Mass and length scale of black holes in quasars and active galactic nuclei

Abstract

The mass M and the length scale R of black holes are deduced from the experimental data of variability times and luminosities of quasars and active galactic nuclei. A simple power law relation, M ~ R ~, emerges. As a possible explanation of the variability and of the relationship of M and R, oscillation of black holes is considered. The large luminosities (1040 ,'~ 1046 erg/sec) and relatively small variability times (102 ,-106 sec) of various kinds of quasars and active galactic nuclei AGN (Seyfert galaxies, x-ray and radio galaxies) suggest massive black holes (102 ,,~ 108M®) as the source of energy. In this article, a relation between the mass and the length scale of black holes is deduced from experimental data (Wandel and Mnshotzky 1986) on the variability time At and the x-ray luminosity Lx and a possible model for a black hole dynamics is put forward. A natural relationship between At, M and R is given by I R3 At = ~ 2GM' (1) where G is Newton's constant and ~ is a constant of order unity. The right hand side of eq. (1) is the time scale for collapse of a mass from a distance R into a heavy mass M. The amount of energy emitted in a variability time, LxAt, can be expressed as a fraction ~ of the available gravitational energy, GM 2 z x A t =,7 R ' (2) where ~ is given by 2AM ~ M A M and A R being the change of M and R in At. From eq. (1) and (2), it follows that A R R , (3) 3 logM = logAt + ~ log Lx + const (4) logR = logAt + 51log Lx + const (s)