The masses of central supermassive black holes and the variability time-scales in gamma-ray loud blazars

The central supermassive black hole (SMBH) masses and variability time-scales of completed EGRET positive γ-ray blazars loud were investigated. We generalized the Elliot-Shapiro relation to the Klein-Nishina regime and derived a corrected form of the relation by taking into account the Klein-Nishina effect, then compiled a small sample including 21 γ-ray loud blazars, the rapid variation time-scales of which in the optical band were well established, to examine whether or not they obey the corrected form of the relation. It is found that the γ-ray luminosity (assuming it is isotropically emitted) at low state (corresponding to the minimum γ-ray flux presented in the Third EGRET Catalog) and the variation time-scales for these sources obey the corrected Elliot-Shapiro relation well. This suggests that the measured variability time-scales are not short enough to require a beaming effect when the Klein-Nishina effect is considered. The γ-ray emissions at low state may be produced in a region close to the central SMBHs, and are unbeamed or weakly beamed. This is quite consistent with Dermer & Gehrels's argument. Thus, taking into account the Klein-Nishina effect, the central SMBH masses and variability time-scales of completed EGRET γ-ray loud blazars were derived with the γ-ray fluxes. The results show that the central SMBH masses range in 10^6.5-10^10.2 M_solar. The mean and the median of the masses are 10^8.9 and 10^9.1 M_solar, respectively. The distribution of the masses exhibits a weak bimodal distribution with peaks at 10^8.2 M_solar and 10^.2 M_solar, and with a valley at 10^8.5 M_solar. This seems to present a signature for classifying these blazars into two groups. Most of the objects (75 per cent) belong to the group of M>= 10^8.5 M_solar, while only about 25 per cent objects are included in the group of M<= 10^8.5 M_solar. We also found that most of the BL Lac objects in the sample belong to the latter group, while most of the quasars belong to the former one. This likely indicates that the masses of the central SMBHs of BL Lac objects are significantly smaller than those of quasars. This is quite consistent with the argument proposed by Ozernoy. The variability time-scale is an observable indicator for examining the reliability of the mass estimate. Our results show that the variability time-scales for these sources range from 10^1.6 s to 10^5.6 s. The variation time-scales show a bimodal distribution too, with two peaks at 10^3.2 s (corresponding to 0.44 h) and 10^4.5 s (corresponding to 8.78 h), and a valley at 10^4.0 s. About 25 per cent of the sources have rapid variability on time-scales of a fraction of an hour, and 75 per cent of the sources have variability on a time-scale of intranight or intraday. The time-scales derived in this work are significantly correlated with observed shortest time-scales. The linear correlation coefficient is 0.76 with a chance probability of 0.0001. These results might indicate that the mass estimate in this work is reliable.