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

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