SDSS-IV MANGA: A Star Formation -- Baryonic Mass Relation at Kpc Scales
J. K. Barrera-Ballesteros, T. Heckman, S. F. Sanchez, N. Drory, I. Cruz-Gonzalez, L. Carigi, R. A. Riffel, M. Boquien, P. Tissera, D. Bizyaev, Y. Rong, N. F. Boardman, P. Alvarez Hurtado, MaNGA team
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SDSS-IV MANGA: A STAR FORMATION - BARYONIC MASS RELATION AT KPC SCALES.
J. K. Barrera-Ballesteros , T. Heckman , S. F. S´anchez , N. Drory , I. Cruz-Gonzalez , L. Carigi , R. A.Riffel , M. Boquien , P. Tissera , D. Bizyaev , Y. Rong , N. F. Boardman , P. Alvarez Hurtado & theMaNGA team ABSTRACTStar formation rate density, Σ
SFR , has shown a remarkable correlation with both components of thebaryonic mass at kpc scales (i.e., the stellar mass density, and the molecular gas mass density; Σ ∗ ,and Σ mol , respectively) for galaxies in the nearby Universe. In this study we propose an empiricalrelation between Σ SFR and the baryonic mass surface density (Σ b =Σ mol , Av + Σ ∗ ; where Σ mol , Av isthe molecular gas – derived from the optical extinction, A V ) at kpc scales using the spatially-resolvedproperties of the MaNGA survey – the largest sample of galaxies observed via Integral Field Spec-troscopy (IFS, ∼ SFR tightly correlates with Σ b . Furthermore, we derivean empirical relation between the Σ SFR and a second degree polynomial of Σ b yielding a one-to-onerelation between these two observables. Both, Σ b and its polynomial form show a stronger correlationand smaller scatter with respect to Σ SFR than the relations derived using the individual componentsof Σ b . Our results suggest that indeed these three parameters are physically correlated, suggestingan scenario in which the two components of the baryonic mass regulate the star-formation activity atkpc scales. INTRODUCTIONUnderstanding what are the physical scenarios that de-scribe the star formation activity in galaxies is funda-mental to explain their evolution throughout their life-time. In turn, depending on the explored spatial scale,there have been mainly two different, yet complement-ing, scenarios that explain the star formation in galaxies.Broadly speaking, one possibility is that the amount ofnewly formed stars in a galaxy is set primarily by thelocal amount of cold gas available to create that newly-born population. On the other hand, star formation canalso be affected by global properties, for instance the dy- Instituto de Astronom´ıa, Universidad Nacional Aut´onoma deM´exico, A.P. 70-264, 04510 M´exico, D.F., M´exico Department of Physics & Astronomy, Johns Hopkins Univer-sity, Bloomberg Center, 3400 N. Charles St., Baltimore, MD21218, USA University of Texas at Austin,McDonald Observatory, 1 Uni-versity Station, Austin, TX 78712, USA Departamento de F´ısica, CCNE, Universidade Federal de SantaMaria, 97105-900, Santa Maria, RS, Brazil Laborat´orio Interinstitucional de e-Astronomia - LIneA, RuaGal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil Universidad de Antofagasta, Centro de Astronom´ıa, AvenidaAngamos 601, Antofagasta 1270300, Chile Instituto de Astrof´ısica, Pontificia Universidad Cat´olica deChile, Santiago, Chile Centro de Astro-Ingenier´ıa, Pontificia Universidad Cat´olica deChile, Santiago, Chile Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349 Chinese Academy of Sciences, National Astronomical Obser-vatories of China, 20A Datun Road, Chaoyang District, Beijing100012, China University of Utah, Department of Physics and Astronomy,115 S. 1400 E., Salt Lake City, UT 84112, [email protected] namical structure of the disk of the galaxy as a whole(Kennicutt & Evans 2012, and references therein). Acomplementary approach suggests that star-formation isself-regulated (e.g., Dopita 1985; Dopita & Ryder 1994;Silk 1997; Ostriker et al. 2010). In this scenario, momen-tum injection from massive stars balances the hydrostaticpressure due to the disk’s weight.Observationaly, empirical scaling relations have beenfundamental to explore the role of the baryonic mass inthe physical processes that yield the amount of newly-born stars in the Universe. The Schmidt-Kennicutt (SK)relation is the most well-known of those relations. It pro-vides a strong correlation between the observed star for-mation rate (SFR) and the amount of cold gas (Schmidt1959; Kennicutt 1998b). When using intensive measure-ments – properties average across a certain area pro-jected in the sky – (SFR surface density, Σ
