The Dearth of Differences between Central and Satellite Galaxies. III. Environmental Dependencies of Mass–Size and Mass–Structure Relations

The observational data, such as galaxy and group catalogs, used in this work are the same as in Paper I. Here, we only briefly describe the sample selection and parameter measurements. The reader is referred to Paper I for details.

The galaxy sample is initially selected from the NYU-VAGC (Blanton et al. 2005) of the SDSS DR7 (Abazajian et al. 2009). Galaxies are selected to have: (1) redshift in the range of 0.01 < z < 0.2, (2) spectroscopic completeness C > 0.7, and (3) magnitude limit r = 17.72 mag. The first two criteria ensure that the sample galaxies are the same as those in the group catalog constructed by Yang et al. (2007), which provides the basic environmental information, such as central-satellite classification and halo mass (M_h).

The stellar masses used here, taken from the group catalog of Yang et al. (2007), are computed using the empirical relation between the stellar mass-to-light ratio and the g−r color as given in Bell et al. (2003), adopting a Kroupa (2001) initial mass function. The overall uncertainty in this stellar mass estimate is about 0.15 dex (Bell et al. 2003). The magnitude and color of galaxies are taken from the NYU-VAGC (Blanton et al. 2005), which is based on SDSS DR4 (Adelman-McCarthy et al. 2006), but includes a set of significant improvements over the original pipelines. The systematic calibration error in the photometry is about 1–2% across the sky.

In Yang et al. (2007), the most massive galaxy in a group/cluster is defined as the central galaxy, and the others are satellites. The halo masses of these galaxy groups/clusters are estimated by abundance matching galaxy groups, rank-ordered by the total stellar mass of all member galaxies with r-band absolute magnitude brighter than −19.5 mag, to dark matter halos rank-ordered by halo mass. Dark matter halos are defined to have a mean overdensity of 180, and the halo mass is defined to be the mass of dark matter enclosed by the radius within which the mean overdensity is 180. The typical uncertainty of halo mass is 0.25 dex (Yang et al. 2007). The center of a group is defined as the luminosity-weighted center of member galaxies. Thus, central galaxies are not always located at the centers of the groups. For each galaxy, we defined a scaled halo-centric radius (R_p/r₁₈₀), which is the projected distance from the galaxy to the host group center scaled by the virial radius of the host halo (Yang et al. 2007).

In this paper, we investigate two structural properties, the size and morphology, of SDSS galaxies. We adopt the SDSS r-band half-light radius (${R}_{{\rm{e}}}$) as the size of a galaxy, and the r-band B/T to represent the morphology. The two parameters are taken from the UPenn photometric catalog¹⁰ (Meert et al. 2015), which contains ∼680,000 galaxies from the SDSS DR7 spectroscopic sample. We adopt the measurement with a Sérsic bulge + exponential disk model. The Sérsic bulge model is more flexible than the de Vacouleurs bulge model, and is likely to be a better choice for both high- and low-mass galaxies.

Our final sample contains 524,852 galaxies, of which 24% are satellite galaxies. Since this is a flux limited sample, we assign each galaxy a weight $w={({V}_{\max }C)}^{-1}$ to correct for the selection effect, where V_max is the co-moving volume between the minimum redshift and the maximum redshift to which the galaxy can be observed in the flux-limited survey (Blanton & Roweis 2007), and C is the spectroscopic completeness.

As already mentioned in Section 1, in this paper we follow the methodology of Paper II, and compare the observational results inferred from the SDSS data with predictions from the latest version of the Munich semianalytical model, L-GALAXIES¹¹ (Henriques et al. 2015, 2017), and the state-of-the-art hydrodynamic simulation, EAGLE¹² (Crain et al. 2015; Furlong et al. 2015; Schaye et al. 2015; McAlpine et al. 2016). This allows us to fairly and directly interpret the observational results in the context of galaxy formation models. In particular, it tests to what extent the environmental dependences of the size and structural evolution of galaxies in EAGLE and L-GALAXIES are compatible with observations.

