Removing biases in resolved stellar mass-maps of galaxy disks through successive Bayesian marginalization
Eric E. Martínez-García, Rosa A. González-Lópezlira, Gladis Magris C., Gustavo Bruzual A
DDraft version October 20, 2018
Preprint typeset using L A TEX style emulateapj v. 01/23/15
REMOVING BIASES IN RESOLVED STELLAR MASS-MAPS OF GALAXY DISKS THROUGH SUCCESSIVEBAYESIAN MARGINALIZATION
Eric E. Mart´ınez-Garc´ıa , Rosa A. Gonz´alez-L´opezlira , Gladis Magris C. , and Gustavo Bruzual A. Draft version October 20, 2018
ABSTRACTStellar masses of galaxies are frequently obtained by fitting stellar population synthesis models togalaxy photometry or spectra. The state of the art method resolves spatial structures within a galaxyto assess the total stellar mass content. In comparison to unresolved studies, resolved methods yield,on average, higher fractions of stellar mass for galaxies. In this work we improve the current methodin order to mitigate a bias related to the resolved spatial distribution derived for the mass. Thebias consists in an apparent filamentary mass distribution, and a spatial coincidence between massstructures and dust lanes near spiral arms. The improved method is based on iterative Bayesianmarginalization, through a new algorithm we have named Bayesian Successive Priors (BSP). We haveapplied BSP to M 51, and to a pilot sample of 90 spiral galaxies from the Ohio State University BrightSpiral Galaxy Survey. By comparing quantitatively both methods, we find that the average fractionof stellar mass missed by unresolved studies is only half than previously thought. In contrast withthe previous method, the output BSP mass-maps bear a better resemblance to near infrared images.
Subject headings: galaxies: fundamental parameters — galaxies: stellar content — galaxies: photom-etry — galaxies: spiral — methods: statistical INTRODUCTIONHow galaxies form and assemble their mass is a primor-dial question in modern astrophysics. Galaxy masses arecrucial for their evolution, and for the evolution of cos-mic structures at all scales. The determination of the stellar mass content of galaxies can help constrain, e.g.,the dark matter fraction, the specific star formation rate(Ψ S , the star formation rate, Ψ, per unit stellar mass),the stellar mass function, and the universe’s stellar massdensity and star formation history (SFH).There are different methods to estimate the mass ofa galaxy, e.g., dynamical or through gravitational lens-ing (see Courteau et al. 2014, for a review). Regardingthe stellar mass component, the use of stellar popula-tion synthesis (SPS) models to estimate mass throughthe stellar mass-to-light ratio, Υ ∗ , has been frequentlyadvocated (e.g. Bell & de Jong 2001; Bell et al. 2003).Notwithstanding their common degeneracies, SPS mod-els can in general yield reliable mass estimates. One noveltechnique is the resolved stellar mass-map method (Zi-betti, Charlot, & Rix 2009, ZCR hereafter), that deliversa map of the stellar mass surface density by photometricmeans. Galaxy masses determined by treating the galax-ies as point sources are often underestimated (and some-times overestimated, see Roediger & Courteau 2015),thus the need to resolve structures (ZCR; Sorba & Saw-icki 2015). Even more, if the stellar mass of each galaxyin a cluster is estimated separately, the total stellarmass fraction is lower than when a constant Υ ∗ is as-sumed (Leauthaud et al. 2012). Throughout this work Υ ∗ refers to the stellar (including rem-nants) mass-to-light ratio in units of M (cid:12) /L (cid:12) , i.e., we do not in-clude dark matter, nor gas mass in Υ ∗ . The resolved stellar mass method is truly powerful,since it can solve not only for the mass, but for otherphysical parameters of the SPS models, based solely onphotometry. Resolved maps of stellar mass are also im-portant for studies aimed at understanding the dynamicsof bars and/or spirals (since gravity is the main driver),and their secular evolution (e.g., Foyle et al. 2010;Mart´ınez-Garc´ıa & Gonz´alez-L´opezlira 2013; Egusa et al.2016). Additionally, they can be used to determine thebaryonic contribution to rotation curves (e.g., Repetto etal. 2013, 2015; McGaugh et al. 2016). The method canalso be extended to higher-redshift studies (e.g., Lanyon-Foster et al. 2007; Wuyts et al. 2012).Despite their potential, the resulting mass-maps maybe biased, in the sense that the stellar mass shows a fila-mentary structure and is concentrated in dust lanes. Inthis paper we aim to understand the origin of this short-coming and improve the method to derive resolved stellarmass-maps. We must also mention that in this researchwe use SPS models that assume a constant metallicityalong the SFH. Gallazzi & Bell (2009) studied the effectsof using a variable metallicity SPS library and found nosignificant biases when estimating Υ ∗ . Nevertheless, Into& Portinari (2013) indicate that the color-mass-to-lightratio relations (CMLR, see e.g. McGaugh & Schombert2014) resulting from an evolving metallicity along a co-herent SFH within an individual galaxy are probably dif-ferent from the CMLR established for the general galaxypopulation. Furthermore, biases in mass determinationsfrom CMLR can be even more significant at high red-shifts than for local studies (see e.g., Mitchell et al. 2013).In this work we do not use CMLR to recover Υ ∗ ; instead,we use a statistically robust Bayesian technique to inferthe predicted Υ ∗ via the comparison of observed colors a r X i v : . [ a s t r o - ph . GA ] D ec Mart´ınez-Garc´ıa et al.with a comprehensive library of SPS models.The paper is organized as follows. In section 2 wedescribe the resolved stellar mass-map method in itspresent form and explain/investigate the source of thebias. We introduce a new method (based on the former)in section 3. In section 4 we apply the new method to thespiral galaxy M 51 (NGC 5194); comparisons with othermethods are also briefly described. In section 5 we applythe new method to a pilot sample of spiral galaxies, anddiscuss and analyze the results. The uncertainties in thestellar mass estimates are discussed in section 6. Finally,we give our conclusions in section 7. RESOLVED MAPS OF STELLAR MASSThe ZCR method uses a Monte Carlo library of SPSmodels obtained from the 2007 version of Bruzual &Charlot (2003; CB07) models with the Chabrier (2003)stellar initial mass function (IMF). The library was builtby adopting prior probability distributions for parame-ters such as the SFH, the dust attenuation (treated as inthe two-component model of Charlot & Fall 2000), and anon-evolving metallicity. By randomly drawing the pa-rameters from the prior distributions (cf. da Cunha etal. 2008), the resulting library consists of ≈ × tem-plates (or models).The ZCR fiducial method is based on surface bright-ness photometry at the g and i Sloan Digital Sky Sur-vey (SDSS) optical bands, and one near-infrared (NIR)filter such as J , H , or K . The method was extendedto include the Spitzer Space Telescope Infrared ArrayCamera (IRAC) 3 . µ m-band by Repetto et al. (2015).Other optical color combinations are possible, with thedisadvantage of having more degeneracy in Υ ∗ , and thusmore uncertain results (see e.g., Repetto et al. 2015; Bell& de Jong 2001, their Figures 1 and 2, respectively).The templates from the SPS library are binned in colors( g − i ) and ( i − H ), using a bin width of 0.05 magnitude(see Figure 1). The median mass-to-light ratio at the H -band, Υ H ∗ , is estimated for each bin. A look-up table canthus be constructed to compare with observed photom-etry on a pixel-by-pixel basis. The Υ H ∗ is the effective mass-to-light ratio, i.e., refers to the light that reachesthe observer, as opposed to the light that is emitted.The effective Υ H ∗ may be affected by extinction (ZCR).Earlier studies concerning pixel-by-pixel spatially re-solved properties of galaxies can be found in Bothun(1986), Abraham et al. (1999), Conti et al. (2003), Es-kridge et al. (2003), Kassin et al. (2003), Lanyon-Fosteret al. (2007), and Welikala et al. (2008).2.1. Application to M 51. A filamentary massstructure?
Now we present results obtained by applying the ZCRmethod to the spiral galaxy M 51. We use g and i -band imaging from the twelfth data release (DR12) ofthe SDSS (Alam et al. 2015), as well as the K s -bandmosaic from Gonzalez & Graham (1996). The NIR im-ages were obtained at Kitt Peak National Observatory(KPNO), with the IR Imager (IRIM) camera on the1.3 meter telescope; the IRIM had a 256 NICMOS3array with a 2 (cid:48)(cid:48) pixel − plate scale. The observations were performed during March 1994, in non-photometricconditions, and the exposures were resampled with sub-pixel accuracy before combining. The final K s -band mo-saic has 0 . × . pixels, and a total exposuretime of 22 minutes; it was photometrically calibrated with the Two Micron All Sky Survey (2MASS, Skrut-skie et al. 2006). The SDSS frames were re-sampledto the resolution of the NIR data, and registered withthe K s -band image. The registration was done with theIRAF (Tody 1993) tasks GEOMAP and
GREGISTER . Nopoint spread function (PSF) match was done to the im-ages, since the data have similar PSFs and the processcan corrupt the noise properties (Zibetti, Charlot, & Rix2009). In Figures 2a (top left panel), and 2b (top rightpanel), we show the K s -band and g -band final images, re-spectively. The foreground stars and background galax-ies were removed and their pixels replaced with valuesfrom the background-subtracted “sky”. With the pur-pose of isolating the disk from the lower signal-to-noise(S/N) background, the final mosaics were treated withthe Adaptsmooth code of Zibetti (2009), as follows. Afirst run of
Adaptsmooth was performed on the the K s -band data (which have a lower S/N ratio than the SDSSimages), with the requirement of a minimum S/N ratioper pixel of 20, a maximum smoothing radius of 10, andthe assumption of background-dominated noise. In orderto homogenize the lower limit of the S/N ratio per pixel,the output smoothing K s -band mask was then used as aninput, in subsequent runs of Adaptsmooth , for the SDSS g and i bands.The SPS library was obtained from the Multi-wavelength Analysis of Galaxy Physical Properties pack-age (MAGPHYS-CB07 library, hereafter) by da Cunhaet al. (2008). The absolute magnitudes of the Sun weretaken from Blanton & Roweis (2007). We assume a dis-tance to M 51 of 9 . ± . i -band image. The color range cov-ered by the observed photometry of M 51 is shown,as a 2-D histogram, in Figure 4. In the left panel weshow the observed colors of the pixels after applying the Adaptsmooth procedure as described earlier. The rightpanel shows the observed colors of the same pixels with-out using the
Adaptsmooth procedure. From the compar-ison of these plots we appreciate the advantage of increas-ing the S/N ratio in the outskirts of the disk, otherwisethe uncertainties in the fits would be quite large. In thesefigures we also demarcate the color range covered by 99%and 68% of the total templates in the MAGPHYS-CB07library, with a blue and a red contour, respectively. Mostof the observed colors fall within the span of the SPS li-brary. The plots are illustrative and do not reflect theobservational uncertainties of the data.One striking thing to notice about the mass-map (Fig- Throughout this work NIR magnitudes are Vega, SDSS mag-nitudes are in the AB magnitude system. IRAF is distributed by the National Optical Astronomy Ob-servatories, which are operated by the Association of Universitiesfor Research in Astronomy, Inc., under cooperative agreement withthe National Science Foundation. EMOVING BIASES IN STELLAR MASS-MAPS 3
Fig. 1.—
Left : decimal logarithm of the effective , i.e., as seen by the observer (cf. ZCR), mass-to-light ratio at the H band, Υ H ∗ , derivedfrom the ( g − i ) vs. ( i − H ) color-color diagram. The data are taken from the MAGPHYS-CB07 Monte Carlo SPS library, and groupedin bins 0 . × .