SFR ; and gasmass surface density, Σ gas ), the SK relation follows asimilar slope across several orders of magnitude includinga wide range of galaxy morphologies and types for nor-mal star-forming galaxies (Gao & Solomon 2004). It canvary when exploring extreme starburst galaxies or galax-ies at high redshift (e.g., Daddi et al. 2010; Genzel et al.2010). Thanks to radio-interferometric surveys, there isalso a wealth of data indicating that the SK relation isalso valid at kpc scales for a large sample of extragalacticstar forming sources (e.g., HERACLES, THINGS, andEDGE-CALIFA surveys Leroy et al. 2008; Walter et al.2008; Bolatto et al. 2017). At a very first order, the a r X i v : . [ a s t r o - ph . GA ] J a n SK law and its counterpart at kpc scales, the resolvedSK relation (rSK) can be describe by the first scenariodescribed above.Another star-forming scaling relation is the one thatshows a tight correlation between the galaxy-integratedSFR and the total stellar mass (M ∗ ). The so-called starformation main sequence (SFMS) has been derived forthousands of galaxies included in the DR7 Sloan DigitalSky Survey (SDSS, e.g., Kauffmann et al. 2003; Brinch-mann et al. 2004). Note that in the SFR-M ∗ plane it isalso observed a cloud of massive galaxies with little SFR(known as the retired sequence of galaxies). In the lastdecade, thanks to Integral Field Unit (IFU) observationsin large samples of galaxies, it has been possible to deter-mine the existence of a local counterpart of the SFMS:the resolved SFMS (rSFMS, Σ SFR vs Σ ∗ ) for a large sam-ple of galaxies (S´anchez et al. 2013; Wuyts et al. 2013).The existence of the rSFMS – as well as the spatially-resolved version of the retired sequence – may also indi-cate a more intimate correlation between the current starformation and the star formation history of a galaxy atkpc scales (e.g., Cano-D´ıaz et al. 2016). Recently, withlarge IFU datasets and the combination with direct ob-servations of spatially resolved molecular gas, differentstudies have established that Σ SFR is mainly correlatedto Σ ∗ than Σ gas (e.g., Dey et al. 2019; Bluck et al. 2020).Similar studies have also noted the significant relation atkpc scales between the molecular mass gas density andthe stellar mass density (e.g., Lin et al. 2019; Barrera-Ballesteros et al. 2020).The above two star-forming relations are not the onlyones that correlate at kpc scales the Σ SFR with the bary-onic mass component in galaxies. Different authors haveexplored the relation between the Σ
SFR and the Σ gas Σ ∗ product or similar relations with the baryonic mass (e.g.,Matteucci et al. 1989; Shi et al. 2011, 2018). Since thecombination of the baryonic densities scales with the diskhydrostatic mid-plane pressure, these studies were aimedto explore whether star-formation follows a self-regulatedscenario. In this scenario, the star-formation is locallyregulated by the interplay between the mid-plane pres-sure produced by the baryonic mass and the momentumflux due to supernovae explosions from the most massivestars (Ostriker et al. 2010; Shetty & Ostriker 2012; Kimet al. 2013).Despite the great advance in our understanding of thestar-formation from these works, there has been a lackof systematic studies exploring the relation between thebaryonic mass (Σ b = Σ ∗ + Σ gas ) and Σ SFR at kpc scales.Such a relation can shed some light regarding the mostlikely physical scenario of star formation at kpc scales.In this study, we take advantage of the SDSS-IV MaNGAsurvey (Wake 2016), the largest IFU dataset to date toexplore this correlation. Furthermore we also investigate the impact of a correlation of the Σ gas with a non-linearcombination of Σ b . This article is structured as follows.In Sec. 2 we present our sample selection as well as anoverview on the main features of the datacube for eachgalaxy included in the MaNGA survey. In Sec. 3 we showthe observables derived from the datacubes required forthis study and our main results, while in Sec. 4 we discussour main findings. The main conclusions of this articleare presented in Sec. 5. SAMPLE AND DATACUBESThe galaxies selected for this study are drawn from thelatest sample of targets observed in the MaNGA survey(Bundy et al. 2015) which is part of the fourth genera-tion of the Sloan Digital Sky Survey (SDSS IV, Blantonet al. 2017). This sample includes galaxies observed fromMarch of 2014 to September of 2019 (8405 datacubes).This sample corresponds to the internal release withinthe collaboration known also as MaNGA product launch(MPL-9). The MaNGA survey has been designed to ob-tain IFS observations for more than 10,000 galaxies. Adetailed description of the selection criteria for this sur-vey is presented in (Wake et al. 2017).The observations of the MaNGA survey are takingplace at the Apache Point Observatory using its 2.5-mtelescope (Gunn et al. 2006). A detailed description ofthe instrumentation of the survey can be found in Droryet al. (2015). For a detailed explanation of the datastrategy the reader is refereed to Law et al. (2016). TheMaNGA reduction pipeline includes wavelength calibra-tion, corrections from fiber-to-fiber transmission, sub-traction of the sky spectrum and flux calibration (Yanet al. 2016). The final product is a datacube with x and y coordinates corresponding to the sky coordinates andthe z -axis corresponds to the wavelength. Its final spaxelsize is 0.5 (cid:48)(cid:48) with a spatial resolution of 2.5 (cid:48)(cid:48) correspondingto a mean physical scale of ∼ ANALYSIS AND RESULTS3.1.