Here, we briefly describe the two models; the details of the two models and how they are used to construct the mock catalogs used here can be found in Paper II. Semianalytic models are phenomenological models that take advantage of empirically motivated prescriptions to describe baryonic processes, such as gas accretion, cooling and heating, star formation, stellar and AGN feedback, and tidal/ram pressure stripping. As the latest version of the Munich model, L-GALAXIES is built upon the Millennium Simulation (Springel et al. 2005) and employs a Markov chain Monte Carlo method to explore the high-dimensional parameter space to match the observed galaxy stellar mass function and quenched fraction as a function of stellar mass from redshift of 0 to 3. In L-GALAXIES, the disk size of a galaxy is determined by the angular momentum of its infalling cold gas, while three different mechanisms are invoked to grow a bulge: major mergers, minor mergers, and disk instability. In particular, during a major merger, the disks of the progenitors are assumed to be completely destroyed and a spheroidal galaxy forms. During a minor merger, the stars from the least massive progenitor are added to the bulge component, while the stellar disk of the more massive progenitor remains unchanged. Note that, in L-GALAXIES, mergers among satellite galaxies are rare; the majority of all mergers are between a central and one of its satellites (Guo et al. 2011; Henriques et al. 2015). Tidal and ram pressure stripping are assumed to affect only the hot gas surrounding satellite galaxies. Hence, these "satellite-specific" processes do not directly affect the stellar distributions of satellite galaxies, but they do suppress their subsequent growth—and thus their size and structural evolution—by quenching star formation. Because of this specific treatment of processes that only operate on galaxies identified by the code as satellites, the structural properties of centrals and satellites in L-GALAXIES are expected to be shaped by different processes.

The EAGLE simulation adopts advanced smoothed particle hydrodynamics and subgrid models for a series of baryonic processes, such as gas cooling, metal enrichment, black hole growth, and stellar and AGN feedback. Free parameters in the feedback models are tuned by matching the galaxy stellar mass function and the stellar mass–black hole mass relation at z ∼ 0 (Crain et al. 2015; Furlong et al. 2015). The luminosities and stellar masses of EAGLE galaxies are obtained from their star formation histories by assuming the Chabrier (2003) initial mass function and the Bruzual & Charlot (2003) stellar population model. In contrast to L-GALAXIES, EAGLE adopts the same subgrid prescriptions for both centrals and satellite, and so the differences between these two populations, if any, must be due to their different environments. For instance, environmental effects such as tidal or ram pressure stripping are treated self-consistently by the gravity and hydrodynamics solvers, without explicitly taking into consideration whether the galaxy is a central or a satellite. Following Paper II, we use the Ref-L100N1504 simulation, which has a box size of 100 Mpc sampled with 2 × 1504³ particles. The simulation contains more than 11,000 dark matter halos with masses above 10¹¹ M${}_{\odot }$, and nearly 10,000 galaxies with masses comparable to or larger than that of the Milky Way.

In both L-GALAXIES and EAGLE, halo mass is defined as the mass within the radius corresponding to an overdensity of 200. This halo mass is only slightly different from that defined in Yang et al. (2007) for halos with an NFW profile (Navarro–Frenk–White; Navarro et al. 1996). The stellar masses (recommended) in EAGLE are measured within a typical aperture of 30 kpc, to avoid the contamination of intercluster/intragroup light. The mass loss due to this effect is negligible for low-mass galaxies. However, for more massive galaxies the aperture reduces the stellar masses somewhat by cutting out intracluster light. At a stellar mass of 10¹¹ ${M}_{\odot }$, the mass loss due to aperture effect is only 0.1 dex. This is only a minor effect for the stellar mass range we are considering, and similar for both centrals and satellites.

The structural parameters for the model galaxies are also taken from the publicly released data. Unfortunately, the publicly available galaxy catalogs L-GALAXIES and EAGLE do not contain both size and bulge-to-total ratio. EAGLE only provides the half-mass radii, while L-GALAXIES only bulge-to-total mass ratios. There is an improved model of the bulge growth anchored on the L-GALAXIES model (Irodotou et al. 2019), which contains the size measurements of L-GALAXIES, while the data is not accessible to the public. We should also keep in mind that the parameters for model galaxies are measured in a very different way from those in the observational data. In principle, one may construct the r-band images of model galaxies to fully mimic the observation, and measure the structural parameters of model galaxies using the same methods as in the observation. However, this is clearly beyond the scope of the present paper. Thus, we can only present a qualitative comparison between the model predictions and observational data.

