The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars

Free access article

Issue		A&A Volume 411, Number 2, November IV 2003


Page(s)		215 - 220
Section		Stellar atmospheres
DOI		http://dx.doi.org/10.1051/0004-6361:20031318

A&A 411, 215-220 (2003)
DOI: 10.1051/0004-6361:20031318

The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars

I. W. Roxburgh^{1, 2} and S. V. Vorontsov^{1, 3}

¹ Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
² LESIA, Observatoire de Paris, place Jules Janssen, 92195 Meudon, France
³ Institute of Physics of the Earth, B Gruzinskya 10, Moscow 123810, Russia

(Received 7 May 2003 / Accepted 14 August 2003)

Abstract
By considering stellar models with the same interior structure but different outer layers we demonstrate that the ratio of the small to large separations of acoustic oscillations in solar-like stars is essentially independent of the structure of the outer layers, and is determined solely by the interior structure. Defining the scaled Eulerian pressure perturbation $\psi_\ell(\omega,t) = r p' /(\rho c)^{1/2}$ we define the internal phase shift $\delta_\ell(\omega,t)$ through the relation $\omega\psi/({\rm d}\psi/{\rm d}t)=\tan(\omega t -\pi\ell/2 + \delta_\ell)$ . The $\delta_\ell$ are almost independent of acoustic radius $t=\int {\rm d}r/c$ outside the stellar core and can be determined as a continuous functions of $\omega$ from partial wave solutions for the interior - that is solutions of the oscillation equations for any $\omega$ that satisfy the Laplace boundary condition at a sufficiently large acoustic radius t_f outside the stellar core. If the $\omega$ are eigenfrequencies then they satisfy the Eigenfrequency Equation $\omega T=(n+\ell/2) \pi+\alpha(\omega)-\delta_\ell(\omega)$ where $\alpha(\omega)$ is the $\ell$ independent surface phase shift (Roxburgh & Vorontsov 2000). Using this result we show that the ratio of small to large separations is determined to high accuracy solely by the internal phase shifts $\delta_\ell(\omega)$ and hence by the interior structure alone. The error in this result is estimated and shown to be smaller than that associated with the errors in the determination of the frequencies ( $\approx$ 0.1-0.3 $\mu$ Hz) from the upcoming space missions MOST, COROT and Eddington.

Key words: stars: oscillations

Offprint request: I. W. Roxburgh, Roxburgh@qmul.ac.uk

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