05 mag . SDSS g and i magnitudes are in the AB magnitude system, H magnitudes are Vega. Right : root mean square(rms) errors of log Υ H ∗ in the left panel. The global median rms error is 0.1 dex. Fig. 2.—
Top left : K s -band mosaic of M 51; grayscale in Vega mag. Top right : g -band mosaic of M 51; grayscale in AB mag. Bottomleft : M 51 stellar mass-map derived with the ZCR method, based on ( g − i ) and ( i − K s ) colors, and K s mass-to-light ratio, Υ K s ∗ ; mass in M (cid:12) . Bottom right : ( g − K s ) color map of M 51. Notice the similarities of the features in this extinction map and in the stellar mass-mapin the bottom left panel (c). Higher extinction is indicated by darker features. North is up, East is to the left. Mart´ınez-Garc´ıa et al.
Fig. 3.—
SDSS i -band mosaic of M 51. Grayscale in AB mag.North is up, East is to the left. ure 2c) is that it does not present a smooth spiral armstructure. There is a well defined two-arm spiral pat-tern, but many filamentary structures are also observed.In addition, a visual comparison of the mass structurewith the optical g -band indicates that, presumably, mostof the structure is coincident with the dust lanes, as in-ferred from optical extinction. This can be seen moreeasily in Figure 2d (bottom right panel), where we showthe ( g − K s ) image. To test the similarities between themass-map and the ( g − K s ) image quantitatively, we usecross-correlation techniques. The Pearson correlation co-efficient is defined as r = (cid:80) j (cid:80) i ( f ij − ¯ f )( g ij − ¯ g ) (cid:114)(cid:80) j (cid:80) i ( f ij − ¯ f ) (cid:114)(cid:80) j (cid:80) i ( g ij − ¯ g ) , (1)where f ij is the intensity of the i th , j th pixel in the firstimage, g ij is the intensity of the i th , j th pixel in the sec-ond image, ¯ f is the mean intensity of the first image,and ¯ g is the mean intensity of the second image. Thecross-correlation function, ( f (cid:63) g )( θ ), is then obtainedby rotating the first image with respect to the secondone, while fixing the center of rotation at the center ofthe object (the nuclei of M 51 in this case). We obtain r ( θ ) from equation 1 by varying θ from − ◦ to 180 ◦ inincrements of 1 ◦ ; we assume that the angle θ increasescounterclockwise. All the M 51 data were deprojectedassuming an inclination angle of 20 ◦ , and a position an-gle of 172 ◦ (Leroy et al. 2008). The result of the cross-correlation between the output mass-map of the ZCRmethod and the intensity ratio in the ( g − K s ) image isshown in Figure 5. By “intensity ratio”, we mean theratio between the intensity in the g -band image and theintensity in the K s -band image. We use this ratio insteadof the ( g − K s ) color because the latter scales logarithmi-cally and cannot be compared with the mass distribution,that scales linearly. Note that we actually take the inten-sity ratio in the minus ( g − K s ) image; this is done withthe purpose of getting positive values of r (when usingequation 1). Error bars were estimated with bootstrapmethods (Bhavsar 1990; Lepage & Billard 1992). We re-place each pixel separately with a random value, drawnfrom a Gaussian probability distribution, and for each θ recalculate equation 1. We repeat this process a total of 30 times and calculate σ cc , the standard deviation of theresulting distribution.There is clearly a peak in the cross-correlation functionnear θ = 0 ◦ , indicating a similarity between the struc-tures. For comparison, we also show the cross-correlationbetween the intensity in the K s -band image and the in-tensity ratio in the ( g − K s ) image. The absolute max-imum in this case occurs around θ = − . ◦ ± .
8, andmarks the angular offset between the spiral arms in the K s -band and the dust lanes in the ( g − K s ) image. Thismeans that if we rotate the spiral arms in the K s -bandby 15 ◦ , clockwise, they will match the spatial location ofthe dust lanes.As is well known, disk galaxies, when studied at differ-ent wavelengths, often show significant differences (e.g.,Block & Wainscoat 1991; Block et al. 1994). Even ifat NIR wavelengths young stars and clusters can con-tribute 20% – 30% to the total radiation in spiral arm re-gions (e.g., Rix & Rieke 1993; Gonzalez & Graham 1996;Rhoads 1998; James & Seigar 1999; Patsis et al. 2001;Grosbøl et al. 2006; Grosbøl & Dottori 2008), most of thelight in the disk comes from evolved giant stars, and mostof the mass is concentrated in low mass main sequencestars. Hence, any structures present in resolved stellarmass-maps should resemble the NIR surface brightnessmorphology to a significant degree. This is not the caseof the stellar mass-map shown in Figure 2c (bottom leftpanel), where we see filamentary structure not present inthe K s light distribution, Figure 2a (top left panel).We perform three other different and independenttests, and compare the resulting stellar mass-maps asdescribed below.1. We do not use the NIR band, and rely only on theoptical SDSS colors, e.g., ( u − i ) and ( g − i ), andon the mass-to-light ratio estimated in the i -band,Υ i ∗ .2. We remove the binning of the models and use thefull 5 × templates of the MAGPHYS-CB07 li-brary in the computations.3. We use a new Monte Carlo SPS (optical-NIR) li-brary taken from the Synthetic Spectral Atlas ofGalaxies (SSAG; Magris et al. 2015). SSAG as-sumes random SFHs according to the Chen et al.(2012) prescription, that includes a burst and atruncation event. Dust is treated as in Charlot& Fall (2000), and metallicity is distributed be-tween 0.02 Z (cid:12) and 2.5 Z (cid:12) , with 95% galaxy tem-plates having Z > . Z (cid:12) . The adopted IMF isChabrier. The library contains 6 . × tem-plates (SSAG-BC03 library henceforth). The rangein these models of the effective mass-to-light ratioin the K s -band, Υ K s ∗ , as determined by a ( g − i )vs. ( i − K s ) color-color diagram, is shown in Fig-ure 6, left panel. For comparison purposes weshow the same diagram for the BC03 version ofthe MAGPHYS library (MAGPHYS-BC03) in theright panel. The MAGPHYS library extends toredder colors due to the different probability distri-bution functions used to model the optical depthin the V -band, τ V (see Figure 7). EMOVING BIASES IN STELLAR MASS-MAPS 5
Fig. 4.— g − i ) and ( i − K s ) colors of M 51’s pixels. The areas inside the blue and red dashed linescontain 99% and 68%, respectively, of the templates in the MAGPHYS-CB07 SPS library corrected for Galactic extinction. Left : afterapplying the
Adaptsmooth procedure as described in the text. The maximum of log(number) occurs near ( i − K s ) ∼ .
29 and ( g − i ) ∼ . Right : without applying the
Adaptsmooth procedure.
Fig. 5.—
Cross correlation functions r ( θ ) (see text). Solid line: between the mass structure resulting from the ZCR method and theintensity ratio in the ( g − K s ) image. The absolute maximum is at θ = 0 ◦ , indicating similarity. Dashed line: between the intensity inthe K s -band image and the intensity ratio in the ( g − K s ) extinctionmap. The maximum occurs at θ ∼ − ◦ (marked by the verticaldotted line), and corresponds to the angular lag between the dustlanes and the stellar arms. The total height of each error bar is2 σ cc . The filamentary structure, and the spatial coincidencebetween mass and dust lanes prevail in all the tests. Asimilar result is obtained for other spiral galaxies as well,and was already noticed although not sufficiently dis-cussed in ZCR (their Figure 8). It is noteworthy thatthis is not a problem of resolution in the SPS libraries,since the mean sampling is ∼ . K s ∗ ; hence, the template set is denselypopulated.In this work we will focus on the structural propertiesof the mass-maps. We will only mention here that boththe local and the integrated stellar masses derived fromSPS models may vary on account of different treatmentsof the thermally-pulsating asymptotic giant branch (TP- AGB; see, e.g., Maraston et al. 2006; Bruzual 2007; Con-roy et al. 2009), and the choice of IMF in the libraries.The mass determinations may also differ if obtained fromdifferent bands, even when using the same models (Mc-Gaugh & Schombert 2014).2.2. The level of accuracy in mass-to-light ratioestimates
Gallazzi & Bell (2009) discuss thoroughly the Υ ∗ ac-curacy that can be achieved by comparing colors withpredictions from a large library of SFHs. Typical accu-racies are of the order of 0.1-0.15 dex. A similar resultis deduced by other authors (e.g., Bell & de Jong 2001;ZCR; Taylor et al. 2011). This level of accuracy is barelyimproved with spectroscopic data (Gallazzi & Bell 2009).To better understand the impact of a limited Υ ∗ ac-curacy on the resolved mass-maps of galaxies, we builda sample of mock galaxies drawn from the MAGPHYS-CB07 Monte Carlo SPS library. Each of the ≈ × templates is used as an individual object in our mock cat-alog. In order to simulate the photometric error, we addto each of the g , i , and K s -band magnitudes in our mocksa random noise component with a Gaussian distribution,having σ mag = 0 .