Spatially-resolved and integrated properties fromthe observed galaxies
The local observables for this study are determined byapplying the
PIPE3D analysis pipeline to the MaNGAdatacubes. A full description of this pipeline, its stellarand emission line fitting, uncertainties estimations, andderivation of physical properties can be found in S´anchezet al. (2016). An overview on how the two-dimensionaldistributions of stellar and gas component are derivedfor MaNGA data can be found in Barrera-Ballesteroset al. (2016, 2018). In particular for this study, we usethe stellar surface mass density (Σ ∗ ), star formation ratedensity (Σ SFR (SSP)), and the specific SFR (sSFR(SSP))derived from the single-population stellar (SSP) anal-ysis. Σ
SFR (SSP) is obtained as the ratio between the mol , Av log(M (cid:12) kpc − ) − − − − Σ S F R l og ( M (cid:12) y r − k p c − ) σ = [0.45, 0.20], r =0.48 This work: a = -5.88, b = 0.54EW(H α ) >
30 ˚A: a = -7.09, b = 0.74Bolatto et al. (2017): a = -9.22, b = 1.0 ∗ log(M (cid:12) kpc − ) − − − − Σ S F R l og ( M (cid:12) y r − k p c − ) σ = [0.31, 0.10], r =0.72 This work: a = -9.50, b = 0.92Bolatto et al., 2017 (EDGE)Cano-Diaz et al., 2019 (MaNGA)
Figure 1 . Distributions of Σ
SFR versus Σ gas (rSK, left panel) and versus Σ ∗ (rSFMS, right panel) for more than 1.1 × spaxels, selected from 2640 galaxies included in the MaNGA survey. In both panels the distributions are color codedaccording to the spaxels density. Black contours enclose 95%, 80%, and 60% of the distributions respectively. Thewhite circles with error bars represent the median Σ SFR for bins of Σ gas and Σ ∗ , respectively. The black lines representthe best fit derived from the medians (with circles). These fits are derived from bins with Σ gas and Σ ∗ larger thanthe gray vertical dashed lines (see Sec.3.2). The slopes from both fits are in good agreement with those derived usingdirect observations of molecular gas in smaller samples of galaxies (Bolatto et al. 2017; Cano-D´ıaz et al. 2019, dashedred and blue lines, respectively).mass of stars formed in the last temporal bin (0.06 Gyr)and the total mass of stars formed across the cosmictime. For this SSP analysis the pipeline adopts a SalpeterIMF. From the emission lines analysis we obtain for eachspaxel the extinction-corrected Σ SFR derived from theH α emission line luminosity, Σ SFR (H α ), and the H α equivalent width, EW(H α ). We follow Kennicutt (1998a)to derive Σ SFR , where a Salpeter IMF is also adopted. InBarrera-Ballesteros et al. (2020) we derived a linear cor-relation between Σ gas (Σ gas = Σ mol + Σ H I ) and the op-tical extinction (A V ). A V was derived from the Balmerdecrement, whereas Σ H was derived from CO resolvedmaps observed within the EDGE survey (Bolatto et al.2017). We found that Σ gas (M (cid:12) pc − ) ∼
26 A V (mag)for radial scales. With the same dataset, we derivedin Barrrera-Ballesteros et al. (in prep.) a linear log-log calibrator for the molecular gas mass density withΣ mol , Av (M (cid:12) pc − ) = 10 . [A V (mag)] . using a leasttrimmed square (LTS) fitting at kpc scales (Cappellariet al. 2013). In comparison to the linear calibrator, itimproves the estimation of Σ mol for high values of A V .Even more, we found that this calibration depends onthe inclination therefore we use it only for low-inclinedgalaxies. All the intensive observables are corrected byinclination following Barrera-Ballesteros et al. (2016).For this study we select those regions (spaxels) thatwe consider bona fide star-forming in low-inclined galax-ies ( b/a < ). A spaxel is consider as star-forming b/a for each MaNGA galaxy is obtained from theextended NASA-Sloan catalog (NSA Blanton et al. 2005): if: (i) the [N II ] / H α and [O III ] / H β emission line fluxratios are below the Kewley et al. (2001) demarcationline in a BPT diagram (Baldwin et al. 1981); (ii) the H α and H β emission lines have a signal-to-noise ratio largerthan 3; (iii) H α /H β > (iv) an EW(H α ) > (cid:38) − . yr − . These criterialead to select more than 1 . × star-forming spaxelslocated in 2640 galaxies included in the MaNGA sam-ple. We should note that due to the spatial resolution ofthis survey ( ∼ (cid:48)(cid:48) ), the number of spaxels overestimatesthe number of independent H II regions detected in thesurvey by at least a factor ∼ Single-variable scaling relation: rSK and rSFMS
In Fig. 1 we show the rSK and the rSFMS derivedfor our sample of galaxies (left and right panels, respec-tively). The outermost black contour encloses 95% ofthe distributions. The solid circles represent the medianΣ
