7 Completeness at the Fornax distance

The completeness of a galaxy catalog depends on the following three parameters: limiting magnitude, limiting surface brightness, and limiting scale length. The surface brightness detection limits of the different fields are given in Table 1. They vary between $24.7 < \mu_{\rm lim} < 25.3$mag for the central fields. The minimum number of connected pixels for a detection, $n_{\rm min} = 5$ results in a limiting radius of about $r_{\rm lim} = 0\hbox{$.\!\!^{\prime\prime}$}6$. Adopting an exponential law for the galaxy profiles, $\mu (r) = \mu_0 + 1.086(r/\alpha)$, the relation of $r_{\rm lim}$ to $\mu_{\rm lim}$ is $r_{\rm lim} = 0.921\cdot \alpha (\mu_{\rm lim}-\mu_0)$.With $r_{\rm eff}=\alpha / 0.5958$ and $\mu_{\rm eff} = \mu_0 + 1.1245$,the selection function in the $\mu_{\rm eff}$, $r_{\rm eff}$ plane is $\mu_{\rm eff}(r_{\rm eff}) = \mu{\rm lim} + 1.1245 - 1.8225 r_{\rm lim}/r_{\rm eff}$. In Fig. 6 all galaxies brighter than $V_{\rm tot} = 23$ mag are plotted in this plane. The selection functions for three typical detection limits are shown. Objects that are located below and left of this functions are not accessible to our survey.

Figure 7 shows our sample of galaxies in a $\mu_{\rm peak}-V_{\rm tot}$diagram. Note that for most galaxies the $\mu_{\rm peak}$is a lower limit compared to the true central surface brightness as shown in the previous section. Also given are the parameters of Local Group dSphs (Mateo et al. 1993) shifted to the Fornax distance. The limiting $\mu_{\rm peak}$ is about 24.0 mag arcsec^-2. For the V magnitude the galaxy counts start to be incomplete for $V_{\rm tot} \gt 22.0$ mag. Concerning dEs in the Fornax cluster which follow the $\mu -V$ relation the completeness starts to drop at even brighter $V_{\rm tot}$. Thus, for the Fornax dEs we are more restricted in surface brightness than in the absolute magnitude. As shown in Fig. 7, several Local Group dSphs would not have been detected due to their low surface brightnesses, even if their total luminosities would have been within our limits. The dSphs And I and And II, for example, would have total magnitudes of about $V_{\rm tot} = 19.6$ mag, but central surface brightnesses of $\mu_{0,V} = 24.5$ mag. On the contrary, the dSph Leo I has the same $V_{\rm tot}$, but a 2 magnitudes brighter $\mu_{0,V}$, which is well within our sample limits.

The resolution limit is given by the seeing conditions. All objects with FWHM larger than $1\hbox{$.\!\!^{\prime\prime}$}5$, or about 130 pc in Fornax distance, appear resolved. Thus, all Local Group dSphs would appear clearly resolved when shifted to the Fornax distance. The dashed lines in Fig. 7 show the limits for different scale lengths of an exponential law in dependence of $V_{\rm tot}$ and $\mu_{\rm peak}$ surface brightness. All objects with scale lengths larger than about $0\hbox{$.\!\!^{\prime\prime}$}5$ appear resolved.

  

Figure 6: The effective surface brightness of all galaxies brighter than V = 23 mag is plotted versus the effective semi-major axis. Circles indicate Fornax members. The dashed lines show the relation of an exponential law for the limit of the total magnitude. The solid lines are the selection functions for our sample for three typical detection limits (see Table 1) and a limiting radius of $0\hbox{$.\!\!^{\prime\prime}$}6$. The regions below and left of these limits are inaccessible to our survey

  

Figure 7: The peak surface brightness of all galaxies is plotted versus the total V magnitude. The dashed lines show the relation of an exponential law for different scale lengths $\alpha$. At V = 22 mag the completeness starts to drop. The limit in peak surface brightness is about $\mu_{\rm limit} = 24.0$ mag. Circles indicate Fornax members. The asterisks are the Local Group dwarf spheroidals shifted to the Fornax distance. None of either category falls in the crowded part of the diagram