ΣSFR–M ∗ Diagram: A Valuable Galaxy Evolution Diagnostic to Complement (s)SFR–M ∗ Diagrams

The specific star formation rate (sSFR) is commonly used to describe the level of galaxy star formation (SF) and to select quenched galaxies. However, since it is a relative measure of the young-to-old population, an ambiguity in its interpretation may arise because a low sSFR can be due to either a substantial previous mass buildup or SF activity that is low. We show, using large samples spanning 0 < z < 2, that the normalization of the star formation rate (SFR) by the physical extent over which SF is taking place (i.e., the SFR surface density, ΣSFR) overcomes this ambiguity. ΣSFR has a strong physical basis, being tied to the molecular gas density and the effectiveness of stellar feedback, so we propose ΣSFR–M * as an important galaxy evolution diagram to complement (s)SFR–M * diagrams. Using the ΣSFR–M * diagram we confirm the Schiminovich et al. result that the level of SF along the main sequence today is only weakly mass-dependent—high-mass galaxies, despite their redder colors, are as active as blue, low-mass ones. At higher redshift, the slope of the “ΣSFR main sequence” steepens, signaling the epoch of bulge buildup in massive galaxies. We also find that ΣSFR based on the optical isophotal radius more cleanly selects both starbursting and spheroid-dominated (early-type) galaxies than the sSFR. One implication of our analysis is that the assessment of the inside-out versus outside-in quenching scenarios should consider both sSFR and ΣSFR radial profiles, because ample SF may be present in bulges with low sSFRs (red color).


Introduction
The rate at which a galaxy is producing stars (Salpeter 1959;Schmidt 1959) is one of its most fundamental physical properties.Integrated over cosmic time (and summed across the progenitors in a merging tree), the star formation rate (SFR) gives the total stellar mass of a galaxy, which is proportional to the current stellar mass (M * ).The ability to constrain how a galaxy changes in SFR over time-its star formation (SF) history-lies at the heart of the study of galaxy evolution.
The physical characterization of galaxies in terms of SFR and M * on a massive statistical scale became possible with the advent of large galaxy spectroscopic surveys, such as the Sloan Digital Sky Survey (SDSS; Strauss et al. 2002) and GAMA (Driver et al. 2011) at low redshift, and DEEP2 (Newman et al. 2013) and zCOSMOS (Lilly et al. 2007) at higher redshifts, and has benefited from the development of modern stellar population synthesis models (e.g., Bruzual & Charlot 2003) and new analysis techniques, including Bayesian parameter estimation (e.g., Kauffmann et al. 2003;Brinchmann et al. 2004;Salim et al. 2005).
Construction of diagrams that involve the SFR and M * of a large number of galaxies led to new insights and new conceptual frameworks, such as the recognition of a relatively narrow range of SFRs at a given stellar mass-the star-forming main sequence-and its turnoff at higher masses (Brinchmann et al. 2004;Bauer et al. 2005;Elbaz et al. 2007;Noeske et al. 2007;Salim et al. 2007).For galaxies on the main sequence, the SFR to first order increases simply because of the scale of the system, so it is useful to normalize it, the usual choice for this being the stellar mass, yielding the specific SFR (sSFR; Bothun 1982;Tully et al. 1982).SFR-M * diagrams, and the related sSFR-M * diagrams (Guzmán et al. 1997;Pérez-González et al. 2003), have emerged as a sort of equivalent of Hertzsprung-Russell (H-R) diagrams for galaxies.
The sSFR is commonly used as an indicator of the current level of SF.High levels suggest a starburst-a galaxy experiencing an increase in SFR as compared to its baseline value-whereas low values, or in some cases the upper limits, indicate a galaxy in which SF has been quenched.Various galaxy colors are also sometimes used as indicators of SF activity.The sSFR, like the color, essentially represents the contrast between young and old stars, so it combines the current state of SF with its past history.Much effort in recent years, both observational (e.g., Fang et al. 2013;Bluck et al. 2014;Yesuf et al. 2014;Woo et al. 2015;Pacifici et al. 2016;Barro et al. 2017;Martin et al. 2017;Rowlands et al. 2018;Belli et al. 2019;Carnall et al. 2020;Moutard et al. 2020;Tacchella et al. 2022;Yesuf 2022) and theoretical (e.g., Ciambur et al. 2013;Sparre et al. 2015;Dubois et al. 2016;Feldmann et al. 2016;Tacchella et al. 2016;Weinberger et al. 2017Weinberger et al. , 2018;;Behroozi et al. 2019;Davé et al. 2019;Donnari et al. 2019;Walters et al. 2022), has been invested to understand both the general SF history of a galaxy while it is actively forming stars and the processes that lead to its quenching.It would be beneficial for such efforts to use a wide array of measures that illuminate relevant processes.The goal of the present study is to point out the conceptual difference between the two roles that the sSFR is used for (current SF activity, including quenching, versus past SF history) and to propose a way to separate them.
Before proceeding, it is worthwhile to discuss the definition of the words "quenching" and "quenched," since we will often refer to them.As pointed out by Belfiore et al. (2018), Donnari et al. (2019), and others, there is no agreed-upon definition of the word "quenching."The definition of quenching is relatively unambiguous for individual SF histories, especially idealized ones.For them, quenching represents a downward change in SFR with respect to some gradual overall trend.For example, Martin et al. (2007), one of the first studies on quenching, defines it as an exponential decline with a variety of rates (efolding times of 0.5-20 Gyr) following a constant SFR.Given the difficulties in constraining the SF histories of individual galaxies, and the fact that their forms in reality contain complex details (Pandya et al. 2017), a definition that is based on the properties of an ensemble of galaxies would be more useful.Thus, the definition that Belfiore et al. (2018) and many other studies use, and the one we will for the most part use here, is that "quenching" refers to a process that moves galaxies below the main sequence.For a galaxy of a certain mass, being below the main sequence implies a different SF history than that of other galaxies of the same mass-one that currently has a lower SFR.
Using the adjective "quenched" to define a galaxy with no SF can be considered an absolute definition.Note that one can alternatively consider a galaxy as quenched when its SFR no longer contributes to its buildup on some relevant timescales, such as the Hubble time at the epoch of observation.Such relative definition is closely tied to the sSFR (Tacchella et al. 2018), since the inverse of the sSFR represents the time needed to double the stellar mass (modulo gas recycling).A galaxy with log sSFR = −10.1 today will take a Hubble time to double its mass, so one can think of it as having finished most of its assembly and thus in a certain way quenched.If one uses this relative definition of being quenched, then the sSFR is, by definition, the only way to identify quenched galaxies.In this paper, we will assume the absolute definition, where "quenched" means no detectable SF.We acknowledge that depending on the science question the relative definition may be more relevant.
"Quenching" implies that the final outcome of the process is a galaxy wherein SF is no longer present, i.e., the galaxy is quenched.By this definition, a galaxy below the main sequence with measurable SF is not yet quenched, so it will be useful to distinguish between a quenching and a quenched galaxy.This distinction is embodied in the green valley galaxies, which are often considered to be quenching, but are in any case not yet quenched (Martin et al. 2007;Salim 2014).Therefore, we will distinguish, when relevant, between galaxies undergoing quenching and ones fully quenched.
Returning to the sSFR for ensembles of galaxies, its ambiguous interpretation can be illustrated using the sSFR-M * diagram of present-day galaxies.Along the main sequence, the average sSFR declines by ∼1 dex over the range of 3 dex in stellar mass.This drop is not usually interpreted to mean that the massive galaxies are collectively being quenched (or more strongly quenched) than the low-mass ones.Rather, this tilt of the sSFR main sequence is equivalent to the wellknown fact that dwarf galaxies and late-type spirals are bluer in optical color than early-type spiral galaxies such as M31 (e.g., Prugniel & Heraudeau 1998).Both the sSFR trend and the (dust-corrected) color trend are manifestations of the fact that the SF histories of low-and high-mass galaxies on the main sequence differ, in the sense that the more massive galaxies have formed a larger fraction of stellar mass earlier (galaxy "downsizing"; Cowie et al. 1996), which is equivalent to saying that the massive star-forming galaxies have older mean population ages.
The ideas regarding the mass-dependent SF histories of starforming galaxies date back to many decades ago (Epstein 1964;Tinsley 1980;Sandage 1986), with subsequent empirical confirmation that the colors or sSFRs of star-forming galaxies are mass-dependent being provided by Gavazzi & Scodeggio (1996), Gavazzi et al. (1996), Kauffmann et al. (2003), andBrinchmann et al. (2004) and further supported by simulations (e.g., Sparre et al. 2015).The confounding aspect of an sSFR for an ensemble of galaxies arises from the fact that in addition to the sSFR systematically decreasing across the main sequence even in the absence of any process that quenches the SF, it obviously also decreases if the galaxy quenches.
In this work, we wish to address this conceptual ambiguity of the sSFR by considering a measure that more closely represents the current SF level of a galaxy.We explore an SFR normalized by the surface over which the SF takes place, i.e., the integrated SFR surface density (Σ SFR ), as such a quantity.The rationale for using the SFR surface density can be illustrated by the following example.Consider two galaxies having the same size (the area over which SF takes place) and the same SFR (or number of H II regions).Intuitively we feel that we should regard the two galaxies as having the same level of SF activity, and Σ SFR being the same for both galaxies supports that notion.On the other hand, if one of the two galaxies has a more massive disk or a more prominent bulge it would have a lower sSFR than the other galaxy, despite their current levels of SF being the same.
A popular alternative to the sSFR in the context of SF levels is the relative SFR, i.e., the logarithmic offset in SFR from the main sequence (Schiminovich et al. 2007;Elbaz et al. 2011;Wuyts et al. 2011).(By definition, the relative SFR and relative sSFR are identical.)The relative SFR has been designated variously as ( ) log sSFR sSFR MS , D log (s)SFR, or log Δ SFR .As a relative measure, its character is different from that of either the sSFR or Σ SFR .Its importance lies in the fact that it is tied to the definition of quenching, as discussed above, and to its resulting practical use, e.g., for identifying starbursts or quiescent galaxies, so we will analyze it alongside the sSFR and Σ SFR .
Although the SFR surface density is a familiar measure, it is primarily used in the context of its relation with gas surface density, i.e., the Kennicutt-Schmidt relation, either in the integrated (e.g., Kennicutt 1989) or in the resolved sense (e.g., Kennicutt et al. 2007).Less often, Σ SFR is discussed as an indicator of the level of SF activity-but rarely so in the context of large samples of galaxies, notable exceptions being Schiminovich et al. (2007) and Wuyts et al. (2011).In any case, the integrated Σ SFR is not as widespread a measure as the sSFR and warrants closer investigation.Thus, the aim of this paper is to provide a comparative assessment of Σ SFR and the sSFR, as well as the SFR relative to the main sequence, and to discuss the implication of using Σ SFR , not just for integrated, but also for resolved galaxy studies.
The paper is organized as follows.In Section 2 we present our samples spanning three redshift ranges and associated galaxy physical parameters obtained from spectral energy distribution (SED) fitting.In particular, in Section 2.4 we discuss the definition of Σ SFR and differences arising from using different types of galaxy radii.In Section 3 we present a comparative analysis of sSFR-M * and Σ SFR -M * diagrams for different types of galaxies at different redshifts.Section 4 explores some additional implications of the results and places them in the context of previous work.We summarize our main findings in Section 5.