SFR for bins of equal width (0.2 dex) on Σ mol , Av andΣ ∗ , respectively. The black solid lines represent a least-square linear fit of those median values in log scales.We use the relation log(Σ SFR ) = a + b log(Σ), where Σrepresents both, Σ mol and Σ ∗ , respectively. FollowingCano-D´ıaz et al. (2019), in order to perform the abovefit for both relations we use those bins where we con-sider that we have reliable estimations of Σ mol , Av andΣ ∗ . For the rSK relation we select those bins with Σ mol , Av > . M (cid:12) kpc − . This limit is motivated bythe gas calibrator used in this study. In Barrera-Ballesteros et al. (in prep.) we show that due to the sen-sitivity of the CO observations (from the EDGE surveyBolatto et al. 2017), the calibrator may not be reliableat Σ mol , Av smaller than this limit. On the other hand,to perform the fitting in the rSFMS Cano-D´ıaz et al.(2019) used bins with Σ ∗ > . M (cid:12) kpc − . They noteda non-physical driven flattening of the rSFMS below thementioned Σ ∗ due to detection limits in each of the ob-servables. Both of these limits are represented as verti-cal dashed lines in Fig.1. In both panels, the horizontaldashed line represent the of H α surface brightness detec-tion limit for the MaNGA survey ( ∼ erg s − kpc − ,Cano-D´ıaz et al. 2019). Finally, we note that the selec-tion criteria based on the EW(H α ) naturally results intight relation since we are considering only regions abovea certain sSFR.Both scaling relations present significant Pearson cor-relation coefficients ( r ). In fact, our rSFMS shows ahigher r coefficient in comparison to previous studiesusing MaNGA data (e.g., Cano-D´ıaz et al. 2019; Linet al. 2019). However, the r coefficient for the rSK rela-tion is smaller in comparison to the value derived for therSFMS relation (0.48 vs 0.72). This could be caused bythe larger scatter observed in the rSK compared to therSFMS. We also note that in comparison to other rela-tions presented in the literature our rSK shows a smallercorrelation factor (e.g., Lin et al. 2019). We compare thebest fit for each relation with those that illustrate thetrends observed from the EDGE survey (Bolatto et al.2017) which makes use of CO maps to estimate Σ gas in asample of 123 galaxies. The best fit for the rSK derivedusing the MaNGA data is flatter in comparison to the re-lation derived by Bolatto et al. (2017). We suggest thatthis flattening is due to the nature of our dataset. Forlow values of Σ SFR we could be sampling physical regionsfor which H α flux emission could be potentially pollutedfrom processes other than young stellar emission (e.g.,diffuse H α emission from old stars, Lacerda et al. 2018).In turn, this causes an underestimation of Σ gas from thecalibrator. By diluting H II regions along with diffuse H α emission, the spatial resolution can also have an impactin the scatter of the rSK (e.g., Vale Asari et al. 2020).However, for large values of Σ SFR , A V recovers the ex-pected value of Σ mol since at high Σ SFR the amount ofdust traces Σ mol . Indeed, when selecting spaxels withEW(H α ) larger than 30 ˚A the best linear fit is steeperto the one derived from the median values (green dashedline) and is close to the linear relation reported usingdirect CO measurements by the EDGE survey (Bolattoet al. 2017). On the other hand, the rSFMS is in verygood agreement with those derived previously using theEDGE and MaNGA datasets (Bolatto et al. 2017; Cano- . . . . . . b = (Σ mol , Av + Σ ∗ ) log(M (cid:12) kpc − ) − − − − Σ S F R l og ( M (cid:12) y r − k p c − ) σ = [0.28, 0.08], r =0.77 This work: a = -9.92, b = 0.97
Figure 2 . The Σ
SFR - Σ b relation for the sample of spax-els included in this study. The description of the linesand symbols is similar as the one presented for panels inFig.1. In comparison to the single-variable scaling rela-tions, Σ b shows a higher correlation factor as well as atighter relation with Σ SFR .D´ıaz et al. 2019).In both panels we also annotate the scatter of the resid-uals, σ . We use two measurements, the standard devi-ation and the variance. By comparing σ between thesetwo relations we note that the rSK shows a larger scatterin comparison to the rSFMS ([0.45, 0.20] vs [0.31, 0.10]).The scatter of the rSK is only ∼ The linear Σ SFR - Σ b relation One of the goals of this study is to probe what is thebest relation of the baryonic mass (Σ b = Σ mol , Av + Σ ∗ )with respect to the Σ SFR . First, we show the direct re-lation between these two parameters in Fig. 2. To ourknowledge, this is the first time that this scaling relationis explored for such a large sample of galaxies at kpcscales. The distribution of this relation is drawn fromthe same regions used in the relations explored in Fig.1.We note that the distribution of spaxels in the Σ
