  We compile a sample of SDSS galaxy clusters with high-quality Chandra X-ray
data to directly study the influence of the dense intra-cluster medium (ICM) on
the quenching of satellite galaxies. We study the quenched fractions of
satellite galaxies as a function of ICM density for low- ($10^9 \lesssim
M_\star \lesssim 10^{10}\,\mathrm{M_\odot}$), intermediate- ($10^{10} \lesssim
M_\star \lesssim 10^{10.5}\,\mathrm{M_\odot}$), and high-mass ($M_\star \gtrsim
10^{10.5}\,\mathrm{M_\odot}$) satellite galaxies with $>\!3000$ satellite
galaxies across 24 low-redshift ($z < 0.1$) clusters. For low-mass galaxies we
find evidence for a broken powerlaw trend between satellite quenched fraction
and local ICM density. The quenched fraction increases modestly at ICM
densities below a threshold before increasing sharply beyond this threshold
toward the cluster center. We show that this increase in quenched fraction at
high ICM density is well matched by a simple, analytic model of ram pressure
stripping. These results are consistent with a picture where low-mass cluster
galaxies experience an initial, slow-quenching mode driven by steady gas
depletion, followed by rapid quenching associated with ram pressure of cold-gas
stripping near (one quarter of the virial radius, on average) the cluster
center.
