The Three Hundred Project: The Influence of Environment on Simulated Galaxy Properties

We study the environment's effect on a galaxy's SFR, luminosity (in the r and g bands), and color (${M}_{g}-{M}_{r}$). How we define and calculate each of these properties is given below:

• 1.  
  SFR and sSFR: We define the SFR of a galaxy as the sum of the SFRs of all gas particles belonging to it. In the simulation, each gas particle has its own star formation rate which is determined by the star formation model within the simulation (Springel et al. 2005). At each step in the simulation, some fraction of the mass of a gas particle is converted into a new star particle according to its SFR. The sSFR(specific star formation rate) is defined as usual, $\mathrm{sSFR}=\mathrm{SFR}/{M}_{* }$.
• 2.  
  Luminosity and Magnitude: The luminosity in any defined spectral band is calculated by applying the stellar population synthesis code STARDUST (see Devriendt et al. 1999, and references therein for more details). This code computes the spectral energy distribution from the far-UV to the radio. The stellar contribution to the total flux is calculated assuming a Kennicutt initial mass function (Kennicutt 1998). Absolute magnitudes are readily calculated from the luminosity. Note dust obscuration is not taken into account. The Three Hundred Project gives the luminosities and magnitudes of galaxies in several bands. For this work we only use those derived from imposing Sloan Digital Sky Survey(SDSS) g band and r band filters.

Observationally, there are three methods commonly used to characterize a galaxy's environment:

• 1.  
  Count the number of neighboring galaxies within a projected ring around each galaxy (e.g., Wilman et al. 2010).
• 2.  
  Count the number of neighboring galaxies within a fixed volume around each galaxy (e.g., Kauffmann et al. 2004; Gallazzi et al. 2009; Grützbauch et al. 2011).
• 3.  
  Find the projected distance r to the $n\mathrm{th}$ nearest neighbor galaxy, with n in the range 3–10. Then, the projected surface density is calculated via ${{\rm{\Sigma }}}_{n}=n/\pi {r}^{2}$ (e.g., Dressler 1980; Hashimoto et al. 1998; Tanaka et al. 2004; Capak et al. 2007).

Further details and a comparison between various environmental estimators are given in Muldrew et al. (2012). As the above methods show, observationally the traditional parameter defining the environmental density is the local galaxy number density. However, as galaxies form a biased tracer of the density field (Mo & White 1996), we might expect to find a more intrinsic environmental dependence if we utilise the underlying dark-matter density field. From observations, this underlying matter density field could be reconstructed from combining the galaxy mass density with a bias factor b. Galaxy mass density comes from weighting each galaxy with their mass in the formula of galaxy number density. However, the bias can be complicated depending on the scale, redshift, and type of galaxies included (Kovač et al. 2010). On the other hand, we can directly measure the actual density (smoothed on some scale) within a simulation.

In this work, we define δ₁, the local overdensity of all matter (including stars, gas, and dark matter) compared with the mean density of the universe, within a sphere of radius $1{h}^{-1}\mathrm{Mpc}$ centered on the galaxy to quantify the local environment. $1{h}^{-1}\mathrm{Mpc}$ is a characteristic radius of dark-matter halos with mass around ${10}^{13}\mbox{--}{10}^{14}{h}^{-1}{M}_{\odot }$. It is also a commonly used distance in the second method above for defining the environmental galaxy number density. This is the region that is thought to be tightly connected with a galaxy (Kauffmann et al. 2004). We have also measured the environment using radii of $2{h}^{-1}\mathrm{Mpc}$ and $3{h}^{-1}\mathrm{Mpc}$ to form δ₂ and δ₃. Such distances are widely used as the smoothing scale when calculating structural features such as nodes, filaments, sheets, and voids (e.g., Hahn et al. 2007; Zhang et al. 2009). We found that δ₁ is linearly related to δ₂ and δ₃. The dependence of a galaxy's properties on δ₂ and δ₃ has almost the same shape as those found for δ₁. Thus, in what follows, we only show the results for δ₁.

