A Semi-Analytical Loss Cone Theory for Tidal Disruption Event Rates Around Kerr Black Holes

Abstract: A tidal disruption event (TDE) occurs when a star is scattered onto a near-radial orbit and is torn apart by a black hole (BH)'s tidal field. The angular momentum threshold for disruption is set by general relativistic tidal dynamics, while the supply of stars to the disruption zone is governed by Newtonian stellar dynamics. A spinning BH breaks the spherical symmetry of the disruption boundary, so a star's survival depends on both the magnitude and the orientation of its angular momentum. Existing treatments either assume a non-spinning BH or rely on numerical simulations of spinning BHs. We develop the first semi analytical framework that incorporates spin-dependent loss cone boundaries into TDE rate theory. Using a novel tidal tensor formalism, we compute inclination-dependent thresholds for tidal disruption and direct capture by the event horizon. We then revisit the classical one dimensional loss cone problem with nested disruption and capture boundaries, deriving a closed form capture fraction valid across all loss cone regimes. Finally, we formulate a two dimensional Fokker--Planck equation describing simultaneous diffusion in angular momentum magnitude and orientation. Through a perturbative treatment, we demonstrate that while the Kerr disruption boundary induces a first-order bias favouring the disruption of retrograde stars, the global TDE rate is remarkably insensitive to black hole spin. This approach offers a tractable route to including spin and orbital inclination in population-level TDE studies.