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2 The mass of the central black hole and the Doppler factor

2.1 Data

Some $\gamma$-ray loud blazars have been observed several times with EGRET while three XBLs have been observed to show TeV radiations. In the following section, we present the $\gamma$-ray loud blazars with available $\gamma$-ray variation timescales. Because the variability timescale corresponds to different variation amplitude for different source and/or different observation period, we use the doubling timescale, $\Delta T_{\rm D} = (F_{\rm initial}/\Delta F) \Delta T$, as the variability timescale.

2.1.1 PKS 0528+134

PKS 0528+134, z = 2.07 (Hunter et al. 1993), is one of the most luminous examples of blazars. It is observed by EGRET, COMPTE and OSSE aboard the CGRO (see Hunter et al. 1993; McNaron-Brown et al. 1995; Mukherjee et al. 1996; Collmar et al. 1997; Sambruna et al. 1997).

During the period of 16-30 May 1991, the source showed F(>100 MeV) = (1.0 $\pm$ 0.2) 10-6 photon cm-2 s-1 with photon spectral index $\alpha_{\gamma} = 2.56\pm0.09$.

During 23-29 March 1993, F(>100 MeV) = (0.23 $\pm$ 0.12 - 3.08 $\pm$ 0.35) 10-6 photon cm-2 s-1; with a photon spectral index $\alpha_{\gamma}$ = 2.21 $\pm$ 0.10. In the 1993 observation, a variation of order 100% over a timescale of $\sim$2 days was detected (see Wagner et al. 1997), which suggests a doubling time scale of $\Delta T_{\rm D}$ = 1 day.

During August, 1994, F(>100 MeV) = (0.32  $\pm$  0.1) 10-6 photon cm-2 s-1; with a photon spectral index $\alpha_{\gamma}$ = 2.70.

There is a clear evidence that the spectrum becomes harder when the $\gamma$-ray flux increases.

2.1.2 PKS 0537-441

PKS 0537-441, z = 0.896, a candidate of gravitational lens (Surpi et al. 1996), is a violently variable object (Fan & Lin 1998). The $\gamma$-ray flux varies from (1.83 $\pm$ 0.91) to (8.98 $\pm$ 1.45) 10-6 photon cm-2 s-1 (Mukherjee et al. 1997). A flare of a factor of $\sim$3 from 0.35 to 2.0 10-6 photon cm-2s-1 over a time scale of $\sim$2 days can be seen from Fig. 3 in Hartman's paper (Hartman 1996). $\Delta T_{\rm D}$ = 16 hr.

2.1.3 1253-055, 3C 279

3C 279 is a well known member of OVV subclass of blazars. it is perhaps the prototypical superluminal radio source (Moffet et al. 1972); and the first quasar detected at the energies of >1 GeV with EGRET/ CGRO. The simultaneous variability in X-rays and $\gamma$-rays (> 100 MeV) suggests for the first time that they are approximately cospatial (M$^{\rm c}$Hardy 1996). The $\gamma$-ray flux varies from 1.28 to 28.7 10-6 photon cm-2 s-1 (Mukherjee et al. 1997). Two $\gamma$-ray flares were detected (see Kniffen et al. 1993; Hartman et al. 1996; M$^{\rm c}$Hardy 1996; Wehrle et al. 1998).

The 16-28 June 1991 flare showed: F(>100 MeV) = (2.8  $\pm$  0.4) 10-6 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 1.89 $\pm$ 0.06. A variation of a factor of 4 over 2 days was obtained.

The January-February 1996 flare showed (see McHardy 1996; Wehrle et al. 1998), F(>100 MeV) = (11.0  $\pm$  1.) 10-6 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 1.97 $\pm$ 0.07. During this flare, a variation of a factor of 4 $\sim$ 5 in a day is observed, $\Delta T = 6$ hrs (Wehrle et al. 1998).

No obvious spectral index variation has been detected when the flux varied.

2.1.4 PKS 1406-074

PKS 1406-074 has been detected to vary with the $\gamma$-ray flux being in the range of 1.54 to 12.76 10-6 photon cm-2 s-1 (Mukherjee et al. 1997). During its $\gamma$-ray flare, a flux of F(>100 MeV) = (5.5  $\pm$  1.4) 10-6 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 2.04 $\pm$ 0.15 and a doubling timescale of shorter than 16 hours has been obtained (see Wagner et al. 1995).