02 mag ( ∼
2% intensity variation). Wethen try to fit the noisy ( g − i ) and ( i − K s ) values of eachsimulated object with the noise-free ( g − i ) and ( i − K s )colors, via χ minimization. Afterwards we compute∆ log[Υ K s ∗ ] = log[Υ K s ∗ ] fit − log[Υ K s ∗ ] true , (2)i.e., the ratio between the fitted Υ ∗ and the true value.The results of this test are shown in Figure 8, where weget a dispersion (standard deviation) σ (∆ log[Υ K s ∗ ]) ∼ .
16 dex, as expected. We carry out the same exercise fordifferent σ mag values and obtain σ (∆ log[Υ K s ∗ ]) for eachone. The results are shown in Figure 9, upper panel.There is a nearly linear decrease of σ (∆ log[Υ K s ∗ ]) withdiminishing σ mag down to σ mag ∼ . σ mag , the shape of the ∆ log[Υ K s ∗ ] distribution We notice that Gallazzi & Bell (2009) obtain a similar plotin spite of neglecting dust corrections, which indicates that dust isnot a decisive factor for Υ ∗ accuracy. Mart´ınez-Garc´ıa et al.
Fig. 6.—
Left : decimal logarithm of the effective mass-to-light ratio at the K s -band, Υ K s ∗ , derived from the ( g − i ) vs. ( i − K s ) color-colordiagram. The data are taken from the SSAG-BC03 Monte Carlo SPS library (Magris et al. 2015), corrected for Galactic extinction towardsM 51. SDSS g and i magnitudes are in the AB magnitude system, K s magnitudes are Vega. The blue/red dashed contour delimits99%/68% of the observed colors for M 51 (see Figure 4, left panel). Right : analogous to left panel, but for the MAGPHYS-BC03 MonteCarlo SPS library.
Fig. 7.—
Probability distribution functions of the V -band opticaldepth of the dust seen by young stars, τ V , used by the SSAG-BC03(solid line), and the MAGPHYS (dashed line) Monte Carlo SPSlibraries, respectively. abruptly begins to change, from nearly Gaussian withkurtosis ∼
3, going through Laplace distributions, andfinally tending to a Dirac delta function with kurtosis → ∞ . This effect can be appreciated in the lower panel ofFigure 9, where we plot the excess kurtosis of ∆ log[Υ K s ∗ ]versus σ mag .As σ mag tends to zero, the dispersion, σ (∆ log[Υ K s ∗ ]),also tends to zero. A (hypothetical) value of σ (∆ log[Υ K s ∗ ]) = 0 would indicate that our adjusted val-ues are equal to the true values (the noise-free models).We can infer that it is not feasible to get accurate Υ ∗ values unless the intrinsic errors of the observations arediminished to zero, i.e., σ mag →
0. Typical photomet-ric calibration errors are of the order of 1 −
2% for the Excess kurtosis is measured with respect to the kurtosis of anyunivariate normal distribution, which equals 3. Therefore, excesskurtosis equals kurtosis minus 3.
Fig. 8.—
Fits to colors of ≈ × mock galaxies (seetext). Noise modeled as a random Gaussian distribution with σ mag = 0 .
02 mag is added to the mock objects before fitting themwith the noise-free templates of the MAGPHYS-CB07 library. Thedifference is quantified as ∆ log[Υ K s ∗ ] = log[Υ K s ∗ ] fit − log[Υ K s ∗ ] true .The standard deviation of ∆ log[Υ K s ∗ ] is σ (∆ log[Υ K s ∗ ]) = 0 .
16 dex.
SDSS (Padmanabhan et al. 2008) and other photometricsurveys. Additionally to this, the degeneracies betweenthe different SPS model parameters (e.g., age-metallicity-reddening) will prevail even when σ mag → χ minimization, will be discrepant from the structuresof NIR surface brightness maps, owing to a limited Υ ∗ accuracy. In this manner, the fit we can obtain for someobserved colors will result in a Υ ∗ value near the statisti-cal mode of similar colors in the SPS library (see also thediscussion in Taylor et al. 2011), and within 0 . − . ∗ will depend on the colors used in the fit.EMOVING BIASES IN STELLAR MASS-MAPS 7 Fig. 9.—
Statistical parameters of the fits to colors of mockgalaxies.
Top: standard deviation, σ (∆ log[Υ K s ∗ ]), vs. σ mag inthe range 0.0-0.2 magnitude (see also Figure 8, where σ mag =0 .
02 magnitude).
Bottom: excess kurtosis (or kurtosis minus 3) of∆ log[Υ K s ∗ ] vs. σ mag . BAYESIAN INFERENCE AIMED AT AN OBJECTIn this section we introduce the Bayesian successivepriors (BSP) algorithm, aimed at an individual object,in order to solve for the mass-map avoiding the bias inthe spatial structure. The idea is to use the previous in-formation regarding the stellar surface mass density asdeduced from the NIR bands. The massive older pop-ulation of a galaxy is mainly traced in the NIR bands,specially the K -band (Rix & Rieke 1993). Having es-tablished this, we can adopt the NIR surface brightnessdistribution as a Bayesian prior, in order to infer the“true” stellar surface mass density. In this work, we willuse the term “prior” in reference to the prior probabilitydistribution function . The Bayesian prior is then directedto a particular galaxy, and not to the entire galaxy pop-ulation. 3.1. Bayes’ theorem
Bayesian probability posits that the best outcome ofany event is found by calculating the probabilities of thevarious hypotheses involved, using the rules of probabil-ity theory (e.g., Loredo 1992, 1995).The ZCR approach uses a method similar to a Bayesian maximum-likelihood estimate by including a uniform (orflat) prior in the fits to the observed colors, regard-less of the SPS library. In the present work a signif-icant improvement is made in the calculation of thestellar mass-maps, by introducing a Bayesian methodwith an informative, non-uniform, prior. Applicationsof Bayesian inference with non-uniform priors have beenused in, e.g., Ben´ıtez (2000), for cosmological redshift es-timates, Rovilos et al. (2014), for AGN sources analysis,and Sch¨onrich & Bergemann (2014), for the determina-tion of stellar parameters.In our case, Bayes’ theorem for the most probable stel-lar mass-to-light ratio Υ ∗ is given by P (Υ ∗ | C ) = P ( C | Υ ∗ ) P (Υ ∗ ) P ( C ) , (3)where P (Υ ∗ | C ) is the posterior probability, i.e., theprobability of having Υ ∗ , for a certain stellar population,if colors C are observed. P ( C | Υ ∗ ) is the likelihood function (or the probabilityof observing colors C given the set of parameters Υ ∗ ): P ( C | Υ ∗ ) ∝ √ π exp (cid:18) − χ (cid:19) , (4) χ = N colors (cid:88) n =1 (cid:18) C obs n − C template n σ col (cid:19) , (5)where C obs n is the observed n th color with σ col photo-metric error, and C template n is the color from a certaintemplate in our SPS library. In our case N colors = 2, forinstance, ( g − i ) and ( i − K s ), hence n = 1 , P (Υ ∗ ) represents the previous knowledge we may haveabout the likely value of the Υ ∗ parameter, and P ( C ) = N templates (cid:88) j =1 P ( C | Υ ∗ j ) P (Υ ∗ j ) (6)is a normalization constant, also called the Bayesian ev-idence (Savage & Oliver 2007). N templates stands for thenumber of templates in our SPS library.3.2. The Bayesian successive priors (BSP) algorithm
The prior probability distribution function
In order to apply the BSP algorithm, we have chosena prior probability distribution function, P (Υ ∗ ), of theform P (Υ ∗ ) = exp − (cid:34) Υ prior ∗ − Υ ∗ σ Υ ∗ (cid:35) , (7)where σ Υ ∗ = (cid:20) ln(10)2 . (cid:21) σ mag Υ prior ∗ . (8)Here, σ mag is the photometric error for a certain pass-band, which is related to σ col in equation 5 through √ σ mag ≈ σ col .Each template in the SPS library corresponds to a sin-gle Υ ∗ . By using equation 7 and Bayes’ theorem (equa-tion 3), we can effectively marginalize the templates fromour SPS library, as we will demonstrate in the followingsections. 3.2.2. Description of the BSP algorithm
The BSP algorithm consists of three iterations that aredescribed below. The algorithm is intended to work witha SPS library and surface photometry in several/variousbands. In the following we assume that these are theoptical g and i bands, and the NIR K s filter. For the Mart´ınez-Garc´ıa et al.library, we use SSAG-BC03 (although the algorithm isdesigned to work independently of the choice of SPS li-brary). The mass-to-light ratio is taken in the K s -band,Υ K s ∗ . Other waveband combinations will be discussedlater. The algorithm is applied on a pixel-by-pixel basis,although in each iteration all pixels are addressed beforemoving to the next iteration.1. In the first iteration we use a uniform prior, i.e., P (Υ ∗ ) = constant, and apply equation 3. Then wecalculate the absolute maximum (which should benear the median) of the posterior probability distri-bution function P (Υ K s ∗ | C ), and the 16th and 84thpercentiles, to account for the corresponding errormap. We estimate the percentiles by progressivelyintegrating the area under the posterior probabilitycurve until we accumulate an area of 0.16 and 0.84(being the total area equal to 1), for the 16th and84th percentiles, respectively. Up to this point the method provides a maximumlikelihood estimate and is similar to the ZCR algo-rithm, with the only difference that the templatesare not binned in our case. We call the unbinnedversion of the ZCR algorithm ZCR (cid:48) from now on.We then use the results of this step for two pur-poses. Firstly, we identify all the pixels for whichthe difference (absolute value) between their ob-served color and the fitted template in the SPS li-brary is smaller than 3 σ col , i.e., | ∆ C n | = (cid:12)(cid:12) C obs n − C template n (cid:12)(cid:12) < σ col , (9)for n = 1 ,
2. The pixels that do not fulfill the3 σ col condition are isolated and flagged. This stepguarantees that we keep only pixels that can bedescribed by our SPS library. Next, we take theresulting Υ K s ∗ values for all the kept pixels and cal-culate the statistical median.
