Asymptotic representation of low-frequency dynamic tides in close binaries

An asymptotic representation of low-frequency dynamic tides in close binaries is developed. The dynamic tides are treated as low-frequency, linear, isentropic, forced oscillations of a non-rotating spherically symmetric star. The asymptotic representation is developed to the second order in the forcing frequency. For the sake of simplification, the star is assumed to be everywhere in radiative equilibrium. As asymptotic approximation of order zero, the divergencefree static tide of which the radial component is solution of Clairaut’s equation, is adopted. In the asymptotic approximation of order two, the oscillatory properties of the star play a role. The asymptotic solutions are constructed by means of a twovariable expansion procedure. The regions near the star’s centre and surface are treated as boundary layers. The Eulerian perturbation of the gravitational potential caused by the star’s tidal distortion is incorporated in the asymptotic treatment. An expression for that perturbation at the star’s surface is derived to the second-order approximation. The expression is determined by the non-oscillatory parts of the asymptotic solutions valid near the star’s surface.