To facilitate the model–data comparison, we construct mock galaxy catalogs for both L-GALAXIES and EAGLE that properly account for various observational effects; for details, see Lim et al. (2017) and Paper II. We use the snapshots at z = 0.1 (the median redshift of SDSS galaxies used in this paper) to construct our mock catalogs. L-GALAXIES uses a simulation box of 480.3h⁻¹ Mpc on a side, while the box size of the EAGLE simulation is only 100 Mpc. Both are significantly smaller than the volume probed by the SDSS, and we therefore stack duplicates of the original simulation boxes side by side to construct a sufficiently large volume. We then choose a location for the observer, and calculate the redshift and the apparent magnitude for each model galaxy based on its luminosity, distance, and velocity with respect to the observer. Finally, we select a flux-limited sample of galaxies from a light cone covering the redshift range 0.01 < z < 0.2, which is similar to that covered by our SDSS group catalog. All comparisons are based on these catalogs unless specified otherwise. Finally, we note that both L-GALAXIES and EAGLE adopt the Planck cosmology (Planck Collaboration et al. 2014b, 2014a), which differs somewhat from the WMAP3 cosmology (Spergel et al. 2007) adopted in the construction of our SDSS group catalog. We note that the different cosmologies are not a concern as long as the results are presented in a self-consistent way, i.e., we use the WMAP3 cosmology for the analysis of the SDSS group catalog, while use Planck cosmology for the analysis of the EAGLE and L-GALAXIES group catalogs.

The left panel of Figure 1 shows the average ${R}_{{\rm{e}}}$ (in logarithmic space) as a function of stellar mass for centrals (red squares), satellites (blue circles) and all galaxies (black line). Here, and throughout the remainder of this paper, the errors are estimated using 1000 bootstrap samples. Note that the M_*–${R}_{{\rm{e}}}$ relations for centrals and satellites are quite similar at ${\mathrm{log}}_{10}({M}_{* }/{h}^{-2}$ ${M}_{\odot }$) > 10.5, while significant differences are evident at lower mass, with centrals being systematically larger, by ∼0.06 dex, than satellites of the same stellar mass.

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These results may reflect the fact that centrals and satellites have experienced different environmental processes. After all, satellites usually reside in more massive halos than centrals of the same stellar mass, and halo mass is known to be one of the primary environmental parameters regulating galaxy formation (in particular, galaxy quenching). To test the role of halo mass in controlling the sizes of galaxies of a given stellar mass, we separate galaxies into six logarithmic bins in halo mass, each with a width of 0.6 dex, and covering the range from 10^11.4h⁻¹ M${}_{\odot }$ to 10¹⁵h⁻¹ M${}_{\odot }$. For each bin, we again compute the M_*–${R}_{{\rm{e}}}$ relations for centrals and satellites separately, the results of which are plotted in the top panels in Figure 2. As one can see, the differences between the two populations shown above are significantly reduced or even eliminated: centrals and satellites follow basically the same M_*–${R}_{{\rm{e}}}$ relation at a given halo mass. This indicates that the environmental processes that affect the sizes of centrals and satellites are similar, strengthening the proposition that the difference in the M_*–${R}_{{\rm{e}}}$ relation for the two populations shown in the left panel of Figure 1 is primarily due to differences in the distributions of halo mass for centrals and satellites.

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Figure 3 shows the size distributions of centrals and satellites in different halo mass bins. To make a fair comparison, the two populations are controlled to have the same distribution in stellar mass. As one can see, the size distributions of centrals and satellites are quite similar. For the two intermediate halo mass bins (${10}^{12.0}\,{M}_{\odot }{h}^{-1}\lt {M}_{{\rm{h}}}\lt {10}^{13.2}\,{M}_{\odot }{h}^{-1}$), the two distributions are almost indistinguishable, as indicated by the probabilities of the Kolmogorov–Smirnov test. For the other three high halo mass bins, there are small differences between the two populations, which may due to the fact that the halo mass distributions of the two populations are not exactly the same. The results for the most massive bin are noisy, as there are only about 100 galaxies in each population.

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Because galaxy size does not seem to depend on whether a galaxy is classified as a central or a satellite, we calculate the average M_*–${R}_{{\rm{e}}}$ relations for galaxies in the six halo mass bins without distinguishing centrals and satellites. The result is shown in the left panel of Figure 4. As can be seen, galaxy size decreases with increasing halo mass at the low stellar mass end. This trend becomes less obvious with increasing stellar mass, and almost disappears at M_* > 10^10.7h⁻² M${}_{\odot }$, as indicated by the vertical dashed line. The fact that galaxies of low stellar mass are smaller in more massive halos may be due to a number of processes. For example, more massive halos may be more effective in quenching star formation, particularly for low-mass galaxies (e.g., Peng et al. 2010; Woo et al. 2015; Wang et al. 2018d), probably due to stronger ram pressure stripping and/or shock-excited heating. Consequently, low-mass galaxies in massive halos may stop growing their sizes due to quenching, and the fading of the disk may cause the half-light radius to shrink (Lilly & Carollo 2016). Tidal stripping and galaxy harassment can strip stars from the outer stellar disk of low-mass galaxies, making the galaxy more compact and smaller. These two processes primarily depend on the mass density of the host halo, which is similar for halos of different mass.

