The Star Formation Rate–Radius Connection: Data and Implications for Wind Strength and Halo Concentration

In this work we use all five fields from the CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011), published in public catalogs by Galametz et al. (2013) for UDS, Guo et al. (2013) for GOODS-S, Barro et al. (2017) for GOODS-N, Nayyeri et al. (2017) for COSMOS, and Stefanon et al. (2017) for EGS.²⁵ Rest-frame photometry and photometric redshifts are calculated using EAZY (Brammer et al. 2008). Spectroscopic redshifts are used if they are available (22% of the sample). The official CANDELS mass catalog includes results from 10 different SED-fitting methods (see Table 1 in Santini et al. 2015). Most of them adopted BC03 (Bruzual & Charlot 2003) stellar templates and minimized the χ² to determine the best fit. We use the median stellar mass because it averages the assumptions of different star formation histories. The median stellar mass is robust with a typical estimated error of ∼0.1 dex.

Star formation rates, galaxy sizes, and their residuals are computed following the methods of Fang et al. (2018). We briefly summarize them here and refer the reader to Fang et al. (2018) for more details.

The SFRs used in this work come from dust-corrected NUV luminosity (2800 Å). In order to obtain a robust value of dust attenuation (in rest-frame V band, ${A}_{V}$), the results from five different SED-fitting methods (labeled 2a, 2d, 12a, 13a, and 14a in Santini et al. 2015) are combined and the median ${A}_{V}$ is selected. These methods were chosen based on their common use of τ models and the Calzetti attenuation law. The typical formal error for the median ${A}_{V}$ is ∼0.1 mag. The Calzetti dust attenuation at 2800 Å, ${A}_{\mathrm{NUV}}$, is 1.8${A}_{V}$, which is used to correct the observed NUV flux. Corrected NUV luminosity is then converted to SFR using the calibration from Kennicutt & Evans (2012): SFR${}_{\mathrm{UV},\mathrm{corr}}\$[M_⊙ yr⁻¹] = 2.59 × 10⁻¹⁰ ${L}_{\mathrm{NUV},\mathrm{corr}}\$[L_⊙]. Although the methods are different, we have verified that our rates from NUV-corrected fluxes are in fact virtually identical to rates derived from full SED fitting.

Because the goal of our study is to analyze the properties of galaxies above and below the SFMS, SFR measurements must be good enough to derive accurate SFR residuals. The formal errors of our SFRs are small, but there might be systematic errors, in part because of the use of τ models and/or the Calzetti law. A separate study (F.S. Liu et al. 2020, in preparation) is testing SED-fitting methods on non-τ star formation histories, with encouraging results. In the meantime, it is desirable to have a separate set of SFRs to compare with. The so-called "hybrid" method (SFR${}_{\mathrm{UV}+\mathrm{IR}}$), which adds together raw UV and IR rates, is thought to be the most reliable (Kennicutt et al. 2009; Hao et al. 2011). We calculate the quantity SFR${}_{\mathrm{UV}+\mathrm{IR}}$ using the formula in Wuyts et al. (2011), where L_IR is determined from 24 μm data using the calibration of Rujopakarn et al. (2013). Data sources and details are given in Fang et al. (2018).

As explained in Fang et al. (2018), the usual method of testing methods by simply plotting one SF indicator against another is not good enough to establish the accuracy of SFR residuals. We accordingly compute residuals using the SFMS ridgelines in different redshifts given in Fang et al. (2018) and compare them in Figure 2. Red points are from GOODS-S, which uses deeper 24 μm data, and the gray points are for all other fields. This figure updates and extends a similar figure in Fang et al. (2018) to all five CANDELS fields.