SFR - Σ b plane does not show the strong flattening observed at lowvalues of Σ b , in the rKS and rSFMS relations (see Fig. 1).The distribution seems to follow a tighter relationthan the single-parameter local star-forming relationsexplored in Fig. 1 (i.e., lower scatter). In fact, the r -coefficient indicates a stronger correlation between Σ SFR and Σ b than the relations derived in Sec.3.2. For in-stance, the r -coefficient is larger for this relation thanfor the rSFMS (0.77 vs 0.72). We follow a similar pro-cedure to derive the best fit relation as in Sec. 3.2. Wechoose a Σ b threshold to select the bins to perform the fit(Σ b (cid:38) . M (cid:12) kpc − , this is the sum of the thresholdsin Sec. 3.2). However, we should note that the resultsare not strongly affected by the implementation of thislimit in the analysis. The best fit parameters (slope andzero-point) are shown in Fig. 2. Both parameters of thebest fit are similar to those derived from the rSFMS (seeFig. 1). This is expected since the main driver in thebaryonic mass density is Σ ∗ . Nevertheless, it is impor-tant to note the role of Σ gas in shaping the Σ SFR - Σ b relation at low values of Σ b as well as increasing the slopeof the best relation making it close to one. This can alsobe noted in the decrease in the scatter of the Σ b -Σ SFR relation in comparison to the rSFMS. Finally, the scat-ter of the residuals with respect to this fit is smaller thanthe rSFMS ( σ = [0 . , .
08] vs σ = [0 . , . b .3.4. Impact of a quadratic term in the Σ b - Σ SFR relation
As we mentioned in Sec. 1, there are different stud-ies exploring scaling relations between the star forma-tion and the Σ gas Σ ∗ product or similar relations. Theserelations have been studied in order to explore the self-regulated star-formation scenario. In this section, weexplore a rather generic approach. We investigate therelation of Σ SFR with a second degree polynomial of Σ b .Using this functional form, we explore both the depen-dence of Σ SFR with respect of mixed quadratic terms(Σ gas Σ ∗ ) as well as quadratic terms of Σ gas 2 and Σ ∗ .In Fig. 3 we plot the relation between the x-axis definedas x new = a Σ b + b Σ b 2 against Σ SFR . The fit betweenx new and Σ
SFR is obtained by using the median values ofΣ b and Σ SFR derived in Fig. 2. This fit yields the valuesof log( a ) = − .
21 and log( b ) = − .
37. The spaxels inthe Σ
SFR − x new plane lies in a well-defined linear trend. In fact, the best linear fit derived from the median Σ SFR for different x new bins yield a one-to-one relation (blacksolid-line in Fig. 3). In contrast to the single-variablescaling relations (rKS, and rSFMS; see Sec. 3.2), theabove relation does not show strong deviations or flat-tening at low values of x new . We perform the same anal-ysis using each of the components of Σ b separately. Wefind that by fitting a quadratic polynomial using Σ ∗ orΣ mol , Av , as independent variables, there is no significantreduction in the scatter and neither it is possible to ob-tain a one-to-one relation from the medians as we findusing Σ b . Therefore, our results indicate that a betterrepresentation between the Σ SFR and the baryonic massis given by a second degree polynomial rather than byconsidering each of its components separately.The best-fit relation between Σ
SFR − x new yields a sim-ilar scatter and a larger r -correlation factor compare tothose derived with Σ b alone. To test how this relationcan be affected by the number of sampled galaxies, wederive the same figures in this analysis using the datasetfrom the previous internal release (MPL8, ∼ SFR -Σ b relation as well as the Σ SFR − x new relation maybe due to the statistical distribution of the observables.We note that despite the linear slope derived including aquadratic term, the best-fit coefficients are small, in par-ticular the factor that multiplies the quadratic term ofΣ b is several orders of magnitude smaller than the factorthat accompanies the linear term of Σ b . The deriva-tion of a slope of one is expected since, as we mention inSec. 3.3, the stellar mass density is the dominant termin Σ b , in particular for regions with large star formationrates.3.5. An independent measure of Σ SFR via SSPs
In the previous sections we derive the scaling rela-tions between the Σ
SFR and the different functions ofthe baryonic mass using the H α luminosity as the ob-servable to derive Σ SFR . Similarly, we estimate Σ gas from the H α /H β emission lines ratio (Balmer decre-ment). Therefore it can be the case that since we areusing similar observables to determine the above scalingrelations we could be inducing such relations. In orderto test this, we use in this section another estimationof Σ SFR . As we mention in Sec. 3.1, the
Pipe3D dataanalysis pipeline allow us, through the fitting of SSPsto the stellar continuum of each spaxel, to determine,among other properties of the stellar component, the av-erage star formation at different cosmic times (this is,their star-formation histories, SFHs; Ibarra-Medel et al.2019). For the purposes of this study we understand theΣ
SFR , SSP as the fraction of the latest stellar burst mea- − − − − b + b Σ ) − − − − Σ S F R l og ( M (cid:12) y r − k p c − ) σ = [0.27, 0.07], r =0.82 This work: a = 0.003, b = 1.00
Figure 3 . A general relation between the Σ