To probe the relationship between the intrinsic matter overdensity, δ₁, and the more observationally motivated galaxy number, N_gal we present Figure 2 here. It shows the median overdensity of $1{h}^{-1}\mathrm{Mpc}$ spheres as a function of the galaxy number in the same region. For each galaxy, we both measure δ₁ and count the number of galaxies with mass above ${10}^{9.5}{M}_{\odot }/h$ contained in the same volume. As Figure 2 shows, once a handful of galaxies are present, the galaxy number density is linearly related to the matter overdensity. The curves of the three sample groups converge as the number of galaxies contained in the volume increases. Divergences appear when N_gal is less than 10. For a region with low galaxy number density, its corresponding overdensity tends to be higher if it belongs to the cluster sample compared with those which belong to the vicinity and field samples. Within our cluster galaxy sample essentially no low overdensity environment exists. This skews the distribution of galaxy number because although sometimes cluster galaxies live in relative isolation this environment is not underdense. Additionally, within this environment, as well as the matter density potentially being increased by the presence of a large dark-matter halo, gas is readily ram-pressure stripped. Thus at the edge of the cluster less gas is available to make stars and consequently fewer (and smaller) galaxies might be formed. Primarily though, out to twice R₂₀₀ matter overdensities much below 30 times the cosmic mean simply do not occur, and this distorts the distribution at low galaxy number.

Zoom In Zoom Out Reset image size

As Figure 2 illustrates, for any single galaxy there is quite a bit of scatter in the δ₁ to N_gal relationship. Generally though, these two quantities are clearly related and we can use the intrinsic quantity δ₁ interchangeably with the observational measure N_gal in what follows. In practice, any observational measure of N_gal would use a different mass limit for the included galaxies and so the correspondence would need to be recalculated for this value. In this case though it is δ₁ that is the fixed quantity, with N_gal changing.

Although in this work we are principally interested in the relationship between a galaxy's properties and the environment it lives in, we first check that a fundamental observed relation, in this case the star formation rate to stellar mass ($\mathrm{SFR}\mbox{--}{M}_{* }$) relation, is recovered. In Figure 3, we show the $\mathrm{SFR}\mbox{--}{M}_{* }$ relation at z = 0 for our three galaxy samples, with cluster galaxies on the top row, vicinity galaxies in the middle and field galaxies at the bottom. The left column of panels are for galaxies within the Gadget-X models, the right column for galaxies from the Gadget-MUSIC simulations. The shading indicates galaxy density as given by the color bar. We overplot best-fit lines for $\mathrm{SFR}({M}_{* })$ derived from Daddi et al. (2007), Elbaz et al. (2007) and Peng et al. (2010). As Figure 3 shows, the Gadget-X runs reproduce galaxies with a $\mathrm{SFR}-M\ast$ relation in good agreement with observations, within the stellar mass range of 10^9.5 to ${10}^{11}{h}^{-1}{M}_{\odot }$. For galaxies more massive than ${10}^{11}{h}^{-1}{M}_{\odot }$ in Gadget-X clusters, the majority are below the observational fits. The SFR in the high mass range is supposed to be suppressed by AGN feedback. A similar suppression may be present in the observations (e.g., see Figure 17 of Brinchmann et al. 2004).

Zoom In Zoom Out Reset image size

The Gadget-X field galaxies possibly show a slightly higher SFR than the fits but this may just reflect an apparent absence of galaxies with low star formation rates within this sample. Note that the main sequence lines from observations are based on samples which are mixtures of galaxies from all our samples, while cluster and vicinity galaxies dominate our total sample. In contrast to the reasonable looking Gadget-X galaxies, the Gadget-MUSIC galaxies have an obviously lower SFR at a given mass than the observed relations. The reason for this low SFR is somewhat counterintuitive in that within the Gadget-MUSIC simulations too many stars have been formed. However, for any given galaxy the measured SFR is always low, even at higher redshift. The contradiction is only apparent: for a matched object between the two simulations the SFR is higher in the Gadget-MUSIC simulation than in the Gadget-X simulation, so the matched galaxy acquires a larger stellar mass. This moves it to the right in Figure 3, giving it an apparently low SFR for a galaxy of this mass. Essentially, Figure 3 demonstrates that the Gadget-MUSIC simulations produce overly massive galaxies on all scales that are significantly more massive than those observed. Because of this for the remainder of this paper we concentrate our results on the more physically reasonable galaxies found within the Gadget-X simulations.