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In this subsection, we focus on the bulge-to-total light ratio to examine how environment impacts bulge formation in centrals and satellites. As in Section 3.1, we first show the mean B/T as a function of stellar mass in the right-hand panel of Figure 1, without separating galaxies into different halo mass bins. As one can see, satellites at the low-mass end tend to harbor more pronounced bulges than centrals. As with the sizes, the differences become weaker with increasing stellar mass and vanish for ${\mathrm{log}}_{10}({M}_{* }/{h}^{-2}$ ${M}_{\odot }$) > 10.5.

To reduce effects caused by the difference in host halo masses between centrals and satellites, we show B/T as a function of stellar mass for centrals and satellites separated in six halo mass bins in the bottom panels of Figure 2. We see again that the significant difference between the two populations shown in Figure 1 disappears when the halo mass is controlled. This indicates that both centrals and satellites have similar B/T when both the stellar mass and halo mass are controlled, indicating again that centrals are not special in comparison with satellites as far as their star formation (Paper I and Paper II) and structural properties are concerned.

Since there is no significant evidence to suggest that centrals and satellites have systematically different B/T, we calculate the average M_*–B/T relation for galaxies in the six halo mass bins without distinguishing centrals from satellites. This is shown in the right-hand panel of Figure 4. Clearly, galaxies appear to have higher B/T in more massive halos for stellar masses less than 10^10.7h⁻² M${}_{\odot }$, but the environmental dependence becomes very weak for galaxies above this stellar mass. This is broadly consistent with previous studies (e.g., Bamford et al. 2009; Bluck et al. 2019; Liu et al. 2019).

The B/T distributions of centrals and satellites, controlled to have the same stellar mass distribution, are shown in the bottom group panels of Figure 3. The subsamples here are exactly the same as those shown in the upper panels. As one can see, centrals and satellites have a similar distribution in B/T for almost all the halo mass bins. Except for the two left panels (${10}^{11.4}\,{M}_{\odot }{h}^{-1}\,\lt {M}_{{\rm{h}}}\lt {10}^{12.0}\,{M}_{\odot }{h}^{-1}$ and ${10}^{13.2}\,{M}_{\odot }{h}^{-1}\lt {M}_{{\rm{h}}}\lt {10}^{13.8}\,{M}_{\odot }{h}^{-1}$ ), the Kolmogorov–Smirnov test, in each of other panels, shows that the difference between the two populations is statistically insignificant. This strengthens the conclusion that centrals and satellites have similar B/T when both stellar mass and halo mass are controlled.

Combined with the result of Section 3.1, we conclude that centrals and satellites have similar sizes and B/T when both stellar mass and halo mass are controlled. We have also examined structural properties other than B/T, including the Sérsic index, the concentration (defined as the ratio of the radii enclosing 90% and 50% of the galaxy light), and the stellar surface densities inside ${R}_{{\rm{e}}}$. All of these show the same trends as B/T. As an example, we present the results for the stellar surface density within ${R}_{{\rm{e}}}$ in the Appendix. Because of this, we therefore only present results based on B/T in the following.

A potential concern is that the results may be contaminated by groups that are not yet relaxed and within which satellites may not have experienced the environmental effects of their current host halos (e.g., Carollo et al. 2013b). A significant fraction of groups (∼6.0% of groups, corresponding to ∼15.5% of galaxies) is found to have centrals that have projected halo-centric distances greater than 0.1r₁₈₀. These groups are more likely to be unrelaxed, and including them may reduce the difference between centrals and satellites. We have re-examined our results by excluding these groups, and found no significant changes in our conclusion.

Various environmental processes discussed in the literature, such as tidal stripping, ram pressure stripping, strangulation, and harassment, are all expected to depend on the location of a galaxy within its host halo. Indeed, recent analyses have revealed that the quenched population of satellite galaxies becomes more dominant toward the halo center (e.g., Weinmann et al. 2006; van den Bosch et al. 2008a; Wetzel et al. 2012; Kauffmann et al. 2013; Woo et al. 2013). This may indicate that the processes associated with quenching become stronger toward the halo center and/or it could reflect the fact that satellites that reside closer toward the center were accreted earlier (e.g., Gao et al. 2004; Contini et al. 2012; van den Bosch et al. 2016) and have therefore been exposed to quenching-inducing processes for a longer duration. Here, we examine how the structural properties depend on the halo-centric distance (R_p). We do not discriminate between centrals and satellites, since the two populations intrinsically have no differences in size and structural properties (see Sections 3.1 and 3.2).

