Dependence of a₂ on ζ and d. Left panel: the spin parameter of M₂ as a function of the accreted mass in units of M_* for the three simulations shown in Figure 2. Here, all of the simulations have ζ = 0.4 and, as a result, the total accreted mass is similar. The solid lines correspond to the values derived from the simulations, while the dashed lines indicate the expected mass and angular momentum accretion from material that remains in the disk (Bardeen 1970; Thorne 1974) when the simulation ends at t = 80 t_*. Middle panel: the dependence of a₂ on ζ. All of the simulations shown have the same separation (d = 3 R_*), but ζ = 0.1, 0.4, 0.5, 0.7, and 0.9. The dashed line represents the spin expected from direct collapse (i.e., ζ = 0). Right panel: the dependence of a₂ on d_* and ζ. The size of the symbols show the final BH mass, ranging from 8.2 M_⊙ to 22.5 M_⊙. The vertical dashed line shows d_τ, the radius for effective tidal synchronization (Zaldarriaga et al. 2018), and the vertical solid line shows d_H, the binary separation required for merging within a Hubble time (28 M_⊙ + 15 M_⊙). Dashed lines depict the spin expected from the direct collapse of different mass fractions, fM_*, of the pre-SN star with f = 0.3, 0.5, 0.8, 1.