As a further check we show the evolution of the cosmic star formation rate density (SFRD) and median specific star formation rate in Figure 4. In the top panel, the star formation rate density in the field (green line) and cluster resimulations (blue line) span the observational fit of Behroozi et al. (2013) (black line). This is to be expected as the cluster re-simulation region by design contains a large cluster and many galaxies while the field region is devoid of any large clusters and contains fewer galaxies than average per unit volume. We have combined the cluster and vicinity galaxy samples into the same blue line as the spatial split between the two makes it complicated to determine the exact volume. To construct this volume at higher redshift we always choose a sphere of co-moving radius $10{h}^{-1}\mathrm{Mpc}$ centered on the middle of the re-simulation volume. In common with the observations, both the cluster and field SFRD curves rise to a peak at intermediate redshift and then the SFRD falls by around an order of magnitude by z = 0.

Zoom In Zoom Out Reset image size

The evolution of the median sSFR for each of our samples is given in the lower panel of Figure 4. At z = 0, galaxies in the field have the highest median sSFR, and cluster galaxies have the lowest median sSFR. The difference among galaxy classes is distinguishable but with large error bars. The difference becomes smaller at higher redshifts. The three median sSFR lines finally converge at about z ≥ 5. The simulated sSFR follows the observational results given by the black symbols at very low redshift, then keeps below the observational symbols from z ∼ 0.1 to z ∼ 2.5 before exceeding the fitting line at high redshifts. The line for cluster galaxies is not within the error bars of black observational data points, but this is reasonable because the observations take all classes of galaxies into account. In common with other hydrodynamical simulations e.g., EAGLE (Furlong et al. 2015), Illustris (Sparre et al. 2015) and semi-analytic models, (Henriques et al. 2015), we do not recover the double-power-law fit indicated by the fit of Behroozi et al. (2013) (see Figure 5 in Davidzon et al. 2017). This lack of any clear break in the median sSFR either indicates that the observational sample is as yet incomplete or that the models are too efficient at forming stars at early times (Asquith et al. 2018).

As we have also shown above in Figure 3, many previous works have indicated a tight relationship between SFR and M_* with M_* expected to be $\propto \mathrm{SFR}$, from both theoretical and observational standpoints (e.g., Davé 2008; Yates et al. 2012; Wang et al. 2013). We can therefore emphasize any deviation from this relationship by calculating the specific star formation rate $\mathrm{sSFR}={\mathrm{SFR}}_{\mathrm{gal}}/{M}_{* }$, which removes the direct element of the mass dependence.

In Figure 5, we show the relationship between the sSFR of our Gadget-X galaxies and the environmental overdensity around them at redshift z = 0. We follow Steinborn et al. (2015) in using a specific star formation rate threshold of $\mathrm{sSFR}=0.3/{t}_{{Hubble}(z)}$ to distinguish between quiescent and star-forming galaxies. Although this definition of a star-forming galaxy is somewhat different from the various observational criteria (see. Peng et al. 2010; Gabor & Davé 2012; Muzzin et al. 2013), the precise division line between active and quenched galaxies varies among observations. As our simulations are not specifically tuned to match the observed $\mathrm{sSFR}\mbox{--}{M}_{* }$ relation, we adopt a simple definition for clarity.

Zoom In Zoom Out Reset image size

In all three of our samples the median sSFR for the all star-forming galaxies (shown as the solid blue line) drops as the overdensity, δ₁, increases. This is principally driven by the changing proportions of galaxies of different stellar masses. In all cases more massive galaxies have a lower sSFR and these objects tend to lie preferentially within volumes of greater local overdensity. This transition between a galaxy population at low overdensity that is dominated by small, relatively high sSFR galaxies to one at high overdensity that has a significant population of massive, low sSFR galaxies leads to the falling trend for the sample as a whole. This same drop in sSFR is also clearly seen in the observations.

