3 Discussion

3.1 Mass

From the high $\gamma$-ray luminosity (assuming isotropic emission) and Eddington-limit, one can derive the central black hole mass expression,

$$M_{10} \geq {\frac{L_{\rm T}}{1.26\ 10^{48}~{\rm erg\ s}^{-1}}}$$ (8)

where, $L_{\rm T}$ is the bolometric luminosity for emission in the Thomson region, M₁₀ is the central black hole mass in units of 10$^{10}~M_{\odot}$. The derived masses are as high as 10$^{11}~M_{\odot}$ for some $\gamma$-ray loud blazars, PKS 0528+134, PKS 1406-074, and PKS 1622-297 for instance (see Col. 10 in Table 1). However, for high energy $\gamma$-ray emission, Klein-Nishina effects must be considered. D&G considered the effect and obtained an expression for the black hole mass, i.e. their Eq. (16b),

$$M_{8}^{\rm KN} \geq {\frac{3\pi d_{\rm L}^{2}(m_{\rm e}c^{2}... ...},\varepsilon_{\rm u})}{1+z}} \ln[2\varepsilon_{\rm l}(1+z)]$$ (9)

where $F(\varepsilon_{\rm l},\varepsilon_{\rm u})$ is the integrated photon flux in units of 10^-6 photon cm^-2 s^-1 between photon energies $\varepsilon_{\rm l}$ and $\varepsilon_{\rm u}$ in units of 0.511 MeV. For the objects considered here, $M_{7}^{\rm KN}$ is obtained and shown in Col. 9 in Table 1. Table 1 shows that the masses obtained from our consideration and those estimated from the D&G method are acceptably similar except for 1622-297 and two low redshift BL Lac objects (Mkn 501 and BL Lacertae). For 1622-297 our value is about 7 times less than that estimated from the D&G method. If we adopt the flux density $2.45 \ 10^{-6}$ photon cm^-2 s^-1 for 1622-297 instead of the peak value as did Muhkerjee et al. (1997) and Fan et al. (1998a), then the isotropic luminosity is $3.87\ 10^{48}$ erg s^-1. This luminosity suggests that the Doppler factor and mass obtained from relations (4) and (7) are respectively 5.01 and 2.41 M₇, and the mass estimated from the D&G is then 3.61 M₇. The two masses are quite similar in this case. For Mkn 501 and BL Lacertae, our value is much greater than that estimated from the D&G method. For 3C 279, our results show that the estimated central black hole masses, 4.74 M₇ and 3.43 M₇ for 1991 and 1996 flares respectively, are almost the same, while the masses estimated from the D&G method are 1.92 M₇ and 7.53 M₇ for 1991 and 1996 flares respectively. Table 1 (also see relation (9)) shows that the mass obtained from the D&G method is sensitive to the flux, variable flux gives different mass for a source. The mass obtained from our method does not depend on the flux so sensitively. For 3C 279 1991 and 1996 flares, masses obtained from our consideration are almost the same while those obtained from the D&G method show a difference of more or less a factor of 4. But our method depends on the timescales (see relation 4). Since we only considered the objects showing short timescales (hours) in the present paper, the masses obtained are in a range of (1 $\sim$ 7) $10^{7}~M_{\odot}$.

To fit 3C 279 multiwavelength energy spectrum corresponding to 1991 $\gamma$-ray flare, Hartman employed an accreting black hole of $10^{8}~M_{\odot}$, our result of $4.74\ 10^{7}~M_{\odot}$ is similar to theirs.

3.2 Beaming factors

To explain the extremely high and violently variable luminosity of AGNs, the beaming model has been proposed. In this model, the Lorentz factor, $\Gamma$, and the viewing angle, $\theta$, are not measurable, but they can be obtained through the measurement of superluminal velocity, $\beta_{\rm app.}$, and the determination of Doppler factor, $\delta$, which are related with the two unmeasurable parameters, $\Gamma$ and $\theta$, in the forms: $\beta_{\rm app}={\frac {\beta_{\rm in} \sin \theta}{(1-\beta_{\rm in} \cos \theta)} }$, $\Gamma={\frac {1}{\sqrt{1-\beta_{\rm in}^{2}}}}$, and $\delta=(\Gamma (1-\beta_{\rm in} \cos \theta))^{-1}$. So, $\Gamma$ and $\theta$ can be obtained from the following relations:

$$\Gamma={\frac {\beta_{\rm app}^{2}+\delta ^{2} +1}{2 \delta}}$$

$$\theta = \tan^{-1}\left({\frac {2 \beta_{\rm app}} {\beta_{\rm app}^{2}+\delta ^{2} - 1}}\right).$$

From our previous work (Fan et al. 1996), we can get superluminal velocities for 3C 279 and BL Lacertae. When the superluminal velocities and the derived Doppler factor are substituted to the above two relations. we found that: For 3C 279, $\Gamma = 2.4-14.4$ and $\theta = 10\hbox{$.\!\!^\circ$}9-15\hbox{$.\!\!^\circ$}6$ for $\delta = 3.37$; and $\Gamma = 2.95-11.20$ and $\theta = 8\hbox{$.\!\!^\circ$}45-9\hbox{$.\!\!^\circ$}7$ for $\delta = 4.89$. For BL Lacertae $\Gamma = 2\hbox{$.\!\!^\circ$}- 4\hbox{$.\!\!^\circ$}$ and $\theta = 25\hbox{$.\!\!^\circ$}-29\hbox{$.\!\!^\circ$}4$.

To let the optical depth ($\tau_{\gamma\gamma}$) be less than unity, Doppler factor in $\gamma$-ray region has been obtained for some objects by other authors. $\delta \geq 7.6$ for Q1633+382 (Mattox et al. 1993); $\delta \geq 6.3 - 8.5$ for 3C 279 1996 flare (Wehrle et al. 1998) and $\delta \geq 3.9$ for 3C 279 1991 flare (Mattox et al. 1993), $\delta \sim 5$ is also obtained by Henri et al. (1993); $\delta \geq 6.6 \sim 8.1$ for PKS 1622-297 (Mattox et al. 1997). Our results in Table 1 are consistent with those results.

3.3 Summary

In this paper, the central mass and Doppler factor are obtained for 8 $\gamma$-ray loud blazars with available short $\gamma$-ray timescales. The mass obtained from relation (4) in the present paper is compared with that obtained from the D&G method, the masses obtained from two methods are similar for 5 out of 8 objects. Our method is not as sensitive to the flux as the D&G method in estimating the central black hole mass. The masses obtained here are in a range of (1 $\sim$ 7) $10^{7}~M_{\odot}$, which stems from the fact that the time scales considered here are in a range of 3.2 to 24 hours. For 3C 279, the masses obtained from the two flares are almost the same. For 3C 279 and BL Lacertae, the Lorentz factor and viewing angle are estimated.

Acknowledgements

The authors thank the referee Dr. S.D. Bloom for his comments and suggestions. J.H.F. thanks Drs. B$\ddot{\rm o}$ttcher, Dermer, Carter-Lewis, Catanese, Ghisellini, Kataoka, and Wagner for their sending him their publications and information and the Education fund of Guangzhou City for support. This work is supported by the National Pan Deng Project of China.