SFR and Σ b .The x-axis (x new ) represents the best fit of a 2nd degreepolynomial of Σ b for the observed Σ SFR which yields aone-to-one relation. As in previous plots, the contoursenclose 95%, 80%, and 60% of the distribution. Thewhite circles show the median Σ
SFR for bins of x new .The black line shows the best fit for these median values.The description of the lines and symbols is similar asthe one presented for panels in Fig.1. The scatter and r coefficient of this one-to-one relation resambles thosevalued derived for the Σ SFR -Σ b relation.sured by the SSP fitting (i.e., the fraction of stars formedin a span of time smaller than ∼
32 Myr, Gonz´alez Del-gado et al. 2016). Besides the selection criteria describein Sec.3.1, for this section we only consider spaxels with asSFR(SSP) > − yr − . As result, the sample for thisexperiment consist of 3.1 × spaxels located in 2098galaxies. In Appendix A we compare the estimation ofΣ SFR using SSPs (Σ
SFR , SSP ) with the one derived usingH α luminosity (Σ SFR , H α ). We note that they stronglycorrelate with each other ( r = 0.73). In particular, theyare similar at large values. In average, Σ SFR , SSP is over-estimated by a factor of 1.3 in comparison to Σ
SFR , H α .Different studies showed that scaling relations at globaland local scales derived using either Σ SFR , SSP or Σ
SFR , H α are similar (e.g., the (r)SFMS; see Gonz´alez Delgadoet al. 2014, 2016; S´anchez et al. 2018).In the left and right panels of Fig. 4 we plot thesame relations from Figs 2 and 3 using Σ SFR , SSP , respec-tively. We find similar results when we derive the Σ
SFR -Σ b relation using Σ SFR , SSP instead of Σ
SFR , H α . How-ever, the distribution is above the relation derived usingΣ SFR , H α (see dashed line). Although the trend is sim-ilar, the slope of this relation is shallower than the onederived using the H α proxy for Σ SFR it also presents ascatter larger than the one derived in Sec. 3.3. In rightpanel of Fig. 4 we plot the Σ
SFR -x new relation followingthe same procedure as in Sec. 3.4. We note that coeffi- cients of the fit for x new are different as those derived inSec.3.4 (a = 2.3 × − , and b = − . × − ). Never-theless, the distribution of this relation is similar to theone derived using Σ SFR , H α with a smaller correlation co-efficient as the relation derived in Sec. 3.4. The medianvalues of Σ SFR , SSP for different bins of x new are in goodagreement with respect to the best relation derived pre-viously in Sec. 3.4 (see dashed line) despite the incrementin the scatter observed for this relation ( σ = [0 . , . SFR -x new relation following thesame procedure as in Sec. 3.4 (see bottom-right panel inFig. 4). We note that coefficients of the fit for x new aredifferent of the ones derived in Sec.3.4 (a = 2.3 × − ,and b = -9.6 × − ). Nevertheless, the distribution ofthis relation is similar as the one derived using Σ SFR , H α with a smaller correlation coefficient as the relation de-rived in Sec. 3.4. The median values of Σ SFR , SSP fordifferent bins of x new are in good agreement with re-spect to the best relation derived previously in Sec. 3.4(see dashed line) despite the increment in the scatter ob-served for this relation ( σ = [0 . , . SFR , SSP issimilar to the one reported using Σ
SFR , H α .In summary, these results suggest that, independentof the observable we use to determine Σ SFR , we obtainsimilar trends when we derive the Σ
SFR - Σ b relation.Also, the inclusion of an extra quadratic term to describethis relation leads to similar results regardless the Σ SFR calibrator. DISCUSSIONIn this article, we present the well-known star-formingscaling relations between each of the components of thebaryonic mass at kpc scales (i.e., the rSK and rSFMS) forthe largest IFU dataset provided by the MaNGA survey( ∼ SFR derived from the SSP analysis.The main result of this work is that Σ b provides a bet-ter correlation with the Σ SFR than using only each of itstwo components (Σ ∗ and Σ mol , Av ). Even more, whenadopting a quadratic polynomial form of Σ b , the log-log relation presents a one-to-one slope (i.e., no power isrequired to match both quantities). We find similar re-sults when using an independent observable to estimatethe star formation rate density. This analysis highlightsthe necessity to consider the full baryonic content, and . . . . . . b = (Σ gas + Σ ∗ ) log(M (cid:12) kpc − ) − − − − Σ S F R , SS P l og ( M (cid:12) y r − k p c − ) σ = [0.37,0.13], r = 0.68 This work: a = -7.4, b = 0.72This work, Σ SFR , H α − − − − new ) = log(a Σ b + b Σ ) − − − − Σ S F R , SS P l og ( M (cid:12) y r − k p c − ) σ = [0.39,0.15], r = 0.68 This work: a = -0.09, b = 0.91This work, Σ SFR , H α Figure 4 . The star-forming scaling relation derived in this study using Σ
SFR , SSP instead of Σ
