Elementary analysis of galaxy clusters: similarity criteria,
  Tully-Fischer, and approximate invariants
  At observations of galaxy clusters luminosity L, size R, mass M, temperature
T$_e$, 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:
$\Pi_1$ and $\Pi_2$. 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 $\le$ z < 1.26 and z $\le$ 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 $\Pi_1$
and $\Pi_2$. With this scatter they may be called approximate invariants. The
luminosity L and universal constant G are forming the dynamical velocity scale
U$_d$, which immediately explains the empirical Tully-Fisher law. The
temperature T$_e$ determines the thermal velocity of the gas plasma particles
U$_T$. The ratio U$_d$/ U$_T$= $\Pi_1$ is used here as a new similarity
criterium which is found to be constant within six per cent for nearly 30
objects cited above: $\Pi_1$=0.163$\pm$0.009 and may be interpreted as the Mach
number. The other criterium $\Pi_2 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 $\le$0.2. It is demanding to expand these results to other
clusters and different objects: singular galaxies, stars and their clouds, etc.
