THE NATURE OF RED DWARF GALAXIES

In the current paradigm of galaxy formation, it is believed that virtually all galaxies initially form as "disks" owing to the cooling of gas with nonzero angular momentum in virialized dark matter halos. This smooth gas accretion dominates the galactic gas supply and hence the fuel for star formation. Galaxies that reside in the centers of lower mass halos, those with masses less than M_halo ≲ 3 × 10¹¹  h⁻¹M_☉, accrete gas through very efficient cold mode accretion, i.e., gas that is never heated (Keres et al. 2005, 2009). The central galaxies that reside in larger halos accrete their gas through the classic, but less efficient, hot mode of accretion where the gas is shock heated to near the virial temperature near the virial radius and then must cool to be accreted by the central galaxy. Hence, the naive expectation would be that dwarf galaxies should be actively star forming and blue.

However, when a small halo is accreted by a larger halo, i.e., when it becomes a subhalo, the central galaxy that formed in the small halo becomes a satellite galaxy and may experience a number of environmental effects that may change its properties. For instance, the diffuse gas originally associated with the subhalo may be stripped, thus removing the fuel for future star formation (e.g., Larson et al. 1980). This process, referred to as strangulation (Balogh & Morris 2000), can result in a gradual decline of the star formation rate in the satellite galaxy, making it redder with the passage of time. If the external pressure is sufficiently high, ram pressure stripping may also be able to remove the entire cold gas reservoir of the satellite (e.g., Gunn & Gott 1972), causing a fast quenching of its star formation. A satellite galaxy is also subject to tidal heating and stripping and galaxy harassment (Moore et al. 1996), which may also cause the satellite to lose its fuel for star formation. These processes are believed to have played an important role in the evolution of satellite galaxies, and to be responsible, to a large extent, for the relation between galaxy properties and their environment. Indeed, satellite galaxies are generally found to be redder and somewhat more concentrated than central galaxies with similar stellar masses (e.g., van den Bosch et al. 2008b; Weinmann et al. 2009; Yang et al. 2009a, 2009b; Guo et al. 2009).

Furthermore, recent investigations based on cosmological N-body simulations have shown that a significant fraction of dark matter halos that are close to, but beyond the virial radius of, a more massive neighboring halo, are physically connected to their neighbor. As shown by Lin et al. (2003) and more recently by Ludlow et al. (2009), some low-mass halos are physically associated with more massive halos, in the sense that they were once subhalos within the virial radii of these more massive progenitors and have subsequently been ejected. This population of halos was found to extend beyond three times the virial radii of their host halos, and represents about 10% of the entire population of low-mass halos (Wang et al. 2008). If galaxies have managed to form in the progenitors of these ejected halos, it is likely that the same environmental processes operating on satellite galaxies may also have affected the properties of these galaxies. In particular, we would expect the presence of a red population of faint galaxies that are closely associated with massive halos that once hosted them.

Galaxies are observed to be bimodal in the color–magnitude plane: red galaxies with very little star formation (the red sequence) and blue star-forming galaxies that are typically disky (the blue cloud; e.g., Kauffmann et al. 2003; Baldry et al. 2004). Extrapolating the observed division line (Yang et al. 2008) to dwarf galaxies, we surprisingly find that for central dwarf galaxies in the Sloan Digital Sky Survey (SDSS), with r-band magnitudes between −14.46 and −17.05, just over 1/4 are red. In this paper, we will investigate the nature of these red dwarf galaxies. Quantifying the spatial distribution of this population of galaxies is clearly important, because it allows us to determine whether or not they can be explained as a population of satellite galaxies that were ejected from larger halos.

In this paper, we use the galaxy group catalog constructed by Yang et al. (2007) from the Sloan Digital Sky Survey Data Release 4 (SDSS DR4; Adelman-McCarthy et al. 2006) to study the distribution of central dwarf galaxies around massive halos. The structure of this paper is as follows. In Section 2, we briefly describe the criteria used to select galaxies and galaxy groups. In Section 3, we study the radial distribution of dwarf galaxies around their nearest neighbor halos and its dependence on galaxy color and concentration. Some systematic effects that may change our results are discussed in Section 4. In Section 5, we use mock catalogs to test the reliability of our results and to quantify their implications. Finally, in Section 6, we present some further discussion regarding our results.