We claim that the falling sSFR is principally driven by the change with mass and that the relative proportions of galaxies with different masses varies with δ. This is supported by the fact that if we narrow the mass range of galaxies contributing to each of the samples shown in Figure 5 the measured sSFR becomes shallower. i.e., for broad mass bands, we again see a declining slope in the sSFR–δ relation.

We also clearly see (Figure 5, bottom right panel) that the fraction of star-forming galaxies within each environment is also falling rapidly as the overdensity increases, with very few star-forming galaxies above overdensities of 100 in any of the samples. In the cluster sample this drop in the number of star-forming galaxies appears at lower overdensities, presumably due to the contamination of this region by back splash galaxies that have passed through the cluster environment and been quenched due to gas stripping. Another feature of this panel is that the fraction of star-forming galaxies actually falls at low overdensity in both the field and vicinity galaxy samples. Such low overdensities simply do not exist in the cluster environment and so no galaxies can possibly reside there. We check components of quenched galaxies in the lowest density bin and find that most of them (over 90%) are gas rich. There is no mechanism of specially heating the gas in low-density regions in our simulations. Combining with the fact that the galaxy number in the lowest density bin is ten times less than that in the secondary lowest density bin, we consider the falling a reasonable fluctuation rather than physical phenomenon.

For galaxies within a specified stellar mass range our simulations indicate that the sSFR tends to be rather flat, particularly in the cluster and cluster vicinities samples. In the field the sSFR for the intermediate mass range appears to fall as the overdensity increases. The apparent constancy is consistent with previous observations (e.g., Peng et al. 2010). The amplitude of the sSFR is also approximately recovered in all environments.

In Figure 5, the flat environmental dependencies of median sSFR and the falling fraction of SFG seem to be at odds with each other as the median sSFR of star-forming galaxies is not suppressed at higher environmental densities while a higher fraction of galaxies are quenched. This reproduces the results found by Peng et al. (2010). Peng et al. (2010) suggested a sharp transition scenario: galaxies continue forming stars at the same rate regardless of their environment, despite the fact that the chance of having been quenched evidently does depend strongly on the environment. This scenario requires instantaneous quenching to occur, and this is the assumption Peng et al. (2010) made in their work. While such instantaneous quenching might appear idealistic, our simulations support this scenario. Examining star formation histories for individual galaxies we often see sharp drops in SFR during a galaxy's lifetime. Other work suggests that galaxies do not undergo an instantaneous quenching scenario. Using subhalo abundance matching (SHAM), Wetzel et al. (2013), Muzzin et al. (2014) suggested rapid quenching for satellite galaxies with a quenching timescale of less than 0.8 Gyr. Hahn et al. (2017) suggested a longer quenching timescale of 0.5–1.5 Gyr for central galaxies. Currently observational evidence on quenching timescales is lacking, and we therefore cannot confirm which hypothesis truly describes a galaxy's SFR evolution.

We further investigate how the median sSFR evolves with redshift in Figure 6, with different line styles indicating our three different samples. All galaxies with stellar mass above ${10}^{9.5}{h}^{-1}{M}_{\odot }$ are included. Figure 6 shows that the fraction of SFG versus δ₁ (top panel) and sSFR − δ₁ (bottom panel) retain a similar shape at all redshifts, with the amplitude decreasing with time. At earlier times three classes of galaxies are in agreement with each other. It is because that at that time the environments are not obviously distinguishable. Essentially, there is a higher fraction of SFG at early times and these galaxies are in general forming stars at a faster rate. There is no sign of any difference between three galaxy classes, but there is some evidence that the dependence of sSFR on overdensity somewhat steepens at early times.