SFR , H α . ( left panel ) TheΣ SFR -Σ b relation. The dashed line represents the best-fit relation derived using Σ SFR , H α (see Fig. 2). ( right panel )The relation between Σ SFR -x new . As in the previous panel, the dashed line represents the best-fit relation as presentedin Fig. 3. Independent of the observable used to determine Σ SFR , the general relation between the baryonic mass andthe star formation at kpc scales holds.not only separate therms of Σ b to properly describe thestar formation at kpc scales. Such a non-linear empir-ical relations have been pointed out as evidence of theimportance of the impact of existing stars in the regula-tion of SFR (e.g., Zaragoza-Cardiel et al. 2019). In thisself-regulated model of star-formation, the hydrostaticpressure of the disk galaxy is balanced by the momen-tum flux injected to the ISM from supernovae explosions(e.g., Cox 1981; Silk 1997; Ostriker et al. 2010). Fromour main result, we argue that a second degree polyno-mial of Σ b provides a better description of Σ SFR since itincludes both the contribution of the amount of gas re-quired to form new stars as well as the non-linear termsthat describe the impact of the hydrostatic pressure ofthe disk (see Sec. 4.1).Along this article we measure the scatter of the differ-ent scaling relations derived for star-formation regionsat kpc scales. For individual components of the baryonicmass we find that the rSFMS yields the smallest scat-ter in agreement with the scatter derived for a smallersample of same MaNGA galaxies (standard deviation, ∼ b have slightly smaller scatterwith stronger correlation coefficients. These results in-dicate that when using this specific combination of thebaryonic components, the main driver to derive Σ SFR in star-forming regions is the stellar mass density. Thishas also been found in recent studies that classify thestrength of correlations among star-formation and otherobservables at kpc scales. Their results suggest that Σ ∗ appears to be the observable that better correlates withΣ SFR (Dey et al. 2019; Bluck et al. 2020). On the con-trary, other explorations, using estimations of the molec- ular gas based on CO observations of a few tens of ob-jects suggest that the strongest and tighter correlation isfound with Σ mol (Lin et al. 2019; Ellison et al. 2020).Overall, molecular gas is essential to form new stars atkpc scales however the regulation of Σ
SFR strongly de-pends on the amount of baryonic matter. In other words,we suggest that locally the gravitational potential is themain regulator of the star formation rate. In Barrera-Ballesteros et al. (submitted) we explore the explicitrelation between the star formation and the mid-planepressure derived from direct estimation of the moleculargas, as well as its interpretation in the context of self-regulation.4.1.
Other non-linear relations: revisiting the extendedSchmidt law
As indicated before, in recent years there have beendifferent studies exploring the relation of a combinationof the components of the baryonic mass with the starformation (e.g., Westfall et al. 2014; Dib et al. 2017;Roychowdhury et al. 2017; Bolatto et al. 2017; de losReyes & Kennicutt 2019; Sun et al. 2020). In partic-ular, Shi et al. (2011, 2018) explored the so-called ex-tended Schmidt law, which correlates the star-formationsurface density with the product of the stellar and gasmass surface density. Shi et al. (2011) suggest thatΣ
SFR ∼ Σ ∗ . Σ gas provides a better relation than theSchmidt law. In other words, the scatter of this ex-tended Schmidt law is reduced in comparison to theΣ SFR -Σ gas relation. Even more, in Shi et al. (2018),they showed that the best relation is slightly super-linear (Σ SFR = 10 − . (Σ ∗ . Σ gas ) . ). In Fig. 5 we ex-plore this extended star-formation law using the current . ∗ Σ gas log((M (cid:12) pc − ) . ) − − − − Σ S F R l og ( M (cid:12) y r − k p c − ) σ = [0.36,0.13], r = 0.41 Σ SFR , H α : a = -3.5, b = 0.67Shi et al., 2018Bolatto et al., 2017 Figure 5 . The extended star-forming law at kpc scalesfor the MaNGA sample. The x-axis is defined by Shiet al. (2011). The distribution of the selected spaxels(i.e., values larger than the vertical dashed gray line) –and their median values of Σ
SFR (white circles) – followsa similar relation as the one derived by Shi et al. (2018)(red-dashed line). Although our sample follows the trendof the proposed extended scaling relation by Shi et al.(2018), its scatter is similar as the one derived for therSFMS (see right panel of Fig. 1).MaNGA data. The slope of the best fit of the median val-ues ( ∼ SFR larger than a threshold in thex-axis (10 . (M (cid:12) pc − ) . , dashed vertical gray line).This threshold considers the limits we use for Σ mol , Av andΣ ∗ . The scatter of this relation is smaller than the onewe derive for the rSK (standard deviation of 0.36 dex,and 0.45 dex, respectively). However, this scatter haslarger dispersion in comparison to the one derived fromthe rSFMS (0.31 dex). Even more, its correlation coef-ficient is significantly smaller than the one derived forthe rSFMS ( r = 0 .
