On the formation of retrograde S-type planets in binaries with a polar circumbinary disk

Abstract

Retrograde S-type planets have been observed in several binary systems, yet their formation pathway remains poorly understood. With high-resolution hydrodynamic simulations, we demonstrate that a polar circumbinary disk around an eccentric, unequal-mass binary can form and sustain a retrograde mini disk around the primary star. This provides a direct in-situ formation channel for retrograde S-type planets. The mini disk forms via a sub-Keplerian accretion stream that is slightly misaligned from the polar disk. The mini disk initially undergoes von Zeipel-Kozai-Lidov (ZKL) oscillations, driving coupled eccentricity and inclination evolution. Rather than oscillating indefinitely, the inner mini disk evolves past the critical ZKL inclination, decouples from the outer disk, and settles into a stable retrograde orbit. This evolution is sensitive to numerical resolution: the retrograde configuration is absent in previous lower-resolution simulations, where the mini disk accretion timescale is too short to sustain ZKL-driven evolution. For a higher disk viscosity, the mini disk remains near-polar due to a shorter accretion timescale. Since protoplanetary disks typically have low viscosity, our results suggest that retrograde S-type planets can form in-situ from retrograde mini disks around polar circumbinary disks, and their occurrence rate may be higher than currently estimated.