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Several conclusions emerge. First, it is apparent that the IR data are highly incomplete at low mass and high redshift, and therefore it would be impossible to carry out the kind of study undertaken in this paper using IR data alone—use of UV-optical SEDs is essential. The second issue is evident in the shaded rectangles, which denote galaxies in the GV below Δlog sSFR $\lt -0.45$ dex (our ridgeline boundary). It is well known that SFRs for GV galaxies are systematically overestimated by 24 μm data according to several studies reviewed by Fang et al. (2018), and the same trend is seen here. If these GV objects are set aside, clear if somewhat noisy correlations are visible in most of the panels. The total scatter for the GOODS-S data (after removing outliers and zero-point offsets) is 0.24 dex (Fang et al. 2018), which, if assigned equally to both measures, implies an error of 0.17 dex in Δlog sSFR${}_{\mathrm{UV},\mathrm{corr}}$.

The last point is the presence of systematic zero-point offsets that tend to be negative at low redshift and positive at high redshift. These are not a concern since our goal is the relative ranking of objects within the SFMS, which is not disturbed by a zero-point shift.

The second important quantity used in this work is galaxy size defined by the half-light optical radii. For this, we use the semimajor axes (${R}_{{\rm{e}},\mathrm{maj}}$ = a) based on GALFIT fits to H-band images by van der Wel et al. (2014). ${R}_{{\rm{e}},\mathrm{maj}}$ is preferred to the circularized radius (${R}_{{\rm{e}},\mathrm{circ}}$ = $\sqrt{{ab}}$) because it is a more stable indicator for inclined disks. Although H-band corresponds to different rest-frame wavelengths at different redshifts, Fang et al. (2018) estimated that, for SF galaxies between redshifts z = 1 and 2, the wavelength-dependent K correction to galaxy size is less than 10%. The residuals of ${R}_{{\rm{e}},\mathrm{maj}}$ are calculated using the mass–size relations from Fang et al. (2018) in different redshift bins. Our use of residuals in bins of mass and redshift makes us insensitive to K-correction errors.

Finally we select only face-on galaxies with b/a > 0.5 in order to minimize the effects of dust on Δlog sSFR${}_{\mathrm{UV},\mathrm{corr}}$ and inclination on radii. A summary of the CANDELS selection cuts is as follows:

• 1.  
  Apparent H-band magnitude < 24.5. This is the limit suggested by van der Wel et al. (2014) for reliable GALFIT measurements.
• 2.  
  Redshift within 0.5 < z < 2.5 and stellar mass within 9.0 < log ${M}_{* }$/M_⊙ <  11.0, to maximize the sample size. We omit log ${M}_{* }$ > 11.0 and z < 0.5 because of few objects in those range.
• 3.  
  PHOTFLAG = 0, CLASS_STAR  <  0.9, and GALFIT flag = 0, to ensure reliable photometric measurements, no foreground stars, and good-quality GALFIT fits. This eliminates merging galaxies and galaxies with peculiar morphologies.
• 4.  
  b/a > 0.5, to minimize dust extinction and radius uncertainties. a and b are the semimajor and semiminor axes of the galaxy from GALFIT measurements.
• 5.  
  Location in the star-forming region of the UVJ diagram. Star-forming ridgeline galaxies must in addition have Δlog sSFR${}_{\mathrm{UV},\mathrm{corr}}$ > −0.45 dex. Galaxies located in the SF region but with Δlog sSFR${}_{\mathrm{UV},\mathrm{corr}}$ < −0.45 dex are retained but classified as GV galaxies.

According to the analysis in Fang et al. (2018), these criteria include nearly all star-forming galaxies in most mass and redshift bins, but the completeness declines to less than 50% for ${M}_{* }$ < 10^9.5M_⊙ and z > 2. See the discussion and Figure 2 in Fang et al. (2018) on completeness.

As a supplement to the high-redshift CANDELS data, we select normal star-forming galaxies from the SDSS DR7 catalog (Abazajian et al. 2009). Spectroscopic redshifts, stellar masses, and emission-line measurements are obtained from the MPA/JHU value-added catalog (Kauffmann et al. 2003b).

SFR and ${A}_{V}$ are taken from Salim et al. (2018), based on UV-optical SED fitting jointly with 22 μm photometry. They compared their SFRs with other published catalogs and found good agreement for star-forming galaxies, which means that our results should not vary with different SFR indicators. The typical error in the star formation rate is about 0.1 dex (Salim et al. 2016, 2018).