Low-redshift Sample
Our principal sample comes from the GALEX-SDSS-WISE Legacy Catalog (GSWLC; Salim et al. 2016) of galaxy physical properties.The catalog contains galaxies corresponding to the SDSS main galaxy spectroscopic survey (0.01 < z < 0.30 and r < 18; Strauss et al. 2002) that were also covered by the Galaxy Evolution Explorer (GALEX) (Martin et al. 2005) UV surveys.GSWLC contains three catalogs, divided according to the depth of GALEX imaging.In this work we use the second release of the medium-deep catalog (GSWLC-M2; Salim et al. 2018), which offers a good combination of the UV depth required to measure SFRs below the main sequence (transitional, green valley galaxies) and a large sky coverage (around half of the SDSS area).
GSWLC-M2 contains ∼361,000 galaxies at 0.01 < z < 0.30, but we restrict our analysis to ∼175,000 galaxies at z < 0.1, in order to ensure greater reliability of the morphological data and higher UV detection rates.The exclusion of type 1 active galactic nuclei and galaxies with poor SED fits removes an additional ∼2000 objects.GSWLC contains galaxies in the GALEX footprint regardless of whether they were detected in the UV.Leaving only the galaxies with detections in either the far-UV (FUV) or the near-UV (NUV) band (∼86,000) ensures more robust SFRs for the transitional galaxies.A requirement that the galaxy have a measured isophotal size and structural parameters (the data sources of which are based on SDSS DR7, rather than DR10 used for GSWLC) results in a final lowredshift sample of ∼77,000 galaxies.
From this sample, for some analyses we focus on a subset of ∼29,000 main-sequence galaxies, which we define as the emission-line galaxies that fall in the star-forming portion of the BPT diagram based on the Agostino et al. (2021) modification of the Kauffmann et al. (2003) demarcation line.We require signal-to-noise ratios (S/Ns) > 2 on the BPT emission lines, but S/N > 10 on Hβ.
We additionally match our final sample to the Nair & Abraham (2010) catalog of visually determined Hubble types, which results in a smaller sample of ∼4000 galaxies (∼2000 main-sequence ones), and to the Darg et al. (2010) catalog of merger pairs, which yields 639 matches (240 main-sequence ones).For merger pairs, we match our sample to each galaxy of the pair.The match is typically the primary galaxy of the pair (∼85% of cases), because only about a quarter of the secondaries have SDSS spectra.

High-redshift Sample
Our high-redshift sample is taken from the CANDELS survey (Grogin et al. 2011;Koekemoer et al. 2011) of the GOODS-S field.We consider two high-redshift windows: z ∼ 1 (0.8 < z < 1.4) and z ∼ 2 (1.5 < z < 2.5).Redshifts are taken from Guo et al. (2013) and are either photometric or spectroscopic when available (see Santini et al. 2015 for details).The sample is limited to galaxies that appear in the van der Wel et al. (2012) catalog of sizes and structural parameters and have H-band magnitude <24.5.The magnitude cut is beneficial to ensuring the reliability of measurements of the Sérsic indices and effective radii.Our final sample size is 1905 galaxies at z ∼ 1 and 1550 galaxies at z ∼ 2, with 27% and 15% of redshifts being spectroscopic, respectively.

