Lighting up stars in chemical evolution models: the CMD of Sculptor
Fiorenzo Vincenzo, Francesca Matteucci, Thomas J. L. de Boer, Michele Cignoni, Monica Tosi
MMNRAS , 1–8 (2016) Preprint 9 September 2018 Compiled using MNRAS L A TEX style file v3.0
Lighting up stars in chemical evolution models: the CMDof Sculptor
F. Vincenzo , (cid:63) , F. Matteucci , , , T. J. L. de Boer , M. Cignoni and M. Tosi Dipartimento di Fisica, Sezione di Astronomia, Universit`a di Trieste, via G.B. Tiepolo 11, 34100, Trieste, Italy INAF, Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34100, Trieste, Italy INFN, Sezione di Trieste, Via Valerio 2, 34100, Trieste, Italy Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA INAF, Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127, Bologna, Italy
Accepted 2016 May 11. Received 2016 May 11; in original form 2016 February 2
ABSTRACT
We present a novel approach to draw the synthetic color-magnitude diagram of galax-ies, which can provide – in principle – a deeper insight in the interpretation andunderstanding of current observations. In particular, we ‘light up’ the stars of chemi-cal evolution models, according to their initial mass, metallicity and age, to eventuallyunderstand how the assumed underlying galaxy formation and evolution scenario af-fects the final configuration of the synthetic CMD. In this way, we obtain a new setof observational constraints for chemical evolution models beyond the usual photo-spheric chemical abundances. The strength of our method resides in the very fine gridof metallicities and ages of the assumed database of stellar isochrones. In this work,we apply our photo-chemical model to reproduce the observed CMD of the SculptordSph and find that we can reproduce the main features of the observed CMD. Themain discrepancies are found at fainter magnitudes in the main sequence turn-off andsub-giant branch, where the observed CMD extends towards bluer colors than thesynthetic one; we suggest that this is a signature of metal-poor stellar populationsin the data, which cannot be captured by our assumed one-zone chemical evolutionmodel.
Key words:
Local Group – galaxies: dwarf – galaxies: stellar content – stars: abun-dances – Hertzsprung-Russell and colour-magnitude diagrams
In this work, we present a novel approach to obtain a syn-thetic color-magnitude diagram (CMD) of galaxies, start-ing from predictions of chemical evolution models. Our new photo-chemical model ‘lights up’ the stars of chemical evolu-tion models, according to their initial mass, metallicity andage; in this way, we can obtain a new set of observationalconstraints for chemical evolution models beyond the usualphotospheric chemical abundances. The method presentedin this work can provide – in principle – a deeper insightin the interpretation of current observations, since we canunderstand how our hypothesis about galaxy formation andevolution can affect the final configuration of the CMD.By solving a set of physically-motivated differentialequations, which take into account the main physical pro-cesses taking place and influencing the evolution of the (cid:63)
E-mail: [email protected] galaxy interstellar medium (ISM), numerical codes of chem-ical evolution are able to provide the galaxy star forma-tion history (SFH) and age-metallicity relation; the evolu-tion of the galaxy stellar and gas mass, and the run of theISM chemical abundances with time. Building up a photo-chemical code consists then in coupling the output of chem-ical evolution models with a database of stellar isochrones,currently available and computed with very high accuracy.Most of the previous works in the literature recover theSFH of galaxies from the observed CMD by adopting so-phisticated fitting techniques (e.g. Harris & Zaritsky 2001;Dolphin 2002; Aparicio & Gallart 2004; Tolstoy et al. 2009;Cignoni & Tosi 2010; Monelli et al. 2010; Hidalgo et al.2011); in particular, they search for the suitable linear com-bination of simple stellar populations (SSPs) with differentage and metallicity, which provides the best agreement withthe observed photometric properties of the galaxy compositestellar population. As a byproduct, this ‘classical’ procedurecan also predict an average galaxy age-metallicity relation. c (cid:13) a r X i v : . [ a s t r o - ph . GA ] M a y F. Vincenzo et al.
Figure 1.
In this Figure, we show the observed CMD of theSculptor dSph (de Boer et al. 2011). The dataset is shown as 2-D histogram, with the bin size in both the x - and y -dimensionsbeing 0 .
