Resonant dynamic tides and apsidal motion in close binaries

The resonance of a dynamic tide with a free oscillation mode in a component of a close binary system of stars is treated by means of a two-time variable expansion procedure. The treatment is developed with respect to a frame of reference corotating with the star. Both the free oscillation mode and the dynamic tide are considered as linear, isentropic perturbations of a spherically symmetric star. At the lowest order of approximation in the small expansion parameter, the resonant dynamic tide corresponds to the tidally excited oscillation mode. Furthermore, the effect of the resonance on the secular apsidal motion is determined. A resonant dynamic tide can contribute to a much larger secular apsidal motion than the static tide and the non-resonant dynamic tides of the same degree can do. The contribution can be oriented in the sense opposite to the orbital motion as well as in the same sense. From numerical applications to polytropic models, it appears that especially resonances of dynamic tides with an f -mode or a lower-order g+-mode bring about larger apsidal motions. In this context, the determination of the contributions to the secular apsidal motion stemming from the static tides and the non-resonant dynamic tides is reviewed. Sterne’s theory (1939) for the effect of the tidal distortion on the apsidal motion in close binary stars, which is used as standard theory, is also reconsidered. The theory is shown to rest on the consideration of non-resonant, low-frequency dynamic tides taken in their lowest-order asymptotic representation.