Hugoniotjump conditions (e.g., Landau & Lifshitz 1959, § 85), if one assumes that all of the dissipated shock energy is thermalized. Consider a small element of the surface of a shock (much smaller than the radius of curvature of the shock, for example). The tangential component of the velocity is continuous at the shock, so it is useful to go to a frame which is moving with that element of the shock surface, and which has a tangential velocity which is equal to that of the gas on either side of the shock. In this frame, the element of the shock surface is stationary, and the gas has no tangential motion. Let the subscripts 1 and 2 denote the preshock and postshock gas; thus, v\= vs is the longitudinal velocity of material into the shock (or alternative, the speed with which the shock is advancing into the preshock gas). Conservation of mass, momentum, and energy then implies the following jump conditions
Here, is the enthalpy per unit mass in the gas, and is the internal energy per unit mass. If the gas behaves as a perfect fluid on each side of the shock, the internal energy per unit mass is given by where 7ad is the ratio of specific heats (the adiabatic index) and is 7ad = 5/3 for fully ionized plasma. The jump conditions can be rewritten as:
where is the shock compression.
If one knew the velocity structure of the gas in a merging cluster, one could use these jump condition to derive the temperature, pressure, and density jumps in the gas. At present, the best X-ray spectra for extended regions in clusters of galaxies have come from CCD detectors on ASCA, Chandra, and XMM/Newton. CCDs have a spectral resolution of >100 eV at the Fe K line at 7 keV, which translates into a velocity resolution of >4000 km/s. Thus, this resolution is (at best) marginally insufficient to measure merger gas velocities in clusters. In a few cases with very bright regions and simple geometries, the grating spectrometers on Chandra and especially XMM/Newton may be useful. However, it is likely that the direct determinations of gas velocities in most clusters will wait for the launch of higher spectral resolution nondispersive spectrometers on Astro-E2 and Constellation-X.
At present, X-ray observations can be used to directly measure the temperature and density jumps in merger shocks. Thus, one needs to invert the jump relations to give the merger shock velocities for a given shock temperature, pressure, and/or density increase. If the temperatures on either side of the merger shock can be measured from X-ray spectra, the shock velocity can be inferred from (Markevitch et al. 1999)
where is the velocity change across the shock, and ¡j, is the mean mass per particle in units of the proton mass mp. The shock compression C can be derived from the temperatures as
Alternatively, the shock compression can be measured directly from the X-ray image. However, it is difficult to use measurements of the shock compression alone to determine the shock velocity, for two reasons. First, a temperature is needed to set the overall scale of the velocities; as is obvious from equation (27), the shock compression allows one to determine the Mach number M but not the shock velocity. The second problem is that temperature or pressure information is needed to know that a discontinuity in the gas density is a shock, and not a contact interface (e.g., the "cold fronts" discussed in § 3.2 below).
X-ray temperature maps of clusters have been used to derive the merger velocities using these relations. Markevitch et al. (1999) used ASCA observations to determine the kinematics of mergers in three clusters (Cygnus-A, Abell 2065, and Abell 3667). Because of the poor angular resolution of ASCA, these analyses were quite uncertain. More recently, possible shocks have been detected in Chandra images of a number of merging clusters (e.g., Abell 85, Kempner et al. 2001; Abell 665, Markevitch et al. 2001; Abell 3667, Vikhlinin et al. 2001b), and the shock jump conditions have been applied to determine the kinematics in these clusters.
The simplest case is a head-on symmetric merger (b = 0 and Mi = M2) at an early stage when the shocked region lies between the two
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