletin|Apr 13, 2006

Stellar, planetary, and galactic studiesPhysics and Astronomy20 references70 citations

TL;DR

This paper presents a non-perturbative method to compute acoustic oscillation modes in rapidly rotating polytropic stars by fully accounting for centrifugal distortion while neglecting the Coriolis force, valid for high radial order p-modes. Key findings include equatorial amplitude concentration, breakdown of asymptotic non-rotating frequency spacing, and significant deviations from perturbative methods at rotation rates above 15% of Keplerian limit.

ABSTRACT

A new non-perturbative method to compute accurate oscillation modes in rapidly rotating stars is presented. In this paper, the effect of the centrifugal force is fully taken into account while the Coriolis force is neglected. This assumption is valid when the time scale of the oscillation is much shorter than the inverse of the rotation rate and is expected to be suitable for high radial order p-modes of $δ$ Scuti stars. Axisymmetric p-modes have been computed in uniformly rotating polytropic models of stars. In the frequency and rotation range considered, we found that as rotation increases (i) the asymptotic structure of the non-rotating frequency spectrum is first destroyed then replaced by a new form of organization (ii) the mode amplitude tends to concentrate near the equator (iii) differences with perturbative methods become significant as soon as the rotation rate exceeds about fifteen percent of the Keplerian limit. The implications for the seismology of rapidly rotating stars are then discussed.

Motivation & Objective

• To develop a non-perturbative method for computing accurate eigenmodes in rapidly rotating stars, where perturbative approaches fail.
• To isolate and analyze the effects of centrifugal force on the equilibrium structure and acoustic modes, neglecting Coriolis forces for high-frequency p-modes.
• To assess the validity limits of perturbative methods in asteroseismology by comparing with non-perturbative results.
• To investigate how centrifugal distortion alters mode frequency spectra, amplitude distribution, and visibility in rotating stars.
• To explore the emergence of regular frequency spacings and equatorial concentration in the acoustic spectrum of rapidly rotating stars.

Proposed method

• Uses uniformly rotating polytropic models as equilibrium states, incorporating centrifugal force effects on the effective gravity and stellar geometry.
• Solves the two-dimensional eigenvalue problem for adiabatic, axisymmetric p-modes under the Cowling approximation, neglecting Coriolis force.
• Employs spectral numerical methods to compute eigenmodes with high accuracy, validated through convergence and consistency checks.
• Focuses on high radial order p-modes in the frequency range relevant to δ Scuti stars, with rotation rates up to 59% of Keplerian limit.
• Analyzes mode structure, frequency spacing, and visibility by integrating over the stellar disk, accounting for inclination and mode symmetry.
• Compares results with perturbative methods to quantify deviations and assess their domain of validity.

Experimental results

Research questions

• RQ1 How does centrifugal distortion alter the asymptotic frequency spacing of p-modes in rapidly rotating stars?
• RQ2 To what extent do mode amplitudes concentrate near the equator due to centrifugal deformation, and how does this affect visibility?
• RQ3 At what rotation rate do deviations from perturbative methods become significant in the computed eigenfrequencies and mode structures?
• RQ4 What is the relationship between the new spectral structure and the equatorial concentration of mode energy?
• RQ5 How does equatorial symmetry (symmetric vs. anti-symmetric modes) influence avoided crossings and mode visibility?

Key findings

• As rotation increases beyond 15% of the Keplerian limit, the asymptotic frequency spacing of non-rotating stars is destroyed and replaced by a new, regular structure.
• Mode amplitudes increasingly concentrate near the equator due to centrifugal distortion, altering disk-integrated visibility and potentially explaining observed amplitude increases with inclination.
• Deviations from perturbative methods become significant at rotation rates above 15% of the Keplerian limit, indicating the breakdown of standard perturbative assumptions.
• Symmetric and anti-symmetric modes exhibit distinct spectral evolution, with stronger avoided crossings in symmetric modes due to greater frequency separation.
• The emergence of regular frequency spacings at high rotation rates suggests a possible near-integrable ray dynamics, contrasting with chaotic behavior expected in non-separable systems.
• The cancellation effect of disk integration on mode visibility is reduced and mode-dependent in rapidly rotating stars, challenging assumptions from non-rotating models.