Astron. Astrophys. 336, 539-544 (1998)

1. Introduction

In a recent investigation, an asymptotic representation of low-frequency, linear, isentropic dynamic tides in components of close binaries has been derived (Smeyers 1997, hereafter referred to as Paper I). The dynamic tide is assumed not to be in resonance with any free oscillation mode of the star. For the sake of simplification, the square of the Brunt-Väisälä frequency is considered to be positive everywhere in the star.

For the construction of the second-order asymptotic solutions for the divergence and the radial component of the tidal displacement, we adopted a method described by Kevorkian and Cole (1981, Sect. 3.3.3; 1996, Sect. 4.3.3). A distinction between various regions in the star is made. At sufficiently large distances from the star's centre and surface, the asymptotic solutions are constructed in terms of the usual radial coordinate, considered as a slow coordinate, and a fast radial coordinate. The regions near the star's centre and the star's surface are treated as boundary layers because of the singularities appearing at these end points.

In the present investigation, we reconsider the constructions of the second-order boundary-layer solutions in order to make them more transparent by the use of a single coordinate in the boundary layer. Moreover, we modify the determination of the constants involved in the boundary-layer solution for the radial component of the tidal displacement that is valid near the surface. As a result of this modification, we arrive at a corrected expression for the second-order Eulerian perturbation of the gravitational potential at the star's surface.

The plan of the paper is as follows. In Sect. 2, we reconsider the construction of the asymptotic expansions that are uniformly valid from the singular point at . Sect. 3 is devoted to the construction of the second-order boundary-layer solutions near the singular point at . In Sect. 4, the continuity of the gravitational potential and its gradient at the star's surface is imposed. The matching of the second-order boundary-layer solutions valid near is performed in Sect. 5. Sect. 6, finally, is devoted to concluding remarks.

© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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