On the apsidal motion in close binaries due to the tidal deformations of the components

The paper is devoted to a confrontation of the apsidal-motion rates in close binaries due to the tidal perturbations of the stellar components that are predicted by Sterne's formula (1939) with the corresponding apsidal-motion rates that are determined in the framework of the theory of the dynamic tides. Sterne's formula is derived in the supposition that the orbital period and the star's rotational period are sufficiently long so that, in accordance with an earlier suggestion of Cowling (1938), the star is almost adjusted to the gravitational field of the companion. From the point of view of the theory of the dynamic tides, the second-degree tide is then approximated at each instant by an appropriate linear combination of three second-degree tides which are considered to be static. In this limiting case, the rate of secular apsidal motion predicted by Sterne's formula agrees, up to large orbital eccentricities, with the rate of secular apsidal motion determined in the framework of the theory of the dynamic tides and depends on the star's central mass condensation. For close binaries with shorter orbital periods, the use of Sterne's formula leads to deviations because of the increasing influence of the compressibility of the stellar fluid and resonances of dynamic tides with lower-order g(+)-modes. The relative deviations may to amount to a few tens of percents for models of zero-age main sequence stars of 5 M-circle dot, 10M(circle dot), and 20M(circle dot).