Planetary migration in precessing discs for S-type wide binaries

The discovery of numerous circumprimary planets in the last few years has brought to the fore the question of planet formation in binary systems. The significant dynamical influence, during the protoplanetary disc phase, of a binary companion on a giant planet has previously been highlighted for wide binary stars. In particular, highly inclined binary companion can induce perturbations on the disc and the planets through the Lidov-Kozai resonance, which could inhibit the formation process. In this work, we aim to study how the disc gravitational potential acting on the planet and the nodal precession induced by the wide binary companion with separation of 1000 au on the disc act to suppress the Lidov-Kozai perturbations on a migrating giant planet. We derive new approximate formulas for the evolution of the disc's inclination and longitude of the ascending node in case of a rigidly precessing disc with a decreasing mass and perturbed by a wide binary companion, which are suitable for N-body simulations. We carry out 3200 simulations with several eccentricity and inclination values for the binary companion. The gravitational and damping forces exerted by the disc on the planet tend to keep the latter in the mid-plane of the former, and suppress the effect of the binary companion by preventing the planet from getting locked in the Lidov-Kozai resonance during the disc phase. We also confirm that because of nodal precession induced by the binary, a primordial spin-orbit misalignment could be generated for circumprimary planets with an inclined binary companion.