2.1. Samples of Galaxy Groups

Our analysis uses the galaxy group catalogs of Yang et al. (2007), which were constructed from the New York University Value-Added Galaxy Catalog (NYU-VAGC, see Blanton et al. 2005b) based on the SDSS DR4 (Adelman-McCarthy et al. 2006). Only galaxies in the Main Galaxy Sample with redshifts in the range 0.01 ⩽ z ⩽ 0.20 and with a redshift completeness ${\cal C}> 0.7$ were used. Three sets of group catalogs were constructed using a modified version of an adaptive halo-based group finder, which was optimized to assign galaxies into groups according to their common dark matter halos (Yang et al. 2005). For our study here, we use group sample II, in which only galaxies with spectroscopic redshifts (either provided by the SDSS or taken from alternative surveys) are used. We have tested, though, that using group sample III, which also includes galaxies that have been missed owing to fiber collisions, does not have a significant impact on any of our results.

For each group in the catalog, Yang et al. (2007) estimated the corresponding halo mass using either the ranking of its characteristic luminosity (this mass is denoted by M_L) or using the ranking of its stellar mass (this mass is denoted by M_S). Throughout this paper, we use M_S as our halo masses. We have also tested that using M_L instead does not change any of our results. As described in Yang et al. (2007), the characteristic luminosity and stellar mass of a group are defined to be the total luminosity and total stellar mass of all group members, respectively, with ^0.1M_r − 5log h ⩽ −19.5. Thus, groups whose member galaxies are all fainter than ^0.1M_r − 5log h = −19.5 cannot be assigned halo masses according to the ranking. For these groups, the halo masses are estimated in the following way. In Yang et al. (2009b), it is shown that the stellar masses of central galaxies are tightly correlated with the masses of their host halos. The mean of this relation is well described by

$$\begin{equation} M_{\ast } = M_0 \frac{ (M_h/M_1)^{\alpha +\beta } }{(1+M_h/M_1)^\beta } \,, \end{equation} \tag{ 1 }$$

where M_* and M_h are the central galaxy stellar mass and the host halo mass of the group, respectively, and (log M₀, log M₁, α, β) = (10.306, 11.040, 0.315, 4.543). For groups that cannot be assigned a halo mass according to the stellar-mass (luminosity) ranking, we use the above relation to obtain M_h through the stellar masses of their central galaxies.

2.2. Galaxy Samples

Group catalog II consists of 369447 galaxies, which are assigned into 301237 groups. The majority, 271420, of the groups contain only one member, i.e., all of them are the central galaxies of the groups. The remaining 98027 galaxies are in groups with more than one member, and 29817 of them are central galaxies (the brightest one in each group). We refer to the other 68210 galaxies as satellites (Yang et al. 2007).

From our galaxy sample, we select three subsamples of dwarf galaxies as follows. We rank order all galaxies according to their absolute magnitudes (in the r band, K- and evolution-corrected to redshift z = 0.1), starting with the faintest galaxy. The 1500 galaxies with the highest rank (i.e., the 1500 faintest galaxies) make up our first sample, called S1, The galaxies with ranks 1501–3000 make up sample S2, and those with ranks 3001–4500 sample S3. Table 1 lists the (sequential) absolute magnitude ranges of all three samples, as well as the numbers of central and satellite galaxies. Note that all the galaxies in S1, S2, and S3 are fainter than ^0.1M_r − 5log h=−17.05. In what follows we refer to all galaxies in these three samples as dwarf galaxies.