The SFMS ridgeline from Speagle et al. (2014) was adopted to calculate SFR residuals for the SDSS sample at fixed stellar mass (Δlog sSFR). On average, we find that the mean sSFR in the Salim catalog is 0.14 dex higher than the ridgeline in Speagle. Because we are only concerned with relative sSFR, we adopt the slope of SFMS in Speagle et al. (2014) and shift the zero point by 0.14 dex to match the SFR in Salim et al. (2016).

Galaxy radii and Sérsic indices are taken from the NYU Value-Added Galaxy Catalog (Blanton et al. 2005). Similar to ${R}_{{\rm{e}},\mathrm{maj}}$ in the CANDELS sample, we use the semimajor axis in z-band images to characterize galaxy size. We further calculate the size residuals according to the mass–size relation for star-forming galaxies in Shen et al. (2003).

A summary of the criteria used to select the SDSS star-forming sample is as follows:

• 1.  
  Redshift in the range 0.02 < z < 0.07, the apparent magnitude within 14   < r < 17.5, and ${M}_{* }$ > 10^9.0M_⊙.
• 2.  
  Single-Sérsic index in the range 0.5 < n < 6. The upper limit excludes galaxies with bad fits.
• 3.  
  Merging galaxies are excluded using the classification from Galaxy Zoo (P_MG < 0.1). (This is analogous to the good GALFIT flag that we required for CANDELS.)
• 4.  
  b/a > 0.5. a and b are taken from the Blanton et al. (2005) catalog by single-Sérsic fitting.
• 5.  
  Main-sequence membership using Δlog sSFR > −0.45 dex after shifting the Speagle et al. (2014) zero point by 0.14 dex (see above).

Galaxies satisfying the above criteria comprise the full SDSS sample, which includes all galaxies on the main sequence and is comparable to the CANDELS sample. In addition, to minimize possible environmental effects on the structure of the lower main sequence and entrance to the GV, we extracted a centrals-only subsample by matching to the group catalog of Yang et al. (2012) and requiring mass rank M_rank = 1. Finally, as another common way to select star-forming galaxies is by their emission lines, we constructed yet another subsample by requiring strong emission (S/N of [O iii] λ5007, Hβ, Hα, [N ii] λ6584 > 5) and location in the BPT diagram in the H ii region of Kauffmann et al. (2003a).

We turn now to correlations between SFR and radius. Before continuing, we note that the rms scatter in log ${R}_{{\rm{e}}}$ at fixed ${M}_{* }$ is only about 0.25 dex (van der Wel et al. 2014), and it might be thought that measurement errors might mask real size differences. To allay that concern, Figure 3 shows color images of large and small galaxies from both samples. In each mass and redshift bin, the two small galaxies on the left are randomly selected from galaxies with Δlog ${R}_{{\rm{e}}}$ < −0.2 dex, and the two large galaxies on the right are randomly selected from galaxies with Δlog ${R}_{{\rm{e}}}$ > 0.2 dex. The SDSS images at low redshift come from gri composites while CANDELS images are generated from HST/ACS F814W, F125W, and F160W. Each image is presented at the same physical scale of 30 kpc on a side. It is seen that galaxies have distinctly different sizes and that ${R}_{{\rm{e}}}$ does a good job of separating small galaxies from large ones. Because surface density is proportional to r⁻², the observed range of galaxy size results in a large change in surface density. A typical difference of ±0.25 dex in radius would cause surface densities to differ by 1 whole dex.

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Figure 4 now plots the residuals Δlog ${R}_{{\rm{e}}}$ versus Δlog sSFR for CANDELS ridgeline galaxies divided into stellar mass and redshift bins. Points are color-coded by the ${A}_{V}$ obtained from SED fitting. The trend predicted by the KS law under the assumption of constant gas fraction (see below) is the gray line in each panel. Ridgeline galaxies with Δlog sSFR ≥ −0.45 dex are in the white areas; GV galaxies are in the shaded regions. Compared to ridgeline galaxies, the latter tend to have smaller radii and lower ${A}_{V}$. However, for ridgeline galaxies only, there is no strong trend for Δlog sSFR to follow the gray lines predicted by the KS relation for constant gas fraction.

