denoted $\langle t_{\mathrm{merge}} \rangle$, for each close pair using the results of @Kitzbichler2008. Using the Millennium Simulation [@Springel2005], they find that the average $t_{\mathrm{merge}}$ for galaxies with $r_{\mathrm{sep}} \leq 50$ kpc and $\Delta v \leq 300$ [kms$^{-1}$]{} can be given by the following: $$\langle t_{\mathrm{merge}} \rangle = 2.2\,\mathrm{Gyr} \, \frac{r_{\mathrm{sep}}}{50 \, \mathrm{kpc}} \, \Big( \frac{M_{\star,\mathrm{com}}}{5.5 \times 10^{10} \, \mathrm{M_{\odot}}}\Big)^{-0.3}  \, \Big(1 + \frac{z}{8}\Big)
\label{eq:t_merge}$$ where $M_{\star,\mathrm{com}}$ is the total stellar mass of the companion at its observed redshift (determined using [kcorrect]{}), $r_{\mathrm{sep}}$ is the physical separation distance (in kpc) and $z$ is the photometric redshift of the cluster. The uncertainties on $\langle t_{\mathrm{merge}} \rangle$ are given by $1\sigma$ standard deviation and are propagated through to the final measured fractional mass growths. Various works have commented that the [@Kitzbichler2008] merger timescales are significantly longer than the estimate from dynamical friction or the orbital period [e.g. @Conroy2007a; @Bertone2009; @Conselice2009a; @Kauffmann2010; @Lotz2011]. The uncertainties on $\langle t_{\mathrm{merge}} \rangle$ in all the redshift bins are large enough to encompass differences with other methods (see Table \[table:redM\_mass\_growth\]).

The merger rate, denoted $R_{\mathrm{merge}}$, of the sample, i.e. the number of mergers per BCG per Gyr, is defined as follows: $$R_{\mathrm{merge}} = \frac{f_{\mathrm{pair}}}{\langle t_{\mathrm{merge}} \rangle}
\label{eq:R_merge}$$ The 