Deprojection of rich cluster images

We consider a general method of deprojecting two-dimensional images to reconstruct the three-dimensional structure of the projected object, assuming axial symmetry. The method consists of the application of the Fourier slice theorem to the general case in which the axis of symmetry is not necessarily perpendicular to the line of sight and is based on an extrapolation of the image Fourier transform into the so-called cone of ignorance. The method is specifically designed for the deprojection of X-ray, Sunyaev-Zeldovich (SZ), and gravitational lensing maps of rich clusters of galaxies. For known values of the Hubble constant H₀ and inclination angle, the quality of the projection depends on how exact the extrapolation in the cone of ignorance is. In the case in which the axis of symmetry is perpendicular to the line of sight and the image is noise-free, the deprojection is exact. Given an assumed value of H₀, the inclination angle can be found by matching the deprojected structure out of two different images of a given cluster, e.g., SZ and X-ray maps. However, this solution is degenerate with respect to its dependence on the assumed H₀, and a third independent image of the given cluster is needed to determine H₀ as well. The application of the deprojection algorithm to upcoming SZ, X-ray, and weak lensing projected mass images of clusters will serve to determine the structure of rich clusters and the value of H₀ and to place constraints on the physics of the intracluster gas and its relation to the total mass distribution. The method is demonstrated using a simple analytic model for cluster dark matter and gas distributions and is shown to provide a stable and unique reconstruction of the cluster three-dimensional structure.