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Astron. Astrophys. 326, 1055-1065 (1997)

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Effects of gravity and density stratification
on the asymptotic representation of p -modes in stars

B. Willems *, T. Van Hoolst ** and P. Smeyers

Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Heverlee, Belgium

Received 11 April 1997 / Accepted 23 May 1997

Abstract

The second-order asymptotic theory for low-degree p -modes in a star developed by Smeyers et al. (1996) is reconsidered, especially for lower-frequency modes. The investigation is undertaken in analogy with an earlier investigation of Roxburgh and Vorontsov (1994), in which a generalization of the first Born approximation for the scattering, by the stellar core, of acoustic waves modified by gravity and buoyancy is applied.

A frequency-dependent velocity of propagation of acoustic waves is introduced that is affected by gravity and density gradient, mainly in the central region of the star. The time needed for an acoustic wave to propagate from the centre of the star to a given radial distance is increased, and, in the first asymptotic approximation, the oscillation frequency of a p -mode is decreased. The differences are larger for lower-frequency p -modes.

The asymptotic theory is applied to a polytropic model with index equal to 3. The relative errors on the scaled frequency separations [FORMULA] for degrees [FORMULA] are reduced in comparison to those resulting from the usual asymptotic theory, but still amount to about 30% for modes of radial order [FORMULA] and to about 18% for modes of radial order [FORMULA].

For a normal solar model, the second asymptotic approximations of the eigenfrequencies do not lead to satisfactory results. The failure is ascribed to the behaviour of the second derivative of the mass density in the partial ionization zone of hydrogen near the solar surface. This behaviour introduces a sharp and high peak in the propagation diagram, which is not taken into account in the present asymptotic analysis.

Key words: stars: oscillations - stars: interiors - Sun: oscillations - methods: analytical

* Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.)
** Postdoctoral Fellow of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.)

Send offprint requests to: P. Smeyers

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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