Table 1. Galaxy Samples

ID (1)	0.1Mr − 5log h (2)	Ntotal (3)	Ncent (4)	Nsat (5)	fred,cent (%) (6)	fred,sat (%) (7)	fbred,cent (%) (8)	fbred,sat (%) (9)	fb(rp/R180) ⩽ 3 (%) (10)
S1	(−14.46, −16.36]	1500	1103	397	13.69	37.53	33.79	53.42	34.60
S2	(−16.36, −16.78]	1500	1081	419	13.69	36.28	22.50	47.02	47.46
S3	(−16.78, −17.05]	1500	1008	492	10.20	40.24	19.51	52.64	47.72
S1 + S2 + S3	(−14.46, −17.05]	4500	3192	1308	12.59	38.15	25.49	51.08	41.63
S4		1500	1080	420	13.51	37.02

Notes. Column 1 indicates the sample ID; Column 2 lists the absolute magnitude range of each sample; Columns 3–5 indicate the number of total, central, and satellite galaxies in each sample, respectively; Columns 6 and 7 list the red fractions among central and satellite galaxies, respectively, where the red galaxies are defined to be the reddest 20% of all the galaxies; Column 8 lists the red fraction of central dwarf galaxies where the red galaxies are defined by extrapolating the division between red sequence and blue cloud galaxies from Yang et al. (2008); Column 9 lists also this red fraction, but for the satellite dwarf galaxies; Column 10 lists the fraction of those (in Column 8) central red dwarf galaxies that have r_p/R₁₈₀ ⩽ 3.

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To study how the spatial distribution of dwarf galaxies depends on galaxy color, we separate each of the samples, S1, S2, and S3, into red and blue subsamples. In particular, we define a color cut

$$\begin{equation} ^{0.1}(g-r) = a + b (\:^{0.1}{M}_r-5\log h)\,, \end{equation} \tag{ 2 }$$

and we adjust the parameters a and b such that samples S1, S2, and S3 roughly have the same fractions of galaxies, f_red, redder than this particular cut. We consider four values for f_red: 20%, 30%, 40%, and 50%, for which we obtain [a, b] = [−0.421, −0.060], [−0.423, −0.055], [−0.375, −0.049], and [−0.313, −0.043], respectively. Thus, if f_red = 20% it means that the red subsamples of S1, S2, and S3 each consist of the 20% reddest galaxies in their particular samples, etc. Figure 1 shows the color–magnitude relations of galaxies in S1, S2, and S3 (delineated by vertical dashed lines). The four panels correspond to the four different values of f_red, as indicated, and the solid line in each panel corresponds to the color cut of Equation (2) used in each case.

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In Table 1 we list, for each sample, the red fractions of central (f_red,cent) and satellite galaxies (f_red,sat). Here, red galaxies are defined to be the reddest 20% of all galaxies (both centrals and satellites) in our sample of dwarf galaxies. Clearly, dwarf galaxies that are satellites have a much higher red fraction than central dwarf galaxies. For comparison, extrapolating the observed division line between the red sequence and the blue cloud from Yang et al. (2008) down to dwarf galaxies, one would find that 33.8%, 22.5%, and 19.5% of the central galaxies in the S1, S2, and S3 samples, respectively, were red. While in total, there are more red central dwarfs than red satellite dwarfs, which is so far not predicted, e.g., by halo occupation models (Brown et al. 2008). Note that different definition of red galaxies may change these fractions (e.g., with respect to the galaxies of similar stellar masses), however, not the general results we find in this paper.

In this section, we investigate how central dwarf galaxies are distributed with respect to their nearest more massive halo (i.e., more massive than their own halo). Since the distances of galaxies based on redshifts suffer from redshift distortions, we separate the distance between a central dwarf galaxy and its nearest more massive halo into two components: π, which is the separation along the line of sight, and r_p, which is the separation in the perpendicular direction.

For each group in the catalog, we use the assigned halo mass, M_S, to estimate its halo radius, $R_{180}=[3M_{S}/(4\pi *180\bar{\rho })]^{1/3}$, which follows from defining the mean mass density within a halo as 180 times the average density of the universe, $\bar{\rho }$. We search around each central dwarf galaxy, within a line-of-sight separation |π| = 15 h⁻¹ Mpc,⁶ for the group that has (1) a halo mass larger than that of the dwarf galaxy in question and (2) the lowest value of r_p/R₁₈₀ (with R₁₈₀ the halo radius of the group). The central galaxy is then said to be at a scaled "distance" r_p/R₁₈₀ from a massive halo, and the halo is referred to as the nearest halo of the galaxy. We use this scaled distance because R₁₈₀ is the only important length scale related to the dynamics of a virialized halo.

