An old topic of dimensional analysis in astrophysics is presented and new results, or quantitative explanations of some observational facts are obtained, in particular, on the base of the supernova, SN, explosions. The presentation starts with the derivation of two similarity criteria for astrophysical objects constructed out of four measurable quantities: mass,
M, luminosity,
Lb, velocity
U, size
R, and gravitational constant
G. The first well known criterium describes the virial principle and the other one seems to be new and is based on the Tully-Fisher observational relationship between luminosity and velocity. The energy generated by SN explosions allows one to estimate well the interstellar turbulent velocities and magnetic field in our Galaxy, resulting in 3 to 4 microgauss. It is found that for z ≥ 0.6 the observed distant galactic clusters are far from virial equilibrium and the degree of disequilibrium is increasing with z. It means that to reach such an equilibrium the cluster age should be of order ten dynamical time scales, see Equation (7). For all considered galaxy clusters the second similarity criterium was found to be constant with a precision of about ten per cent. Therefore it could be considered as a general law, though for different classes of objects the numerical coefficient may vary. Some scales are proposed and two of them are tested for galactic clusters by finding numerical coefficients with accuracies of about 20 percent or better: e.g. observed luminocities of clusters are
W=Lb=a1(M/R)5/2G3/2 with for the first eleven objects from the Table for which the virial equilibrium is found with the same accuracy. The square root of the two criteria ratio
3=(
2/
1)
1/2=U(WG)
-1/5 explains the Tully-Fisher law and is constant for all 32 available clusters from [1,2] and is equal to 1.8 ± 0.2. This is because
3 has not global values of total mass and size.