Foreseeing the era of high spatial resolution measurements of the Sunyaev-Zel'dovich effect (SZE) in clusters of galaxies, we present a prototype analysis of this sort combined with Chandra X-ray data. It is applied specifically to RX J1347-1145 at z = 0.451, the most X-ray-luminous galaxy cluster known, for which the highest resolution SZE and X-ray images are currently available. We demonstrate that the combined analysis yields a unique probe of complex structures in the intracluster medium, offering determinations of their temperature, density, and line-of-sight extent. For a subclump in RX J1347-1145, previously discovered in our SZE map, the temperature inferred after removing the foreground and background components is well in excess of 20 ke V, indicating that the cluster has recently undergone a violent merger. Excluding the region around this subclump, the SZE signals in submillimeter to centimeter bands (350, 150, and 21 GHz) are all consistent with those expected from Chandra X-ray observations. We further present a temperature deprojection technique based on the SZE and X-ray images, without any knowledge of spatially resolved X-ray spectroscopy. The methodology presented here will be applicable to a statistical sample of clusters available in the future SZE surveys.
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