In this subsection, we first use our measured CLFs to infer the conditional stellar mass functions and then use the results to study the stellar mass contents of dark matter haloes. To convert luminosity into stellar mass, one typically uses a mass-to-light relation based on galaxy colour (e.g. Bell et al. 2003). This requires robust colour estimates. In our case, galaxies with r ∼ 21 in the SDSS photometric sample have typical error in the (u − r) colour of about 1 mag, mostly due to uncertainty in the u-band photometry. This error will propagate into the stellar mass estimates and can bias the stellar mass function, leading to an overestimate at the high-mass end.7 To reduce such bias, we estimate stellar masses using the observed mean colour–magnitude relations for blue and red galaxies separately. The details of this procedure are described in Appendix C. As our final goal is to estimate the global baryon fractions in stars, the use of average values as opposed to full colour distributions is not a severe limitation. Following Bell et al. (2003), we convert the observed luminosity and colour into stellar mass using

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\begin{eqnarray}
\log \bigg [\frac{M_{{\ast }}}{\mathrm{M}_{\odot }}\bigg ] = -0.223+0.299\,(u-r) -0.4\,(M_{\rm r}-4.64) -0.1,\nonumber\\
\end{eqnarray}

where (u − r) is the mean colour of a blue or red galaxy at a given absolute magnitude Mr. The constant, 4.64, is the r-band magnitude of the Sun in the AB system (Blanton & Roweis 2007) and the −0.1 offset corresponds to the choice of the Kroupa initial mass function (Kroupa 2001). Using this light-to-mass relation, we convert the global best-fitting luminosity functions into the corresponding stellar mass functions. The two left-hand panels in Fig. 10 show the estimated conditional stellar mass functions for blue and red satellites as a function of halo mass, respectively. Since a fixed $M^{{\ast }}_{{\rm b}}$ is applied to satellite galaxies for all halo masses, a slight overestimate of the stellar mass occurs at the massive ends for small haloes. As the stellar masses of central galaxies are obtained using individual observed (u − r) colours, the overestimate of the stellar mass of satellites can sometimes cause the stellar mass of a satellite galaxy to exceed that of the central. The dashed lines in the left-hand panels indicate the ranges where such situation is present. In order to estimate the total stellar mass in haloes of a given halo mass, we integrate the inferred conditional stellar mass functions down to low masses. We find that using 107 M⊙, which is about the minimum stellar mass reachable by the sample used here or zero lead to similar results. The results are shown in the right-hand panel of Fig. 10. The coloured dashed lines show the stellar mass to halo mass ratios for blue and red satellites, respectively. The grey dashed line is the total stellar mass of satellite galaxies to halo mass ratio, while the grey solid line is the stellar to dark matter mass ratio of central galaxies. The total ratio is shown as the black line. For haloes with M200  1013 M⊙, the total stellar mass is dominated by the central galaxies; in contrast, for more massive haloes, it is dominated by red satellites. The contributions from red and blue satellites are comparable for haloes with M200 ∼ 1012 M⊙, and the contribution from blue satellites appears to increase towards lower halo masses. Note that, although there are marked upturns in the stellar mass functions at the low-mass ends for red galaxies, the low-mass galaxies in the upturns (M*  108 M⊙) contribute little to the total stellar mass. Our results are qualitatively consistent with estimates based on data with more limited dynamical ranges (e.g. Leauthaud et al. 2012a,b; Kravtsov, Vikhlinin & Meshscheryakov 2014).
Citation: Blanton & Roweis 2007
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