02 dex; the color coding in the figure corresponds to thenumber of stars within each grid element. We consider in ouranalysis only clean isolated stellar detections and the Sculptormember stars with an elliptical radius r ell ≤ . Nevertheless, no underlying approximate physical model isassumed in these works for the galaxy formation and evolu-tion.In this paper, the first of a series of future works,we focus on reproducing the CMD of the Sculptor dwarfspheroidal galaxy (dSph). In particular, we investigatewhether the best chemical evolution model for Sculptor –reproducing the galaxy stellar metallicity distribution func-tion (MDF) – is able to predict a synthetic CMD whichagrees with the observed one.This work is organized as follows: in Section 2 we sum-marize the main properties of the Sculptor dSph and de-scribe the observed set of data used in this work for thecomparison with our models; in Section 3 we present themain characteristics of our photo-chemical model and themethods we employ to fairly compare the synthetic with theobserved CMD; in Section 4 we show our results, and inSection 5 we draw some conclusions.
The Sculptor dwarf galaxy was discovered by Shapley(1938). Although it might appear simple at first glance, fromthe study of the kinematical, chemical and spatial distribu-tion of its stellar populations, Tolstoy et al. (2004) were ableto disentangle in this galaxy an inner, kinematically ‘cold’,metal-rich stellar population from an outer ‘hot’ metal-poorone, later on confirmed by Battaglia et al. (2008) and Walker& Pe˜narrubia (2011). Other studies based on photometricdatasets also were able to identify (or confirm) the existenceof stellar populations distinct in metallicity (Majewski etal. 1999), age and spatial distribution (de Boer et al. 2011,2012).McConnachie (2012) reported for Sculptor an averageV-band surface brightness µ V = 23 . ± . − , Figure 2.
In this figure, we show how the stellar lifetimes wehave derived from the PARSEC stellar evolutionary tracks varyas functions of the stellar mass and metallicity. The dashed blackcurve corresponds to the stellar lifetimes of Padovani & Matteucci(1993). an half-light radius r h = 283 ±
45 pc, and an absolute V-band magnitude M V = − . ± . µ = 19 . ± .
04 mag derived by Mart´ınez-V´azquez et al. (2015).The observed CMD is taken from de Boer et al. (2011,see Fig. 1), which were able to resolve stars down to the old-est main sequence turn-off (MSTO) of the Sculptor dSph,by taking advantage of the deep wide-field photometry ofCTIO/MOSAIC. In order to avoid a non-negligible contam-ination of foreground MW disc field stars, we consider onlystars along the line-of-sight to the Sculptor dSph with ellipti-cal radius r ell ≤ . r ell > . . MNRAS000
04 mag derived by Mart´ınez-V´azquez et al. (2015).The observed CMD is taken from de Boer et al. (2011,see Fig. 1), which were able to resolve stars down to the old-est main sequence turn-off (MSTO) of the Sculptor dSph,by taking advantage of the deep wide-field photometry ofCTIO/MOSAIC. In order to avoid a non-negligible contam-ination of foreground MW disc field stars, we consider onlystars along the line-of-sight to the Sculptor dSph with ellipti-cal radius r ell ≤ . r ell > . . MNRAS000 , 1–8 (2016) he photo-chemical evolution of the Sculptor dSph Figure 3.
In this figure, we compare the observed Sculptor stellarMDF (Romano & Starkenburg 2013, grey histogram with blueerror bars) with the predictions of our best Sculptor chemicalevolution model, having star formation efficiency ν = 0 .
04 Gyr − ,wind efficiency λ wind = 3 . − , infall mass M inf = 2 . × M (cid:12) , and infall time-scale τ inf = 0 . σ = 0 . We make use of the PARSEC stellar isochrones (Bressan etal. 2012; Tang et al. 2014; Chen et al. 2015), as computedfor the following grid of stellar ages and metallicities, byassuming a Reimers mass loss with efficiency η = 0 . Z = 1 . × − , from a minimum metallicity Z min = 1 . × − to a maximum metallicity Z max = 3 . × − .(ii) The step in age between two adjacent isochrones is∆ log( t/ yr) = 0 .