3.1. Color Dependence

Figure 2 shows the fraction N_red/N_total of central dwarf galaxies that are red as a function of the scaled distance, r_p/R₁₈₀, to their nearest halos. Here, N_total is the total number of dwarf galaxies in each sample (S1, S2, or S3) in that bin of r_p/R₁₈₀, and N_red is the number of those galaxies that are red according to the criterion used. The numbers are shown in the small window in each panel (the black histogram for N_total and the red, hatched histogram for N_red). The four panels show the results for f_red = 20%, 30%, 40%, and 50%. In each panel, the different lines show the results obtained for the three samples, S1, S2, and S3, as indicated in the upper-left panel. The error bars are obtained using 100 bootstrap resamplings (Barrow et al. 1984; Mo et al. 1992). Galaxies are counted in bins specified by N − 0.5 ⩽ r_p/R₁₈₀ ⩽ N + 0.5 (for N = 2, 3, ..., 7) and 0 ⩽ r_p/R₁₈₀ ⩽ N + 0.5 for N = 1. For comparison, the data point at r_p/R₁₈₀ = 0 indicates the fraction of red satellite galaxies within the same luminosity range as the central galaxies. Note that the results obtained for S1, S2, and S3 are almost identical, indicating that the spatial distribution of dwarf galaxies around massive halos does not depend on their luminosities. However, if we consider much brighter galaxies, e.g., at ^0.1M_r − 5log h∼−18.5, the radial dependence starts to level off.

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There is a clear trend that the fraction of red central dwarf galaxies increases with decreasing scaled distance to the nearest halo. The fraction of the 20% reddest population at r_p/R₁₈₀ ≳ 4 is around 5%–10%, increases systematically to ∼25% at r_p/R₁₈₀ ∼ 1, and to ∼40% at r_p/R₁₈₀ = 0 for the satellite galaxies. For the other three cases (with f_red = 30%, 40%, and 50%), the fraction also decreases with r_p/R₁₈₀, but reaches a higher level at large r_p/R₁₈₀. This indicates that the less red galaxies in these subsamples are not strongly associated with massive halos. We quantify the association of red dwarf galaxies with massive halos by fitting the data obtained from S3 shown in the four panels simultaneously with a function f = a + b × exp(−x/2), where x = r_p/R₁₈₀. In the fit, we subtract a constant of 0.1, 0.2, and 0.3, from the data for the 30%, 40%, and 50% reddest subsamples, respectively, to account for the component that is not closely associated with massive halos. The best-fit results, with a = 0.045 and b = 0.356, are shown in each panel of Figure 2 as the green, long-dashed lines. We have also checked the radial distribution of the central dwarf galaxies when using f_red = 15% to define the subsample of red dwarfs. In this case, we find less than a 5% decrease in the red fraction at large r_p/R₁₈₀, indicating that the 5% least red galaxies in the 20% reddest subsample are not randomly but more closely distributed relative to the massive halos. Thus, the overall results suggest that the 15%–20% reddest dwarf galaxies are quite distinct from the other dwarfs, in that they reveal a clear preference to reside close to their nearest more massive dark matter halo. In addition, the nonzero red fraction at large r_p/R₁₈₀ indicates that there is a ∼5% tail of red dwarfs randomly distributed throughout the background population, especially in the voids, due to some processes that shut down the star formation. A similar trend has also been found by Cooper et al. (2007) from the DEEP2 survey, however, for more massive galaxies in low-density regions at higher redshifts. In Section 5, we use mock galaxy redshift surveys constructed from cosmological N-body simulations to quantify this connection.

3.2. Concentration Dependence

Apart from a color dependence, we also check whether the radial distribution of dwarf galaxies with respect to their nearest more massive dark matter halos depends on their surface brightness profiles. To this end, we split our dwarf galaxies into two subsamples according to the value of their concentration parameter C = r₉₀/r₅₀. Here, r₉₀ and r₅₀ are the radii that contain 90% and 50% of the Petrosian r-band flux, respectively. As shown in Strateva et al. (2001), C is a reasonable proxy for the Hubble type, with C>2.6 corresponding to early-type galaxies. We, therefore, separate galaxies into high (C>2.6) and low (C ⩽ 2.6) concentrations, as illustrated in the lower-right panel of Figure 3. Roughly, 20% of the dwarf galaxies thus end up in the high-concentration subsample.