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The lack of any significant correlation is confirmed by the Pearson correlation coefficient r and the p values, which are listed in the top-right corner of each panel. The coefficients r are close to zero in most panels, indicating no correlation between the two variables. The p value indicates the probability of an uncorrelated system producing relations that have a Pearson correlation as large as the one computed from these data sets. In all panels, the correlation coefficients are low (r < 0.28). For low-mass galaxies with ${M}_{* }$ < 10¹⁰ M_⊙, the signs of r vary randomly and p values are mostly not significant (>0.05). At higher masses ${M}_{* }$ > 10¹⁰ M_⊙, there is a slight trend to see positive slopes from the white points, and half of the p values are significant. In any case, there is no sign of the systematically negative slopes that are predicted by the KS relation.

To gain further insight, we plot medians of Δlog sSFR at fixed Δlog ${R}_{{\rm{e}}}$ (white points) and medians of Δlog ${R}_{{\rm{e}}}$ at fixed Δlog sSFR (red points). In all panels, the two sets of lines outlined by these points are orthogonal or nearly so, which is another classic signature of little or no correlation. It is interesting to note that the mean slopes indicated by the white points above ${M}_{* }$ > 10¹⁰ M_⊙ are weakly positive. This trend follows the numbers of galaxies near the bottom of the ridgeline and in the GV, which are also increasing with mass. Because these galaxies have both small radii and low SFR, their presence tends to create a positive slope. We note that the trend for galaxies well below the ridgeline to have smaller radii was also seen by Brennan et al. (2017) for CANDELS, but we have now shown that this trend is correlated with the number of GV galaxies. An inference might be that a significant fraction of star-forming galaxies below the ridgeline at high mass is actually en route to the GV, i.e., that they are not bobbing temporarily below the ridgeline and are about to return. However, the bigger picture is that even the largest slope of ∼0.2 is small, amounting to a total variation of only 0.1 dex (=25%) from small (−0.25 dex) to large (+0.25 dex) galaxies. In addition, by comparing the slopes in different redshift bins, we do not see significant evolutionary trends with time.

We referred above to the prediction of the KS law (gray line), which says that if gas fractions are all the same for galaxies in a given stellar mass bin, large, diffuse galaxies should have lower SFRs than compact, dense ones. The prediction is computed in the cartoon in Figure 1, which compares two galaxies of the same stellar mass, one large and one small. The left panel describes the situation with identical gas fractions. Spreading the same amount of gas over a wider area reduces its ability to form stars owing to the exponent 1.4 in the KS law, which exceeds unity. The predicted slope is −0.80 for Δlog sSFR versus Δlog ${R}_{{\rm{e}}}$, which is shown by the gray lines in Figure 4.

For comparison, the case of constant star formation rate is shown in the right panel of Figure 1, where it is shown that the larger galaxy must have more gas in order to make the same amount of stars. The predicted slope in that case is +0.57 for Δlog M_gas versus Δlog ${R}_{{\rm{e}}}$ (we show this trend in Figure 8).

A new version of the KS law has recently been proposed called the extended Kennicutt–Schmidt (eKS) law, ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ ∝ ${{\rm{\Sigma }}}_{\mathrm{gas}}$ ${{\rm{\Sigma }}}_{* }\$^0.5 (Shi et al. 2011), which takes stellar surface density into account and improves the relation for low-surface-brightness galaxies (Shi et al. 2018). Using the method of Figure 1, the predicted slope for eKS is −1.00 for Δlog sSFR versus Δlog ${R}_{{\rm{e}}}$. Both predicted relations are far steeper than the data. Hence, we conclude that, if these galaxies obey either the KS law or the eKS law, larger galaxies must have larger gas fractions. This is confirmed for local galaxies in Section 4 below.