Figure 5 shows the M_*–${R}_{{\rm{e}}}$ and M_*–B/T relations for galaxies in three halo-centric distance bins: R_p/r₁₈₀ < 0.3 (red lines), 0.3 < R_p/r₁₈₀ < 0.6 (green lines), and 0.6 < R_p/r₁₈₀ < 0.9 (blue lines), where r₁₈₀ is the halo virial radius (see Yang et al. 2007). Both the size and B/T depend on R_p/r₁₈₀ over the whole halo mass range.

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There are two interesting phenomena. The first one is that the R_p/r₁₈₀ dependences seem to be driven mainly by the innermost bin. This suggests the existence of a critical halo-centric radius, beyond which the mass–size (or mass–B/T) relation does not depend on the halo-centric radius. We search for the critical halo-centric radius (if any) by dividing galaxies into more halo-centric bins. We find that the mass–size (and the mass–B/T) relation does not depend on the halo-centric radius at R_p/r₁₈₀ above ∼0.5, at least for the three highest halo mass bins. The existence of a critical halo-centric radius is not surprising, since most of the galaxies beyond this radius may have not fallen into the inner region of the halos and therefore have not yet been affected by the environmental effects.

The second phenomenon is that the dependence of the size on R_p/r₁₈₀ is different for low- and high-mass galaxies. Except for the lowest halo mass bin, low-mass galaxies appear to have larger size in the outer region of halos than those of the same mass in the inner region. This trend is reversed at the high M_* end. Similar results can be seen in B/T: at low M_*, galaxies at smaller projected halo-centric distances tend to have larger B/T. At the high M_* end, though, this dependence of B/T on R_p vanishes. This motivates us to define a "transitional" stellar mass, ${M}_{* ,{\rm{t}}}$, above which the dependence of the size and B/T on R_p is absent or becomes opposite to that at the low stellar mass end. As is evident from Figure 5, ${M}_{* ,{\rm{t}}}$ increases with halo mass, consistent with the fact that some of the environmental processes are expected to be stronger in more massive halos. Furthermore, comparing the upper and lower sets of panels shows that the size and B/T have similar transitional masses for a given halo mass, indicating that the changes in size and B/T may be driven by similar processes.

In Paper I, we found that the quenched fraction of low-mass galaxies strongly depends on R_p, and the dependence becomes weak at the high-mass end (see also Wang et al. 2018d). The transitional stellar masses for star formation quenching, taken from Paper I, are shown by the vertical dotted line in each panel of Figure 5. It is interesting to see that the ${M}_{* ,{\rm{t}}}$ for star formation quenching is very similar to those for the size and B/T. This provides strong support that the star formation quenching, size, and structural evolution of galaxies may be connected. We have also examined the dependence of the inner stellar mass surface density within ${R}_{{\rm{e}}}$ (Σ_Re) on the scaled halo-centric radius at given stellar mass (see Appendix). The behavior of Σ_Re is quite similar to that of B/T (Figure 5), with a similar transitional stellar mass for each halo mass bins.

The pentagrams in Figure 6 show the transitional stellar mass as a function of halo mass, with the error bars indicating the halo mass bins. Grey and blue dots indicate centrals and satellites from a sample of 20,000 galaxies randomly selected from the SDSS group catalog, while the red, dashed curve indicates the central mass–halo mass relation obtained by Yang et al. (2009), which is given by

$$\begin{eqnarray}&&{M}_{* ,{\rm{t}}}={M}_{0}\displaystyle \frac{{({M}_{{\rm{h}}}/{M}_{1})}^{\alpha +\beta }}{{(1+{M}_{{\rm{h}}}/{M}_{1})}^{\beta }},\end{eqnarray} \tag{ 1 }$$

with ${\mathrm{log}}_{10}{M}_{0}=10.31$, ${\mathrm{log}}_{10}{M}_{1}=11.04$, α = 0.3146, and β = 4.5427. For comparison, the red, solid curve is the same relation but shifted downward by 0.7 dex (i.e., with ${M}_{0}\to 0.2{M}_{0}$), which is a reasonably good fit to the ${M}_{* ,{\rm{t}}}\mbox{--}{M}_{{\rm{h}}}$ relation, indicating that ${M}_{* ,{\rm{t}}}$ is approximately one-fifth of the stellar mass of the average central galaxy at the corresponding halo mass. Interestingly, a similar transitional mass was identified by Wang et al. (2018d), who found that the dependence of quenched fraction on halo-centric distance becomes insignificant for galaxies with stellar masses greater than one-fifth of the masses of their centrals. From this figure, we can clearly see that the transition behavior is totally determined by satellite galaxies, as central galaxies are all above and far away from the transition curve.