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The lower-right panel of Figure 4 shows the fraction of galaxies in this high-concentration subsample as a function of the scaled distance to the nearest more massive halo. Unlike the reddest galaxies, the most concentrated galaxies have a radial distribution that is similar to that of the total population of central dwarf galaxies. Note, however, for brighter galaxies, especially in and around clusters, there is a so-called morphology–radius relation (e.g., Dressler et al. 1997), which according to Park & Hwang (2008) may be largely induced by the interaction of the target galaxy with its nearest (early-type) neighbor galaxy.

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Before we proceed to study the origin of the central red dwarf galaxies, there are a number of issues that need to be addressed. An obvious worry is that the group finder is not perfect and has misclassified a number of satellite galaxies as central galaxies. We will discuss this issue in more detail with the help of mock galaxy and group catalogs in Section 5. In this section, we discuss systematics that can be addressed without the need for mock catalogs.

4.1. Stellar-Mass Dependence

4.2. Dependence on the Mass of the Nearest Neighbor

In our analysis above, the "nearest more massive halo" of a central dwarf galaxy is defined as the halo with a line-of-sight separation |π| ⩽ 15 h⁻¹ Mpc which has (1) a mass that is more massive than that of the dwarf galaxy and (2) the smallest value of r_p/R₁₈₀ (see Section 3). This implies that some of these nearest more massive halos may have masses that are only slightly larger than that of the dwarf galaxy under consideration.

In what follows we use M_h to refer to the halo mass of the central dwarf galaxy, and M_n to refer to the mass of its nearest more massive halo. For our combined sample (S1 + S2 + S3), the average value of M_h is about 10^10.9 h⁻¹M_☉. Figure 5 shows the ratio M_n/M_h as a function of the scaled distance r_p/R₁₈₀ for all central dwarf galaxies in S1 + S2 + S3. Here, the results for the 20% reddest galaxies are shown as red triangles, while the other 80% are indicated by blue crosses. Note that there is a very large amount of scatter in M_n/M_h, ranging from unity to well in excess of 1000.

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It is interesting to investigate whether the color dependence of the radial distribution of central dwarfs with respect to their nearest more massive halo depends on M_n. This can provide valuable insight into the actual origin of this color dependence. We therefore proceed as follows. We first combine samples S1 + S2 + S3, and then calculate the fraction of central red dwarf galaxies that belong to the 20% reddest subsample as we did in Section 3. However, now we only select systems for which M_n/M_h is restricted to [1,2], [2,4], [4,8] or log [M_n/ h⁻¹M_☉] ⩾ 12.0, respectively. The results are shown in the four panels of Figure 6 as indicated. For comparison we also show, in each panel, the results obtained for all systems (i.e., M_n/M_h>1). A comparison of all four panels shows that there is a clear, albeit somewhat weak, dependence of the trend on M_n/M_h. Overall, the colors of central dwarf galaxies are most strongly affected by nearest neighbor halos that are more massive. In the case of 1 < M_n/M_h ⩽ 2 (upper-left panel), the central dwarf galaxies have an N_red/N_total that is almost independent of r_p/R₁₈₀, and much lower than that of the dwarf satellites. On the other hand, the central dwarfs that are distributed around halos more massive than 10^12.0 h⁻¹M_☉ (lower-right panel), have a radial dependence that is somewhat stronger than that for all systems.

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4.3. Survey Edge Effect

Since the SDSS is not a full-sky survey, and since our group catalog is constructed using only galaxies with redshifts 0.01 ⩽ z ⩽ 0.2, our results may be influenced by edge effects of the survey: for central dwarf galaxies near an edge of the survey, there is an enhanced probability that it is actually a satellite (or central) galaxy in a more massive group (halo), but for which all other members just happen to lie beyond the edges of the survey. Although we tried to take these effects into account when assigning halo masses to our groups (see Yang et al. 2007 for details), it could still be that a significant fraction of our central dwarf galaxies are in reality misclassified centrals or satellites owes to the survey geometry.

