tokens: ( 75 ) and ( 78 )) normalized by the total number of mergers. In these two examples, we set χ 1 = χ 2 . The top panel is for a 0 = 100 au (see Figure 19 ) and we include merging systems with the number of mergers is 673 for e out = 0, 790 for e out = 0.3, 1159 for e out = 0.6, and 2828 for e out = 0.9, so that N max = 5450. The lower panel is for a 0 = 20 au, and we include systems with the number of mergers is 146 ( e out = 0), 180 ( e out = 0.3), 227 ( e out = 0.6), and 411 ( e out = 0.9), so that N max = 964. The other parameters are the same as in Figure 19 . If the distributions of and are uncorrelated, as we may expect to be the case for , when the octupole effect is significant (see Figure 9 and the discussion in the last paragraph of Section 4.2 ), the distribution of χ eff can be derived directly from and , the distribution functions of and . Define where is the maximum possible value of χ eff for given m 1 χ 1 and m 2 χ 2 (this maximum is achieved at ). Then Equation ( 75 ) becomeswhere and . Note that and . Given P 1 ( μ 1 ) and P 2 ( μ 2 ), the distribution function of is In the special case when μ 1 and μ 2 are uniformly distributed, we have P 1 = P 2 = 1/2, and Equation ( 80 ) giveswhere we have assumed without loss of generality. Thus, even for uniform distributions of and (see the case of e out = 0.9 in Figure 19 ), the effective spin parameter χ eff is preferentially distributed around χ eff = 0 (see Figure 20 ). Note that the spin–orbit misalignment distribution and χ eff distribution obtained above refer to relatively wide BH binary systems ( a 0 ≳ 10–100 au) that experience a merger due to large LK eccentricity excitation. Such systems necessarily have . For BH binaries with smaller separations ( a 0 ≲ 1 au) and not much less than unity, the spin–orbit misalignment distribution can be quite different (see Liu Lai 2017 ). Antonini et al. ( 2018 ; see also Rodriguez Antonini 2018 ) have carried population studies of BH mergers in triple systems (based on DA secular equations) and have found a similar peak around χ eff = 0 in the χ eff distribution. Rodriguez Antonini ( 2018 ) also showed an example of the final spin–orbit misalignment distribution with a peak around 90°, in qualitative agreement with our result. They did not distinguish the difference in the spin–orbit misalignment distributions between largely quadrupole systems (small ε oct ) and strong octupole systems (large ε oct ). We do not agree with the reason(s) they gave for the peak in the χ eff distribution. In particular, it is important to recognize that, during the orbital decay, the adiabaticity parameter (Equation ( 61 )) transitions from ≪1 to ≫1, and this transition determines . 5. Summary and Discussion In this paper we have studied BH binary mergers in triple systems. A sufficiently inclined tertiary companion excites large eccentricity in the BH binary orbit through gravitational perturbations (the LK mechanism), significantly shortening its
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