2. In the second iteration this median value of Υ K s ∗ ,from iteration number 1, is used as a constant pa-rameter in equation 7, i.e.,Υ prior ∗ = constant (10)for all pixels in the disk. The prior, P (Υ ∗ ), isnot uniform in this case, and adopts the functionalform of equation 7. Now we compute the maximumin P (Υ K s ∗ | C ), and the respective 16th and 84thpercentiles. Similarly to iteration number 1, weidentify all the pixels where the difference betweenthe observed colors and the fitted library templatesis smaller than ασ P , i.e., These values are equivalent to − σ and 1 σ , respectively, in anormal distribution. These include elements recording emission from AGN activity. The number separating the lower and higher value halves ofΥ K s ∗ . A refinement of the method could be achieved by separatingthe bulge from the disk of the galaxy, and treating them as objectswith different median Υ K s ∗ (Portinari et al. 2004). | ∆ C n | < ασ P , (11)for n = 1 ,
2. The value of σ P is determined fromthe resulting ∆ C n (no absolute value) pixel distri-bution by calculating its 16th and 84th percentiles,P and P , respectively, and then using σ P = (P − P ) / , (12)for each color. After some tests (see Appendix A),we have found that α = 1 . C n ∼
0. In a hypothetical case,having ∆ C n = 0 would indicate that our observedcolors match perfectly the fitted library templates.The | ∆ C n | < ασ P pixels will be the “backbone”of our mass-map, and represent the locations inthe disk where the K s -band is a reliable tracer ofthe stellar mass surface density, considering theΥ prior ∗ = constant condition. The | ∆ C n | > ασ P pixels belong mainly to luminous red stars in theasymptotic giant branch, red supergiants, low sur-face brightness regions in the outskirts of the disk,and high extinction regions where Υ K s ∗ does nothave the constant (median) value we assumed ear-lier. We then need to provide a new Υ K s ∗ valuefor these | ∆ C n | > ασ P pixels. For this purposewe use the information from the “backbone” pix-els. We interpolate the stellar mass surface densityto fill the places where we need a new Υ K s ∗ value.The interpolation is done in the 0 ◦ , 45 ◦ , 90 ◦ , and135 ◦ directions, and then an average is taken. Af-ter the interpolation, we visually inspect the result-ing maps to determine whether a minor smoothingis needed. The smoothing is only applied to the | ∆ C n | > ασ P pixels, and is performed by replacingeach pixel value with the average of the neighbor-ing pixels. There are other interpolation techniquesthat could be used (see, e.g., Gumus & Sen 2013),but for the present work we will apply the abovementioned procedure to all objects. Having estab-lished this, the new Υ K s ∗ values are estimated asthe ratio of the interpolated mass-map and the ob-served K s photometry.3. The third and last iteration is intended to deal onlywith the | ∆ C n | > ασ P pixels, identified in itera-tion number 2. For each pixel, we use the Υ K s ∗ value also estimated in iteration number 2 to repre-sent Υ prior ∗ in equation 7, and calculate the absolutemaximum of the posterior probability distributionin equation 3. Before this, we may also update theuncertainty in Υ ∗ , in equation 7; such uncertaintynow reads σ Υ ∗ = (cid:115)(cid:18)(cid:20) ln(10)2 . (cid:21) σ mag Υ prior ∗ (cid:19) + β , (13)where β accounts for the propagation of uncertain-ties arising from the previous iteration (e.g., theEMOVING BIASES IN STELLAR MASS-MAPS 9 BSP algorithm
Photometry(Optical, NIR)+SPS library P (Υ ∗ | C ) ∝ exp (cid:16) − χ (cid:17) Identify | ∆ C n | < σ col pixelsGet median Υ NIR ∗ P (Υ ∗ | C ) ∝ exp (cid:16) − χ (cid:17) exp (cid:18) − h Υ prior ∗ − Υ ∗ σ Υ ∗ i (cid:19) Use Υ prior ∗ = median Υ NIR ∗ , as constant for all pixelsIdentify | ∆ C n | < ασ P pixelsInterpolate MassGet new Υ NIR ∗ for | ∆ C n | > ασ P pixels P (Υ ∗ | C ) ∝ exp (cid:16) − χ (cid:17) exp (cid:18) − h Υ prior ∗ − Υ ∗ σ Υ ∗ i (cid:19) Use Υ prior ∗ = new Υ NIR ∗ , individually for each pixelResolvedpixel-by-pixelstellar mass map Fig. 10.—
Bayesian successive priors (BSP) flowchart. Photom-etry in some optical bands (e.g., SDSS g and i ) and one NIR band(e.g., K s ), as well as a SPS library, are required. P (Υ ∗ | C ) denotesthe posterior mass-to-light ratio probability distribution; ∆ C n isthe difference between the observed n th color of a pixel, and thecolor of the fitted template in the SPS library; σ col is the photo-metric error in the n th color; σ P is determined from the ∆ C n pixeldistribution in iteration number 2 (see equation 12). Each one ofthe three rectangular white boxes stands for an iteration of theBSP algorithm (see text). mass surface density interpolation from neighbor-ing pixels). Using bootstrap methods we have es-timated that β ≈ . K s ∗ map we then obtain thestellar mass surface density to complete our mass-map.As an optional last step, the flagged pixels from it-eration number 1 that belong to the inner disk can beinterpolated in mass with the information about the sur-rounding pixels provided by all three iterations. For theexternal disk pixels, the interpolation is more uncertain.We find that adding more iterations does not leadto any further improvement in the mass-maps. Theflowchart of the BSP algorithm is shown in Figure 10.For the BSP algorithm to work properly, the require-ment of NIR data with high S/N ratio is essential; oth-erwise, any noisy and patchy features will be transferredto the mass-map. A minimum S/N ratio of ∼ − Adaptsmooth code, or alternatively with Voronoi two-dimensional binning (Cappellari & Copin 2003).In this investigation we have adopted only two colors,( g − i ) and ( i − K s ), and thus N colors = 2. The ben-efits of using the g and i SDSS data together with one NIR band are an excellent spatial resolution per element(pixel), and extensive spatial coverage (of the entire ob-ject). Nevertheless, the BSP algorithm can also be ap-plied by using N colors >
2, with the only requirement ofthe inclusion of one NIR band as described earlier. Ina separate publication we will explore the use of the al-gorithm to fit optical IFU observations, for instance, theCalar Alto Legacy Integral Field Area survey (CALIFA,S´anchez et al. 2012), and the Mapping Nearby Galaxiesat Apache Point Observatory survey (MaNGA, Bundyet al. 2015). APPLICATION OF BSP TO M 51We apply the BSP algorithm to M 51 employing thesame data described in section 2.1. We calculate σ mag on a pixel-by-pixel basis assuming that σ mag ≈ (cid:113) σ + σ , (14)where σ flux is the random error in the flux per pixel,which we assume to be dominated by the uncertainty inthe background (see also, Mentuch Cooper et al. 2012),and σ calib is the calibration uncertainty, or zero point er-ror, for which we assume σ calib ∼ .
01 mag for the SDSSimages, and σ calib ∼ .
03 mag for the K s image (Jar-rett et al. 2003). We compute σ flux in mag by using σ flux = 1 . ∗ σ back flux , where σ back is the standard de-viation in the background (in a sky-subtracted image).We compute σ back by sampling the background statis-tics in different boxes near the edges of the images. Toaccount for the use of the Adaptsmooth procedure we di-vide σ back by √ n pix , where n pix is the number of pixelsused to increase the S/N of the corresponding pixel by Adaptsmooth .Without taking into account correlation betweenbands, we compute σ col by summing in quadrature the σ mag values of each band involved in the color determi-nation.In Figures 11 and 12, we show the results of adoptingthe SSAG-BC03 and MAGPHYS-CB07 libraries, respec-tively. In both figures, the top left panels (a) show themask obtained after iteration number 1. White regionsrepresent the pixels where the observed colors are within3 σ of at least one SPS-library template (see Figures 4or 6). In the respective top right panels (b), we showthe masks obtained after iteration number 2. For thesemasks, the gray regions represent the pixels where thecolor difference (absolute value) between the models andthe observations, | ∆ C n | , is greater than ασ P , with α = 1(see section 3.2, and Appendix A), assuming a constantΥ K s ∗ . These regions will be interpolated in mass withthe information of neighboring pixels. We can also ap-preciate that the SSAG-BC03 library does a better job atmodeling the outskirts of the disk than the MAGPHYS-CB07 library. To investigate the cause of this behaviorwe obtain a mass map by using MAGPHYS-BC03. Weobtain very similar masks to those from the SSAG-BC03library (Figure 11, top panels). With this in mind, mostof the differences between BC03 and CB07 mass-mapsin our results are mainly due to the distinct treatmentsof the TP-AGB stage. To a lesser extent, we also no-tice an improvement when SSAG-BC03 is used, instead0 Mart´ınez-Garc´ıa et al.of MAGPHYS-BC03. We attribute this to the fact thatSSAG covers a wider range of possible star formationhistories.In the bottom left panels (c) of Figures 11 and 12,we show the resulting stellar mass surface density mapafter iteration number 3. The filamentary structure isno longer present, and the maps show greater resem-blance to the features in NIR bands, as expected. Fi-nally, the bottom right panels (d) of both figures showthe “residuals”; these are the result of subtracting thefinal output (iteration 3) mass-map using BSP, from amass-map that assumes a constant Υ K s ∗ (the median Υ ∗ after iteration number 1). The dark/white regions rep-resent positive/negative mass differences, i.e., where Υ ∗ has been overestimated/underestimated. For example,the Υ ∗ may be overestimated when young luminous redstars are mixed with older populations, and underesti-mated due to extinction in the NIR bands. This is dif-ferent from the “outshining bias” (Maraston et al. 2010;Sorba & Sawicki 2015), where the light from young starseclipses the old population and the amount of stellarmass is underestimated. In our case we overestimate themass (by using a constant Υ K s ∗ ) because we are assum-ing, mistakenly but for convenience, that all the lightcomes from old stars.4.1. Isolating the old massive disk
We will now discuss in more detail the positive massdifferences in the residuals. In Figure 13 we plot a 2-Dhistogram of the colors of the pixels for which the massdifference is > × M (cid:12) . This cut in the mass waschosen in order to isolate most of the positive residualsnear the spiral arms. We have excluded the pixels fromthe bulge region. We note that most points gather in agroup with a maximum near ( i − K s ) ∼ . g − i ) ∼ .
