Approximations to the spin of close Black-hole-Wolf-Rayet binaries

Population synthesis studies of binary black-hole mergers often lack robust black-hole spin estimates as they cannot accurately follow tidal spin-up during the late black-hole-Wolf-Rayet evolutionary phase. We provide an analytical approximation of the dimensionless second-born black-hole spin given the binary orbital period and Wolf-Rayet stellar mass at helium depletion or carbon depletion. These approximations are obtained from fitting a sample of around $10^5$ detailed MESA simulations that follow the evolution and spin up of close black-hole--Wolf-Rayet systems with metallicities in the range $[10^{-4},1.5Z_\odot]$. Following the potential spin up of the Wolf-Rayet progenitor, the second-born black-hole spin is calculated using up-to-date core collapse prescriptions that account for any potential disk formation in the collapsing Wolf-Rayet star. The fits for second-born black hole spin provided in this work can be readily applied to any astrophysical modeling that relies on rapid population synthesis, and will be useful for the interpretation of gravitational-wave sources using such models.