  The black hole (BH) binaries in active galactic nuclei (AGN) are expected to
form mainly through scattering encounters in the ambient gaseous medium. Recent
simulations, including our own, have confirmed this formation pathway is highly
efficient. We perform 3D smoothed particle hydrodynamics (SPH) simulations of
BH scattering encounters in AGN disks. Using a range of impact parameters, we
probe the necessary conditions for binary capture and how different orbital
trajectories affect the dissipative effects from the gas. We identify a single
range of impact parameters, typically of width $\sim0.86-1.59$ binary Hill
radii depending on AGN disk density, that reliably leads to binary formation.
The periapsis of the first encounter is the primary variable that determines
the outcome of the initial scattering. We find an associated power-law between
the energy dissipated and the periapsis depth to be $\Delta E\propto r^{-b}$
with $b=0.42\pm0.16$, where deeper encounters dissipate more energy. Excluding
accretion physics does not significantly alter these results. We identify the
region of parameter space in initial energy vs impact parameter where a
scattering leads to binary formation. Based on our findings, we provide a
ready-to-use analytic criterion that utilises these two pre-encounter
parameters to determine the outcome of an encounter, with a reliability rate of
>90\%. As the criterion is based directly on our simulations, it provides a
reliable and highly physically motivated criterion for predicting binary
scattering outcomes which can be used in population studies of BH binaries and
mergers around AGN.