Data
GSWLC-2 provides stellar masses and dust-corrected SFRs derived from the energy-balance SED fitting based on photometry from GALEX, SDSS, and the Wide-field Infrared Survey Explorer (WISE).The models were generated and the fitting was performed using CIGALE (Boquien et al. 2019).The models feature two-component SF histories, as described in Salim et al. (2016), and flexible dust attenuation curves, as described in Salim et al. (2018).
Stellar masses and dust-corrected SFRs for the high-redshift sample are taken from Osborne et al. (2020), derived using SED modeling consistent with that for the low redshift, with some adjustments needed to account for the difference in lookback times.The principal difference is that the highredshift SED fitting did not include dust emission constraints, because IR observations are not available for most of the sample.
All stellar masses and SFRs are based on Bruzual & Charlot (2003) stellar population synthesis models and a Chabrier initial mass function (Chabrier 2003), and are given in units of solar mass and solar mass per year, respectively.
Isophotal sizes for the low-redshift sample are taken from the official SDSS DR7, and are based on 25 mag arcsec −2 AB isophotes.We use r-band minor and major axes, but find that other bands produce equally robust results.The isophote at 25 mag arcsec −2 AB in r is roughly 0.7 mag deeper than the 25 mag arcsec −2 Vega isophote in B. Subsequent SDSS data releases have omitted isophotal sizes, claiming they were unreliable.We find no evidence of any issues.We also use effective (half-light) radii and Sérsic indices from the Meert et al. (2015) catalog of structural parameters, derived using the GALFIT routine and assuming a single Sérsic profile.
Effective radii and Sérsic indices for the CANDELS GOODS-S high-redshift sample are taken from van der Wel et al. (2012), also derived using GALFIT.We use the effective radii and Sérsic indices estimated from the H-band (F160W) imaging, which is roughly comparable to the r band at z ∼ 0. We use the isophotal area from the Guo et al. (2013) photometric catalog ISOAREAF, also based on F160W, to which we apply a redshift and a zero-point correction (Section 2.5).
Visual morphological classification into Hubble types is taken from the catalog of Nair & Abraham (2010), based on SDSS images.The redshift coverage of the catalog is the same as that of our principal sample (0.01 < z < 0.10), but it contains far fewer objects (∼14,000) because it targets brighter galaxies (g < 16), and is based on DR4, which has 60% of the area of DR7.We also utilize the sample of ∼3000 merger pairs from Darg et al. (2010), identified based on the Galaxy Zoo visual classification of SDSS DR6 images (85% of the DR7 area).The parent sample for this catalog has essentially the same redshift and magnitude selections as our primary sample.
Emission-line data needed for the selection of low-redshift main-sequence galaxies come from the MPA/JHU processing of SDSS DR7 spectra, which follows the methodology of Tremonti et al. (2004).We verify some of our results using the MPA/JHU SFRs and stellar masses, which were derived following the approach of Brinchmann et al. (2004).

SFR Surface Density
SFR surface density is defined as where A is the physical (not the projected) area of the galaxy.In the case of an effective radius, the area is taken to be where R eff,a corresponds to the semimajor axis (a) of the ellipse containing half of the galaxy light, and the factor of 2 is meant to qualitatively take into account that the effective radius includes half of the galaxy light.Note that many studies, including Meert et al. (2015), define effective radius as the geometric mean of the semimajor (a) and semiminor (b) axes, the so-called circularized effective radius (R eff ).In that case p R eff 2 is not the physical area but the projected area (abπ), and we can introduce a projection factor ( f proj ) to go from the projected to the physical area: We determine f proj empirically for the SDSS sample to be 1.26 (0.1 dex).
Similarly, for Σ SFR based on the isophotal ellipse we use where the (circularized) isophotal radius (R iso ) is the geometric mean of the apparent isophotal major and minor axes (given as isoA and isoB in SDSS DR7) and is converted from angular to physical size using scale s(z): The rationale for using circularized sizes is that the linear extent (a) is less robustly constrained than ab .For galaxies without well-defined disks, such as irregular galaxies and post-mergers, using the isophotal ellipse may not be the most accurate way to establish the area.In such cases one can obtain the area as the sum of the area of the pixels that lie above some surface brightness threshold.Photometry software SExtractor (Bertin & Arnouts 1996) provides such measurements as ISOAREAF and ISOAREA.This area should replace p R iso 2 in Equation (4).

Obtaining Isophotal Size from Effective Size and Stellar Mass
As we will show, Σ SFR is a more precise indicator of SF activity if derived using the isophotal size (or area), rather than the effective (half-light) size (Section 4.1).However, the isophotal size or area is often not reported in photometric catalogs.Or, if it is, it may be based on different thresholds.To overcome these practical issues, we devise a transformation from effective to isophotal radius, which we calibrate using our low-redshift, SDSS sample, and validate using CANDELS 0.8 < z < 2.5 data.
The left panel of Figure 1 shows that the correlation between isophotal and effective physical radii is strongly nonlinear and has a nonuniform scatter, suggesting that a direct conversion from effective to isophotal radii would be neither accurate nor precise.Fortunately, there is a way to improve matters using the stellar mass.The middle panel of Figure 1 shows that the isophotal radius is also correlated with the stellar mass, though not much better than with the effective radius, with the formal scatter around the best fit being similar-0.10versus 0.11 dex.However, it turns out that R iso is very well correlated with the combination of the effective radius and stellar mass, with a scatter of just 0.05 dex: where the sizes are in kiloparsecs and the stellar mass is in units of solar mass.The isophotal size corresponds to 25 mag AB in r.The coefficients in Equation ( 6) are determined from a linear regression.In other words, stellar mass, isophotal radius, and effective radius form a relatively tight 3D plane that contains both early-and late-type galaxies.The right panel of Figure 1 shows that R iso predicted from this relation are reasonably unbiased.We test expanding the calibration to include additional parameters (the Sérsic index or SFR), but further gains are very small (2% reductions in scatter).We expect the observed isophotal size to be affected by cosmological surface brightness dimming.Indeed, when applying Equation (6) to CANDELS, we find the difference between the predicted and observed isophotal radii to be strongly redshift dependent.This allows us to construct a relation to correct the observed isophotal size of a galaxy at redshift z to its value at z ∼ 0: Note that the relation is not tested at z > 2.5.Finally, we check the validity of our calibration for higher redshifts by comparing the predictions from Equation (6) to the actual, redshift-corrected isophotal sizes in CANDELS (0.8 < z < 2.5).The scatter between real and predicted sizes is 0.09 dex, in contrast to the 0.17 dex between the isophotal and effective radii.Comparison reveals a small zero-point offset (0.033 dex), arising from the different surface brightness thresholds used in SDSS and CANDELS.
One potential concern is that some of the reduction in scatter in the Σ SFR main sequence when using isophotal sizes derived from Equation (6) as compared to effective sizes could be the product of covariances introduced by the calibration itself.To test this, we compare the width of the SDSS Σ SFR main sequence when using the isophotal sizes derived from Equation (6) and when using the real Σ SFR .The latter is smaller (0.31 dex versus 0.28 dex), suggesting that covariances do not affect it.

Results
In this section we explore, via sSFR-M * and Σ SFR -M * diagrams, the differences between sSFR and Σ SFR as indicators of the degree of SF activity or quiescence.The analysis first focuses on the low-redshift sample and its various subsamples before turning to higher redshifts and the comparison to low redshift.

