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A HR-like diagram for galaxies: the M_BH versus M_G sigma^2 relation

We show that the relation between the mass of supermassive black holes located in the center of the host galaxies and the kinetic energy of random motions of the corresponding bulges is a useful tool to study the evolution of galaxies. In the form log[M_BH] = b + m log[M_G sigma^2/c^2], the best-fitting results for a sample of 64 galaxies of various morphological types are the slope m=0.80 and the normalization b=4.53. We note that, in analogy with the H-R diagram for stars, each morphological type of galaxy generally occupies a different area in the M_BH - (M_G sigma^2)/c^2 plane. In particular, we find elliptical galaxies in the upper part of the line of best fit, the lenticular galaxies in the middle part, and the late-type galaxies in the lower part, the mass of the central black hole giving an estimate of the age, whereas the kinetic energy of the stellar bulges is directly connected with the temperature of each galactic system. Finally, the values of the linear correlation coefficient, the intrinsic scatter, and the chi^2 obtained by using the M_BH - M_G sigma^2 relation are better than the corresponding ones obtained from the M_BH - sigma or M_BH - M_G relation.