01, from a minimum age log( t min / yr) = 6 . t max / yr) = 10 . τ m ( Z ) = A( Z ) × exp (cid:104) B( Z ) m − C( Z ) (cid:105) , (1)where A( Z ), B( Z ) and C( Z ) are the fitting parameters,provided with the corresponding 1- σ errors in the supple-mentary material, as functions of the metallicity Z . In Fig.2 we compare our derived stellar lifetimes with the onesof Padovani & Matteucci (1993), which do not depend onmetallicity and are extensively used in chemical evolutionmodels. The numerical code of chemical evolution is the same asthe one adopted in Vincenzo et al. (2014, 2015) – where weaddress the reader for details – for the study of the classicaland ultra-faint dSph galaxies. We make use of an updatedversion, by assuming the stellar yield compilation of Romano
Figure 4.
In this figure, we show the predicted SFH (top panel)and age-metallicity relation (bottom panel) as predicted by ourbest chemical evolution model for Sculptor. Our best model forSculptor predicts that ∼
99 per cent of the stars observable atthe present time are formed within the first 2 .
16 Gyr of the galaxyevolution; this time corresponds to the vertical dashed blue line inthe figures. Furthermore, the number of stars with initial metal-licity
Z < . × − is roughly ∼ .
72 per cent of the totalnumber of stars alive at the present time. et al. (2010, their model 15) and the stellar lifetimes derivedfrom the PARSEC isochrones.We assume the galaxy to assemble by accreting pristinegas from an external reservoir, until an infall mass – givenby M inf – is accumulated at t G = 14 Gyr. The infall rate isassumed to follow a decaying exponential law, with typicaltime-scale τ inf . We assume for the star formation rate theSchmidt-Kennicutt law, namely SFR( t ) = νM gas ( t ), where ν is the so-called star formation efficiency (SFE) and M gas isthe galaxy gas mass. The run of the intensity of the SFR withtime is crucially regulated by the various physical processesacting on M gas , namely inflows and outflows of gas, returnedmatter from dying stars and supernovae, astration due to thestar formation activity itself.A fundamental role in the evolution of dSphs is playedby the galactic outflows, which are predicted to occur verysoon in these galaxies because of their shallow potential well;the intensity of the outflow rate is assumed to be directlyproportional to the SFR. On the one hand, if the efficiencyfor the gas removal is high (typically λ wind ≈
10 Gyr − ),then the galaxy gas mass suddenly decreases and hencethe SFR rapidly drops to zero; on the other hand, if thegalactic wind has a relatively lower efficiency (typically λ wind ≈ − ), then the decrease in the SFR is smootherand it drops to zero on longer typical time-scales. We have explored the parameter space, by running a largenumber of chemical evolution models. The best parametersfor Sculptor are found by minimizing the χ figure of merit,with the best model being the one reproducing the shape ofthe observed stellar MDF, which represents the most reliableobservational constraint to any slight variation of the free pa- MNRAS , 1–8 (2016)
F. Vincenzo et al. rameters. We vary the SFE in the range ν = 0 . . − ,the wind efficiency in the range λ wind = 2 . − , andthe infall time-scale in the range τ inf = 0 . . r ell ≤ . ν = 0 .