Low-redshift Main Sequence
The left panels of Figure 2 show the sSFR-M * diagram for the low-redshift sample.The upper panel contains all the galaxies in the sample, spanning 3.5 dex in stellar mass ( * < < M 8 log 11.5) and some 5 dex in sSFR (−13.5 < log sSFR < −8.5).Higher-mass star-forming galaxies tend not to reach sSFR values that are as high as the sSFRs of low-mass galaxies.This tilt of the main sequence can be seen more clearly in the lower left panel, which shows only the galaxies selected as star-forming in the BPT diagram.The tilt in the sSFR-M * main sequence, i.e., that α < 1 in SFR * µ a M , is a robust feature that exists irrespective of the choice of SFR indicator (Appendix).
As discussed in Section 1, the tilt of the main sequence is an indication of mass-dependent SF histories and not of any change that would suggest quenching (a downward departure from an overall trend in SF history).However, an actual quenching would also lower the sSFR, introducing an ambiguity in the interpretation of the sSFR.Σ SFR does not have this ambiguity and it should tell us the current level of SF activity.Therefore, we now take a look at the lower panels of Figure 2, allowing us to contrast the appearance of the main sequence in sSFR-M * and Σ SFR -M * diagrams.The two panels span the same dynamic range (6 dex) in the y-direction.The surface area used to normalize the SFR is based on isophotal sizes.This choice will be discussed in Section 4.1.Most notably, we see that the main sequence no longer has a downward tilt, and is quite flat (the standard deviation of the mass-binned averages of Σ SFR is 0.07 dex and the maximum amplitude of the binned averages is 0.21 dex).There are possible breaks at * = M log 9.0 and 10.4, which, by the way, are not at the same exact masses as the breaks in the sSFR-M * main sequence ( * = M log 9.1 and 10.0).These subtle features aside, the first conclusion we draw in this study is that the SF level of present-day star-forming galaxies is remarkably constant across the stellar mass.
The defining feature of the sSFR-M * (or, equivalently, the SFR-M * ) main sequence is that it is relatively tight.We find the widths of both the Σ SFR -M * and the sSFR-M * main sequence to be the same (the average of the standard deviations in the mass bins is 0.28 dex).This is despite the fact that the measurement error of Σ SFR must be greater than that of the sSFR because it includes the error on galaxy size.

Low-redshift Starbursts
Next we explore how well Σ SFR identifies starburst galaxies compared to the sSFR.To answer this, we need a sample of starbursts that are not identified using any SFR-related measure, because such reasoning would obviously be circular.That includes selecting by the birth parameter b = SFR/ <SFR>, which is equivalent to an sSFR selection for a fixed galaxy age.Instead, here we utilize the fact that galaxy interactions can lead to an enhancement of SF and therefore such galaxies are more likely to be starbursting (e.g., Barton et al. 2000;Patton et al. 2013).
In Figure 3 we overplot on the general main-sequence sample the subsample of such galaxies identified visually in Darg et al. (2010) as merging pairs.We include all pairs regardless of the merging stage (separated, interacting, and approaching post-merger).The great majority of this sample are identified as being in the interacting stage.The left panel shows the sSFR-M * diagram.We see that the merging galaxies are indeed offset from the main sequence, with a mass-binned average difference of 0.29 dex (a factor of 2.0 enhancement, the same as that found in Osborne et al. 2020 for galaxies out to z ∼ 2, and in Patton et al. 2013 for pair separations <30 kpc).The plot shows that what makes a starburst a starburst is the relative enhancement in SF-selecting starbursts on some fixed sSFR value is clearly not justified.The offset between the interacting galaxy main sequence and the overall main sequence is not strongly mass dependent.From this it is immediately clear why the relative (s)SFR provides a much more meaningful way to select starbursts than any sSFR cut.The width of the interacting galaxy main sequence (0.36 dex) is not much larger than that of the general main sequence (0.28 dex), suggesting a relatively uniform degree of SFR enhancement, having a standard deviation of 0.22 dex.Note that this standard deviation of SFR enhancement is likely The lower panels in both figures show the subset of galaxies that form the main sequence.The character of the main sequence changes when going from sSFR to Σ SFR -it becomes flatter, while remaining tight (width, in dex, given in the panels).Once the measure of SF activity is decoupled from the past SF history by using Σ SFR , the SF level is not strongly mass dependent.The merger pairs serve as proxies for starbursts and come from Darg et al. (2010).The merging galaxies have a similar average offset (blue line) from the ridge of the overall main sequence (red line) in both cases; however, because there is no mass dependence, selecting starbursts by the SFR surface density is obviously much cleaner than doing so based on the sSFR.partially suppressed by the SFR averaging timescale that our measure of SFR employs (∼100 Myr) being longer than the starburst timescale (∼50 Myr; Wuyts et al. 2009;Tacchella et al. 2020).
Turning our attention to the right panel of Figure 3, we see that the Σ SFR main sequence of interacting galaxies is flat, and that it shows a similar offset with respect to the general main sequence, with a mass-binned average difference of 0.21 dex (a 60% enhancement).The width of the interacting galaxy Σ SFR main sequence is even more similar to that of the overall Σ SFR main sequence (0.32 dex versus 0.28 dex), implying a standard deviation of the enhancement in Σ SFR of just 0.13 dex.From the flatness of the interacting galaxy Σ SFR main sequence we conclude that the selectivity of Σ SFR to starbursts must be at least as good as that when using the relative SFR.We confirm this more directly by comparing ( ) log sSFR sSFR MS to log Σ SFR , and finding that the Darg et al. (2010) star-forming mergers form an even sharper lower boundary in SFR surface density (at log Σ SFR = −3.0)than in the relative SFR.Selection of starbursts using a threshold in Σ SFR is not a new thing (Kennicutt & Evans 2012), but its preference with respect to other methods is not universally recognized.

Low-redshift Transitional/Quenched Galaxies
Before becoming fully quenched, galaxies must transition, so that their sSFRs are significantly lower than on the main sequence, but not yet entirely devoid of SF.It should be noted that observationally establishing a complete absence of SF is challenging.Although not formally having an SFR of zero (which the SF history parameterization used in the SED fitting does not even allow), galaxies with log sSFR < −12 typically show no evidence of star-forming regions in UV images and can be considered as fully quenched for all practical purposes.
We next ask how well Σ SFR separates actively star-forming galaxies from those that have or are experiencing quenching, i.e., transitional galaxies.As in the case of starbursts, we cannot define quenching galaxies using any of the measures that we wish to evaluate (sSFR, relative SFR, and Σ SFR ).Instead, we will take advantage of the fact that transitional/quenching galaxies, unlike those that are not quenched, all have prominent spheroidal components.Qualitatively, this means that we expect transitional galaxies to be dominated by early-type galaxies (ellipticals, lenticulars, and early-type spirals).Quantitatively, we expect the profiles of the galaxies to be more highly concentrated, as reflected in their Sérsic indices being higher.
In Figure 4   S0s, with the former being more dominant at higher mass.Interestingly, the main-sequence spirals can be as massive as any but the most massive of the early-type galaxies.A trend similar to that with the Hubble type is seen with respect to the Sérsic index, where we see a significant increase in a typical Sérsic index starting just below the main sequence.Note, however, that at any point in the sSFR-M * plane there exists a range of Sérsic indices and Hubble types, i.e., the trends are there, but are not very tight.This confirms the finding of Wuyts et al. (2011) that galaxies are not just a two-parameter (SFR, M * ) family.
We can see that the "threshold" for the morphological transition in sSFR-M * is somewhat tilted, following the tilt of the main sequence.This suggests that, as in the case of starbursts, it is the relative SFR that provides a cleaner morphological distinction than the sSFR, as previously pointed out by Wuyts et al. (2011).On the other hand, in Σ SFR -M * (Figure 4, upper right panel) the transition in Hubble types is essentially flat, i.e., it happens at a fixed Σ SFR .This transition in Σ SFR is as sharp as that for the relative SFR.We confirm this quantitatively by determining the interval over which the fraction of early-type galaxies (T 0) increases from the level typical for the main sequence (12%) to the level typical among quiescent galaxies (93%).This interval is 1.0 dex for both log Σ SFR and ( ) log sSFR sSFR MS , compared to 1.2 dex for log sSFR.We also fit a logistic function to the fraction of earlytype galaxies versus the parameter, and the slope is steepest with respect to Σ SFR .Similar results are found regarding how sharply the fraction of galaxies with a high Sérsic index (n > 2.5, corresponding to classical bulges; Drory & Fisher 2007) rises as a function of decreasing sSFR, relative SFR, or Σ SFR .To conclude, Σ SFR produces a cleaner separation between late-and early-type galaxies than the sSFR, and as clean a separation as that produced by the relative SFR.

