The asymptotic representation of higher-order g-modes in stars with a convective core

¹ Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 300 D, 3001 Leuven, Belgium e-mail: Paul.Smeyers@ster.kuleuven.be
² LESIA, UMR 8109, Observatoire de Paris-Meudon, France
³ Present address: Instituto de Astrofísica de Andalucía, CSIC, PO Box 3004, 18080 Granada, Spain

Received: 8 August 2006
Accepted: 4 December 2006

Abstract

Aims.A first-order asymptotic representation of higher-order non-radial g⁺-modes in spherically symmetric stars with a convective core is constructed from the full fourth-order system of governing equations. Stars are considered that, besides their convective core, also contain a radiative envelope, or both an intermediate radiative zone and a convective envelope. At the same time, the earlier asymptotic theory of Willems et al. (1997, A&A, 318, 99) relative to stars consisting of a convective core and a radiative envelope is made more transparent.

Methods.As in the asymptotic theory of Smeyers (2006, A&A, 451, 223) for low-degree, higher-order p-modes, two-variable expansion procedures and boundary-layer theory are applied to the fourth-order system of differential equations established by Pekeris (1938, ApJ, 88, 189).

Results.Eigenfrequency equations are derived in terms of the radial order n of the g⁺-mode. The first  nodes of the radial component of the Lagrangian displacement coincide with the  nodes of the divergence of the Lagrangian displacement, and are situated in the radiative envelope or in the intermediate radiative zone according to the type of star considered. The radial displacement displays an nth node near the surface. In stars containing an intermediate radiative zone and a convective envelope, the nth node is situated in the envelope.

Conclusions.As well as for higher-order p-modes of spherically symmetric stars, the divergence of the Lagrangian displacement plays a basic role in the development of the asymptotic theory.