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Abstract We explore three-body binary formation (3BBF), the formation of a bound system via gravitational scattering of three initially unbound bodies (3UB), using direct numerical integrations. For the first time, we consider systems with unequal masses, as well as finite-size and post-Newtonian effects. Our analytically derived encounter rates and numerical scattering results reproduce the 3BBF rate predicted by Goodman & Hut for hard binaries in dense star clusters. We find that 3BBF occurs overwhelmingly through nonresonant encounters and that the two most-massive bodies are never the most likely to bind. Instead, 3BBF favors pairing the two least-massive bodies (for wide binaries) or the most- plus least-massive bodies (for hard binaries). 3BBF overwhelmingly favors wide-binary formation with superthermal eccentricities, perhaps helping to explain the eccentric wide binaries observed by Gaia. Hard-binary formation is far rarer, but with a thermal eccentricity distribution. The semimajor axis distribution scales cumulatively asa³for hard and slightly wider binaries. Although mergers are rare between black holes when including relativistic effects, direct collisions occur frequently between main-sequence stars—more often than hard 3BBF. Yet, these collisions do not significantly suppress hard 3BBF at the low-velocity dispersions typical of open or globular clusters. Energy dissipation through gravitational radiation leads to a small probability of a bound, hierarchical triple system forming directly from 3UB. 

ABSTRACT We present a novel, few-body computational framework designed to shed light on the likelihood of forming intermediate-mass (IM) and supermassive (SM) black holes (BHs) in nuclear star clusters (NSCs) through successive BH mergers, initiated with a single BH seed. Using observationally motivated NSC profiles, we find that the probability of an $${\sim }100\hbox{-}\mathrm{M}_\odot$$ BH to grow beyond $${\sim }1000 \, \mathrm{M}_\odot$$ through successive mergers ranges from $${\sim }0.1~{{\ \rm per\ cent}}$$ in low-density, low-mass clusters to nearly 90  per cent in high-mass, high-density clusters. However, in the most massive NSCs, the growth time-scale can be very long ($$\gtrsim 1\,$$ Gyr); vice versa, while growth is least likely in less massive NSCs, it is faster there, requiring as little as $${\sim }0.1\,$$Gyr. The increased gravitational focusing in systems with lower velocity dispersions is the primary contributor to this behaviour. We find that there is a simple ‘7-strikes-and-you’re-in’ rule governing the growth of BHs: Our results suggest that if the seed survives 7–10 successive mergers without being ejected (primarily through gravitational wave recoil kicks), the growing BH will most likely remain in the cluster and will then undergo runaway, continuous growth all the way to the formation of an SMBH (under the simplifying assumption adopted here of a fixed background NSC). Furthermore, we find that rapid mergers enforce a dynamically mediated ‘mass gap’ between about $${50\!-\!300 \, \mathrm{M}_\odot }$$ in an NSC.