High-redshift Main Sequence
Our attention is now turned to higher-redshift samples and the evolution between z ∼ 2 and z ∼ 0. In the upper panels of Figure 5 we show the sSFR-M * diagrams at z ∼ 0, 1, and 2. We see that the fraction of high-redshift (z ∼ 1 and 2) galaxies below the main sequence is much smaller than that in SDSS, so for clearer comparison we only show the main-sequence sample for low redshift.We first note that there is not much evolution in sSFR-M * between z ∼ 2 and z ∼ 1-the slope and the normalization of the main sequence are similar.Actually, the slope of the main sequence is not very different at low redshift either (−0.34 at z ∼ 0 compared to −0.27 at z ∼ 1 over the same mass range, * < < M 9 log 11), as noted already in Noeske et al. (2007).However, as expected, the normalization For low-redshift samples we show only the main-sequence galaxies.In the sSFR-M * diagrams the slope of the main sequence stays more or less the same, whereas the Σ SFR -M * diagrams reveal a change-there was higher SF activity of massive galaxies compared to less massive ones in the past, consistent with a bulge buildup of those systems.Dashed lines in the lower panels signify the upper boundary of the low-redshift transition region.At higher redshift this boundary is higher and would have encompassed today's main-sequence galaxies.The dotted lines represent the threshold for full quenching, which is independent of the redshift.
today is significantly lower-0.9dex with respect to z ∼ 1.So, based on the invariant slopes of the main sequence in the sSFR (or, alternatively, just the SFR) one might conclude that SF is less active at all masses by a similar degree.
The picture looks different with Σ SFR replacing the sSFR (lower panels of Figure 5).Again, there seems to be little evolution between z ∼ 2 and z ∼ 1 in terms of the slope and the normalization of the Σ SFR main sequence.However, unlike the case of sSFR-M * , we see a change between low and high redshift, not only in the normalization, but also in the slope of the main sequence.The slope is slightly positive at z ∼ 0 (0.11), whereas it is 0.25 at z ∼ 1.So, based on the main sequence in Σ SFR we conclude that the level of SF activity since "cosmic noon" has dropped more for massive galaxies (9× for * < < M 10.6 log 11.0) than for low-mass ones (5× for * < < M 9.0 log 9.4).

High-redshift Transitional/Quenched Galaxies
Here we address a question: Can the distinction between a non-quenched and a transitional galaxy be based on a redshiftinvariant parameter?We know that the sSFR does not provide this, because the normalization of the main sequence changes, and the lower panels of Figure 5 show that neither does Σ SFR -the Σ SFR main sequence also changes in redshift, as discussed in the preceding section.In other words, at a given stellar mass a typical high-redshift galaxy will have both a higher SFR and a higher SFR per surface area compared to a present-day galaxy.
To further explore if it is justified to consider a galaxy with the same M * and the same SFR (or Σ SFR ) as quenching at one redshift and not quenching at another, we again look at the structural properties.Figure 6 presents, for the same three redshift bins, the Sérsic index as a function of sSFR (upper panels) and Σ SFR (lower panels).At z ∼ 0, the average Sérsic index for a galaxy at log sSFR = −10.2(main-sequence at that redshift) is n = 1.3-an exponential disk.On the other hand, the average Sérsic index at that same sSFR at z ∼ 1 is n = 3.0, typical of early-type galaxies today.A similar result is obtained if considering Σ SFR .From this we confirm the Wuyts et al. (2011) conclusion that the transitional/quenching status is indicated by the SFR (and in our case also by Σ SFR ) relative to the main sequence at that redshift, and not by any absolute threshold in either the sSFR or Σ SFR .
Whereas the criterion for the onset of quenching remains tied to the position with respect to the main sequence for either the (s)SFR or the Σ SFR main sequence, the achievement of full quiescence is still meaningful as a fixed, redshift-independent Figure 6.Trends of the Sérsic index as a function of sSFR (upper panels) and Σ SFR (lower panels), at different redshifts.Galaxies with today's values of sSFR or Σ SFR characteristic of the main sequence would be found below the main sequence at high redshift and, like transitional galaxies today, would have, on average, a high Sérsic index.This suggests that morphological transformation is associated with a relative suppression of sSFR or Σ SFR , but to values that would still be considered quite high by today's norms.The red dashed line corresponds to an n = 4 de Vaucouleurs profile, and the blue line corresponds to n = 2.5, an approximate dividing line between galaxies with and without a central spheroid.Shown are galaxies with * < < M 10 log 11.Curves represent the 16th percentile, the mean, and the 84th percentile in each x-axis bin.Curves at z ∼ 1 and 2 are smoothed to mitigate low numbers of objects.
threshold in Σ SFR , but not in sSFR or relative SFR.The reason for this is again that unlike the sSFR, Σ SFR is not the young-toold population contrast, but an absolute measure of the young population.Similar ambiguity regarding the threshold for full quiescence would apply to a color, or Hα equivalent width, as both contrast the young and old populations.Locally, the bottom boundary of the transitional region appears to be around log Σ SFR = −4.5.Establishing the lowest Σ SFR for transitional galaxies is difficult because of the difficulties involved with measuring very low levels of SF, so we consider this threshold as provisional.If we take this threshold to be redshift independent, Figure 5 shows that very few galaxies at z ∼ 1 and no galaxy at z ∼ 2 fall below it.It should be noted however that constraining these low Σ SFR values, especially at high redshift, is challenging and sensitive to the assumptions regarding the SF histories used in the SED fitting, so it is difficult to know for sure if any of them fall below the log Σ SFR = −4.5 threshold.
We should also point out that the Σ SFR radial profiles of galaxies typically decline toward the outskirts (e.g., Gil de Paz et al. 2007), which means that any threshold that we want to attach to full quiescence will depend on the size used for Σ SFR .If we were to use the effective radii instead of the isophotal ones, the proposed threshold for quiescence would be ∼1 dex higher (log Σ SFR = −3.5).Similarly, the Σ SFR defining the lower envelope of the main sequence (the threshold for the onset of the transitional region) will be different depending on the size of the aperture used to obtain Σ SFR .
There are several additional things one can infer from Figure 6.
1.At all redshifts, galaxies with high Sérsic indices are common on the main sequence, but the reverse is not true -there are few low-n transitional/quenched galaxies.Indeed the lower threshold for the Sérsic index off the main sequence (n ≈ 2.5) coincides with the demarcation between galaxies that contain a central spheroid (a classical bulge) and ones that do not (Drory & Fisher 2007).This confirms previous results regarding the structure of quenching or quenched galaxies (Bell 2008;Mosleh et al. 2017) and that compaction starts on the main sequence (Cheung et al. 2012;Barro et al. 2017).2. The low-redshift sample shows that the galaxies above the main sequence are on average more compact (having 1.25 times higher n) than those on the main sequence.A similar trend has been reported in Schiminovich et al. (2007) and Wuyts et al. (2011), which is consistent with a compaction proceeding via a starburst stage (Schiminovich et al. 2007;Tacchella et al. 2016;Lapiner et al. 2023).

The transitional region has a 68th percentile range of
Sérsic indices similar to that of the main sequence or the quiescent region, i.e., it is inconsistent with being a mix of the tails of two populations, as such mixing would widen the distribution in the transitional region.

