  By tracing the orbital and star formation histories of galaxies within the
Coma cluster, we seek to understand the role of the high-density cluster
environment in quenching these galaxies. We combine star-formation histories
extracted from high-signal-to-noise spectra of 11 early-type galaxies around
the center of the Coma cluster with probability distributions for their orbital
parameters obtained using an N-body simulation to connect their orbital and
star formation histories. We find that all 11 galaxies likely quenched near
their first pericentric approach. Higher stellar mass galaxies
($\log(M_\star/\mathrm{M}_\odot)>10$) had formed a higher fraction of their
stellar mass (more than $\sim90\%$) than their lower mass counterparts ($\sim
80$-$90\%$) by the time they fell into the cluster (when they cross
$2.5r_\mathrm{vir}$). We find that the expected infall occurred around
$z\sim0.6$, followed by the first pericentric passage $\sim4\,\mathrm{Gyr}$
later. Galaxies in our sample formed a significant fraction of their stellar
mass, up to $15\%$, between infall and first pericenter, and had assembled more
than $\sim98\%$ of their cumulative stellar mass by first pericenter. Unlike
previous low-redshift studies which suggest that star formation continues until
about first apocenter or later, the high percentage of stellar mass already
formed by first pericenter in our sample galaxies points to star formation
ceasing within a gigayear after the first pericentric passage. We consider the
possible physical mechanisms driving quenching and find that our results
resemble the situation in clusters at $z\sim1$, where active stripping of gas
(ram-pressure or tidally driven) seems to be required to quench satellites by
their first pericentric passage. However, a larger sample will be required to
conclusively account for the unknown fraction of pre-processed satellites in
the Coma cluster.