2.1.5 PKS 1622-297

For PKS 1622-297, z=0.815, we have very little information in lower energy bands. But it is one of the most luminous objects in the $\gamma$-ray region. A peak flux of (17 $\pm$ 3) 10-6 photon cm-2 s-1 (E>100 MeV) and a flux increase by a factor of 2 in 9.7 hours were observed (Mattox et al. 1997).

2.1.6 Q1633+382, 4C 38.41

Quasar 1633+382, z=1.814, is an LPQ ($P_{\rm opt}=2.6\%$,Moore & Stockman 1984). During 1992 November 17 - December 1 period, it was detected to show a flux of F(>100 MeV) = (0.30 $\pm$ 0.06) 10-6 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 1.87 $\pm$ 0.07. The flux varied by it a factor of 1.5 within 24 hr, $\Delta T_{\rm D}$ = 16 hrs, while the spectral index did not change. The $\gamma$-ray luminosity is at least two orders of magnitude larger than the maximum ever observed in any other band (see Mattox et al. 1993).

2.1.7 B2 1652+399, Mkn 501

Mkn 501, z=0.033, together with other two XBLs are three known TeV $\gamma$-ray sources detected by the Whipple group (see Quinn et al. 1996; Catanese et al. 1997a; Samuelson et al. 1998; Kataoka et al. 1998).

During 1995 March-July period, Mkn 501 was observed to show a flux of F(>300 GeV) = (8.1  $\pm$  1.4) 10-12 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 2.2. A variation of a factor of 4 over one day is also detected, $\Delta T_{\rm D}$ = 6 hrs. The upper limit corresponds to a flux of F(>100 MeV) = 1.5 10-7 photon cm-2 s-1 (see Quinn et al. 1996). During the 1996 multiwavelength campaign, Mkn 501 was detected with EGRET a flux of F(>100 MeV) = $(0.32~\pm~.13)$ 10-6 photon cm-2 s-1 with a photon index of 1.6 $\pm$ 0.5 (see Kataoka et al. 1998). During 1997 April 9-19 observation, Catanese et al. (1997a) obtained F(>300 GeV) = (40.5  $\pm$  9.6) 10-11 photon cm-2 s-1, $\alpha_{\gamma}$ = 2.5, the April 9-15 flux corresponds to a flux of F(>100 MeV) < 3.6 10-7 photon cm-2 s-1.

The TeV observations show that the spectrum softens when the source brightens.

2.1.8 2200+420, BL Lacertae

2200+420 is the prototype of BL Lacertae class. It is variable in all wavelengths (see Fan et al. 1998b, 1998c; Bloom et al. 1997; B$\ddot{\rm o}$ttcher & Bloom 1998; Madejski et al. 1998). A 14-year period was found in the optical light curve (Fan et al. 1998b). During 1995 January 24 - February 14, BL Lacertae showed a flux of F(>100 MeV) = (40  $\pm$ 12) 10-8 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 2.2 $\pm$ 0.3, the up limit flux in higher energy is F(>300 GeV) < 0.53 10-11 photon cm-2 s-1 (Catanese et al. 1997b). During 1997 January 15/22 observation period, it was detected a flux of F(>100 MeV) = (171  $\pm$ 42) 10-8 photon cm-2 s-1 with a photon spectral index $\alpha_{\gamma}$ = 1.68 $\pm$ 0.16 and a dramatic factor of 2.5 increase within a timescale of 8hrs, $\Delta T_{\rm D}$ = 3.2 hrs. Besides, simultaneous optical and $\gamma$-ray flares were observed ruling out external scattering models (see Bloom et al. 1997).

The observations from the object show that the spectrum of BL Lacertae hardens when the $\gamma$-ray flux increases.

2.2 The central black hole mass and the Doppler factor

The objects discussed here show variability time scale of hours to days. The variability could be directly related to shock processes in a jet, far from the accretion disk (we thank Dr. S.D. Bloom to point out this for us). If we take the variability timescale as the measurements of the size, R, of the emission region, then the R in the jet obeys to the inequality,
R \leq c \Delta T_{\rm D} {\frac { \delta }{(1+z)}} {\rm cm}\end{displaymath} (1)
where c is the speed of light, $\delta$ the Doppler factor, z the redshift of the source, and $\Delta T_{\rm D}$, in units of second, the doubling time scale.

For an object with a mass M, the Eddington limit gives (Frank et al. 1985)
L_{\rm Edd.} \approx 1.26\ 10^{38}\left({\frac {M}{M_{ \odot}}}\right)\ {\rm erg\ s}^{-1}.\end{displaymath} (2)
So, we have that the intrinsic luminosity, $L^{\rm in.}$ of a source with a mass of M should satisfy $L^{\rm in}\leq L_{\rm Edd.}$.

