Galaxy clusters, similarity parameters, and ratios between measurable characteristics

The study of galaxy clusters provides insights into the different stages of the evolution of the Universe. Cluster observations measure luminosity, size, temperature, and mass. What binds a cluster into a single entity is gravity, its force being proportional to the Newtonian constant of gravitation $G$. Because all five of these quantities are measured in units of mass, length, and time, two nondimensional parameters, commonly known as similarity parameters, can be argued to characterize the system. One of these is the well-known virial ratio of kinetic to potential energies. The velocities of galaxy clusters are not measured, however. The luminosity $L$ and the constant $G$ can be combined to introduce the dynamic velocity scale $U_\alpha=(LG)^{1/5}$. The ratio of this scale to the particle thermal velocity gives the similarity parameter $\varPi _1$, which is constant to within about $10 \%$ for all 30 objects studied, allowing the virial similarity parameter $\varPi _2$ to be evaluated for 31 objects. For nearby objects with a red shift of $z \le 0.2$, the parameter $\varPi _2$ is of order 10 and decreases with increasing $z$, i.e., with decreasing age. To test the quality of the data, the value of $G$ was determined using other measured quantities and found to be equal to its true value to within $\le 6 \%$ and $28 \%$ for close and distant objects, respectively. A number of other ratios between measured quantities have been proposed and checked, showing a scatter of $10$–$20 \%$ from constancy on the linear scale in the numerical coefficients involved. Older clusters are, on average, larger in mass and size, implying that smaller clusters can be absorbed by large ones. The results obtained can be valid for clusters with a temperature of $T > 1$ keV, i.e., in the X-ray range of the spectrum. The claster mass reduction with increasing $z$, i.e., with decreasing age, is also traced, on the average, in other spectral regions. It is shown that by knowing the temperature and the received X-ray intensity, the possibility arises to estimate the distance to the cluster.