Relevant Measure of Galaxy Size for Σ SFR
It is common in the literature to obtain the global SFR surface density either using the effective (half-light) radius or using the isophotal radius.We find that the choice of measure is very important.In Figure 7 we show the Σ SFR -M * diagram for our low-redshift samples in which Σ SFR is based on the effective radius (Equation ( 3)).This figure is to be compared to the right panel of Figure 2, which is based on the isophotal radius (Equation ( 4)).As expected, the absolute values of Σ SFR are higher, because the isophotal sizes are about 3 times larger on average than the effective ones (Figure 1).Nonetheless, the relatively flat shape (and even the details of the inflections) of the Σ SFR main sequence remains.
What has drastically changed is the the width of the Σ SFR main sequence.Its scatter around the mean is 0.50 dex, compared to 0.28 dex for the main sequence based on the isophotal radius.Furthermore, Σ SFR based on the effective radius is less well able to distinguish between early-and latetype galaxies.Overall, it is a much noisier measure of the level of SF activity than Σ SFR based on the isophotal radius.This worse performance cannot be due to the differences in precision of the measurements of two sizes.If anything, measurements of the effective size are more robust than those  2, which uses the isophotal sizes, our nominal choice.Effective-radius-based Σ SFR are significantly noisier, leading to a broader main sequence (width, in dex, given in the lower panel).The effective radius, being dependent on the galaxy light profile, is not as good an indicator of the extent over which SF is (or was) taking place as is the isophotal radius.
of the isophotal one (Chamba 2020;Trujillo et al. 2020).The effective radii used in this exercise are from the Meert et al. (2015) catalog.An alternative source of SDSS effective sizes, the catalog of Simard et al. (2011), yields similar results (mainsequence scatter of 0.48 dex).Looking at yet other measures of galaxy size, both the Meert et al. (2015) and Simard et al. (2011) catalogs provide estimates for the scale lengths of the disk components alone.However, Σ SFR based on these disk scale lengths (which are proportional to the disk effective radii) produce small or no improvement in terms of the mainsequence scatter as compared to the effective radii of full galaxies.
We propose that the fundamental reason why isophotal sizes produce more meaningful Σ SFR lies in the fact that they better reflect the extent over which the SF is taking place.Isophotal and effective sizes are actually very different measures.To first order the isophote reflects a certain mass density threshold, and is therefore related to the extent over which SF happens.On the other hand, the effective radius is sensitive to the galaxy's profile, i.e., its structure.This can be illustrated by the following example.Take two identical bulgeless disk galaxies.Their isophotal areas are the same.Now let us place in one of them a massive compact bulge with no ongoing SF.The effective radius of that galaxy shrinks, leading to a higher Σ SFR if it is based on the effective size, even though nothing has changed in terms of the extent over which SF takes place.The isophotal size and Σ SFR based on it, however, are not affected by this addition of a bulge.
Recently, Trujillo et al. (2020) have argued against the effective size, on the grounds that it depends on the light profile, and illustrated their point with an example similar to the one above.Instead, they proposed an "isomass" measure of size based on the mass density threshold of 1 M e pc −2 .Isophotal sizes are a good proxy for isomass sizes, as confirmed by Tang et al. (2020), who found that the optical colors of different galaxies at 25 AB mag arcsec −2 in the g band are quite uniform, implying, via the color-M/L relation, that such an isophote is a good tracer of stellar mass surface density.Our nominal isophotal size is based on 25 AB mag arcsec −2 in the r band, which is somewhat deeper than the traditional 25 Vega mag arcsec −2 in the B band.We find that the isophotal sizes based on the u and g bands produce almost identical results to r in terms of the width of the main sequence and the ability to distinguish early-type galaxies, even though they correspond to somewhat smaller (0.22 dex and 0.06 dex, respectively) physical sizes than the r-band isophote.Conceptually, it is not clear that one should aim to use the area based on very short wavelengths.Tying the SFR surface density to an area over which SF is or could be taking place (if gas is present), which is achieved by using optical sizes, allows us to incorporate both actively star-forming and quiescent galaxies (for which SFRs are essentially upper limits and UV sizes would be meaningless) into a single scheme.
We conclude that any sort of "iso" size (isophotal or isomass radii) will be more appropriate as a basis for Σ SFR as an indicator of SF activity than an effective size.Indeed, the original global Kennicutt-Schmidt relation is based on the Σ SFR from the isophotal size (Buat et al. 1989;Kennicutt 1989), and the rationale for this choice given in Kennicutt (1998) was that the isophotal radius is comparable to the extent of the active SF disk in Hα.For these and other reasons that suggest that isophotal sizes are better behaved and provide tighter scaling relations (e.g., Saintonge & Spekkens 2011;Tang et al. 2020), future surveys should aim to include them in their catalogs.If that is not possible (for example due to the difficulties arising from cosmological dimming), a viable alternative would be to estimate the isophotal size from the combination of the effective size and the stellar mass (Section 2.4).Indeed, using this calibration to infer isophotal sizes essentially recovers the tightness of the Σ SFR main sequence (0.30 dex, versus 0.28 dex with the actual isophotal size).

