terms of the distance $r_t\approx 2r$ for a galaxy with given $v$ (see Menci et al. 2003, 2004).

The fraction of cold gas accreted by the BH in each interaction event is computed in terms the variation $\Delta j$ of the specific angular momentum $j\approx Gm/v_d$ of the gas, to read (Menci et al. 2003) $$f_{acc}\approx {1\over 8}\,
\Big|{\Delta j\over j}\Big|=
%{1\over 6}\Big\langle {Gm'\,r_d\over V\,b}/{G\,m\over v_d}\Big\rangle  = 
{1\over 8}\Big\langle {m'\over m}\,{r_d\over b}\,{v_d\over V_{rel}}\Big\rangle\, .$$ Here $b$ is the impact parameter, evaluated as the average distance of the galaxies in the halo. Also, $m'$ is the mass of the partner galaxy in the interaction, and the average runs over the probability of finding such a galaxy in the same halo where the galaxy $m$ is located.

The average cold gas accreted during an accretion episode is thus $\Delta 
m_{acc}=f_{acc}\,m_c$, and the duration of an accretion episode, i.e., the timescale for the QSO to shine, is assumed to be the crossing time for the destabilized cold gas component, $\tau=r_d/v_d$.

The bolometric luminosity so produced by the QSO hosted in a given galaxy is then given by $$L(v,t)={\eta\,c^2\Delta m_{acc}\over \tau} ~.$$ We adopt an energy-conversion efficency $\eta= 0.1$ (see, e.g., Yu & Tremaine 