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To conclude, our results indicate that environmental effects on the size and structural properties of galaxies, as well as on star formation quenching, depend on where the galaxies are located on the ${M}_{* }\mbox{--}{M}_{{\rm{h}}}$ diagram: galaxies located above the ${M}_{* ,{\rm{t}}}\mbox{--}{M}_{{\rm{h}}}$ relation (the solid red line in Figure 6), have sizes, quenched fractions and structural properties that are all independent of their location with their host halos. For galaxies below the ${M}_{* ,{\rm{t}}}\mbox{--}{M}_{{\rm{h}}}$ relation, however, significant dependences on R_p emerge. We will discuss the implications of these findings in Section 5.

The parameter B/T is highly model-dependent. In our analysis above, we have adopted the measurements based on the Sérsic bulge + exponential disk model of the r-band image. In their original paper, Meert et al. (2015) have performed the two-dimensional modeling with other models, such as the de Vacouleurs bulge + exponential disk model, and found that the decomposition results can differ from that based on the Sérsic bulge + exponential disk model. In Paper I, we adopted the B/T measurements based on the de Vacouleurs bulge + exponential disk model from Simard et al. (2011). In order to quantify whether or not our results are sensitive to the different measurements, we have conducted tests using the B/T measurements of both Simard et al. (2011) and Meert et al. (2015) based on the de Vacouleurs bulge + exponential disk model. Our tests clearly showed that our results are insensitive to the use of different decomposition measurements.

Another important issue is the seeing effect. Even though the size and B/T are measured from the point-spread function (PSF) deconvolved light profiles, the seeing effect may still influence the accuracy of the measurements of both the size and B/T, especially for small galaxies. To check this effect quantitatively, we have made the same plots shown in Sections 3.1–3.3 using a subsample limited to z < 0.1, where the half-width at half-maximum of the SDSS PSF (07) corresponds to 1.3 kpc. We found little change in the results. This is expected, as the seeing effect affects centrals and satellites in a similar way, and for most of the halo mass bins, the comparisons between centrals and satellites are for relatively massive galaxies (M_* > 10^10.0h⁻² M${}_{\odot }$).

Figure 7 shows the half-mass–radius of centrals and satellites from EAGLE as a function of stellar mass in six halo mass bins. Note that we use the same symbol, ${R}_{{\rm{e}}}$, to denote the half-mass–radius as used previously to indicate the half-light radius of SDSS galaxies. Note that, because of the relatively small volume of the EAGLE simulation, there are too few massive halos to be able to plot any meaningful results for the most massive halo mass bin ($14.4\lt {\mathrm{log}}_{10}{M}_{{\rm{h}}}/{h}^{-1}\,{{\rm{M}}}_{\odot }\lt 15.0$). We caution that this also implies that the error bars for the massive EAGLE galaxies are severely underestimated (see Section 2.2 and Paper II).

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Overall, Figure 7 reveals good qualitative agreement with the SDSS results. In particular, for galaxies residing in halos with a mass less than 10^13.2h⁻¹ M_⊙, the M_*–${R}_{{\rm{e}}}$ relation of EAGLE galaxies is nearly parallel to that of SDSS galaxies, albeit with an overall offset of ∼0.3 dex. The overall offset may be due to different measurement methods and the relatively bright surface brightness limit of the SDSS survey with respect to the EAGLE simulation, as demonstrated in Furlong et al. (2017). In massive halos, the offset becomes larger at both the low and high stellar mass ends. The larger offset at the low-mass ends may be due to the inadequate resolution, which blurs the boundaries of the low-mass galaxies, while the larger offset at the high-mass ends may be due to the inclusive of intercluster stellar components in the identification of EAGLE galaxies in groups or clusters. More importantly, centrals and satellites in EAGLE appear to have similar M_*–${R}_{{\rm{e}}}$ relations over the whole halo mass range, with one noteworthy exception: the low-mass centrals in halos with $13.2\lt {\mathrm{log}}_{10}{M}_{{\rm{h}}}/{h}^{-1}\,{M}_{\odot }\lt 13.8$, which seem to have larger sizes than their corresponding satellites. However, there are only seven unique (i.e., nonrepeating) centrals in these two data points combined, indicating that the true statistical errors are much larger than those estimated from the mock data set, which uses many repetitions of the same simulation box. Hence, this apparent difference between centrals and satellites is not significant.