In the relativistic beaming frame, the observed luminosity is $L^{\rm ob.}=\delta^{(4+\alpha)}L^{\rm in.}$, $\alpha$ is the energy spectral index, which follows that
L^{\rm ob.} \leq\delta^{(4+\alpha)}L_{\rm Edd.}
 \end{displaymath} (3)
Ghisellini & Madau (1996) obtained that the $\gamma$-rays are emitted within the BLR region, which is 1017-18 cm far from the central source. Hartman et al (1996) obtained that the $\gamma$-rays are produced at a distance of $\sim$100 $R_{\rm g}$. Recently, Celotti & Ghisellini (1998) argued that the $\gamma$-rays are from a region of some hundreds of Schwarzschild radii from the center. From our previous paper, a distance of 205 $R_{\rm g}$ is obtained for the $\gamma$-rays from Mkn 421. In the sense of the theory of accretion (Sunyaev 1975). When $R < 200~ R_{\rm g}$, the electrons in the accretion flow become ultrarelativistic. On the other hand, the mixture of relativistic electrons and nonrelativistic protons has an adiabatic index $\gamma < {\frac {5}{3}}$, with such an adiabatic index the transition to supersonic accretion regime is possible in the region $R < 200~ R_{\rm g}$ (Sunyaev 1975). So, the 200 $R_{\rm g}$ is perhaps an important critical point. If we assume that the $\gamma$-rays are from this place then relations (1), (2), and (3) give

{\frac {M}{M_{\odot}}}= 5\ 10^{2}{\frac{\delta}{1+z}} \Delta T_{\rm D}
 \end{displaymath} (4)

L^{\rm ob.} \leq 6.3 \ 10^{40} {\frac { \delta^{(5+\alpha)}}{(1+z)}}
\Delta T_{\rm D}~{\rm erg\ s}^{-1}.
 \end{displaymath} (5)
It is a common property of the EGRET-detected AGNs to show that their $\gamma$-ray flux is dominant over the flux in lower energy bands but this is not always the case (Mukherjee et al. 1997). For PKS 0528+134 and 3C 279, their $\gamma$-ray luminosity, $L_{\gamma}$, is 0.80 $L_{\rm bol.}$ and 0.5 $L_{\rm bol.}$, respectively (see Sambruna et al. 1997; Hartman et al. 1996). Because we consider the flare states of the selected objects, we can take the $\gamma$-ray luminosity to stand for half of the bolometrical luminosity approximately, i.e. $L_{\gamma} \sim 0.5 \ L_{\rm bol.}$. So, we have
L_{\gamma} \leq 3.15 \ 10^{40} {\frac {\delta^{(5+\alpha)}}{(1+z)}}
\Delta T_{\rm D}~{\rm erg\ s}^{-1}\end{displaymath} (6)
which gives
\delta \geq \left[{\frac{L_{\gamma}(1+z)}{3.15\ 10^{40}~{\rm erg\ s}^{-1}\Delta
T_{\rm D}}}\right]^{({\frac{1}{5+\alpha}})}.\end{displaymath} (7)
So, from the available $L_{\gamma}$ and $\Delta T_{\rm D}$, we can obtain the central black hole mass and the Doppler factor from relations (4) and (7) and $\alpha = \alpha_{\gamma} - 1$, $\alpha_{\gamma}$ is the photon spectral index. They are shown in Table 1, in which Col. 1 gives the name; Col. 2, the redshift; Col. 3. the flux F(>100 MeV) in units of 10-6 photon cm-2 s-1, $\sigma$ is the uncertainty; Col. 4, the photon spectral index, $\alpha_{\gamma}$ = 2.0 is adopted for 0537-441 (see Fan et al. 1998a); Col. 5, the doubling time scale in units of hours; Col. 6, $\gamma$-ray luminosity (assuming isotropic emission) in units of 1048 erg s-1; Col. 7, Doppler factor estimated from Eq. (7); Col. 8, the central black hole mass in units of $10^{7}~M_{\odot}$; Col. 9, the central black hole mass estimated from Dermer & Gehrels (1996), thereafter D&G, in units of $10^{7}~M_{\odot}$; Col. 10, the mass estimated directly from Eddington limit in units of $10^{10}~M_{\odot}$.

Table 1: Mass and Doppler factor for $\gamma$-ray loud blazars

 Name & $z$\space & $ F...
 ...(0.42) & 1.68 & 3.2 & 0.019 & 2.04 & 1.09 & 0.02 &
0.015\\  \hline \end{tabular}

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