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Internal Structure and Apsidal Motions of Polytropic Stars in Close Binaries

We consider a synchronized, circular-orbit binary consisting of a polytrope with index n and a point-mass object, and use a self-consistent field method to construct the equilibrium structure of the polytrope under rotational and tidal perturbations. Our self-consistent field method is distinct from others in that the equilibrium orbital angular velocity is calculated automatically rather than being prescribed, which is crucial for obtaining apsidal motion rates accurately. We find that the centrifugal and tidal forces make perturbed stars more centrally condensed and larger in size. For n=1.5 polytopes with fixed entropy, the enhancement factor in stellar radii is about 23% and 4-8% for mu=1 and sim0.1-0.9, respectively, where mu is the fractional mass of the polytrope relative to the total. The centrifugal force dominates the tidal force in determining the equilibrium structure provided mu > 0.13-0.14 for n > 1.5. The shape and size of rotationally- and tidally-perturbed polytropes are well described by the corresponding Roche models as long as n > 2. The apsidal motion rates calculated for circular-orbit binaries under the equilibrium tide condition agree well with the predictions of the classical formula only when the rotational and tidal perturbations are weak. When the perturbations are strong as in critical configurations, the classical theory underestimates the real apsidal motion rates by as much as 50% for n=1.5 polytropes, although the discrepancy becomes smaller as n increases. For practical uses, we provide fitting formulae for various quantities including the density concentration, volume radius, and effective internal structure constant, as functions of mu and the perturbation parameters.