Figure 4 is an opportunity to show the locations of the compact star-forming galaxies identified by Barro et al. (2014) in relation to other objects at the same redshift. These so-called "blue nugget" galaxies were identified as having particularly small sizes compared to typical main-sequence ridgeline objects. We select them using the same definition log(${M}_{* }/{R}_{e}^{1.5})\gt 10.45$ M_⊙ kpc^−1.5 used by Barro et al. (2014) and plot them with black circles in Figure 4. Most compact star-forming galaxies in our sample appear at redshifts z > 1.5 and either lie on the main sequence or below it, consistent with the scenario that blue nuggets are in the process of quenching and will soon evolve into red nuggets by z = 1.5 (Barro et al. 2014).

Figure 5 shows Δlog ${R}_{{\rm{e}}}$ versus Δlog sSFR for the SDSS sample. The first row shows the full sample, which follows the same selection criteria as CANDELS. The second row shows central galaxies only, while the third row selects strongly star-forming galaxies using the emission-line criterion (see Section 2.2). The correlation coefficient r and the p values are calculated for each panel as in Figure 4. The medians of both the X and Y directions are shown in each panel. The predicted slopes according to the KS law are the dashed lines.

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As in Figure 4 for CANDELS, the correlation coefficients are low, the medians in the X and Y directions are quite orthogonal, and the choice of sample also has little effect. The second row shows central galaxies only. Removing the satellites appears to have removed some of the small galaxies below the ridgeline at low masses, and the star-forming sample looks a bit cleaner. However, the correlation coefficients and the slopes in X and Y directions are basically unaffected. The emission-selected sample in the third row has a tail of low-SFR galaxies with small sizes. This population resembles a similar population in CANDELS and seems stronger in the emission sample than the other samples. As before, however, the correlation coefficients are all small.

In conclusion, the result from SDSS agrees with CANDELS in showing that smaller star-forming ridgeline galaxies must contain less gas at fixed ${M}_{* }$ if galaxies obey either the KS or eKS star-forming laws.

We note in passing that the data points are colored by ${A}_{V}$ in Figures 4 and 5. Even though both samples are deliberately restricted to face-on galaxies with b/a > 0.5 in order to minimize dust effects, nevertheless, two trends are evident. The stronger one is that more compact galaxies have higher ${A}_{V}$ than larger galaxies (this effect looks more prominent in SDSS, but note the compressed color range compared to CANDELS—CANDELS is just noisier). On the face of it, this is puzzling because we have just shown that more compact galaxies have less gas, so why do they have higher ${A}_{V}$? The answer is that ${A}_{V}$ varies as the surface density of the gas, not the gas fraction. Compact galaxies evidently produce larger ${A}_{V}$ on account of their smaller area even with less total gas. The second trend is that galaxies with high Δlog sSFR have higher ${A}_{V}$ at fixed Δlog ${R}_{{\rm{e}}}$. This trend is plausible because a higher SFR at fixed size implies more gas, and thus more dust. Both trends will be explored in future papers.

Figures 6 and 7 summarize the preceding data by plotting log sSFR as a function of position in the mass–size diagrams. Stellar mass and galaxy size are binned and colored by the median log sSFR in each pixel. The CANDELS sample in Figure 4 is a bit noisy, and the contours do not vary smoothly, probably due to the smaller sample size or limitation of the signal-to-noise ratio. Overall, however, they are roughly vertical but are slightly tilted at intermediate mass, consistent with the small positive or negative slopes in Figure 4. In the SDSS sample (Figure 7), both the full and central samples show nearly vertical contours, but the emission-selected contours are more tilted, consistent with the trend for this sample in Figure 5.