4. Their ( g − i ) color is relatively blue whencompared with all the colors observed (delimited by theblue dashed contour). We also note a cluster of pointswith redder colors, near ( i − K s ) ∼ . g − i ) ∼ . r -band light-weighted age and for Υ K s ∗ , obtained for M 51 using theMAGPHYS-CB07 library. The dashed-dotted green linecorresponds to the, previously described, “positive massdifferences” in the residuals, while the blue solid linerefers to the whole disk, both results after BSP. Interest-ingly, the excess mass regions are younger (age ∼ K s ∗ (by 30%) than most of the pixelsin the disk. Together with the bluer ( g − i ) color, theabove characteristics indicate that these regions containrelatively young stars, that mix with the old stellar pop-ulation in star forming regions. These were effectivelyisolated by BSP!The red dashed line in Figure 14 shows the probabilitydistributions for the whole disk after applying the ZCR (cid:48) approach. The light-weighted age yields a larger fractionof younger pixels with ZCR (cid:48) . As expected from our pre-vious assumptions, the values of Υ K s ∗ are more narrowlyconfined with BSP, around Υ K s ∗ = 0 . ± . K s ∗ for the whole disk, we recover a median Υ K s ∗ = 0 . K s ∗ = 0 . ± . | ∆ C n | < ασ P pixels we have Υ K s ∗ =0 . ± . | ∆ C n | > ασ P pixels weobtain Υ K s ∗ = 0 . ± . K s ∗ , derived with MAGPHYS-CB07 and SSAG-BC03, are consistent (within 3 . σ ) withthe result derived by Just et al. (2015) for the solar cylin-der from star counts (Υ K s ∗ = 0 . K s ∗ = 0 . Integrated mass estimates
With respect to the total resolved mass, defined as M resolved ∗ = (cid:88) j (cid:88) i M ∗ ij , (15)where M ∗ ij is the stellar mass of the i th , j th pixel, wefind the following results. By using the MAGPHYS-CB07 library we obtain for M 51 a total stellar mass of M resolved ∗ = 3 . × M (cid:12) with ZCR (cid:48) , and M resolved ∗ =3 . × M (cid:12) with BSP. The SSAG-BC03 library, mean-while, leads to M resolved ∗ = 6 . × M (cid:12) with ZCR (cid:48) , and M resolved ∗ = 5 . × M (cid:12) with BSP. The discrepancybetween the SSAG-BC03 and MAGPHYS-CB07 mass es-timates is mainly due to the different treatments of theTP-AGB phase (Bruzual 2007). In Figure 15, we showthe azimuthally averaged surface mass density vs. radiusfor M 51 obtained with SSAG-BC03. For most of thedisk, the BSP method yields smaller mass estimates thanZCR (cid:48) , resulting in a ∼
10% decrease in the total mass.To complement the analysis, we show in Figures 16a and16b (top left and top right panels) the Υ g ∗ maps obtainedwith the ZCR (cid:48) method and the BSP algorithm, respec-tively. Figures 16c and 16d (bottom left and bottom rightpanels) present the Υ K s ∗ maps from ZCR (cid:48) and BSP, re-spectively. Figure 17 shows the azimuthally averaged Υ ∗ for the g , i , and K s bands, as a function of radius. Asexpected, the K s profile is virtually constant, while the g and i profiles show variations with radius, with lowervalues at the outskirts of the disk, as a result of a lowersurface brightness and bluer colors (de Jong 1996; Bell& de Jong 2001).In Figure 18, we show the azimuthally averaged stel-lar metallicity, Z/Z (cid:12) ; similar results are obtained forboth BSP and ZCR (cid:48) . In this figure we also plot themetallicity abundance gradients for M 51 from Mous-takas et al. (2010). From ancillary data, Moustakas etal. (2010) estimate radial oxygen abundance gradientsfor 75 galaxies in the Spitzer Infrared Nearby Galax-ies Survey (SINGS, Kennicutt et al. 2003), using boththe Kobulnicky & Kewley (2004; KK04) and the Pi-lyugin & Thuan (2005; PT05) calibrations. We trans-form Moustakas et al. (2010) oxygen abundance gradi-ents in units of 12 + log(O / H), to units of
Z/Z (cid:12) , adopt-ing (e.g., Mart´ınez-Garc´ıa et al. 2009)log(
Z/Z (cid:12) ) (cid:39) .
12 + log(O / H) . (16)The stellar metallicity we recover with SSAG-BC03 fallsbetween the two curves of Moustakas et al. (2010). Men-EMOVING BIASES IN STELLAR MASS-MAPS 11 Fig. 11.—
Application of the BSP algorithm to the spiral galaxy M 51. The Monte Carlo SPS library used is SSAG-BC03.
Top left :resulting mask after iteration number 1. White regions have observed colors within 3 σ of at least one template in the library. Top right :resulting mask after iteration number 2. Gray regions represent pixels where the assumption of a constant Υ K s ∗ for the whole disk isnot fulfilled by the observed colors. Bottom left : resulting mass-map after iteration number 3.
Bottom right : residuals after subtractingthe mass-map obtained at the end of the BSP algorithm (iteration 3), from a mass-map that assumes a constant Υ K s ∗ (the median afteriteration 1). Dark/white regions represent positive/negative mass differences. tuch Cooper et al. (2012) obtain a similar result for theWhirlpool galaxy, from optical and infrared photometry.4.3. Other filter combinations
In this section we discuss the application of the BSPalgorithm with other filter combinations. By using onlyoptical filters, e.g., the ( g − i ) color and Υ i ∗ , the methodis not able to recover a spatial structure consistent withthe one obtained with optical-NIR combinations. This isdue to the fact that the information of the prior spatialstructure is missing, as it can only be provided by theNIR bands. The Υ i ∗ cannot be assumed to be constantthrough the entire disk (see Figure 17); besides, dustlanes can still be noticed near spiral arms, even at theredder optical wavelengths (see Figure 3).For the case when the u filter is included, we wereunable to fit the data satisfactorily. We have quantifiedthe mean S/N ratio of the imaging data for the entire diskof M 51 (without applying the Adaptsmooth procedure),and obtain a value of 2 .
8, 23 .
2, and 29 . u , g ,and i bands respectively. Taking this into account wecan deduce that the issues we encounter when trying to fit the u -band SDSS data with our methods are mainlydue to their low S/N ratio. This shortcoming can beremedied with deeper data. We should also mention thatΥ ∗ is more degenerate at shorter wavelengths.We also applied the BSP algorithm including theSpitzer-IRAC 3 . µ m band. We used the colors ( g − i ) and( i − . µ m), and Υ . µ m ∗ . We computed pixel-by-pixel σ mag errors as in section 4, assuming σ calib ∼ .
01 magfor the SDSS images, and σ calib ∼ .
03 mag for the 3 . µ mband (Reach et al. 2005). We corrected for Galactic ex-tinction as in Schlafly & Finkbeiner (2011), and Chap-man et al. (2009). The results with the MAGPHYS-BC03 library are shown in Figure 19. It can be noticedthat the residuals, i.e., the difference between a mass-map that assumes a constant Υ . µ m ∗ and the outputmass-map from BSP (Figure 19d, bottom right panel),are significantly different from the ones obtained whenusing the K s -band (see Figures 11d and 12d, bottomright panels). We attribute this to polycyclic aromatichydrocarbons (PAHs) and continuum dust emission at3 . µ m. To corroborate this we compare our result to the2 Mart´ınez-Garc´ıa et al. Fig. 12.—
Like Figure 11, but for the MAGPHYS-CB07 Monte Carlo SPS library.
0 1 2 3 4 (i-K s ) -0.50.00.51.01.52.0 ( g - i ) l og ( nu m b e r) Fig. 13.— one derived through the Independent Component Anal-ysis (ICA) method of Meidt et al. (2012, 2014). Thismethod separates the stellar emission from the dust emis-sion; Querejeta et al. (2015) applied it to the SpitzerSurvey of Stellar Structure in Galaxies (S G, Sheth et
Fig. 14.—
M 51 probability ( p ) distributions with theMAGPHYS-CB07 library. Left: r -band light-weighted age (yr); right: mass-to-light ratio Υ K s ∗ . Green dashed-dotted line:
BSP ob-served “positive mass differences” in the residuals (see text); bluesolid line:
BSP results for the whole disk; red dashed line:
ZCR (cid:48) output for the entire disk. al. 2010). We compare quantitatively the residuals fromBSP with the non-stellar (dust) component from ICA forM 51, by following the same cross-correlation procedureEMOVING BIASES IN STELLAR MASS-MAPS 13
Fig. 15.—
Azimuthally averaged mass surface density ( M (cid:12) pc − ) vs. radius, R (kpc), for M 51 with the SSAG-BC03 library.Results are for deprojected mass-maps. Blue solid line:
BSP; reddashed line:
ZCR (cid:48) . as in section 2.1 (equation 1). The results of this testare shown in Figure 20. We find that there is a strongspatial correlation between the ICA dust component andthe BSP residuals, indicated by the sharp peak at θ = 0in Figure 20. We also compare the BSP residuals to thestellar component obtained by ICA, and find no spatialcorrelation at θ = 0. Although our adopted SPS librarydoes not include the emission from dust in the 3 . µ mband , the BSP algorithm was able to isolate much ofit, together with that of red luminous young stars.A discussion of the differences between ICA and BSPwould require further analysis and comparisons using alarger sample of galaxies. This goes beyond the scope ofthe present work, and will be investigated in a separatepublication. PILOT TEST WITH OTHER GALAXIESIn order to better understand the differences betweenusing the BSP algorithm of section 3.2, and adopting theZCR (cid:48) method (i.e., a maximum likelihood estimate) toobtain resolved maps of stellar mass, we analyzed 90 ob-jects with H -band imaging from the Ohio State Univer-sity Bright Spiral Galaxy Survey (OSUBSGS, Eskridgeet al. 2002). The main statistical results from this sampleshould hold for other surveys, such as SINGS and S G.Our sample comprises all objects in the OSUBSGS forwhich SDSS g and i data are available (see Table 1). Abar chart of the Hubble types of our OSUBSGS sampleis shown in Figure 21. We subtracted the H -band data“sky offset” (see also Kassin et al. 2006) with either aconstant or a plane, depending on the object, and thencalibrated the resulting frames with 2MASS. We took op-tical g and i bands frames from the eighth release (DR8)of the SDSS (Aihara et al. 2011), and mosaicked them In principle the emission from dust could be included becauseit is predicted by MAGPHYS. Nevertheless, the number of tem-plates increases from 5 × to ∼ . × , and CPU time wouldbe ∼ × times larger. with the SWarp software (Bertin 2010). SDSS mosaicswere registered and re-sampled to the (lower resolution) H -band data with the aid of foreground stars. All fore-ground stars and background objects were then removedand replaced with random values from the background.The Adaptsmooth code was then used to increase theS/N ratio at the outskirts of the disk, while maintainingthe relatively higher S/N ratio for the inner disk pixels.We adopt a minimum S/N ratio per pixel of 10, and amaximum smoothing radius of 10.Together with the OSUBSGS sample, we also analyzedM 51b (companion of M 51, aka NGC 5195) using thesame data presented in section 4.5.1.