To check the impact of these edge effects, we follow Yang et al. (2007) by measuring the edge parameter f_edge. For each central dwarf galaxy in S1 + S2 + S3, we randomly distribute 500 points within a radius 1 h⁻¹ Mpc. Next, we apply the SDSS DR4 survey mask and remove those random points that fall outside of the region where the completeness ${\cal C} > 0.7$. For each central dwarf galaxy, we then compute the number of remaining points, N_remain, and we define f_edge = N_remain/500 as a measure for the volume around the central dwarf galaxy that lies within the survey edges. To test the impact of edge effects on our measurements in Section 3, we remove those central dwarf galaxies with f_edge ⩽ 0.8 (about 13%) and recalculate the radial distribution of the remaining central galaxies. The result is shown in Figure 7 (dashed line), compared to the results for all the central dwarfs, independent of their value of f_edge (solid line). Clearly, the two curves are almost indistinguishable, indicating that our results are not an artifact of survey edge effects.

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One potential problem with the results presented above is that the group finder used to identify galaxy groups is not perfect. Hence, some of the dwarf galaxies classified as central galaxies may, in fact, be satellite galaxies. To test the severity of such effects and to quantify the true association between central dwarf galaxies and their nearby massive halos, we apply the same analysis to mock samples and compare the results with the observational data that we have obtained. Here, we use the mock SDSS DR4 galaxy and group catalogs that are constructed by Yang et al. (2007) to test the performance of the group finder. Following Yang et al. (2004), the mock galaxy catalog is constructed by populating dark matter halos in numerical simulations of the standard ΛCDM model with galaxies of different luminosities, using the conditional luminosity function (CLF) model of Cacciato et al. (2009). The cosmological parameters adopted here are consistent with the three-year data release of the WMAP mission: Ω_m = 0.238, Ω_Λ = 0.762, n_s = 0.951, h = 0.73, and σ₈ = 0.75 (Spergel et al. 2007). This CLF describes the halo occupation statistics of SDSS galaxies, and accurately matches the SDSS luminosity function, as well as the clustering and galaxy–galaxy lensing data of SDSS galaxies as a function of their luminosity. Next, a mock redshift survey is constructed mimicking the sky coverage of the SDSS DR4 and taking detailed account of the angular variations in the magnitude limits and completeness of the data (see Li et al. 2007 for details). Finally, we construct a group catalog from this mock redshift survey, using the same halo-based group finder as for the real SDSS DR4.

To test the impact of contamination on our observational results obtained above, we consider three models for the distribution of red dwarf galaxies.

• 1.  
  Case I: Here, we assume that a fraction, f_red,sat, of true satellite dwarf galaxies are red, but that all true central dwarf galaxies are blue.
• 2.  
  Case II: Same as Case I, but here we assume that a fraction, f_red,cent, of true central dwarf galaxies are also red and have the same spatial distribution as blue central dwarfs.
• 3.  
  Case III: Similar to Case II, but here we assume that f_red,cent depends on the distance of the central galaxy to its nearest more massive halo, according to f_red,cent(r) = a + b × exp(−(y − 1)/2). Here a and b are constants and y = r/R₁₈₀.

These models are used to assign a color to each of the mock dwarf galaxies according to their positions in real space and their true membership of host halos.

As for the observational data, we select the 4500 faintest galaxies from the mock group catalog, using the same criteria as described in Section 2.2. We choose the observational result for the central galaxies in S1 + S2 + S3 (shown as the dots with error bars in Figure 6 to compare with our models). As discussed in the previous section, this result is representative of the distribution of red dwarf galaxies with respect to their nearest more massive halos. For all cases (I, II, and III), we adopt f_red,sat = 0.38 so that the red fraction of the satellite galaxies is consistent with the SDSS data (repeated in Figure 8 as a solid line). In Case II, we set f_red,cent = 0.07, so that the red fraction of the central galaxies at large projected distance, r_p/R₁₈₀ ⩾ 4, is roughly the same as the observational data. The corresponding results obtained from Cases I and II are shown in the upper-left and upper-right panels of Figure 8, respectively. It is clear that in Case I, in which the only red dwarfs are satellites, the fraction of false central galaxies in the mock catalog is too small too match the observational data at r_p/R₁₈₀ ≳ 1. For Case II, although by construction the fraction of red central galaxies matches the observational results at large r_p/R₁₈₀ ⩾ 4, the model underestimates the fraction of red central galaxies at intermediate r_p/R₁₈₀. These results indicate that (1) not all red dwarf galaxies are satellites and (2) the central red dwarf galaxies have a different distribution than the total central dwarf population.