We have used EAGLE to examine the size distributions of centrals and satellites controlled in stellar mass for a set of halo mass bins. We found that the two populations have a similar distribution for halo masses less than 10^12.6h⁻¹ M_⊙, while the two size distributions have a small offset for higher halo masses. This is consistent with the result shown in Figure 7.

Figure 8 compares the B/T as a function of stellar mass for centrals (red squares) and satellites (blue circles) in L-GALAXIES to the SDSS results (dotted curves, taken from Figure 2). Clearly, the agreement is extremely poor; although L-GALAXIES roughly reproduces the increasing trend of B/T with stellar mass, at least for galaxies with M_* > 10^9.7h⁻² ${M}_{\odot }$, the trend predicted by L-GALAXIES for less massive galaxies is opposite to that in the data. In addition, typically there are large offsets between the average B/T predicted by L-GALAXIES and that observed for SDSS galaxies. Most importantly, in halos with mass greater than 10^12.0h⁻¹ ${M}_{\odot }$, centrals and satellites in L-GALAXIES typically have very different bulge-to-total ratios at fixed stellar and halo mass, inconsistent with the observational findings of Section 3.2.

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We emphasize that, for SDSS galaxies, the size and B/T measurements are based on the light distribution, while for the model galaxies, they are based on the stellar mass distributions. Although this could explain some of the offsets between the model and data, we are primarily concerned with the differences between centrals and satellites, and these differences should largely be unaffected by such offsets. We therefore conclude that, as a whole, EAGLE successfully reproduces the similar mass–size relations for centrals and satellites, while L-GALAXIES fails to reproduce the similar B/T of the two populations.

We now focus on the dependence of the M_*–${R}_{{\rm{e}}}$ and M_*–B/T relations on halo-centric radius, and investigate whether current galaxy formation models can reproduce the transitional stellar mass found in Section 3.3. Figure 9 shows the M_*–${R}_{{\rm{e}}}$ relation of EAGLE galaxies in three bins of normalized halo-centric distance, ${R}_{{\rm{p}}}/{r}_{180}$. For comparison, we also show the results for SDSS galaxies as dotted lines. Similar to Figure 7, the sizes of EAGLE galaxies are overall ∼0.3 dex higher than those of SDSS galaxies, almost over the entire stellar and halo mass ranges. Interestingly, the dependence on R_p/r₁₈₀ for EAGLE galaxies is very similar to that for SDSS galaxies: galaxies at smaller halo-centric distances are smaller at the low-mass end and larger at the high-mass end. Most remarkably, the corresponding transitional stellar masses for EAGLE galaxies are almost identical to those for SDSS galaxies over the entire halo mass range probed.

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For a given halo mass bin, the difference in the mass–size relation between the two high halo-centric radius bins is insignificant, consistent with the result of SDSS galaxies shown in Figure 5. As in Section 3.3, we have also tried to detect a possible critical halo-centric radius in the EAGLE sample. Such a critical radius was found, at least for halo masses above 10^13.2 M${}_{\odot }$h⁻¹, and its value is about 0.5r₁₈₀, comparable to that observed for SDSS galaxies.

Figure 10 shows the B/T of L-GALAXIES galaxies as a function of stellar mass at different R_p/r₁₈₀. For comparison, the M_*–B/T relations for SDSS galaxies, taken from Figure 5, are also plotted. In addition to the discrepancy in the overall trend between L-GALAXIES and SDSS galaxies discussed in Section 4.1, the dependence of the M_*–B/T relation on R_p/r₁₈₀ for L-GALAXIES is also different from that of SDSS galaxies. The B/T of L-GALAXIES galaxies is almost independent of R_p/r₁₈₀ at low M_*, and decreases with increasing R_p/r₁₈₀ at high M_*, which is opposite to the trend seen in the observational results.

