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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.