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To summarize, we have compared Δlog sSFR versus Δlog ${R}_{{\rm{e}}}$ for star-forming ridgeline galaxies over a wide range of stellar mass and redshift. Neither CANDELS nor SDSS shows a large correlation between the star formation rate and galaxy radius at fixed stellar mass. This result does not depend on the SFR indicator used, nor does it appear to vary much with sample selection (in SDSS). The assumption of a constant gas fraction at fixed stellar mass would predict a large negative trend between Δlog sSFR and Δlog ${R}_{{\rm{e}}}$, which does not appear in the real data. If galaxies obey a density-dependent star-forming law like the KS law or its relatives, these results indicate that more compact galaxies have lower gas fractions.

We return now to the discussion of previous work that was initiated in the Introduction. The basic question is whether the properties of galaxies are correlated with their position above or below the SFMS. Our approach here has been to measure the correlation between the main-sequence star-forming residual Δlog sSFR and the radius residuals Δlog ${R}_{{\rm{e}}}$. This is the same approach used by Whitaker et al. (2017) for CANDELS galaxies and by Omand et al. (2014) for SDSS galaxies. These papers also found no significant correlation if the sample is restricted to ridgeline galaxies, and we agree.

An alternative approach is to use the main-sequence residual Δlog sSFR as the basic variable and look for trends versus that. This is the approach used by Wuyts et al. (2011) for SDSS and CANDELS galaxies and by Brennan et al. (2017) for GAMA and CANDELS galaxies. As is well known, if there is significant scatter between two quantities X and Y, the median of X on Y can behave differently from the median of Y on X. To facilitate comparison with Wuyts et al. (2011) and Brennan et al. (2017), Figures 4 and 5 cut the samples horizontally in slices of Δlog sSFR and show the median value in each slice (red circles). The SDSS points reproduce closely the trend found by Brennan et al. (2017) using GAMA galaxies, showing the largest value Δlog sSFR on the ridgeline and declines amounting to ∼0.2 dex above and below it. This also agrees with Wuyts' analysis of SDSS although the trends there were slightly larger. In CANDELS, none of the three works reports any significant trend in Δlog ${R}_{{\rm{e}}}$ across the main sequence, all trends both above and below the SFMS being ≤0.1 dex. The data in Figure 4, though noisy, agree with this. However, galaxies well below the ridgeline but still with Δlog sSFR  > − 0.45 dex appear to be a little smaller in all works, and we have wondered whether this population is slowly quenching and moving toward the GV.

In summary, the lack of any significant correlation between Δlog ${R}_{{\rm{e}}}$ and Δlog sSFR for star-forming ridgeline galaxies is now well established from several different studies using different samples of galaxies at different redshifts. Our study has divided galaxies by mass and redshift and plotted each mass–redshift bin as individual points. This has revealed a probable population of compact galaxies below the SFMS that may be en route to the GV and shown that this population is stronger at high masses. We return to this population briefly in Section 5. Our use of central galaxies and exclusion of mergers and galaxies with bad GALFIT fits have also removed any concern that improper sample selection might have colored previous results. Finally, color-coding by ${A}_{V}$ has revealed systematic trends for stronger reddening in galaxies above the SFMS and in galaxies with smaller radii. These trends will be followed up in future papers.

We turn to interpret these results by assuming the equilibrium bathtub model. The basic equation for the model can be written (Dekel & Mandelker 2014; Rodríguez-Puebla et al. 2016) as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}={\dot{M}}_{* }+{\dot{M}}_{\mathrm{out}}+{\dot{M}}_{\mathrm{ISM}}=(1+\eta ){\dot{M}}_{* }+{\dot{M}}_{\mathrm{ISM}}\end{eqnarray} \tag{ 1 }$$

or

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{ISM}}={\dot{M}}_{\mathrm{in}}-(1+\eta ){\dot{M}}_{* }=0.\end{eqnarray} \tag{ 2 }$$

