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 10⁵ detailed MESA simulations that follow the evolution and spin up of close black hole–Wolf–Rayet systems with metallicities in the range . 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.

双黑洞合并的种群综合研究通常缺乏可靠的黑洞自旋估计，因为它们无法准确地跟踪黑洞-沃尔夫-拉叶演化后期的潮汐自旋。考虑到双星轨道周期和氦耗尽或碳耗尽时的沃尔夫-拉叶恒星质量，我们提供了无量纲次生黑洞自旋的解析近似。这些近似值是通过拟合大约 10 ^{5 个}详细 MESA 模拟的样本而获得的，这些模拟遵循金属丰度范围内的封闭黑洞-Wolf-Rayet 系统的演化和自旋. 随着沃尔夫-拉叶前身的潜在旋转，第二个黑洞自旋是使用最新的核心坍缩处方计算的，该处方解释了坍缩沃尔夫-拉叶星中任何潜在的圆盘形成。这项工作中提供的对二胎黑洞自旋的拟合可以很容易地应用于任何依赖于快速人口合成的天体物理模型，并将有助于使用此类模型解释引力波源。