Toward a More Physical Measure of SF Activity
Being tied to the gas densities and the effectiveness of stellar feedback, Σ SFR may be considered as a move toward a more physical measure of current SF activity/quiescence.In that sense, a switch to Σ SFR aims to provide further conceptual and practical improvements, similar to those that the sSFR has with respect to the use of optical color.As illustrated in the left panel of Figure 8, the optical color of actively star-forming galaxies is even more strongly affected by the age of the stellar populations than the sSFR, and produces a steep tilt when plotted against the mass.Furthermore, the optical color has very poor sensitivity to low relative levels of SF (e.g., Kauffmann et al. 2007), which results in an inability to distinguish between transitional and fully quenched galaxies (Salim 2014).As a result, even massive main-sequence galaxies have optical colors nearly as red as those of earlytype galaxies (Cortese 2012), affecting the quenched fraction estimates based on optical color (Figure 8, left panel) and making it appear as if there were no, or very few, massive starforming galaxies.UV-optical colors overcome many of the limitations of optical colors (middle panel of Figure 8), but are subject to dust and metallicity effects (which can somewhat be mitigated by combining the UV-optical color with an opticalnear-IR color, e.g., in an NUV-r-J diagram; Arnouts et al. 2013).The sSFR determined from the SED fitting effectively utilizes a range of UV-optical-near-IR colors but, being constrained using models that include dust and metallicity effects, is not subject to them.As a result, it provides a cleaner separation between star-forming, transitional, and quenched galaxies (Figure 8).Finally, Σ SFR improves over the sSFR by removing the age effect, which through M * , or the red optical band, is present in all previous measures discussed here.Kennicutt & Evans (2012) considered Σ SFR as one of two ways to normalize the SFR, the other being the sSFR.They also commented that the range of Σ SFR for normal (nonstarburst) galaxies is relatively small and in that sense similar to the range of sSFRs.More quantitatively, we see that the observed Σ SFR main sequence has the same scatter as the sSFR main sequence (0.28 dex, Section 3.1), and intrinsically, the Σ SFR main sequence may be even narrower than the sSFR one.Namely, since we measure the scatter in small mass bins, the only contributor to sSFR measurement uncertainty is that of the SFR.On the other hand, Σ SFR has measurement uncertainties from both the SFR and the isophotal area.This is an indication that Σ SFR may be a more physical measure of current SF activity than the sSFR.
Furthermore, replacing the sSFR with Σ SFR almost entirely removes the downward tilt of the main sequence at low redshift and even results in a slight upward trend (Figure 2).The downward tilt of the main sequence in sSFR-M * is essentially the result of SF histories being dependent on the mass, and is unrelated to the current SF level (see Section 1).On the other hand, the main sequence based on the SFR surface density makes the character of the ongoing SF more uniform for galaxies of different masses.The relative constancy of Σ SFR across the main sequence was first pointed out by Schiminovich et al. (2007), who called the result "intriguing."That result has not received much attention and, as far as we are aware, has not been the focus of any theoretical work.We now find that at higher redshifts Σ SFR actually rises with mass.This is consistent with the rapid growth of the central mass concentration (the bulge) in more massive galaxies, but less so in present-day dwarfs and late spirals.The slight upward tilt of the Σ SFR main sequence at z ∼ 0 may suggest that there is still some in situ bulge buildup in massive star-forming galaxies.We agree with Schiminovich et al. (2007), who concluded that the redshift evolution lies fundamentally in Σ SFR .Indeed, a galaxy that maintains a constant SFR will have a progressively dropping sSFR by definition.Considering that Σ SFR provides complementary information to the (s)SFR, we propose that the Σ SFR -M * scaling relation be included among the benchmarks for galaxy simulations.
A flattening of the main sequence can to some degree be produced by the normalization of the SFR not by the total stellar mass, but by only the disk stellar mass, as proposed and shown by Abramson et al. (2014).An underlying assumption behind this modification is that the bulge represents an inert component that is not associated with the current SF.We confirm with our low-redshift sample that replacing the nominal sSFR with the disk-only sSFR, obtained by multiplying our total stellar mass by the disk-to-(disk+bulge) mass ratio from the n = 4 + 1 decompositions of Mendel et al. (2014), reduces the tilt of the main sequence from −0.35 to −0.20 ( * > M log 8).The disk sSFR main sequence is broader than the nominal sSFR one (0.36 dex versus 0.28 dex), most likely because of the greater uncertainties in deriving the stellar mass of the disk component compared to those of the total stellar mass.Disk-bulge decompositions are especially challenging based on SDSS images.By using the disk mass to normalize the SFR, this measure becomes partially decoupled from the past SF history and therefore achieves similar aims as Σ SFR .One conceptual advantage of Σ SFR is that it allows purely spheroidal galaxies with no disk to be encompassed by the scheme.
Many studies nowadays use the relative SFR as a principal variable of the analysis.The relative SFR will by construction flatten the sSFR main-sequence tilt.It was introduced by Schiminovich et al. (2007) for the very reason of eliminating the dependence of the sSFR on M * .There is no ambiguity that in the relative sense (for galaxies of fixed mass) galaxies with high relative SFRs can be considered as experiencing a current burst, whereas galaxies with low relative SFRs have experienced or are experiencing quenching and have a diminished current capacity to form stars.Our analysis shows that the relative SFR has an ability to identify starbursts and early-type galaxies that is comparable to that of Σ SFR .Its nonoptimal aspect is that it is defined relative to a main sequence that needs to be observationally established, which is by no means unambiguous, especially at the massive end, where the main sequence blends with the turnoff.More importantly, by referring to SFRs in relative terms, we are in a way giving up on the idea that there is a physical quantity that describes the SF level.Interestingly, Schiminovich et al. (2007) said that the physical basis for the introduction of the relative SFR is its correlation with Σ SFR .
The global (integrated) SFR surface density is not commonly considered in studies of galaxy evolution outside of the context of the Kennicutt-Schmidt relation.For example, SF history is usually defined as the change in SFR over time.Lehnert et al. (2014), on the other hand, discuss the evolution of the Milky Way in terms of the change in Σ SFR (which they call SF intensity; like Lanzetta et al. 2002 andBoquien et al. 2010).They note that it is Σ SFR that determines the role of stellar feedback in outflows and in the mass-metallicity relation.
Likewise, Σ SFR -M * featuring the global SFR density is a rarely used diagram.Kelly et al. (2014) and Lunnan et al. (2015) used it to compare SF properties of long gamma-ray bursts and superluminous supernova hosts to those of other supernova hosts (and found them to be elevated.)Tran et al. (2017) used it to compare field and cluster galaxies at z ∼ 2, and described Σ SFR as the intensity of SF.Förster Schreiber et al. (2019) showed the Σ SFR -M * diagram of 600 galaxies at 0.6 < z < 2.7 color-coded by incidence of outflows.The incidence follows Σ SFR remarkably well (as pointed out in Heckman 2002;Newman et al. 2012), and somewhat better than the main-sequence offset (Figure 7 of Förster Schreiber et al. 2019).Interestingly, their Σ SFR -M * main sequence shows an upward tilt similar to what we see in the z ∼ 1 panel of Figure 5. Panels from left to right demonstrate the successive improvements in the ability to separate these categories afforded by moving from optical to UV-optical color and then to sSFR.Specific SFRs provide a good separation between these categories, but the use of Σ SFR (based on which these categories have been defined) goes an additional step by eliminating the effect of age on the main sequence (its tilt).

Implications for Studies of Resolved SF
The advent of integral field unit (IFU) spectrographs and associated surveys, such as MaNGA (Bundy et al. 2015), CALIFA (Sánchez et al. 2016), andSAMI (Bryant et al. 2015), has shifted the focus from general considerations of the global SF level to trying to understand the processes of SF regulation on spatially resolved scales.The most common aspect of IFU studies concerns the radial profile of SF activity and the question of the dynamics of the quenching process, such as the inside-out versus the outside-in scenarios (e.g., Tacchella et al. 2015;Belfiore et al. 2018;Lin et al. 2019).
One can imagine making two types of radial profiles that involve SFR-related quantities.One is the sSFR radial plot, where the SFR in a radial bin is divided by the stellar mass in that bin (sometimes designated as Σ sSFR ), and another is the Σ SFR radial plot, where the SFR is divided by the physical area of the radial bin.Here we wish to point out that the sSFR radial profile is subject to the same ambiguities as the global sSFR in the sense that the sSFR depends on both the current SF level and the past SF history.Consider an example of a galaxy with a prominent bulge, i.e., a large stellar mass concentration.The profile of such a galaxy could be redder in the bulge area, corresponding to a dip in the sSFR profile.However, a lower sSFR because of the substantial mass does not imply that SF levels are suppressed.SF may actually be present in the bulge at the same or higher levels than those further out in the disk (Tacchella et al. 2018).There is no question that in such a case the current SF in the central region contributes relatively little to the stellar mass compared to when the bulge was being built up, but that relative change does not imply that any active quenching is taking place now, rather than just a gradual decline.Thus, for galaxies that have red (low-sSFR) centers but a significant amount of SF, the more neutral term may be inside-out "buildup" (e.g., Lilly & Carollo 2016;Nelson et al. 2016;Belfiore et al. 2018), rather than inside-out "quenching." We illustrate this point with a specific example in Figure 9, where we take advantage of the sSFR and Σ SFR determined from dust-corrected Hα within the SDSS spectroscopic fiber to probe central quantities.The galaxy on the left (LEDA 1793) has a very high central sSFR, appearing as a blue compact dwarf, whereas the sSFR is significantly lower for the galaxy on the right (UGC 5965).UGC 5965 has a distinctly red bulge in the SDSS image.The UV images from GALEX paint a different picture.UGC 5965 reaches the highest UV brightness in the bulge.As a matter of fact, the UV surface brightness appears to be similar in UGC 5965 and LEDA 1793, as corroborated by the nearly identical Σ SFR in the fiber.