Mass-maps results
We adopt the SSAG-BC03 SPS library for all mass es-timates for this sample. For simplicity we assume that σ mag ∼ .
02 mag for every band and pixel. The short-coming of using a constant σ mag (and consequently a con-stant σ col ) for every band and pixel is that some of the fit-ted values could give slightly ( ∼ .
3% for individual pix-els) different results when compared to the case where in-dividual errors are computed for every pixel. The reasonfor this is the use of equation 4 together with equation 5.In our case we adopt two colors, hence equation 4 can beseen as the product of two Gaussian functions (one foreach color). In the case where σ col differs for each color,it can be easily demonstrated that this product resultsin another Gaussian function with different characteris-tics, including a distinct maximum, when compared tothe case of two equal Gaussian functions. Also, the un-certainties in the fitted values will be different. Despitethis, the overall results for each object will be practicallythe same (a ∼ .
1% difference for the resolved total massestimate).Some examples of the mass-maps from both the ZCR (cid:48) approach and BSP are shown in Figure 22. The differ-ence in spatial structures is clearly evident: whereas theZCR (cid:48) method gives noisy maps, BSP mass-maps beara greater similarity to the structures in the NIR-bands.Also shown in this figure are two extreme cases, wheredust extinction affects our mass estimates considerably.NGC 7814 is an edge-on spiral with a prominent mid-plane dust lane. From the first BSP iteration, the colorsof the pixels belonging to the dust lanes are identified(and flagged) as outside of the range available in the SPSlibrary. A similar phenomenon occurs with M 51b, sincethe dust lanes of one of the arms of M 51 are projecteddirectly on it. Consequently, a substantial number ofpixels are excluded after the first iteration of BSP. Nev-ertheless, our recovered stellar mass value for M 51b (seeTable 1), obtained via BSP, is ∼ half of the one derivedfor M 51. The same result was obtained by MentuchCooper et al. (2012).As a result of the application of the BSP algorithmto our pilot sample, we identify a trend of the medianΥ H ∗ (after BSP iteration number 1) with Hubble type, aspredicted by Portinari et al. (2004) and consistent withmore recent star formation/more constant SFHs for laterHubble types. A strong linear inverse (or negative) cor-relation with Hubble type is shown in Figure 23, with acorrelation coefficient (Bevington 1969), r xy = − . The value of r xy varies from 0, for no correlation, to ±
1, when
Fig. 16.— Υ ∗ maps for M 51 with the SSAG-BC03 library. Top left : Υ g ∗ obtained with ZCR (cid:48) method. Top right : Υ g ∗ with BSP algorithm. Bottom left : Υ K s ∗ , ZCR (cid:48) method. Bottom right : Υ K s ∗ , BSP algorithm. Darker pixels indicate higher Υ ∗ . Fig. 17.—
Azimuthally averaged Υ ∗ as a function of radius, R (kpc). Solid lines:
BSP; dashed lines:
ZCR (cid:48) . Dark blue: g -band; green: i -band; dark red: K s -band. Results are for deprojectedmaps of M 51 with the SSAG-BC03 library. Thus, potential biases are introduced when the same Υ ∗ is used for a sample of galaxies with different Hubbletypes. there is a full correlation. Generally, | r xy | (cid:39) . | r xy | ≈ . | r xy | ≈ . Fig. 18.—
Azimuthally averaged stellar metallicity,
Z/Z (cid:12) . Bluesolid line:
BSP; red dashed line:
ZCR (cid:48) . Results are for depro-jected maps of M 51 with the SSAG-BC03 library. For comparisonwe show the metallicity abundance gradients of Moustakas et al.(2010), using the KK04 (black dashed line) and PT05 (black dottedline) calibrations.
The total resolved stellar masses, M resolved ∗ (equa-tion 15), obtained, respectively, with the BSP algorithm, M BSP ∗ , and with the ZCR (cid:48) approach, M ZCR (cid:48) ∗ , are givenin Table 1. In Figure 24 we display the behavior of theEMOVING BIASES IN STELLAR MASS-MAPS 15 Fig. 19.—
Application of BSP algorithm to M 51 with ( g − i ), ( i − . µ m), Υ . µ m ∗ , and the MAGPHYS-BC03 SPS library. Panelsorganized as in Figures 11 and 12. Fig. 20.—
Cross-correlation functions (see text) for BSP resid-uals using the 3.6 µ m-band. Solid line: with the non-stellar (dust)emission from ICA (Meidt et al. 2012, 2014; Querejeta et al. 2015);the absolute maximum is at θ = 0 ◦ . Dashed line: with the stellaremission from ICA. Height of error bars is 2 σ cc . Fig. 21.—
Bar chart of Hubble types for our galaxy sample (90objects). Embedded images from the Digitized Sky Survey, DSS(blue). ratio M BSP ∗ /M ZCR (cid:48) ∗ vs. M BSP ∗ . From these data we findthat BSP mass estimates are on average ∼
10% lower6 Mart´ınez-Garc´ıa et al.
Fig. 22.—
Resolved maps of stellar mass.
Columns 1 and ZCR (cid:48) ; columns 2 and BSP. From left to right, in pairs: NGC 157,NGC 1042, NGC 4254, NGC 4051, NGC 4548, NGC 7606, NGC 7814, and M 51b. than those derived from ZCR (cid:48) , similarly to the M 51result. We also investigate possible trends of the ratio M BSP ∗ /M ZCR (cid:48) ∗ with Hubble Type; with the ratio of ma-jor to minor galaxy axes a/b ; with star formation rate, Ψ;and with V -band optical depth, τ V . We find no strong ormoderate correlations with these parameters, except forthe star formation rate, having r xy = − .
061 for Hub-ble Type, r xy = − .
297 for galaxy axial ratio (excludingthe edge-on object NGC 7814), and r xy = 0 .
082 for themedian τ V for the entire disk, obtained via BSP. We com-puted the star formation rate averaged over the last 10 yr from the parameters of the fitted templates as (cid:104) Ψ (cid:105) = (cid:82) tt − t last Ψ( t (cid:48) )d t (cid:48) t last , (17)where time t corresponds to the current Ψ, and t last =10 yr. We calculate (cid:104) Ψ (cid:105) on a pixel-by-pixel basis andthen sum over all pixels (in the same way as the resolved mass estimate). We also estimate the specific star for-mation rate averaged over the last 10 yr: (cid:104) Ψ (cid:105) S = (cid:82) tt − t last Ψ( t (cid:48) ) M ∗ ( t (cid:48) ) d t (cid:48) t last ≈ (cid:104) Ψ (cid:105) M ∗− , (18)where M ∗ is the current stellar mass. In this manner weobtain the resolved (cid:104) Ψ (cid:105) , and (cid:104) Ψ (cid:105) S , for the correspondingobject. In Figure 25 we show the ratio M BSP ∗ /M ZCR (cid:48) ∗ vs.the resolved (cid:104) Ψ BSP (cid:105) S for the whole disk. The correla-tion coefficient is r xy = − .
335 indicating a weak inversecorrelation. In the case of the resolved (cid:104) Ψ (cid:105) we obtaina correlation coefficient of r xy = − . M ZCR (cid:48) ∗ ,when compared to M BSP ∗ , is weakly related to the starformation rate over the disk.For completeness, we show in Figure 26 the resolvedgalaxy “main sequence” of star formation (see e.g.,Noeske et al. 2007; Daddi et al. 2007; Elbaz et al. 2007;EMOVING BIASES IN STELLAR MASS-MAPS 17 Fig. 23.—
Median Υ H ∗ vs. Hubble T-type after BSP iterationnumber 1, applied to the OSUBSGS pilot sample. Although withsome scatter, a strong ( r xy = − . H ∗ decreases with increasing T (later Hubble type). Fig. 24.—
Comparison of total resolved stellar mass esti-mates, log( M BSP ∗ /M ZCR (cid:48) ∗ ) vs. log( M BSP ∗ ). Horizontal error barsfor M BSP ∗ represent the propagated uncertainty in the distance tothe objects. Salim et al. 2007), i.e., the relationship between re-solved (cid:104) Ψ (cid:105) and M resolved ∗ . We find that this correlationis stronger with BSP ( r xy = 0 . (cid:48) ( r xy = 0 . Comparison with unresolved mass estimates
We also obtain for each object an unresolved mass es-timate, M unresolved ∗ . To this end, we fit the global ( g − i )and ( i − H ) colors of the object to all templates, and getthe optimum one via equation 4. Global magnitudes arecalculated by summing the intensities of all the pixels:mag global = − . (cid:88) j (cid:88) i f ij + zp, (19) Fig. 25.—
Decimal logarithm of ( M BSP ∗ /M ZCR (cid:48) ∗ ) vs.log (cid:104) Ψ BSP (cid:105) S , with Ψ S in yr − . The resolved specific star forma-tion rate is obtained as the sum of all pixels in the disk using BSP.A weak correlation is observed with negative correlation coefficient r xy = − . Fig. 26.—
The resolved “main sequence” of star forming galaxiesfor the OSUBSGS pilot sample with the SSAG-BC03 library.
Leftpanel (red triangles):
ZCR (cid:48) ; right panel (blue dots): BSP. Resolvedstar formation rate, (cid:104) Ψ (cid:105) , in units of M (cid:12) yr − , and resolved stellarmass, M resolved ∗ , in units of M (cid:12) . where f ij is the intensity of the i th , j th pixel at a certainband, and zp is the appropriate zero point. The samenumber of pixels is used in all mass estimates for thesame object.We compare in Figure 27 M unresolved ∗ with M resolved ∗ .The results for ZCR (cid:48) are shown in the left panel, andthose for BSP are presented on the right. On average wefind that, for our sample of galaxies, unresolved valuesunderestimate masses by ∼
20% compared to ZCR (cid:48) , butonly by ∼
10% relative to BSP. We also find, however,that for a fraction of the objects (15% when comparing toZCR (cid:48) and 25% vis-`a-vis BSP) the unresolved mass esti-mates are actually larger than those determined from re-solved studies. The estimate we can get for an unresolvedmass depends on how each pixel contributes to the globalcolors. Pixels that contain relatively young star forming8 Mart´ınez-Garc´ıa et al.
Fig. 27.—
Comparison of total unresolved and resolved stellarmass estimates.
