3 Mass estimates derived in the context of high energy emission models

Following a quite different approach, the BH mass of Mkn 501 could also be estimated in the framework of high energy emission models:

(1) With respect to the high energy emission, Fan et al. (1999) have recently determined the central black hole masses for several $\gamma$-ray loud blazars (including Mkn 501) by assuming that the observed $\gamma$-rays are produced at $\simeq 200~R_{\rm g}$. Accordingly, the central BH mass obeys the relation $M_H/M_{\odot}\simeq 500~\delta~ \Delta t$, where $\Delta t$ denotes the doubling timescale in seconds and $\delta$ the Doppler factor. For Mkn 501 they obtained a central BH mass of $M_H \simeq 0.9 \times 10^7 ~M_{\odot}$, using $\Delta t =6$ hrs and $\delta \simeq 0.9$. However, current evidence indicates that the relevant timescale $\Delta t$ might be substantially smaller. Recent observations reveal considerable sub-hour variability (e.g. Ghosh et al. 2000; Catanese & Sambruna 2000) on a timescale as low as $\Delta t = 1200$ s (Sambruna et al. 2000). Such a low value for the observed timescale could possibly be accommodated by assuming a high doppler boosting factor $\delta \simeq 18$. Hence, it appears that the more crucial point in this derivation is the assumption that the $\gamma$-rays dominating the emission are produced at $\sim$200 Schwarzschild radii. Indeed, at least in the case of the TeV-blazars, the variable, high energy emission is usually regarded as produced by moving knots or shocks in the jet far from the accretion disk (for a review, cf. Mannheim 1997; Aharonian & Völk 2001 and references therein). While instructive, the derived estimate should thus not be considered as a robust, general limit.

(2) A further mass estimate for Mkn 501 has been derived by DeJager et al. (1999) following an approach developed by Hayashida & Miyamoto et al. (1998). Assuming the variation in the accretion process to drive the X-ray and TeV variation in the jet via the dynamo effect, their result yields a central BH mass of $M_H=(1{-}6) \times 10^7~M_{\odot}$ for $\delta=(10{-}15)$. However, due to the absence of a physical basis for the required scaling of the Fourier spectrum and due to the assumption of a linear proportionality between variability timescale and BH mass (cf. Hayashida et al. 1998), which is probably not valid for the blazar class (Kataoka et al. 2001), this estimate again does not appear to be robust.

(3) The estimates (1) and (2) which suggest a BH mass less than $\sim$ $6 \times 10^7~M_{\odot}$, are strongly model-dependent as shown above. We may illustrate this in more detail by comparing them with results derived in the context of another, high energy emission model. Bednarek et al. (1996) for example, have developed a special model for the origin of the high energy particles in TeV blazars like Mkn 421 and Mkn 501, assuming the electrons responsible for the high energy emission to be accelerated rectilinearly in an electric field. In this model, the mass of the central BH is expected to be limited by $M_H \ga10^8 M_{\odot}~(E_{\rm max}/34~ {\rm TeV})^{2.5}/l_{\rm Edd}^{1/2}$, where $E_{\rm max}$ denotes the maximum photon energy and $l_{\rm Edd}$ the disk luminosity in units of the Eddington luminosity. There is strong evidence for a sub-Eddington accretion mode in BL Lacs in general (e.g. Cavaliere & D'Elia 2002) and particularly for the TeV emitting blazars (e.g. Celotti et al. 1998). Thus, using characteristic values, i.e. $l_{\rm Edd}=(0.01$-0.001) and $E_{\rm max}=20$ TeV (cf. Samuelson et al. 1998; Konopelko 1999), we arrive at a mass $M_H \ga(2.46$- $14.66) \times 10^8~ M_{\odot}$, which is up to ten times larger than the estimates (1) and (2).