[Paper Review] Acoustic oscillations of rapidly rotating polytropic stars. II. Effects of the Coriolis and centrifugal accelerations

D. R. Reese, F. Lignières+1 more|ArXiv.org|May 19, 2006

Stellar, planetary, and galactic studiesPhysics and Astronomy31 references105 citations

TL;DR

This study develops a high-accuracy 2D spectral numerical method to compute acoustic pulsation modes in rapidly rotating polytropic stars, fully accounting for both Coriolis and centrifugal forces. It demonstrates that perturbative methods fail beyond $v imes ext{sin}i = 50 ext{ km·s}^{-1}$ for a $1.9 ext{M}_igodot$, $2.3 ext{R}_igodot$ star, with centrifugal distortion being the dominant source of error in such approximations.

ABSTRACT

Context: With the launch of space missions devoted to asteroseismology (like COROT), the scientific community will soon have accurate measurements of pulsation frequencies in many rapidly rotating stars. Aims: The present work focuses on the effects of rotation on pulsations of rapidly rotating stars when both the Coriolis and centrifugal accelerations require a non-perturbative treatment. Method: We develop a 2-dimensional spectral numerical approach which allows us to compute acoustic modes in centrifugally distorted polytropes including the full influence of the Coriolis force. This method is validated through comparisons with previous studies, and the results are shown to be highly accurate. Results: In the frequency range considered and with COROT's accuracy, we establish a domain of validity for perturbative methods, thus showing the need for complete calculations beyond v.sin i = 50 km/s for a R = 2.3 R_\odot, M = 1.9 M_\odot polytropic star. Furthermore, it is shown that the main differences between complete and perturbative calculations come essentially from the centrifugal distortion.

Motivation & Objective

• To develop a non-perturbative numerical method that fully includes Coriolis and centrifugal forces in computing acoustic pulsation modes of rapidly rotating stars.
• To establish the domain of validity for perturbative methods in the context of upcoming asteroseismic missions like COROT.
• To quantify the relative contributions of Coriolis and centrifugal forces to mode frequency shifts and mode structure deformation.
• To provide a high-accuracy reference for mode identification and stellar structure inference in rapidly rotating stars.
• To assess the necessity of full non-perturbative calculations for stars with high rotation rates and high-order pulsation modes.

Proposed method

• A 2D spectral numerical approach is employed using a surface-fitting spheroidal coordinate system based on Bonazzola et al. (1998), enabling accurate representation of centrifugally distorted stellar equilibria.
• The method solves the full 2D eigenvalue problem for acoustic modes, incorporating both the Coriolis force and the centrifugal distortion in the equations of motion and equilibrium structure.
• The spectral method, adapted from Canuto et al. (1988), ensures high accuracy (6–7 digits) by using global basis functions and collocation techniques.
• The equilibrium model is computed by solving the nonlinear equations of hydrostatic equilibrium in the rotating frame, including the centrifugal potential $\frac{1}{2}\Omega^2 s^2$.
• The perturbation equations are derived from the linearized equations of motion and continuity, with the full Coriolis and centrifugal terms retained.
• Validation is performed through comparisons with previous studies (e.g., Saio 1981; Clement 1984; Christensen-Dalsgaard & Mullan 1994), variational principle tests, and sensitivity analysis of numerical parameters.

Experimental results

Research questions

• RQ1 At what rotation rate does the perturbative treatment of rotation become invalid for acoustic modes in rapidly rotating polytropic stars?
• RQ2 How do the Coriolis and centrifugal forces individually affect the frequency spectrum and mode structure of stellar pulsations?
• RQ3 What is the relative importance of centrifugal distortion versus Coriolis force in introducing errors in perturbative calculations?
• RQ4 Can a non-perturbative spectral method achieve sufficient accuracy to serve as a benchmark for asteroseismic modeling of fast rotators?
• RQ5 How does the mode structure of high-order acoustic modes change under strong centrifugal deformation, and what implications does this have for mode identification?

Key findings

• For a $1.9 ext{M}_igodot$, $2.3 ext{R}_igodot$ polytropic star, perturbative methods lose validity beyond $v \times \text{sin}i = 50\text{ km·s}^{-1}$ when considering COROT’s primary target precision ($0.08\,\mu\text{Hz}$).
• The domain of validity for perturbative methods extends to $v \times \text{sin}i = 75\text{ km·s}^{-1}$ for secondary targets with $0.6\,\mu\text{Hz}$ precision.
• The main source of error in perturbative calculations arises from the neglect of centrifugal distortion, not the Coriolis force.
• At a rotation rate of $0.59\Omega_K$, the perturbative frequency spectrum differs significantly from the complete non-perturbative solution, invalidating mode identification based on perturbation theory.
• The effect of centrifugal distortion increases with mode frequency, amplifying errors in high-order modes, which necessitates full non-perturbative treatment even for moderately rotating stars.
• The method achieves 6–7 digit accuracy, validated through comparisons with analytical and numerical benchmarks, variational principle checks, and parameter sensitivity tests.