2021MNRAS.501..507O

We investigate the possible origin of the transiting giant planet WD 1856+534 b, the first strong exoplanet candidate orbiting a white dwarf, through high-eccentricity migration (HEM) driven by the Lidov-Kozai (LK) effect. The host system's overall architecture is a hierarchical quadruple in the '2 + 2' configuration, owing to the presence of a tertiary companion system of two M-dwarfs. We show that a secular inclination resonance in 2 + 2 systems can significantly broaden the LK window for extreme eccentricity excitation (e >= 0.999), allowing the giant planet to migrate for a wide range of initial orbital inclinations. Octupole effects can also contribute to the broadening of this 'extreme' LK window. By requiring that perturbations from the companion stars be able to overcome short-range forces and excite the planet's eccentricity to e ≃ 1, we obtain an absolute limit of a₁ ≳8 au (a₃/ 1500 au)^6/7 for the planet's semimajor axis just before migration (where a₃ is the semimajor axis of the 'outer' orbit). We suggest that, to achieve a wide LK window through the 2 + 2 resonance, WD 1856 b likely migrated from 30 au≲a₁ ≲60 au, corresponding to ∼10-20 au during the host's main-sequence phase. We discuss possible difficulties of all flavours of HEM affecting the occurrence rate of short-period giant planets around white dwarfs.