Standoff Distance of Bow Shocks in Galaxy Clusters as Proxy for Mach Number

X-ray observations of merging clusters provide many examples of bow shocks leading merging subclusters. While the Mach number of a shock can be estimated from the observed density jump using Rankine-Hugoniot condition, it reflects only the velocity of the shock itself and is generally not equal to the velocity of the infalling subcluster dark matter halo or to the velocity of the contact discontinuity separating gaseous atmospheres of the two subclusters. Here we systematically analyze additional information that can be obtained by measuring the standoff distance, i.e. the distance between the leading edge of the shock and the contact discontinuity that drives this shock. The standoff distance is influenced by a number of additional effects, e.g. (1) the gravitational pull of the main cluster (causing acceleration/deceleration of the infalling subcluster), (2) the density and pressure gradients of the atmosphere in the main cluster, (3) the non-spherical shape of the subcluster, and (4) projection effects. The first two effects tend to bias the standoff distance in the same direction, pushing the bow shock closer to (farther away from) the subcluster during the pre- (post-)merger stages. Particularly, in the post-merger stage, the shock could be much farther away from the subcluster than predicted by a model of a body moving at a constant speed in a uniform medium. This implies that a combination of the standoff distance with measurements of the Mach number from density/temperature jumps can provide important information on the merger, e.g. differentiating between the pre- and post-merger stages.