A simple and accurate prescription for the tidal disruption radius of a star and the peak accretion rate in tidal disruption events

A star destroyed by a supermassive black hole (SMBH) in a tidal disruption event (TDE) enables the study of SMBHs. We propose that the distance within which a star is completely destroyed by an SMBH, defined r_t,c, is accurately estimated by equating the SMBH tidal field (including numerical factors) to the maximum gravitational field in the star. We demonstrate that this definition accurately reproduces the critical β_c = r_t/r_t,c, where r_t = R_⋆(M_•/M_⋆)^1/3 is the standard tidal radius with R_⋆ and M_⋆ the stellar radius and mass, and M_• the SMBH mass, for multiple stellar progenitors at various ages, and can be reasonably approximated by β_c ≃ [ρ_c/(4ρ_⋆)]^1/3, where ρ_c (ρ_⋆) is the central (average) stellar density. We also calculate the peak fallback rate and time at which the fallback rate peaks, finding excellent agreement with hydrodynamical simulations, and also suggest that the partial disruption radius - the distance at which any mass is successfully liberated from the star - is β_partial ≃ 4^-1/3 ≃ 0.6. For given stellar and SMBH populations, this model yields, e.g. the fraction of partial TDEs, the peak luminosity distribution of TDEs, and the number of directly captured stars.