Left panel (red triangles):
ZCR (cid:48) ; right panel (bluedots): BSP.
Fig. 28.—
Ratio of unresolved to resolved stellar mass estimatesvs. galaxy axial ratio, a/b (from RC3, de Vaucouleurs et al. 1991).
Left panel (red triangles):
ZCR (cid:48) ; right panel (blue dots): BSP. regions will lead to global bluer colors, and consequentlya lower global Υ ∗ (see Figure 1 or 6). On the other hand,pixels that contain extinction regions, due to dust, willlead to global redder colors and therefore a higher globalΥ ∗ . In spite of these possible effects the error bars forlog( M unresolved ∗ /M resolved ∗ ) > M unresolved ∗ /M resolved ∗ ) ∼ M unresolved ∗ /M resolved ∗ withHubble type ( r xy = 0 .
064 for BSP, and r xy = 0 . (cid:48) ), global ( g − i ) color ( r xy = 0 .
057 for BSP, r xy = − .
025 for ZCR (cid:48) ), or median τ V ( r xy = 0 . r xy = 0 .
116 for ZCR (cid:48) ). The correlation testwas also negative for galaxy inclination (see Figure 28),with r xy = 0 .
114 for BSP, and r xy = − .
090 for ZCR (cid:48) .When comparing the resolved (cid:104) Ψ (cid:105) S for each object withthe ratio M unresolved ∗ /M resolved ∗ , we find a weak positivecorrelation ( r xy = 0 . r xy = 0 . (cid:48) . In Figure 29 we show the ratio M unresolved ∗ /M resolved ∗ vs. resolved (cid:104) Ψ (cid:105) . The correlationcoefficients are r xy = 0 .
336 for BSP (right panel), and r xy = 0 .
212 for ZCR (cid:48) (left panel) indicating a weak cor-relation in our case test.
Fig. 29.—
Ratio of unresolved to resolved stellar mass estimatesvs. resolved star formation rate (cid:104) Ψ (cid:105) ( M (cid:12) yr − ). Left panel (redtriangles):
ZCR (cid:48) ; right panel (blue dots): BSP. UNCERTAINTIES IN THE STELLAR MASSESTIMATESAll the stellar mass estimates given in Table 1 arefor the SSAG-BC03 library; if, instead, the MAGPHYS-CB07 library is used, the masses will be smaller ( ∼ ∼
27% with ZCR (cid:48) ,and of ∼
3% with BSP. The reduction of the uncertaintyin BSP is due to the inclusion of equation 7 in the cal-culations. The random errors in the total resolved massestimates, on the other hand, are rather small, given thevery large number of pixels involved in the calculations( ∼ × and ∼ × pixels, for M 51 and the OS-UBSGS objects, respectively). For an object with n pix pixels the relative uncertainty ( σ mass /mass) decreases as ∼ √ n pix . Hence the random uncertainties in the totalresolved mass estimates tend to be less than 0.1%. Therandom uncertainties in the median Υ ∗ (after iterationnumber 1), and (cid:104) Ψ (cid:105) are also relatively small due to thelarge number of pixels involved in the calculations. Withregard to the systematic uncertainty due to the zero pointerror σ calib , we estimate a 3% relative error in the re-solved mass estimates, and the median Υ ∗ . However,this systematic error dominates the relative uncertaintiesin M unresolved ∗ (see equation 19), which have a median of ∼
22% (see Table 1).Another source of systematic error is the uncertaintyin the distance to the objects, σ dist . Propagating σ dist leads to a ∼
14% uncertainty in the mass, and (cid:104) Ψ (cid:105) ,for all galaxies in our pilot sample, with the excep-tion of NGC 3319, NGC 4051, and NGC 4212, forwhich the uncertainty in the mass is ∼ M BSP ∗ /M ZCR (cid:48) ∗ or M unresolved ∗ /M resolved ∗ ), since all mass estimates areequally affected. Equivalently, (cid:104) Ψ (cid:105) S is not affected by σ dist .Regarding the choice of the IMF, our defaultis Chabrier (2003). Stellar masses can be ∼ . ± . ∼ . ± .
03 times larger with the Kroupa (2001) IMF.We also have quantified that using only a constant Υ ∗ (i.e., skipping iteration number 3) yields masses per pixel ∼
1% higher on average, and up to ∼
30% larger inlocalized regions.6.1.
Dependence on disk inclination
Stellar mass is an intrinsic property of galaxies, inde-pendent of inclination to the line of sight. Stellar massdeterminations from broad-band colors, however, are in-dependent of inclination only as surface brightness atdifferent wavelengths is independent of it. Maller etal. (2009) study the effects of inclination on mass esti-mates, by comparing a statistically significant sample ofedge-on ( a/b ≥ .
33) and face-on ( a/b ≤ .
18) SDSSgalaxies. They find no statistical difference for massesderived from K -band photometry by Bell et al. (2003)but, on the other hand, point out the very importantcorrections with inclination that are necessary for the B -band (Driver et al. 2007).We remind the reader that all our calculations arebased on the effective Υ ∗ . Extinction effects may intro-duce biases with inclination. In subsequent publicationswe will address this issue in more detail. CONCLUSIONSWe have demonstrated quantitatively that resolvedmaps of stellar mass obtained by the maximum likeli-hood estimate (as in ZCR) yield biased spatial struc-tures. The bias consists in a filamentary morphology,and a spatial coincidence between dust lanes and pur-ported stellar mass surface density. The bias is due to alimited Υ ∗ accuracy ( ∼ . − .
15 dex) arising from un-certainties inherent to observations, and to degeneraciesbetween templates of similar colors in the SPS libraries.Similar observed colors will yield the mode Υ ∗ . Here,we have succeeded in mitigating the bias with the BSPalgorithm we have developed. We have applied the newalgorithm to M 51 and a pilot sample of 90 spirals. BSPeffectively identifies and isolates the old stellar popula-tion, and the output mass-maps bear more resemblanceto NIR structures.The results also indicate that total resolved mass es-timates obtained by adding up the pixel-by-pixel contri-butions are on average ∼
10% lower with BSP than withthe ZCR (cid:48) approach. Hence, unresolved stellar mass es-timates for our pilot sample underestimate the mass by ∼
20% when compared to the resolved ZCR (cid:48) results, butonly by ∼
10% vis-`a-vis BSP.The fact that the same SPS libraries can produce, ornot, filamentary structures where the mass is suppos-edly organized indicates that such structures are merelyan artifact of the method, and not real massive featurespresent in disk galaxies.An additional advantage of using a spatial structureprior for mass estimates is its independence of SPS modelparameters (e.g., SFH, metallicity, dust, age, etc.) or in-gredients (e.g., TP-AGB phase, or IMF). Galaxy massesdetermined from SPS models should be compared to re-sults of independent studies, e.g., the Disk Mass Sur- -0.2 0 0.2 0.4 0.6-0.4-0.20.00.2 01234 l og ( nu m b e r) Fig. 30.— C = ( g − i ) obs − ( g − i ) template ,and ∆ C = ( i − K s ) obs − ( i − K s ) template distributions beforeapplying the | ∆ C n | < ασ P condition at iteration number 2 of BSP.Data pixels for M 51, adopting the MAGPHYS-CB07 library. Fig. 31.— “Skewness curves” of the ∆ C and ∆ C distributionsafter applying the | ∆ C n | < ασ P condition for different α values.Same data as in Figure 30. vey (DMS, Bershady et al. 2010) . Systematic un-certainties may be constrained through these compar-isons (see also de Jong & Bell 2007).We acknowledge the referee for her/his commentsand suggestions that significantly improved the qual-ity of the manuscript. We appreciate discussions withand comments from Margarita Rosado, Ivˆanio Puer-ari, Bernardo Cervantes-Sodi, Sebasti´an S´anchez, Fabi´anRosales-Ortega, Olga Vega, Edgar Ram´ırez, and WilliamWall. We thank Alfredo Mej´ıa-Narv´aez for useful discus-sions about the SSAG parameters.EMG acknowledges support from INAOE during theinitial stages of this research, and from IRyA during thedevelopment of the project; he gives special thanks to his DMS uses measurements of the vertical velocity dispersionof disk stars as a dynamical constraint on the mass surface densityof spiral disks. https://archive.stsci.edu/cgi-bin/dss_form .This work has made use of the adaptive smoothingcode
Adaptsmooth , developed by Stefano Zibetti andavailable at the URL .APPENDIX
DETERMINATION OF THE α PARAMETER FOR BSP.
The last step of iteration number 2 is to identify the pixels which satisfy the condition | ∆ C n | < ασ P . From thedefinition of ∆ C n = C obs n − C template n , we have:∆ C = ( g − i ) obs − ( g − i ) template , (A1)and ∆ C = ( i − K s ) obs − ( i − K s ) template , (A2)for the ( g − i ), and ( i − K s ) colors, respectively. The value of σ P is computed from equation 12. In Figure 30 we showa plot of ∆ C vs. ∆ C for the case of the MAGPHYS-CB07 SPS library, before applying the | ∆ C n | < ασ P conditionto the pixels of M 51 (see section 4). From these ∆ C n distributions we obtain σ P = 0 . σ P = 0 . g − i ) and ( i − K s ) colors, respectively. The purpose of applying the | ∆ C n | < ασ P condition is to isolate the pixelsthat deviate significantly from the value ∆ C n ∼
0. In Figure 31 we show a plot of the skewness (a measure of thedegree of asymmetry) of the ∆ C and the ∆ C distributions, after applying the | ∆ C n | < ασ P condition for different α values. The “skewness curves” have extrema near α ∼
1, a minimum for the ∆ C curve and a maximum for the ∆ C curve. These extrema values indicate a transition of the shape of the ∆ C n distributions. Similar plots are obtainedfor the MAGPHYS-BC03 and the SSAG-BC03 libraries, which also have extrema near α ∼
1. The ∆ C n distributionsbecome extremely asymmetric for α >
1. Therefore, in a statistical manner, applying the condition | ∆ C n | < ασ P ,with α ∼
1, fulfills our purposes.
PROBABILITY DISTRIBUTION FUNCTIONS FOR DISK PARAMETERS.
