On the non-dissipative tidal evolution of the misalignment between spin and orbital angular momenta

We extend our previous work on the evolution of close binary systems with misaligned orbital and spin angular momenta resulting from non-dissipative tidal interaction to include all physical effects contributing to apsidal motion. In addition to tidal distortion of the primary by the compact secondary these include relativistic Einstein precession and the rotational distortion of the primary. The influence of the precession of the line of nodes is included. The dependence of the tidal torque on the apsidal angle $\hat\varpi$ couples the apsidal motion to the rate of evolution of the misalignment angle $\beta$ which is found to oscillate. We provide analytical estimates for the oscillation amplitude $\Delta\beta$ over a wide range of parameter space confirmed by numerical integrations. This is found to be more significant near critical curves on which $d{\hat \varpi } /dt=0$ for a specified $\beta$. We find that to obtain $0.1 < \Delta\beta < \sim 1,$ the mass ratio, $q > \sim1$ the initial eccentricity should be modest, $\cos \beta < 1/\sqrt{5},$ with $\cos\beta <0$ corresponding to retrograde rotation, initially, and the primary rotation rate should be sufficiently large. The extended discussion of apsidal motion and its coupled evolution to the misalignment angle given here has potential applications to close binaries with anomalous apsidal motion as well as transiting exoplanets such as warm Jupiters.