Galaxy Groups in the SDSS DR4: II. halo occupation statistics

We investigate various halo occupation statistics using a large galaxy group catalogue constructed from the SDSS DR4 with an adaptive halo-based group finder. The conditional luminosity function (CLF) is measured separately for all, red and blue galaxies, as well as in terms of central and satellite galaxies. The CLFs for central and satellite galaxies can be well modelled with a log-normal distribution and a modified Schechter form, respectively. About 85% of the central galaxies and about 80% of the satellite galaxies in halos with masses $M_h\ga 10^{14}\msunh$ are red galaxies. These numbers decrease to 50% and 40%, respectively, in halos with $M_h \sim 10^{12}\msunh$. For halos of a given mass, the distribution of the luminosities of central galaxies, $L_c$, has a dispersion of about 0.15 dex. The mean luminosity (stellar mass) of the central galaxies scales with halo mass as $L_c\propto M_h^{0.17}$ ($M_{*,c}\propto M_h^{0.22}$) for halos with masses $M\gg 10^{12.5}\msunh$, and both relations are significantly steeper for less massive halos. We also measure the luminosity (stellar mass) gap between the first and second brightest (most massive) member galaxies, $\log L_1 - \log L_2$ ($\log M_{*,1}-\log M_{*,2}$). These gap statistics, especially in halos with $M_h \la 10^{14.0}\msunh$, indicate that the luminosities of central galaxies are clearly distinct from those of their satellites. The fraction of fossil groups, defined as those groups with $\log L_1 - \log L_2\ge 0.8$, ranges from $\sim 2.5$ for groups with $M_h\sim 10^{14}\msunh$ to 18-60% for groups with $M_h\sim 10^{13}\msunh$. Finally, we measure the fraction of satellites, which changes from $\sim 5.0$ for galaxies with $\rmag\sim -22.0$ to $\sim40$ for galaxies with $\rmag\sim -17.0$. (abridged)