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Now, let us look at Case III. We have experimented with different values for a and b, and found that the following set of parameters matches the observational data reasonably well: (a, b) = (0.05, 0.36). The results obtained from the mock catalog using this set of model parameters are shown in the lower-left panel of Figure 8. This shows that the central red dwarf galaxies are correlated with massive halos on scales given by (y − 1) ≲ 2 (i.e., r ≲ 3R₁₈₀).

Finally, as we did for the observational sample, we can also obtain f_red,cent(r) for dwarf galaxies near halos of different masses using the mock sample Case III. As an illustration, we consider the case with M_n ⩾ 10^12.0 h⁻¹M_☉. The model for f_red,cent(r) that best matches the observational result has (a, b) = (0.05, 0.45) and is slightly steeper than that obtained for the case without any restrictions on M_n. The model prediction obtained from the mock sample is shown in the lower-right panel of Figure 8 along with the corresponding observational data.

In the mock catalog considered above, the distribution of satellite galaxies in individual halos is assumed to be spherically symmetric and to follow the Navarro–Frenk–White (NFW; Navarro et al. 1997) profile (see Yang et al. 2004 for details). In reality, the distribution of satellite galaxies in individual halos may not be spherical, which may cause further contaminations of the group memberships selected by the group finder. To test this, we have constructed a mock catalog assuming that the distribution of satellite galaxies in individual halos is triaxial, with axis ratios given by the model of Jing & Suto (2002) for cold dark matter (CDM) halos. We found a slightly higher level of contamination in this new mock catalog, but it does not change any of our results significantly. As an illustration, in each panel of Figure 8 we also show the results for nonspherical halos using symbols connected with dot-dashed lines. We obtain for Case III slightly weaker radial dependence with (a, b) = (0.05, 0.34) and (a, b) = (0.05, 0.43), for the cases shown in the lower-left and lower-right panels, respectively.

We set out to understand why about 1/4 of central dwarf galaxies, those with r-band magnitudes between −14.46 and −17.05, are red when one defines a dwarf galaxy to be red by extrapolating the division between red sequence galaxies and the blue cloud, as determined by Yang et al. (2008) for brighter galaxies, down to dwarf magnitudes. Current models of galaxy formation would naively expect such galaxies to be blue since they would be efficiently accreting gas through cold mode accretion and rapidly converting it into stars.

In a recent study, Ludlow et al. (2009; see also Lin et al. 2003) analyzed the properties of subhalos in galaxy-sized CDM halos using a suite of cosmological N-body simulations. The subhalos in their definition refer to the whole population of subhalos physically associated with the main system, including both subhalos that are found within the virial radius of the host halo at the present time and halos that were once within the virial radius of the main progenitor of the host and have survived as self-bound entities until z = 0. They found that such populations can extend beyond three times the virial radius, and contain objects on extreme orbits, with some approaching the nominal escape speed from the system. On average the subhalos identified within the virial radius represent only about one-half of all associated subhalos, and many relatively central halos may have actually been ejected in the past from a more massive system. Since galaxies are assumed to form in dark matter halos, it is interesting to see if the results we obtain here can be understood in terms of galaxy formation in this population of subhalos.

According to the current theory of galaxy formation, satellite galaxies can experience various environmental effects that can quench their star formation and make them red (e.g., van den Bosch et al. 2008a and references therein). Because the galaxies in ejected subhalos have also been satellite galaxies, at least for some period of time, they are likely to have been subjected to similar environmental effects, and thus to have experienced some quenching of their star formation rates. It is thus likely that the association of red dwarf galaxies with massive halos presented here is produced by the association of ejected subhalos with their (former) hosts.

