How empty is an empty loss cone?

Abstract

We consider two body relaxation in a spherical system with a loss cone. Considering two-dimensional angular momentum space, we focus on "empty loss cone" systems, where the typical scattering during a dynamical time $j_{d}$ is smaller than the size of the loss cone $j_{\rm lc}$. As a result, the occupation number within the loss cone is significantly smaller than outside. Classical diffusive treatment of this regime predict exponentially small occupation number deep in the loss cone. We revisit this classical derivation of occupancy distribution of objects in the empty loss cone regime. We emphasize the role of the rare large scatterings and show that the occupancy does not decay exponentially within the loss cone, but it is rather flat, with a typical value $\sim [(j_d/j_{\rm lc})]^2\ln^{-2}(j_{\rm lc}/j_{\min})$ compared to the occupation in circular angular momentum (where $j_{\min}$ is the smallest possible scattering). Implication are that although the loss cone for tidal break of Giants or binaries is typically empty, tidal events which occurs significantly inside the loss cone ($\beta\gtrsim 2$), are almost as common as those with $\beta\cong 1$ where $\beta$ is the ratio between the tidal radius and the periastron. The probability for event with penetration factor $>\beta$ decreases only as $\beta^{-1}$ rather than exponentially. This effect has no influence on events characterized by full loss cone, such as tidal disruption event of $\sim 1m_\odot$ main sequence star.