where ${\dot{M}}_{\mathrm{in}}$ is the accretion rate of pristine gas into the halo, ${\dot{M}}_{* }$ is the star formation rate, ${\dot{M}}_{\mathrm{out}}$ is the outflow rate, and ${\dot{M}}_{\mathrm{ISM}}$ is the rate of mass accumulation in the ISM. Using the mass-loading factor η for the wind results in the second line. Here we have assumed that all gas that falls into the halo finds its way soon into the galaxy, i.e., that gas is not accumulating in the halo. We have also assumed that no wind gas falls back in, i.e., that it either escapes the halo or is inert. Dekel & Mandelker (2014) showed that the equilibrium solution under these circumstances is ${\dot{M}}_{\mathrm{ISM}}=0$, which is reached asymptotically over time. This is explained by the feedback in the sign of ${\dot{M}}_{* }$, which varies negatively with ${M}_{\mathrm{ISM}}$. Errors in M_ISM are therefore self-correcting, and the star formation rate is self-regulating. The equilibrium solution is obtained provided the response time for changes in ${\dot{M}}_{* }$ is short compared to variations in ${\dot{M}}_{\mathrm{in}}$. Dekel & Mandelker (2014) adopt for t_* the local crossing time in the galaxy, or R_d/V_d, while t_infall is the crossing time of the halo, or R_vir/V_vir, which is ∼10 times longer. The needed inequality is therefore satisfied.

The next step notes that, if ${\dot{M}}_{\mathrm{ISM}}=0$, then $(1+\eta ){\dot{M}}_{* }\,={\dot{M}}_{\mathrm{in}}$, i.e., that the star formation rate is proportional to the halo gas accretion rate. Equation (2) says that there are only two ways of increasing the star formation rate: a larger halo accretion rate (${\dot{M}}_{\mathrm{in}}$) or a weaker wind (smaller η). In particular, increasing the gas surface density (by, say, reducing galaxy radius) or increasing the local star formation efficiency (by, say, raising the coefficient in the KS law) does not make more stars—it cannot because the total gas supply is limited. The only consequence of raising the local star formation efficiency is to reduce the mass of the ISM that it takes to support the same star formation rate. In other words, the gas density is adjusting itself to accommodate the halo accretion rate, and the proper way to read the KS law is backward, from the Y-axis to X-axis, as suggested in the Introduction.

We now use the equilibrium bathtub model to see how galaxies respond to varying halo concentration. The focus on concentration is motivated by an analysis of galaxy radii from the VELA and NIHAO simulations by Jiang et al. (2019), who find that

$$\begin{eqnarray}&&{R}_{{\rm{e}}}=0.02{R}_{\mathrm{vir}}{(C/10)}^{-0.7},\end{eqnarray} \tag{ 3 }$$

where ${R}_{{\rm{e}}}$ is the 3D mass-weighted half-mass radius and C is the halo concentration. The trend with C reflects the fact that galaxies in higher-concentration halos are smaller because a larger fraction of their mass is accreted early, making a smaller and denser galaxy. As a result, halos with higher concentration at fixed M_vir and fixed epoch have lower ${\dot{M}}_{\mathrm{in}}$. This is illustrated in Figure 9, which plots the bivariate distributions of specific halo accretion rates, ${\dot{M}}_{\mathrm{in}}/{M}_{\mathrm{vir}}$, versus halo mass from the Rockstar halo catalog (Behroozi et al. 2019) based on the Bolshoi–Planck dark-matter simulation (Klypin et al. 2016; Rodríguez-Puebla et al. 2016; Lee et al. 2017). The data are in panels binned by halo mass and redshift, and the lines show the mean and ±1σ contours. A good fit to the average instantaneous specific accretion rate is

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}/{M}_{\mathrm{vir}}\sim {\left(C/\langle C\rangle \right)}^{-0.5},\end{eqnarray} \tag{ 4 }$$

where C is the instantaneous concentration. The negative trend reflects the fact that high-concentration halos accreted more of their mass early and thus accrete less mass later. Putting the two equations together yields the prediction

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{in}}/{M}_{\mathrm{vir}}\sim {R}_{{\rm{e}}}^{0.7}.\end{eqnarray} \tag{ 5 }$$

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Hence, the halo mass accretion rate should be lower in compact galaxies. At fixed ${M}_{\mathrm{vir}}$, ${M}_{* }$, and η, Equations (2) and (5) imply Δlog sSFR/Δlog ${R}_{{\rm{e}}}$ = +0.7. In the context of Figures 4 and 5, this would be a strong trend and easily detected if present. Because no trend is seen, there must be another offsetting effect, but the only other knob in the model is to turn down the wind strength. This could plausibly work in the right direction as more compact galaxies have higher V_esc (at fixed ${M}_{* }$) and would thus have weaker winds.