Zoom In Zoom Out Reset image size

While our simulations appear to do a reasonable job at reproducing many of the features of the observed sSFR–δ₁ relationship we re-iterate that our simulations were not tuned to fit this, although many properties are consistent with observations (see. Rasia et al. 2015; Biffi et al. 2017, 2018; Truong et al. 2018). One issue is the diversity among observational claims. Patel et al. (2009, 2011) found a declining trend for the sSFR—local density relation of star-forming galaxies at 0.6 < z <  0.9 and Muzzin et al. (2012) found the sSFR of star-forming galaxies increases with the distance from the cluster center at z ∼ 1. These results are consistent with our work. However, there are also works which show no correlation between these properties at high redshift: Grützbauch et al. (2011), Scoville et al. (2013) did not found any correlation between the sSFR of all galaxies and the local overdensity at z > 1. Darvish et al. (2016) found that the SFR and sSFR of star-forming galaxies are not related to the environmental overdensity from z = 0.1 to z = 3. As Darvish et al. (2016) claimed, the result that no dependence on environment is found at z ≥ 1 may be partly due to the larger photo-z uncertainties at higher redshift and the lack of extremely dense regions in the COSMOS field. Moreover, Elbaz et al. (2007), Cooper et al. (2008), Welikala et al. (2016) found an ascending trend of the SFR-density relation at high redshift, which is opposite to what Patel et al. (2009) found or other SFR-density relations at low redshifts. Popesso et al. (2011) attributed such a reversal in the relation to the contamination of AGN.

First we present the r band luminosity function for three bands of environmental overdensity, δ₁ for each of our three different sample groups in Figure 7. The Luminosity function for galaxies within a specific density bin is measured following the method described in Section 2.5 of (McNaught-Roberts et al. 2014) The dashed lines indicate best fits to the SDSS luminosity function (red) and galaxies within under- (blue) and over- (green) dense environments within the GAMA survey. The green solid lines in each panel indicate galaxies with the highest local overdensity, δ₁. There are lots of these in the cluster sample and they are generally brighter and more massive than the galaxies in less overdense environments. Even though we have removed the central galaxies from each of the 324 clusters (these are hard to distinguish from their surrounding intracluster light) there remain some extremely bright objects. This indicates that the treatment of large central galaxies within the Gadget-X model could be improved. The excess here is due to central galaxies within large infalling structures. This effect for the most overdense galaxies persists into the vicinities (middle panel) and occurs for the same reason: there are some large groups with dominant central galaxies here. In the field sample there are no large groups by construction and so there are very few galaxies in the most overdense environment. For the orange and blue solid lines indicating galaxies in intermediate and low overdensity local environments the luminosity function in all three of our sample regions is not unrealistic when compared with the observational best fits. The shape is well recovered in all cases with, as expected, the amplitude of the underdense luminosity function rising as we move from the cluster to vicinities to field samples. The observed luminosity function is built up from galaxies within all three of our sample regions and is a combination of them with unknown weights. Essentially, unless a very large unbiased survey has been constructed the precise amplitude of the luminosity function curves will depend strongly on the sample selection.

Zoom In Zoom Out Reset image size

Second, we check the color–δ₁ relation in Figure 8. We use the difference between the magnitude in the g and r bands, ${M}_{g}-{M}_{r}$ to evaluate the color of each galaxy and display this as a function of overdensity, δ₁. Contours of different colors indicate the galaxy number density at z = 0 within each of our three sample regions as indicated in the legend. The horizontal dotted red line at g − r = 0.6 indicates the rough division between red and blue galaxies at redshift z = 0. Clearly, within the cluster sample the vast majority of the galaxies are red at z = 0, with almost no blue galaxies. There are also no galaxies with low environmental overdensity because such an environment does not exist for this sample. The field sample contains examples of both red and blue galaxies and, as previously seen, does not contain any high overdensity environment (and therefore has no high overdensity galaxies). The cluster vicinities sample contains galaxies with both colors and environments, although it does not probe either the highest or the lowest overdensities. The clear bimodality between blue and red galaxies in the observed galactic population is well recovered by the Gadget-X simulations.

