2009ApJ...698..715S

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 (SCF) method to construct the equilibrium structure of the polytrope under rotational and tidal perturbations. Our SCF 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 polytropes with fixed entropy, the enhancement factor in stellar radii is about 23% and 4%-8% for µ = 1 and ∼0.1-0.9, respectively, where µ is the fractional mass of the polytrope relative to the total. The centrifugal force dominates the tidal force in determining the equilibrium structure provided µ ≳ 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 the density concentration, volume radius, coefficient of the mass-radius relation, moment of inertia, spin angular momentum, critical rotation parameter, effective internal structure constant, etc., as functions of µ and the perturbation parameters.