Dynamical Stability of Polar Circumbinary Orbits and Planet Formation in the Planetary Disk of 99 Herculis

A possible polar-ring debris disk, the dynamics of which can be described by the outer hierarchical restricted three-body problem, has been detected in 99 Herculis. An empirical formula for the minimum radius beyond which test particles in polar orbits can keep stable within 107 binary periods is provided through numerical fitting, applying to the binary eccentricity ${e}_{1}\in \left[0,0.8\right)$ and the mass ratio of the binary $\lambda \in \left[0.1,1\right]$, where λ = m0/m1 (m0 and m1 represent the masses of the two binary stars). The polar planetary disk has the lowest statistical accretion efficiency and a moderate impact frequency of collisions among planetesimals (with radii of 1–10 km) compared to those in a circumbinary coplanar disk and a standard disk around the single host star. The collision timescale in the circumbinary disk (both polar and coplanar configurations) is longer than 107 yr, exceeding the dissipation timescale of the gas disk. Stochastic simulations show that successive collisions cannot make planetesimals grow, which may explain the formation of the debris disk observed in 99 Herculis.