Model-Independent Limits on the Line-of-Sight Depth of Clusters of
  Galaxies Using X-ray and Sunyaev-Zel'dovich Data
  We derive a model-independent expression for the minimum line-of-sight extent
of the hot plasma in a cluster of galaxies. The only inputs are the 1-5 keV
X-ray surface brightness and the Comptonization from Sunyaev-Zel'dovich (SZ)
data. No a priori assumptions regarding equilibrium or geometry are required.
The method applies when the X-ray emitting material has temperatures anywhere
between 0.3 keV and 20 keV and metallicities between 0 and twice
solar---conditions fulfilled by nearly all intracluster plasma. Using this
method, joint APEX-SZ and Chandra X-ray Observatory data on the Bullet Cluster
yield a lower limit of 400 +/- 56 kpc on the half-pressure depth of the main
component, limiting it to being at least spherical, if not cigar-shaped
primarily along the line of sight.
