The fraction of cool-core clusters in X-ray vs. SZ samples using Chandra
  observations
  We derive and compare the fractions of cool-core clusters in the {\em Planck}
Early Sunyaev-Zel'dovich sample of 164 clusters with $z \leq 0.35$ and in a
flux-limited X-ray sample of 100 clusters with $z \leq 0.30$, using {\em
Chandra} observations. We use four metrics to identify cool-core clusters: 1)
the concentration parameter: the ratio of the integrated emissivity profile
within 0.15 $r_{500}$ to that within $r_{500}$, and 2) the ratio of the
integrated emissivity profile within 40 kpc to that within 400 kpc, 3) the
cuspiness of the gas density profile: the negative of the logarithmic
derivative of the gas density with respect to the radius, measured at 0.04
$r_{500}$, and 4) the central gas density, measured at 0.01 $r_{500}$. We find
that the sample of X-ray selected clusters, as characterized by each of these
metrics, contains a significantly larger fraction of cool-core clusters
compared to the sample of SZ selected clusters (44$\pm$7\% vs. 28$\pm$4\% using
the concentration parameter in the 0.15--1.0 $r_{500}$ range, 61$\pm$8\% vs.
36$\pm$5\% using the concentration parameter in the 40--400 kpc range,
64$\pm$8\% vs. 38$\pm$5\% using the cuspiness, and 53$\pm$7\% vs. 39$\pm$5\%
using the central gas density). Qualitatively, cool-core clusters are more
X-ray luminous at fixed mass. Hence, our X-ray flux-limited sample, compared to
the approximately mass-limited SZ sample, is over-represented with cool-core
clusters. We describe a simple quantitative model that uses the excess
luminosity of cool-core clusters compared to non-cool-core clusters at fixed
mass to successfully predict the observed fraction of cool-core clusters in
X-ray selected samples.
