Tidal perturbations of linear, isentropic oscillations in components of circular-orbit close binaries I. Synchronously rotating components

The effects of the tidal force exerted by a companion on linear, isentropic oscillations of a uniformly rotating star that is a component of a circular-orbit close binary are studied. In contrast to an earlier perturbation method, which is almost only applicable to polytropic models, the procedure starts from an arbitrary physical model of a spherically symmetric equilibrium star. The tidal field and the nonspherical tidally perturbed star are supposed to be determined by means of the theory of dynamic tides, in which the tides are treated as forced, linear, isentropic oscillations of a nonrotating spherically symmetric star. The equations governing linear, isentropic oscillations of a tidally perturbed star are established in the domain instantaneously occupied by the star and are transformed into equations defined in the domain of the spherically symmetric star, so that usual perturbation methods can be applied. The procedure is developed for the general case in which the starʹs rotation is not necessarily synchronous with the orbital motion of the companion. The second part of the paper is devoted to the case in which the star rotates synchronously and is subject to an equilibrium tide. The eigenfrequencies of radial modes are shown to remain unaffected by the tidal perturbation at the lowest order of approximation. For the lowest degrees l=1,2,3, the degeneracy of the eigenvalue problem of the linear, isentropic oscillations of a spherically symmetric star is lift partially, so that a (2l+1) -fold eigenfrequency is split up into (l+1) eigenfrequencies. A main result is that the eigenfrequencies of the modes belonging to a given degree l are shown to be all split up according to the same pattern. Attention is paid to the linear combinations of eigenfunctions that have to be adopted at order zero when the polar axis of the spherical harmonics of the angular coordinates coincides with the starʹs rotation axis perpendicular to the orbital plane. The solutions of order zero are also considered in terms of spherical harmonics of angular coordinates for which the polar axis coincides with the tidal axis.