Conclusions
Replacing the (s)SFR with Σ SFR in the galaxy "H-R diagram" has given us a different perspective regarding the character of SF on and off the main sequence, and its evolution.The main findings are as follows: 1.The SFR surface density (Σ SFR ) is largely insensitive to past SF history and thus provides a measure of the current global star-forming level of a galaxy tied to its molecular gas density.This is in contrast to the sSFR, a relative measure of young-to-old populations.2. Σ SFR provides a cleaner separation of likely starbursts and spheroid-dominated (early-type) galaxies than the sSFR.Its selection power is comparable to that of the SFR offset relative to the main sequence.3. The Σ SFR of main-sequence galaxies at low redshift is essentially mass-independent.Dwarfs and highmass spiral (disk) galaxies have very similar levels of star-forming activity.This was first pointed out in Schiminovich et al. (2007).4. The Σ SFR -M * main sequence at z  1 is tilted upward: high-mass galaxies have ∼2× higher SF levels than lower-mass galaxies, possibly reflecting a rapid buildup of a central mass concentration (bulge).Such a trend is not seen in SFR-M * or sSFR-M * diagrams, the slope of which does not evolve much between z = 0 and z = 2.Because of this complementarity, we propose that the Σ SFR -M * scaling relation be included among the benchmarks for galaxy simulations.5. We confirm that galaxies that fall below the main sequence at high redshift are structurally similar to quenched galaxies today (Wuyts et al. 2011).However, the Σ SFR values of many such high-redshift galaxies are as high as those of main-sequence galaxies today.A highredshift galaxy can drop more than 2 dex below the main sequence and still not be fully quenched.6.While the threshold for the onset of quenching is redshiftdependent for either Σ SFR or the sSFR, the former allows one to define an absolute threshold for full quiescence that is independent of the redshift.We tentatively propose defining full quiescence as log Σ SFR < −4.5 (in units of M e yr −1 kpc −2 ) when using r = 25 isophotal sizes, or log Σ SFR < −3.5 when using effective radii.7. The use of Σ SFR radial profiles allows us to distinguish galaxies where bulges are red (a central dip in sSFR) but SF still proceeds at high levels from cases where SF is suppressed in the centers.8.The ability of Σ SFR to serve as a precise measure of SF activity is severely affected if the area is based on the effective (half-light) radius, rather than on the isophotal one.This is because the isophotal radius, being tied to the physical mass and gas density thresholds, defines the extent over which SF takes place, whereas the effective radius depends strongly on the galaxy light profile/ concentration.9.The isophotal radius can be obtained from a combination of the effective size and the stellar mass with an error of just 0.05 dex, thus facilitating the use of the isophotal-like Σ SFR in cases where isophotal sizes are not available.
The main takeaway message from this study is that the use of the sSFR (or SFR), especially in the context of (s)SFR-M * diagrams, should be critically assessed depending on the context, and where appropriate be complemented with plots involving Σ SFR .

Figure 1 .
Figure1.Relationship between the isophotal radius (in kiloparsecs) and (a) the effective radius (also in kiloparsecs), (b) the stellar mass (in solar masses), and (c) a combination of the effective size and stellar mass that minimizes scatter (from left to right).Isophotal size is poorly correlated with either the effective size or the stellar mass, but is tightly correlated with the linear combination of the two (in logarithm), providing a way to precisely predict the isophotal radius if it is not available (see Equation (6)).Standard deviations of the residuals around the best linear fit (dashed line) are given in the panels.Solid lines show a 1:1 relation where applicable.All of the low-redshift samples in this and subsequent figures come from GSWLC-M2.

Figure 2 .
Figure 2. Low-redshift galaxy samples shown on the sSFR-M * (left) and Σ SFR -M * diagrams (right).The lower panels in both figures show the subset of galaxies that form the main sequence.The character of the main sequence changes when going from sSFR to Σ SFR -it becomes flatter, while remaining tight (width, in dex, given in the panels).Once the measure of SF activity is decoupled from the past SF history by using Σ SFR , the SF level is not strongly mass dependent.

Figure 3 .
Figure 3. Main-sequence merger pairs (interacting galaxies) overplotted on the general low-redshift main-sequence population.The same galaxies are shown on the sSFR-M * (left) and Σ SFR -M * diagrams (right).The merger pairs serve as proxies for starbursts and come fromDarg et al. (2010).The merging galaxies have a similar average offset (blue line) from the ridge of the overall main sequence (red line) in both cases; however, because there is no mass dependence, selecting starbursts by the SFR surface density is obviously much cleaner than doing so based on the sSFR.
we again show sSFR-M * (left panels) and Σ SFR -M * diagrams (right panels) of galaxies matched to the Nair & Abraham (2010) catalog and color-code them by the Hubble types provided in that catalog (upper panels), and by the Sérsic indices from Meert et al. (2015; lower panels).A clear trend is present on the sSFR-M * plots-the main sequence is dominated by late-type spirals (Sd at * < M log 9.8, Sc and Sb types above that mass), whereas the region below the main sequence is dominated by ellipticals and

Figure 4 .
Figure 4. Subsample of low-redshift galaxies with Hubble types from Nair & Abraham (2010), shown on sSFR-M * (left) and Σ SFR -M * diagrams (right).Upper panels are color-coded by the Hubble type, and lower panels by the Sérsic index (from Meert et al. 2015).Galaxies on the main sequence are mostly later types and have lower Sérsic indices.The SFR surface density provides a cleaner demarcation between early and late types than the sSFR.

Figure 5 .
Figure5.Evolution of galaxies in sSFR-M * (upper panels) and Σ SFR -M * diagrams (lower panels).For low-redshift samples we show only the main-sequence galaxies.In the sSFR-M * diagrams the slope of the main sequence stays more or less the same, whereas the Σ SFR -M * diagrams reveal a change-there was higher SF activity of massive galaxies compared to less massive ones in the past, consistent with a bulge buildup of those systems.Dashed lines in the lower panels signify the upper boundary of the low-redshift transition region.At higher redshift this boundary is higher and would have encompassed today's main-sequence galaxies.The dotted lines represent the threshold for full quenching, which is independent of the redshift.

Figure 7 .
Figure7.The effects on the Σ SFR -M * relation arising from defining the SFR surface density using the effective (half-light) radius.This figure is to be compared to the right panel of Figure2, which uses the isophotal sizes, our nominal choice.Effective-radius-based Σ SFR are significantly noisier, leading to a broader main sequence (width, in dex, given in the lower panel).The effective radius, being dependent on the galaxy light profile, is not as good an indicator of the extent over which SF is (or was) taking place as is the isophotal radius.

Figure 8 .
Figure8.The placement of galaxies separated into star-forming, transitional, and quenched (quiescent) categories on various diagrams with stellar mass on the x-axis.Panels from left to right demonstrate the successive improvements in the ability to separate these categories afforded by moving from optical to UV-optical color and then to sSFR.Specific SFRs provide a good separation between these categories, but the use of Σ SFR (based on which these categories have been defined) goes an additional step by eliminating the effect of age on the main sequence (its tilt).

Figure 9 .
Figure 9. Differences in inferences regarding a galaxy's SF based on whether the sSFR or Σ SFR is considered.The left panels correspond to LEDA 1793, whereas the right panels show UGC 5965.The upper panels present optical images from SDSS and the lower panels are UV images from GALEX.UGC 5965 has a red center, but its central SFR surface density is as high as that of LEDA 1793, which is blue and has a high central sSFR.Quantities are based on dust-corrected Hα and correspond to the 3″ SDSS fiber, shown as a circle.The SDSS images are taken from SkyServer, whereas the GALEX images (FUV plus NUV composites) are from the Legacy Survey Sky Browser.