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Paper: Asymptotic Representation of p-modes Modified for Effects of Gravity and Density Stratification
Volume: 135, A Half Century of Stellar Pulsation Interpretations: a Tribute to Arthur N. Cox
Page: 87
Authors: Van Hoolst, T.; Willems, B.; Smeyers, P.
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, mostly in the stellar core. In the first asymptotic approximation, the time needed for an acoustic wave to propagate from the centre of the star to a given radial distance is increased, and the oscillation frequency of a p- mode is decreased. The increase of the propagation time of an acoustic wave and the associated decrease of the oscillation frequency are larger for lower-frequency p-modes. For a polytropic model with index equal to three, the relative errors on the first asymptotic approximations of the eigenfrequencies are sensibly reduced by the use of the frequency-dependent velocity of propagation of the acoustic waves. Also, the phase shifts between the asymptotically derived eigenfunctions and the exact eigenfunctions are much smaller. In the second asymptotic approximation, the improvement is less pronounced as gravity and density stratification is also incorporated in the usual asymptotic theory from that approximation on. We also applied the modified asymptotic theory to a normal solar model. As for the polytropic model, the relative errors of the first asymptotic approximations of eigenfrequencies are reduced, but to a lesser extent. The second asymptotic approximations of the eigenfrequencies do not lead to satisfactory results. The main cause seems to be that the reflection of waves at the outer boundary of the acoustic cavity in the sun is not adequately described by the asymptotic theory. The second derivative of the density, which is one of the ingredients determining the reflection (see, e.g., Tolstoy 1973), shows a very high peak in the solar outer layers due to the partial ionization of hydrogen, and does not appear in the asymptotic theory developed up to the second order. References: Roxburgh, I.W., Vorontsov, S.V., 1994, MNRAS 267, 297 Smeyers, P., Vansimpsen, T., De Boeck, I., Van Hoolst, T., 1996, A&A 307, 105 Tolstoy, I., 1973, Wave Propagation, McGraw-Hill Book Company, New York
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