As shown in Table 1, about 30% of the dwarf galaxies are satellite galaxies. According to the results obtained by Ludlow et al. (2009), there should thus also be a significant fraction of dwarf galaxies that are physically associated with nearby more massive halos out to about three times the virial radius. If these associated galaxies (now outside their hosts) have properties similar to the satellite galaxies, we would expect an enhanced fraction of red dwarf galaxies that are distributed outside massive halos. This is qualitatively consistent with our findings presented above and also as shown in Table 1. However, since the observational data are obtained in redshift space and based on galaxy groups that may contain interlopers and may be incomplete, a detailed comparison between the data and the models requires the construction of mock catalogs that make use of the subhalo population and contain all the observational selection effects.

The fact that central dwarf galaxies have concentrations that are independent of their distances to the nearest massive halos indicates that the process that causes them to become red does not have a significant impact on their structure. This is similar to the results obtained by van den Bosch et al. (2008b) who found that the transformation mechanisms operating on satellites affect color more than structure (see also Kauffmann et al. 2004; Blanton et al. 2005a; Ball et al. 2008; Weinmann et al. 2009). Once again, this similarity between red dwarfs that are satellites and those that are centrals suggests that both populations may have experienced similar kinds of environmental effects.

However, as shown in Table 1, in the S1 + S2 + S3 sample less than 42% of the red dwarf central galaxies have r_p/R₁₈₀ ⩽ 3. In other words, more than 58% of the red dwarf central galaxies are not close enough to a larger halo so that they could have been preprocessed there, becoming red and then subsequently being ejected. The origin of this population of central red dwarf galaxies, which is almost 10% of the combined S1 + S2 + S3 dwarf sample, still remains a mystery within the standard paradigm of galaxy formation. Furthermore, if we had used stellar mass instead of r-band magnitude to define our dwarf sample, the percentage of galaxies in this population would likely increase. This population of isolated red dwarfs is not merely a dust reddened star-forming population seen near edge-on because their axis ratios are consistent with a randomly oriented population. Although as Croton & Farrar (2008) probed the origin of the red dwarfs in voids using the semianalytical models, they only found ∼0.4% of the dwarfs in the total population are red centrals in voids. While here we find that ∼10% dwarfs are red centrals without close neighbors with r_p/R₁₈₀ ⩽ 3.

An outstanding problem for all galaxy formation models concerns the low-mass slope of the galaxy mass function. CDM models in general predict too many low-mass dark matter halos compared to the number of low-mass galaxies. The mass function of dark matter halos, n(M), scales with halo mass roughly as n(M) ∝ M⁻² at the low-mass end. This is in strong contrast with the observed luminosity function of galaxies, Φ(L), which has a rather shallow shape at the faint end, with Φ(L) ∝ L⁻¹. To reconcile this difference, one usually invokes some form of feedback within these low-mass halos. If the feedback mechanism were to prevent gas from entering these halos at late times, such galaxies would appear red. For example, the preheating mechanism of Mo et al. (2005), where gas is preheated by gas shocks within the forming large-scale structures in which the low-mass dark matter halos themselves are forming, has this feature. Therefore, this population of isolated, red, central dwarf galaxies could represent the tail of the process that prevents the vast majority of low-mass dark matter halos from forming galaxies and their further study could shed new light on the mechanism responsible.

We thank the referee Darren Croton for helpful comments that improved the presentation of this paper. Y.W. acknowledges the support of China Postdoctoral Science Foundation. This work is supported by the One Hundred Talents project, Shanghai Pujiang Program (No. 07pj14102), 973 Program (No. 2007CB815402), the CAS Knowledge Innovation Program (grant KJCX2-YW-T05) and grants from NSFC (10533030, 10673023, 10821302). H.J.M. likes to acknowledge the support of NSF AST-0607535, NASA AISR-126270, and NSF IIS-0611948. N.S.K. and D.H.M. like to acknowledge the support of NASA LTSA NAG5-13102.