Elementary analysis of galaxy clusters: similarity criteria, Tully-Fischer, and approximate invariants

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At observations of galaxy clusters luminosity L, size R, mass M, temperature Tₑ, sometimes velocities are usually measured. These four quantities and the gravity constant G are determined by three measurements units: mass M, length L and time T. Therefore one can form two non-dimensional similarity criteria: Π₁ and Π₂. Any chosen observable can be formed as a function of the other three ones. The author has at hand the data by Vikhlinin (2002) and Vikhlinin et al. (2006), rather more complete than any other. This material consists of more than thirty clusters at 0.4 ≤ z < 1.26 and z ≤ 0. This material gives a possibility to test the derived dimensional relationships and to determine the dimensionless numerical coefficients at them. These coefficients are found with a scatter less than 30 per cent in the data above and could be considered as other similarity criteria but functions of Π₁ and Π₂. With this scatter they may be called approximate invariants. The luminosity L and universal constant G are forming the dynamical velocity scale Ud, which immediately explains the empirical Tully-Fisher law. The temperature Tₑ determines the thermal velocity of the gas plasma particles Uₜ. The ratio Ud/ Uₜ= Π₁ is used here as a new similarity criterium which is found to be constant within six per cent for nearly 30 objects cited above: Π₁=0.163±0.009 and may be interpreted as the Mach number. The other criterium Π₂ is the virial one. It is found to be a function of the cluster age. At z>0.5 the mean cluster mass is five times less, that at small z ≤0.2. It is demanding to expand these results to other clusters and different objects: singular galaxies, stars and their clouds, etc.