Zoom In Zoom Out Reset image size

Finally we examine the evolution of the color–δ₁ plane in Figure 9. Three panels show via colored contours the change in the distribution of galaxy number density on the g − r versus δ₁ plane as a function of redshift. Each panel displays results from a different sample region as indicated. As before, the red dotted horizontal line at g − r = 0.6 indicates the rough division between red and blue galaxies at redshift z = 0. Clear evolution in the color of the galactic population in all three samples is seen and this mirrors the expected trend. In all three samples an obvious red sequence is visible and in-place at z =  1 and color bimodality is also evident. At z ∼ 2.5 the majority of galaxies in all environments are blue, even if the evolution of the dividing line between red and blue galaxies is taken to evolve with redshift. The bottom right panel displays the evolution of the median values within each sample (distinguished via line type) as a function of redshift (distinguished via line color). The median color as a function of overdensity within all three of our samples is largely indistinguishable, indicating that to a large extent the apparent differences in color as a function of δ are largely driven by how well the overdensity range is spanned by the sample rather than via any difference in the galaxy's color due to its environment.

Zoom In Zoom Out Reset image size

One critical issue is what environmental measure is most closely related to a galaxy's properties. Based on the theory of hierarchical structure formation, it is quite natural to think that a galaxy's properties should be closely linked to their host virialized dark-matter halo. On the other hand, many observations found a close relationship between local overdensity and galactic properties. Our simulations support the observational result, as we find that local overdensity is a more direct measure of a galaxy's properties than its host halo mass.

An intuitive way to quantify the relationship between two variables is by calculating Spearman's rank correlation coefficient. Spearman's rank correlation coefficient r_s is defined as

$$\begin{eqnarray}&&{r}_{s}=\displaystyle \frac{\mathrm{cov}({{rg}}_{X},{{rg}}_{Y})}{{\sigma }_{{{rg}}_{X}}{\sigma }_{{{rg}}_{Y}}}.\end{eqnarray} \tag{ 1 }$$

Here, rg_Xi is the rank of ith element in the array of variable X. The value of r_s indicates how well the relationship between two variables can be described using a monotonic function. The range of r_s is [−1, 1], r_s = 0 indicating no relationship between the two variables. r_s = 1 indicates a perfect positive relationship, while r_s = −1 indicates a perfect negative relationship. The larger the value of $| {r}_{s}|$, the better the relationship is.

In Table 2, we tabulate the value of r_s between galaxy properties and three environmental measures, namely ${\delta }_{1}$, the host halo mass and the galaxy stellar mass. We calculate this measure for all three of our galaxy samples, as indicated in the table. The value of ${r}_{s}({\delta }_{1})$ is quite consistent among our three samples. As we have already seen above: the fraction of SFG is strongly negatively correlated with δ₁; the sSFR of SFG is slightly negatively correlated with δ₁; the color, g − r, is positively correlated with δ₁. The ${r}_{s}({M}_{\mathrm{halo}})$ shows a quite different relationship, as well as changing significantly across the three galaxy samples. This variation in ${r}_{s}({M}_{\mathrm{halo}})$ implies that the influence of host halo differs in different large-scale environments. Thus halo mass is a worse indicator of local environment than local overdensity δ₁. This should not necessarily be a surprise, as a large galaxy cluster contains many thousands of galaxies with a wide variation in properties and so the single halo mass should not be expected to correlate particularly well on this level.

Table 2.  Spearman's Correlation Coefficient r_s for the Relationship between Galaxy Properties and Local Overdensity, Halo Mass, and Stellar Mass

Cluster
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.932	−0.099	0.311
Mhalo	0.809	−0.280	−0.104
M*	0.924	−0.321	0.388
Vicinity
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.962	−0.161	0.454
Mhalo	0.528	−0.041	0.009
M*	−0.364	−0.209	0.398
Field
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.946	−0.178	0.373
Mhalo	0.502	−0.111	−0.293
M*	−0.490	−0.250	0.421

Note. r_s for our cluster, vicinity, and field galaxy samples are listed in separate tables.