41 vs 0.72). We also note that eventhough the slope from the best fit is sub-linear, the Σ
SFR derived for large values of Σ gas Σ ∗ . is in agreement withthe relations derived in the literature suggesting that forregions with intense star formation, Σ gas and Σ ∗ playsan important role describing Σ SFR . In further studies weexplore the explicit relation between the Σ
SFR and thebaryonic mass in the context of the self-regulation of starformation in order to quantify the role of the mid-planepressure in shaping the star-formation rate at kpc scales(Barrera-Ballesteros et al., submitted).Recently, Lin et al. (2019) explored the functional form of the extended KS law (Σ gas Σ ∗ β ). Using a homogeneousdataset, they found that the exponential that yields thesmallest scatter in this relation is β ∼ − .
30. Even whenusing this exponential the scatter in comparison to therSK or rSFMS is not reduced as expected from Shi et al.(2011, 2018). As these authors, we do not find a strongreduction of the scatter for the star-formation when usingthe functional form described by Shi et al. (2018). Fromour analysis, we conclude that although a relation suchas the extended Schmidt law – which is derived in thecontext of self-regulation of star formation – is necessaryto describe the Σ
SFR it may also needed to include othercontributions of the baryonic mass such as the seconddegree polynomial relation presented in Sec.3.4. SUMMARY AND CONCLUSIONSUsing a sample of more than 1.1 × spatial elements( ∼ × independent regions) of kpc size located in2640 galaxies drawn from the MaNGA survey – thelargest IFU survey up to date –, we present a scalingrelation between the star formation rate surface den-sity (Σ SFR ) and the baryonic mass surface density (Σ b =Σ gas + Σ ∗ ). Σ gas is obtained by using as proxy the opticalextinction. Our results can be summarized as follows: • We reproduce the well-known star-forming scalingrelations for individual components of Σ b : the re-solved Schmidt-Kennicutt law (rSK) and the re-solved star-formation main sequence (rSFMS). Bymeasuring their scatter and correlation factors, wefind that the rSFMS yields the tighter and strongerrelation with respect to Σ SFR . • We derive a scaling relation between the Σ
SFR ,Σ b and a second degree polynomial of Σ b . Theserelations show a strong correlation and a smallerscatter than those derived from individual compo-nents of Σ b . In particular, the second one natu-rally yields a one-to-one relation. We find similartrends using two independent indicators of Σ SFR :the H α emission line luminosity and a stellar de-composition using single-stellar population fittingof the stellar continuum. • We contrast these new relations with other empir-ical star-forming scaling relations such as the ex-tended Schmidt law proposed by Shi et al. (2011,2018). We find that the Σ
SFR -Σ b relation yields astronger correlation and has a smaller scatter incomparison to the extended Schmidt law.We conclude that these star-forming scaling relationsquantify the strong impact of the baryonic mass as awhole in the conditions of formation of newly born starsat kpc-scales. Even more, besides the evident role thatΣ gas − . − . − . − . − . − . SFR , H α log(M (cid:12) yr − kpc − ) − . − . − . − . − . − . Σ S F R , SS P l og ( M (cid:12) y r − k p c − ) one-to-one relationbest linear fit − y/x axis ratio N u m b e r o f S p a x e l s Figure A6:. A comparison between the star formationderived from the SSP analysis, Σ
SFR , SSP , against theone derived using the H α luminosity Σ SFR , H α . As inprevious plots, the contours enclose 90%, 80%, and 60%of the distribution. The white circles show the medianΣ SFR , SSP for bins of Σ
SFR , H α . The solid lines shows thebest fit of these bins whereas the dashed lines representsthe one-to-one relation. Both estimations of Σ SFR aresimilar to each other.Wisconsin, Vanderbilt University, and Yale University.APPENDIX A. Σ SFR
ESTIMATIONS
In Sec. 3.5 we use the Σ
SFR derived from the SSP toestimate the star-forming scaling relations derived usingthe H α luminosity as proxy of Σ SFR . In this appendixwe compare these two independent estimations of thisquantity. In Fig. 6 we compare the Σ
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