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

Abstract

Abstract. 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 , 10 M , and 20 M .

New results on the apsidal-motion test to stellar structure and evolution including the effects of dynamic tides

“…In binaries with shorter orbital periods, deviations from the classical formula arise due to the effects of the stellar compressibility and due to the occurrence of resonances between dynamic tides and free oscillation modes (Smeyers & Willems 2001;Willems & Claret 2002). When the expansion of the tide-generating potential is restricted to the dominant = 2 terms, the formula for the rate of secular apsidal motion taking into account the effects of stellar compressibility and resonances is given bẏ…”

“…For details on the quantities G 2,m,k and F 2,m,k , we refer to Smeyers & Willems (2001) and Willems & Claret (2002). In the context of this investigation, it is furthermore convenient to rewrite Eq.…”

“…up to high orbital eccentricities (Smeyers & Willems 2001), the constant k 2,dyn can be considered as a weighted mean of the coefficients F 2,m,k with the weights depending on the orbital eccentricity. In order to determine rate of secular apsidal motion within the framework of the theory of dynamic tides, the contribution of each term in Eq.…”

“…For instance, the assumption that the star readjusts its shape instantaneously during the revolution around the center of mass of the system is only valid when the orbital period is much larger than the periods of the low-order modes of oscillation. Several authors investigated the importance of resonances and stellar compressibility (Papaloizou & Pringle 1980;Quataert et al 1996; and more recently Smeyers & Willems 2001;Willems & Claret 2002). Unfortunately, these results were never applied systematically to the apsidal-motion scenario and their contribution could not be quantified.…”

Tidal and rotational effects in the perturbations of hierarchical triple stellar systems

“…(We mainly refer to the apsidal motion in close, eccentric binaries.) While most recent studies (e.g., Claret 1999;Witte & Savonije 1999;Smeyers & Willems 2001;Claret & Willems 2002) in this field are based on the most recent stellar models, they do not consider the possible perturbative effects of a close third companion which can cause e.g. some precession.…”