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The SDSS groups are identified using the halo-based group finder of Yang et al. (2007), which carries its own uncertainties in central-satellite classification, group member identification and the assignment of halo mass (e.g., Campbell et al. 2015). In Paper II, we therefore carried out a detailed investigation of how errors due to the group finder impact a comparison of centrals and satellites at a given stellar and/or halo mass. Here, we examine potential errors in the M_*–${R}_{{\rm{e}}}$ (for EAGLE) and M_*–B/T (for L-GALAXIES) relations of centrals and satellites caused by imperfections of the group finder. To this end, we apply the group finder to the two mock galaxy catalogs and obtain two mock group catalogs, to which we refer as L-GALAXIES+GF and EAGLE+GF, respectively. For the two catalogs, halo masses are assigned to individual groups on the stellar mass of galaxies, consistent with the halo mass estimate used for SDSS groups.

The results are shown in Figures 11–14. After applying the group finder, the results for EAGLE change only very slightly. As before, centrals and satellites reveal similar behavior (see Figure 7), and the transitional stellar masses are apparent and very similar to those in Figure 9. Furthermore, the overall trend in the mass–size relation is broadly consistent with that for SDSS galaxies. In contrast, in the case of L-GALAXIES, the application of the group finder drastically reduces the differences between centrals and satellites seen in Figure 8. Moreover, the dependence of B/T on halo-centric distance apparent in Figure 10 also disappears, but the upturn in the mass–B/T relation at the low-mass end remains (Figure 10).

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Similar results were also obtained in Paper II, where we demonstrated that the similarity in the quenched fraction between centrals and satellites in EAGLE remains after applying the group finder, while the significant differences in the quenched fraction between centrals and satellites in L-GALAXIES were largely reduced. This indicates that the performance of the group finder depends on the galaxy formation models. Indeed, further investigation shows that the application of the group finder to EAGLE yields more accurate halo mass assignments and better central-satellite separations than in the case of L-GALAXIES (see Paper II). Specifically, for EAGLE, the group finder is able to reproduce the true halo mass with an overall offset of 0.03 dex and an overall scatter of 0.2 dex. For L-GALAXIES, on the other hand, the group finder reproduces the true halo mass with a larger offset, 0.09 dex, and a larger scatter, 0.4 dex (see Figure 8 in Paper II).

This leaves us with the somewhat uncomfortable situation that there are two possible explanations for the observed similarity between centrals and satellites in the SDSS: either there indeed is no intrinsic difference (at fixed stellar and halo mass), or the group finder has artificially erased an existing difference. Because it is unknown a priori whether L-GALAXIES or EAGLE is a better representation of the real universe, it is unclear which interpretation is more appropriate for SDSS galaxies. In Paper II, we addressed this conundrum by proposing an independent test to check the performance of the group finder. The test calculates the clustering of galaxy groups identified by the group finder as a function of their assigned halo mass and examines whether or not the resulting halo bias is consistent with theoretical predictions. If the group finder performs poorly, the assigned halo masses and member identifications should have large errors, causing the inferred "halo bias" to deviate from theoretical expectations. This test can be applied not only to galaxy formation models such as L-GALAXIES and EAGLE, but also to the actual SDSS data. Our examination in Paper II clearly shows that the resulting halo bias for EAGLE+GF agrees well with the prediction of Sheth et al. (2001), while the halo bias inferred from L-GALAXIES+GF differs strongly from these theoretical expectation. This is consistent with a direct examination of halo mass assignments: for L-GALAXIES, the halo masses assigned by the group finder carry large errors; in the case of EAGLE, the errors are significantly smaller. Applying the "halo-bias" test to the SDSS group catalog, we obtain results that are again in good agreement with the theoretical prediction (Wang et al. 2008). These results suggest that EAGLE is likely to be more reminiscent of the real universe and that the group finder is expected to work well for SDSS galaxies. Although not a watertight argument, this does suggest that centrals and satellites really are very similar at fixed stellar and halo mass.

When applying the group finder to EAGLE and L-GALAXIES, we correct the spectroscopic incompleteness due to the magnitude limit by using the observed luminosity function to account for the contribution of the missing galaxies. However, we do not include the incompleteness due to the fiber collisions¹³ and the photometric uncertainties (or stellar mass uncertainties). Including these would undoubtedly worsen the group finder's performance. We have examined our results by using a halo mass assignment based on the r-band luminosity for EAGLE+GF, and found that the results shown in Figures 11 and 13 are not affected. This suggests that the uncertainties in converting r-band luminosity to stellar mass should not change our results. This is consistent with the finding in Yang et al. (2007) that halo masses based on the r-band luminosity and the stellar mass agree with each other (see their Figure 10).