In this section we explain the method we use to obtain the probability distributions for the disk parameters shownin Figure 14. After applying either ZCR (cid:48) or BSP to a given object, we obtain a set of templates which were fittedto a group of pixels (e.g., for the whole disk we use the pixels shown in Figure 12a, top left panel). We then usea Gaussian kernel density method (Keen 2010) to estimate the probability density function. In essence, the kernelmethod produces a smoothed version of a histogram. First, we build a grid for each parameter (e.g., age) within therange of values given by the SPS library. The grid contains 512 bins and has a distinct bin width of size b par w foreach parameter. A single parameter has bins of equal width, b par w , which is estimated as the difference between thehighest value minus the lowest value, divided by 512. Then we count how many pixels fall into each bin, i.e., we builda histogram of the pixel population given a certain parameter. 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TABLE 1Galaxy parameters
Name RC3 type T-type Dist (Mpc) Υ H ∗ M BSP ∗ ( M (cid:12) ) M ZCR (cid:48) ∗ ( M (cid:12) ) M unresolved ∗ ( M (cid:12) )M 51 SA(s)bc pec 4.0 9.9 a ± K s ) 5.56 × × (9.02 ± . × M 51b I0 pec 90.0 9.9 a ± K s ) 2.96 × × (3.26 ± . × NGC 157 SAB(rs)bc 4.0 22.6 ± × × (3.96 ± . × NGC 428 SAB(s)m 9.0 15.9 ± × × (3.80 ± . × NGC 488 SA(r)b 3.0 30.4 ± × × (3.19 ± . × NGC 779 SAB(r)b 3.0 18.5 ± × × (2.38 ± . × NGC 864 SAB(rs)c 5.0 20.9 ± × × (1.85 ± . × NGC 1042 SAB(rs)cd 6.0 18.1 ± × × (1.32 ± . × NGC 1073 SB(rs)c 5.0 16.1 ± × × (6.90 ± . × NGC 1084 SA(s)c 5.0 18.6 ± × × (2.17 ± . × NGC 1087 SAB(rs)c 5.0 20.1 ± × × (1.30 ± . × NGC 1309 SA(s)bc: 4.0 28.3 ± × × (1.13 ± . × NGC 2775 SA(r)ab 2.0 21.4 ± × × (1.33 ± . × NGC 2964 SAB(r)bc: 4.0 23.2 ± × × (2.59 ± . × NGC 3166 SAB(rs)0/a 0.0 22.0 ± × × (5.87 ± . × NGC 3169 SA(s)a pec 1.0 19.9 ± × × (5.07 ± . × NGC 3227 SAB(s)a pec 1.0 20.3 ± × × (3.27 ± . × NGC 3319 SB(rs)cd 6.0 3.3 ± × × (9.68 ± . × NGC 3338 SA(s)c 5.0 23.2 ± × × (1.87 ± . × NGC 3423 SA(s)cd 6.0 14.1 ± × × (4.42 ± . × NGC 3504 (R)SAB(s)ab 2.0 27.8 ± × × (5.65 ± . × NGC 3507 SB(s)b 3.0 15.0 ± × × (8.79 ± . × NGC 3583 SB(s)b 3.0 35.7 ± × × (3.77 ± . × NGC 3593 SA(s)0/a 0.0 5.6 ± × × (4.69 ± . × NGC 3596 SAB(rs)c 5.0 22.5 ± × × (1.62 ± . × NGC 3646 RING 4.0 65.2 ± × × (1.36 ± . × NGC 3675 SA(s)b 3.0 14.3 ± × × (5.34 ± . × NGC 3681 SAB(r)bc 4.0 24.9 ± × × (2.02 ± . × NGC 3684 SA(rs)bc 4.0 22.8 ± × × (8.70 ± . × NGC 3686 SB(s)bc 4.0 22.6 ± × × (2.36 ± . × NGC 3705 SAB(r)ab 2.0 13.2 ± × × (1.04 ± . × NGC 3810 SA(rs)c 5.0 10.7 ± × × (5.54 ± . × NGC 3877 SA(s)c: 5.0 17.8 ± × × (2.72 ± . × NGC 3893 SAB(rs)c: 5.0 19.4 ± × × (2.33 ± . × NGC 3938 SA(s)c 5.0 15.5 ± × × (2.13 ± . × NGC 3949 SA(s)bc: 4.0 15.8 ± × × (6.06 ± . × NGC 4030 SA(s)bc 4.0 26.4 ± × × (5.54 ± . × NGC 4051 SAB(rs)bc 4.0 2.9 ± × × (6.34 ± . × NGC 4062 SA(s)c 5.0 10.4 ± × × (6.03 ± . × NGC 4100 (R’)SA(rs)bc 4.0 21.5 ± × × (2.74 ± . × NGC 4123 SB(r)c 5.0 27.3 ± × × (1.78 ± . × NGC 4136 SAB(r)c 5.0 6.7 ± × × (4.42 ± . × NGC 4145 SAB(rs)d 7.0 20.3 ± × × (1.01 ± . × NGC 4151 (R’)SAB(rs)ab: 2.0 20.0 ± × × (3.30 ± . × NGC 4212 SAc: 4.5 16.3 c ± × × (1.76 ± . × NGC 4254 SA(s)c 5.0 16.5 d ± × × (3.70 ± . × NGC 4293 (R)SB(s)0/a 0.0 14.1 ± × × (3.61 ± . × NGC 4303 SAB(rs)bc 4.0 13.6 ± × × (2.32 ± . × NGC 4314 SB(rs)a 1.0 17.8 ± × × (4.34 ± . × NGC 4388 SA(s)b: sp 3.0 41.4 ± × × (1.55 ± . × NGC 4394 (R)SB(r)b 3.0 14.1 ± × × (1.74 ± . × NGC 4414 SA(rs)c? 5.0 9.0 ± × × (1.67 ± . × NGC 4448 SB(r)ab 2.0 7.0 ± × × (4.66 ± . × NGC 4450 SA(s)ab 2.0 14.1 ± × × (4.66 ± . × NGC 4457 (R)SAB(s)0/a 0.0 13.6 ± × × (1.91 ± . × NGC 4490 SB(s)d pec 7.0 9.2 ± × × (4.69 ± . × NGC 4496A SB(rs)m 9.0 13.6 ± × × (2.62 ± . × NGC 4527 SAB(s)bc 4.0 13.5 ± × × (3.85 ± . × NGC 4548 SB(rs)b 3.0 3.7 ± × × (2.73 ± . × NGC 4568 SA(rs)bc 4.0 13.9 ± × × (2.19 ± . × NGC 4571 SA(r)d 6.5 2.6 ± × × (2.72 ± . × NGC 4579 SAB(rs)b 3.0 13.9 ± × × (6.19 ± . × EMOVING BIASES IN STELLAR MASS-MAPS 23
TABLE 1 — Continued
Name RC3 type T-type Dist (Mpc) Υ H ∗ M BSP ∗ ( M (cid:12) ) M ZCR (cid:48) ∗ ( M (cid:12) ) M unresolved ∗ ( M (cid:12) )NGC 4580 SAB(rs)a pec 1.0 13.6 ± × × (5.60 ± . × NGC 4618 SB(rs)m 9.0 8.8 ± × × (1.64 ± . × NGC 4643 SB(rs)0/a 0.0 27.3 ± × × (1.64 ± . × NGC 4647 SAB(rs)c 5.0 13.9 ± × × (1.66 ± . × NGC 4651 SA(rs)c 5.0 14.0 ± × × (1.05 ± . × NGC 4654 SAB(rs)cd 6.0 13.9 ± × × (1.09 ± . × NGC 4665 SB(s)0/a 0.0 13.5 ± × × (4.80 ± . × NGC 4666 SABc: 5.0 27.5 ± × × (1.91 ± . × NGC 4689 SA(rs)bc 4.0 14.0 ± × × (1.07 ± . × NGC 4691 (R)SB(s)0/a pec 0.0 17.0 ± × × (1.25 ± . × NGC 4698 SA(s)ab 2.0 13.7 ± × × (3.12 ± . × NGC 4699 SAB(rs)b 3.0 22.9 ± × × (1.28 ± . × NGC 4772 SA(s)a 1.0 13.3 ± × × (1.32 ± . × NGC 4900 SB(rs)c 5.0 9.1 ± × × (2.38 ± . × NGC 5005 SAB(rs)bc 4.0 19.3 ± × × (1.83 ± . × NGC 5334 SB(rs)c 5.0 24.2 ± × × (8.32 ± . × NGC 5371 SAB(rs)bc 4.0 42.8 ± × × (1.29 ± . × NGC 5448 (R)SAB(r)a 1.0 35.2 ± × × (5.67 ± . × NGC 5676 SA(rs)bc 4.0 36.5 ± × × (1.20 ± . × NGC 5701 (R)SB(rs)0/a 0.0 26.7 ± × × (6.32 ± . × NGC 5713 SAB(rs)bc pec 4.0 31.3 ± × × (3.96 ± . × NGC 5850 SB(r)b 3.0 41.6 ± × × (1.38 ± . × NGC 5921 SB(r)bc 4.0 26.2 ± × × (4.40 ± . × NGC 5962 SA(r)c 5.0 34.2 ± × × (6.85 ± . × NGC 6384 SAB(r)bc 4.0 29.2 ± × × (7.26 ± . × NGC 7217 (R)SA(r)ab 2.0 16.5 ± × × (1.02 ± . × NGC 7479 SB(s)c 5.0 33.7 ± × × (7.82 ± . × NGC 7606 SA(s)b 3.0 31.3 ± × × (1.52 ± . × NGC 7741 SB(s)cd 6.0 12.5 ± × × (2.82 ± . × NGC 7814 SA(s)ab: sp 2.0 15.7 ± × × (6.21 ± . × Note . — Col. 1: galaxy name. Col. 2: RC3 type (de Vaucouleurs etal. 1991). Col. 3: T Hubble type (de Vaucouleurs et al. 1991). Col. 4:distance to object in Mpc, from NED (Virgo + GA + Shapley), unlessotherwise indicated. Col. 5: median Υ H ∗ after BSP iteration number1. For M 51 and M 51b the median Υ Ks ∗ is tabulated instead of Υ H ∗ .Col. 6: total resolved stellar mass obtained from the BSP algorithm, M BSP ∗ , in solar units. Col. 7: total resolved stellar mass obtainedfrom ZCR (cid:48) , M ZCR (cid:48)∗ , in solar units. Col. 8: unresolved stellar mass, M unresolved ∗ , in solar units. All the masses given in this table havebeen calculated using the SSAG-BC03 SPS library. The uncertaintiesin M unresolved ∗ correspond to the propagation of the systematic errordue to the zero point calibration, which affects the values of M BSP ∗ and M ZCR (cid:48)∗ by only ∼∼