We give a short summary of the fitting procedures here, a detailed description can be found in Molnar & Birkinshaw (1996). We assumed an isothermal model distribution centered at A3558 for the intra-supercluster gas with a cut off at a few core radius. We used the galactic contamination subtracted COBE DMR map provided by Al Kogut (Kogut et al. (1996)) to fit the SZ effect model. It was not necessary to do point source subtraction since the DMR experiment has a large beam dilution and is relatively insensitive to point sources. We used a HEAO-1 A2 instrumental background subtracted map provided by Keith Jahoda (Jahoda & Mushotzky (1989)) and the publicly available ROSAT All-Sky Survey maps (Voges et al. (1996)) to fit the thermal brehmsstralung model.
We assumed a linear dependence of the smooth galactic X-ray component with galactic latitude and the distance from the galactic center. We used published positions of X-ray point sources (clusters, AGNs etc.). We determined the HEAO and ROSAT point spread functions, the amplitudes of our models, galactic models and point sources in an iterative process.
We minimized to fit values for a set of reasonable model parameters ( and ) within windows with and radii for the COBE and X-ray data respectively (our results are not sensitive the cut-off and window parameters as long as they are large). We found that the value of is largely insensitive to the model parameters: thus we concluded that no significant SZ or X-ray signals of the modelled form are present.
Although we used truncated isothermal beta models to describe the the structure of the SSC gas, the assumptions on which these models are based, such as constant temperature and hydrostatic equilibrium, and an arbitrary cut-off radius, are questionable, we note that from a mathematical point of view we fit a set of models with two parameters, which makes it possible to find intracluster gas centered at the center of the Shapley concentration, even if the physical assumptions of the model break down.