An approximate theory for substructure propagation in clusters

Aims. The existence of dark matter can be proved in an astrophysical context by the discovery of a system in which the observed baryons and the inferred dark matter are spatially segregated, such as the bullet cluster (1E0657-558). The full descriptions of the dark matter halo and X-ray gas substructure motions are necessary to forecast the location of the dark halo from X-ray maps, which can be confirmed by the detection of a galaxy concentration or by gravitational lensing. Methods. We present an analytical hydrodynamic model to determine the distance between the X-ray and dark-matter components and the Mach number of the merger shock. Results. An approximate solution is given for the problem of the substructure propagation in merging clusters. A new method to predict the position of a dark matter halo in clusters, where there is a separation between the X-ray gas and the dark halo, is proposed and applied to the clusters 1E0657-558 and Abell 1763.