  Central galaxies make up the majority of the galaxy population, including the
majority of the quiescent population at $\mathcal{M}_* >
10^{10}\mathrm{M}_\odot$. Thus, the mechanism(s) responsible for quenching
central galaxies plays a crucial role in galaxy evolution as whole. We combine
a high resolution cosmological $N$-body simulation with observed evolutionary
trends of the "star formation main sequence," quiescent fraction, and stellar
mass function at $z < 1$ to construct a model that statistically tracks the
star formation histories and quenching of central galaxies. Comparing this
model to the distribution of central galaxy star formation rates in a group
catalog of the SDSS Data Release 7, we constrain the timescales over which
physical processes cease star formation in central galaxies. Over the stellar
mass range $10^{9.5}$ to $10^{11} \mathrm{M}_\odot$ we infer quenching
e-folding times that span $1.5$ to $0.5\; \mathrm{Gyr}$ with more massive
central galaxies quenching faster. For $\mathcal{M}_* =
10^{10.5}\mathrm{M}_\odot$, this implies a total migration time of $\sim
4~\mathrm{Gyrs}$ from the star formation main sequence to quiescence. Compared
to satellites, central galaxies take $\sim 2~\mathrm{Gyrs}$ longer to quench
their star formation, suggesting that different mechanisms are responsible for
quenching centrals versus satellites. Finally, the central galaxy quenching
timescale we infer provides key constraints for proposed star formation
quenching mechanisms. Our timescale is generally consistent with gas depletion
timescales predicted by quenching through strangulation. However, the exact
physical mechanism(s) responsible for this still remain unclear.
