4 Mass estimates derived in the context of a binary black hole scenario

In a recent contribution (Rieger & Mannheim 2000, hereafter RMI; 2001), we have shown that the periodicity of $P_{\rm obs}=23$ days, observed during the 1997 high state of Mkn 501, could be plausibly related to the orbital motion in a BBHS, provided the jet, which dominates the observed emission, emerges from the less massive (secondary) BH. If such an interpretation (henceforth called the standard scenario) is appropriate, we may derive a third estimate for the central mass in Mkn 501. We have demonstrated in RMI that, due to relativistic effects, the observed period appears drastically shortened, so that for the intrinsic Keplerian orbital period one finds $P_{\rm k} = (6{-}14)~~{\rm yrs}~$. Taking into account that the observed emission is periodically modulated by differential doppler boosting due to the orbital motion, one may derive a simple equation for the required mass dependence in the standard scenario (cf. RMI, Eq. (8)):

 

$$\frac{M}{(m+M)^{2/3}} = \frac{P_{\rm obs}^{1/3}}{(2~\pi~[1+z]~ G)... ...\alpha)}-1}{f^{1/(3+\alpha)}+1}~ \left(1-\frac{v_z}{c}~\cos i\right)^{2/3}\cdot$$ (1)

Here $\alpha$ denotes the spectral flux index, z the redshift, v_z the outflow velocity, $i\simeq 1/\Gamma_{\rm b}$ the angle of the jet axis to the line of sight, $\Gamma_{\rm b} \simeq 10{-}15$ (e.g. Mannheim et al. 1996; Spada et al. 1999) the bulk Lorentz factor and f the observed flux ratio between maximum and minimum (for the TeV range $f \sim 8$), while m and M denote the masses of the smaller and larger BH, respectively.

In order to break the degeneracy in this mass ratio, we may utilize an additional constraint by assuming that the current binary separation d corresponds to the separation at which gravitational radiation becomes dominant (cf. RMI). Such a constraint yields an upper limit for the allowed binary masses and might be associated with the key aspect that BL Lac objects are old, more evolved and underluminous sources, i.e. they might be close binaries, probably settled above or near the critical gravitational separation, because the possibility of removing further angular momentum has been almost terminated as a result of declining gas accretion rates. We can specify the corresponding gravitational separation $d_{\rm g}$ by equating the timescale $\tau_{\rm grav}=\vert d/\dot{d}\vert = (5~ c^5/64~ G^3) ~d^4/(M~m~[m+M])$ on which gravitational radiation shrinks the binary orbit, with the dynamical timescale $\tau_{\rm gas}$ for gas accretion (cf. Begelman et al. 1980; note that compared with RMI, this estimate for $\tau_{\rm grav}$ is a factor 2.5 more precise, cf. Rieger & Mannheim 2001). A characteristic measure for $\tau_{\rm gas}$ is given by the Eddington limit $\tau_{\rm gas}\simeq 3.77 \times 10^7~(\eta/0.1)$ yrs, assuming a canonical $10\%$ conversion efficiency (cf. Krolik 1999). Using $\tau_{\rm grav}$ and $\tau_{\rm gas}$, one finally arrives at

$$d_{\rm g} \simeq 3.50 \times 10^{16}~M_8^{1/4}~m_8^{1/4}~ (m_8+M_8)^{1/4} ~~~{\rm cm}.$$ (2)

If we combine this expression with the relevant expression for the binary separation (i.e. Eq. (7) of RMI), we obtain a second constraint on the allowed mass ratio given by

 

$$\frac{(m_8 + M_8)^{3/4}}{M_8^{5/4}~m_8^{1/4}} = 7.3 \times 10^{6}~ \frac{f^{1/(3+\alpha)}+1}{f^{1/(3+\alpha)}-1}~ \frac{(1+z)}{P_{\rm obs}}~\sin i,$$ (3)

assuming the jet to arise from the less massive BH (for the reverse case the BH masses should be interchanged in Eqs. (1) and (3)). By using Eqs. (1) and (3), we may determine the binary masses permitted for a BBHS with separation near that for which gravitational radiation becomes dominant. In Fig. 1 we have plotted the allowed range for the case of $\Gamma _{\rm b}=15$ and $\alpha \simeq (1.2{-}1.7)$ (e.g. Aharonian et al. 1999). The run of the curves is determined by Eq. (1) with the upper bounds given by Eq. (3).

Figure 1: Allowed central mass dependence as a function of the secondary mass assuming a BBHS, where the observed periodicity of 23 days is related to the orbital motion of a jet, which emerges from the less massive BH. The allowed mass range lies inside the curves and is calculated using $\Gamma _{\rm b}=15$.

We note that the upper bounds depend on the presumed timescale for the binary evolution, i.e. on the estimate of the gravitational separation. In particular, the derived upper bounds may be larger, if, for example, disk-driven migration (e.g. Armitage & Natarajan 2002) might be relevant. Figure 1 reveals that (a) the typical combined mass for a BBHS is expected to be less than $\sim$ $3 \times 10^8 ~M_{\odot}$, and that (b) a BBHS with, e.g. $m \sim 6 \times 10^7 ~M_{\odot}$ and combined central mass $(M+m) \simeq (1.5{-}2) \times 10^8 ~M_{\odot}$ appears well-conceivable given the current limits.

On the other hand, if a high central mass of $\sim$ $10^9~ M_{\odot}$ will be established by further research, the proposed binary scenario appears to be ruled out. We note, however, that even in this case a binary scenario may be still possible provided that the jet, which dominates the emission, is produced by the primary BH. To illustrate the implications in this case, let us consider a (combined) central mass of $\simeq$ $0.9 \times 10^9~M_{\odot}$ by demanding the primary to be in the range $(5{-}6) \times 10^8~M_{\odot}$ (see Table 1). The mass of the secondary BH then is determined by Eq. (1) with the masses interchanged, the current separation d by Eq. (3) of RMI (again with the masses interchanged) and the gravitational separation $d_{\rm g}$ by Eq. (2). The results as shown in Table 1 imply a close binary system with $d \la d_{\rm g}$. If the optically bright QSO stage thus occurs during the binary evolution and the applied doppler factors are considered as typical, phases of super-Eddington accretion and/or with decreased conversion efficiency seem to be necessary for the binary to be above its gravitational separation.

 

 

Table 1: Binary masses for a central core mass of $\simeq$ $0.9 \times 10^9~M_{\odot}$, currently expected separation d and corresponding gravitational separation $d_{\rm g}$. The calculations have been done using $\alpha =1.2$ (in brackets: $\alpha =1.7$). The jet is assumed to be produced by the more massive BH.

$i=1/\Gamma_{\rm b}$	1/10	1/15
$m~[10^8~M_{\odot}]$	4.22     (3.62)	3.85     (3.34)
$M~[10^8~M_{\odot}]$	5.00     (5.00)	6.00     (6.00)
$d~[10^{16}~\rm {cm}]$	4.87     (4.76)	8.54     (8.39)
$d_{\rm g}~[10^{16}~\rm {cm}]$	13.1     (12.4)	13.6     (12.9)