A quantitative estimate of the lifetime impact of concentration variations on star formation rates is available from previously unpublished data from the Dutton SAM (Dutton et al. 2010), which divides disk galaxies into annuli and calculates the local mass-loading factor η at each radius. Their winds are weaker in deeper potential wells and scale either as $\eta \sim {V}_{\mathrm{esc}}^{-1}$ (momentum-driven winds) or as $\eta \sim {V}_{\mathrm{esc}}^{-2}$ (energy-driven winds). Data are available from the model for a collection of halos having energy-driven winds in a statistically realistic distribution of halo concentrations that are evolved appropriately over time. The scatter in ${R}_{{\rm{e}}}$ at fixed ${M}_{* }$ is ∼0.25 dex, a good match to observations, but there is no trend in the results for smaller galaxies to have lower star formation rates, at either z = 3 or z = 0. Smaller galaxies originate from higher-concentration halos, as expected, but their weaker winds fully cancel the effect of lower halo infall. This is a valuable simulation because it attempts to model the effects of wind and concentration over an entire galaxy's lifetime. The lesson learned is that wind differences must be strong—a parallel collection of galaxies with momentum-driven winds, which vary less with galaxy size, shows remaining correlated residuals in Δlog sSFR versus Δlog ${R}_{{\rm{e}}}$.

This discussion brings us back to the EAGLE simulations, which exhibit many of the expected effects of halo concentration (and formation time) differences. Matthee et al. (2017) studied scatter about the stellar-mass–halo-mass relation, which they find to be strongly correlated with halo concentration: more-concentrated halos form stars more rapidly, have weaker winds, and higher stellar mass at fixed halo mass (see also Wechsler et al. 2002; Zhao et al. 2009; Jeeson-Daniel et al. 2011; Ludlow et al. 2014; Correa et al. 2015; Kulier et al. 2019). Matthee et al. (2017) note that initial concentration differences are amplified by the presence of baryons, which rapidly collect in the central region, further deepening the central potential. Residuals about the SFMS relation are therefore even bigger when baryons are included. Davies et al. (2019) and Oppenheimer et al. (2020) studied the effects of concentration on the gas content of halos in EAGLE. More tightly bound halos have a higher concentration, earlier formation time, bigger black holes, lower gas content due to higher black hole feedback, and thus earlier quenching times. Finally, Furlong et al. (2017) studied residuals about the star formation main sequence and found that smaller galaxies at fixed mass have lower star formation rates today. This agrees with the predicted lower accretion rates in high-concentration halos (see Figure 9) but disagrees with our data showing no trend. Perhaps the EAGLE wind prescription does not weaken winds enough in high-concentration halos. We have not been able to find a discussion in the EAGLE literature explicitly treating the joint effects of concentration on galaxy radii and winds.

Finally, we remind readers once again of the correlation between galaxy size and concentration in the NIHAO and VELA simulations (Jiang et al. 2019). This motivated Chen et al. (2020) to posit concentration as the second halo parameter driving residuals in ${R}_{{\rm{e}}}$ versus ${M}_{* }$. Smaller (i.e., denser) galaxies make bigger black holes in their picture, which causes them to quench earlier in a manner similar to the EAGLE simulations. The notion that halo concentration modulates black hole mass goes back to Booth & Schaye (2010), who noted that bigger black holes form in halos with higher binding energy in their simulations. The cause in their case was weaker AGN feedback, not stellar feedback, but the same idea was present, namely, that halo concentration is a powerful second parameter influencing the life histories of galaxies.