The second-order asymptotic representation of higher-order non-radial p-modes in stars revisited

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

Received: 18 November 2005
Accepted: 23 January 2006

Abstract

Aims. We improve the second-order asymptotic theory of higher-order non-radial p-modes in spherically symmetric stars that was developed by Smeyers et al. (1996) as an alternative for Tassoul's approach (1990).

Methods. Like the previous authors, we use asymptotic methods appropriate for singular perturbation problems, i.e. expansion procedures in terms of two variables and boundary-layer theory. However, in contrast with them, we no longer adopt boundary-layer coordinates near the singular boundary points that are identical to the fast variable used in the asymptotic expansions at larger distances.

Results. By our definitions of the boundary-layer coordinates, the matchings of the boundary-layer expansions to the asymptotic expansions valid at larger distances from the boundary points, and the constructions of the uniformly valid asymptotic expansions are more transparent.

Conclusions. The present asymptotic theory confirms that the application of expansions in terms of two variables and boundary-layer theory to the fourth-order system of differential equations established by Pekeris (1938, ApJ, 88, 189) is particularly appropriate for the construction of the asymptotic representation of higher-order p-modes in spherically symmetric stars. For these modes, the divergence of the Lagrangian displacement is the basic function, and the radial component of the Lagrangian displacement is of one order higher in the small expansion parameter. In the lowest-order asymptotic approximation, the divergence of the Lagrangian displacement obeys a second-order differential equation of the Sturm-Liouville type. This property explains that the eigenfunction that is associated with the nth eigenfrequency displays nodes, with