Download table as:  ASCIITypeset image

For comparison we also calculate Spearman's correlation coefficient for galaxy properties and stellar mass. In this case, for the cluster sample, r_s has a value of 0.924 for the fraction of star-forming galaxies, which indicates a significant positive relationship. This relationship is opposite to that observed due to the essential lack of low-mass SFG in the cluster sample. Except for this, the values of the other ${r}_{s}({M}_{* })$ meet our expectations well. By comparing ${r}_{s}({M}_{* })$ and ${r}_{s}({\delta }_{1})$, we find that the fraction of SFG is more influenced by the local overdensity, while the sSFR of SFG is more influenced by the stellar mass. For galaxy color, it seems that both δ₁ and M_* have similar weights.

The accuracy of r_s could suffer due to the incompleteness of our samples. Thus we divided our galaxies into different stellar mass bins to test the sensitivity of r_s to our galaxy sample. As Table 3 shows, the r_s for the low-mass and high mass ends changes a lot compared with Table 2, while the r_s for the median stellar mass range keeps the same trends and similar values. Except for massive galaxies, the strong influence of environment, δ₁, on the fraction of SFG is stable, which further prove the important link between environment and the SFG fraction. The negative correlation between M_* and sSFR of SFG is still clear, although it becomes weaker in the low and high stellar mass range. The ${r}_{s}({M}_{\mathrm{halo}})$ becomes variable when the different stellar mass ranges are applied. The r_s between color g − r and δ₁, M_*, M_halo is very similar to Table 2, except for massive galaxies. ${r}_{s}({M}_{\mathrm{halo}},g-r)$ shows negative relation in most places, which is again counterintuitive. In summary, we consider that the influence of δ₁ is much less sensitive to mass completeness than M_halo.

Table 3.  The Same as Table 2, But the Samples are Divided into three Stellar Mass Bins

Cluster	${M}_{* }\lt {10}^{10}{h}^{-1}{M}_{\odot }$			${10}^{10}{h}^{-1}{M}_{\odot }\lt {M}_{* }\lt {10}^{11.5}{h}^{-1}{M}_{\odot }$			${M}_{* }\gt {10}^{11.5}{h}^{-1}{M}_{\odot }$
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.883	0.082	0.446	−0.933	0.076	0.326	0.939	−0.013	−0.400
Mhalo	0.345	0.085	−0.718	0.285	−0.162	−0.304	0.867	−0.172	−0.738
M*	0.818	−0.076	0.283	0.552	−0.143	0.102	0.564	−0.095	−0.731
Vicinity	${M}_{* }\lt {10}^{10}{h}^{-1}{M}_{\odot }$			${10}^{10}{h}^{-1}{M}_{\odot }\lt {M}_{* }\lt {10}^{11.5}{h}^{-1}{M}_{\odot }$			${M}_{* }\gt {10}^{11.5}{h}^{-1}{M}_{\odot }$
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.964	−0.080	0.638	−1.0	−0.171	0.396	0.127	−0.026	−0.456
Mhalo	0.818	0.134	−0.750	0.152	−0.025	−0.088	0.033	−0.106	−0.458
M*	⋯	−0.082	0.013	−0.685	−0.199	0.169	0.762	0.033	−0.499
Field	${M}_{* }\lt {10}^{10}{h}^{-1}{M}_{\odot }$			${10}^{10}{h}^{-1}{M}_{\odot }\lt {M}_{* }\lt {10}^{11.5}{h}^{-1}{M}_{\odot }$			${M}_{* }\gt {10}^{11.5}{h}^{-1}{M}_{\odot }$
rs	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r	${\mathrm{Frac}}_{\mathrm{SFG}}$	${\mathrm{sSFR}}_{\mathrm{SFG}}$	g − r
${\delta }_{1}$	−0.927	−0.078	0.698	−0.988	−0.216	0.310	⋯	−0.335	0.335
Mhalo	0.964	−0.132	−0.811	0.133	−0.129	0.270	⋯	−0.415	0.340
M*	⋯	−0.082	0.136	−0.358	−0.317	0.266	⋯	−0.400	0.137

Download table as:  ASCIITypeset image