tokens: This channel is more common for those models with strong mass segregation effects, i.e., m BBH ≫ m ⋆ . Figure 7 shows an example of the evolutions of the BBH’s inner and outer orbits in model M2. The outer orbit of the BBH first gradually migrates from the outer parts of the cluster ( a 2 ∼ 10 5 au) into the inner regions ( a 2 ∼ 10 3 au). Along the way the binary–single encounters become more and more frequent. Each binary–single encounter changes the inner orbit slightly, as the incoming field star is much lighter than the components of the BBHs, i.e., m A , m B ≫ m ⋆ . From the top right panel of Figure 7 , we can see that the encounters gradually increase the e 1 of the BBH up to ∼0.8–0.9, and the SMA of the BBH also increases up to about 2 au. When the pericenter of the BBH’s outer orbit approaches to about tens of r t , the KL effects become important, which can increase e 1 rapidly. Note that simultaneously the binary–single encounters also change the inner orbits. Under these two effects, the inner orbit of the BBH finally reaches to a very high eccentricity, and eventually the BBH merges owing to GW radiation. Figure 7. Example of evolutions of BBH’s inner and outer orbits. Left panels: evolutions of the outer orbit of the BBH. The dashed cyan line in the top left panel shows the tidal radius for the BBH with a 2 = 2 au. In the bottom left panel, the evolutions of Ω 2 and I 2 are mainly driven by vector RR. Right panels: evolutions of the inner orbit of the BBH. The dashed lines in the top right panel show the GW timescales. “BySn” means the position after the binary–single encounter; “KL” means the position after the KL oscillations. The filled blue square symbol marks the starting position of the BBHs in each panel. The filled blue star symbol in all panels marks the position where the BBH is merged. The initial condition of the model is given in model M2. The bottom panels of Figure 7 show the evolutions of the orbital orientations of both the inner and outer orbits of the BBH. The orientation of the outer orbit evolves under vector RR and the KL effect (bottom left panel of Figure 7 ). The orientation of the inner orbit is changed cumulatively during multiple binary–single encounters and becomes quite rapid during the KL precesses (bottom right panel of Figure 7 ). The rapid oscillations of the inner orbital orientations can help to trigger the merging of BBHs by the KL effect. For example, before merging, I 2 ∼ π and I 1 ∼ π /2, the relative inclination between the inner and outer orbits is close to π /2, which can help to enhance the eccentricity oscillations and merging of the BBHs (Wen 2003 ). Figure 8 shows another example of the BBH’s evolutions in model M2. In Figure 8 , the SMA of the BBH always tends to decrease owing to multiple binary–single encounters. This is because the BBH is mostly remaining in the hard binary regions, and the binary–single encounters always decrease the SMA for a hard binary. A hard binary
ner_tags: O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O