04 Gyr − ;(ii) wind efficiency λ wind = 3 . − ;(iii) infall time-scale τ inf = 0 . M inf,ref =1 . × M (cid:12) , as in Vincenzo et al. (2014), for which we pre-dict a present-day total stellar mass M (cid:63), ref = 8 . × M (cid:12) .The infall mass of the best model is then obtained by rescal-ing our results for the reference model so as to have the samenumber of stars in the synthetic and observed CMD. We findfor our best model an infall mass M inf,best = 2 . × M (cid:12) ,giving rise to a present-day total stellar M (cid:63), best = 1 . × M (cid:12) , larger than the value M (cid:63) = (1 . ± . × M (cid:12) estimated by de Boer et al. (2012), but of the same order ofmagnitude.According to the fitting formula of Faucher-Gigu`ere etal. (2011), which is assumed in many recent works to mimica cosmologically motivated infall in galaxy formation andevolution models, dwarf galaxies with M halo = 10 M (cid:12) musthave accreted almost 63 per cent of their cumulative infallmass (which turns out to be M inf ≈ . × M (cid:12) ) in thefirst ∼ . z = 6to redshift z ∼
4, a larger timescale than the one found byour best model ( τ inf = 0 . < z < σ = 0 . / H] abundances. The width of the MDFis mostly determined by the wind efficiency; in particular,the lower the λ wind parameter, the wider is the bulk of thegalaxy star formation activity and hence also the MDF.In Figure 4a) we show the predicted SFH of our bestmodel, while in Figure 4b) we show the corresponding age-metallicity relation. In summary, the length of the bulk ofthe galaxy star formation activity can be regulated in ourmodel by suitably varying the main parameters determin-ing the star formation and outflow intensity and the galaxygas accretion rate; these parameters are the SFE, which de- termines the intensity of the SFR and the rate of thermalenergy injection by supernovae, the wind efficiency, whichdetermines the slope with which the SFR drops to zero, andthe infall time-scale, which crucially determines the evolu-tion with time of the galaxy potential well. The stepwise structure of the photo-chemical model is thefollowing.(i) We sample the galaxy SFH, as predicted by our bestchemical evolution model for Sculptor, to randomly extractan age for the formation of a given star.(ii) We sample the assumed initial mass function (IMF) torandomly assign a mass to the star. In this work we assumethe Salpeter (1955) IMF for simplicity.(iii) We use the age-metallicity relation of our best Sculp-tor chemical evolution model to find the initial metallicityof the star.(iv) Given the age, mass and metallicity of the star, wecheck whether the star is alive or not at the present time,by assuming the metallicity dependent stellar lifetimes wehave derived from the PARSEC stellar evolutionary tracks(see Section 3.1).(v) If the star can be observed at the present time, westore the photometric properties of the synthetic star, tolater draw it in the synthetic CMD.On the one hand, the strength of our method resides inthe very fine grid of the assumed isochrone database; more-over, in our approach, we start from the predictions ofchemical evolution models, assuming ab initio an underlyinggalaxy formation and evolution scenario, which is physically-motivated. On the other hand, the main shortcoming of ourmodel is due to the fact that the lowest available metallicityin the PARSEC database is Z min = 1 . × − . We assumethat all the stars with Z < Z min have the same photomet-ric properties as the stars with Z = Z min . This fact canintroduce a systematic error. By looking at Figs. 4a) andb), according to our best model, the galaxy spends its first122 Myr at metallicity Z < . × − ; the number of starswith initial metallicity Z < . × − is roughly ∼ .
72 percent of the total number of stars alive at the present time.
To get a fair comparison with the observed CMD, we con-volve the synthetic CMD with the distribution of the ob-served photometric errors, by assuming that the latter areGaussian. In particular, we first divide the observed CMDin an uniform grid and, for each grid element ij , we com-pute the average V- and I-band observed photometric errors, σ ij (V , I), which we adopt as the standard deviations of thephotometric noise in the ij -th grid element. Hence, for anygiven k -th synthetic star residing in the ij -th grid element,we add the following noise to its predicted V- and I-bandmagnitudes: σ k (V , I) = r k × σ ij (V , I), where r k is a randomnumber, drawn according to the standard normal distribu-tion. In this way, the model ‘spreads out’ according to theerrors in the data and we can fairly compare the syntheticwith the observed CMD. MNRAS000
To get a fair comparison with the observed CMD, we con-volve the synthetic CMD with the distribution of the ob-served photometric errors, by assuming that the latter areGaussian. In particular, we first divide the observed CMDin an uniform grid and, for each grid element ij , we com-pute the average V- and I-band observed photometric errors, σ ij (V , I), which we adopt as the standard deviations of thephotometric noise in the ij -th grid element. Hence, for anygiven k -th synthetic star residing in the ij -th grid element,we add the following noise to its predicted V- and I-bandmagnitudes: σ k (V , I) = r k × σ ij (V , I), where r k is a randomnumber, drawn according to the standard normal distribu-tion. In this way, the model ‘spreads out’ according to theerrors in the data and we can fairly compare the syntheticwith the observed CMD. MNRAS000 , 1–8 (2016) he photo-chemical evolution of the Sculptor dSph Figure 5.
In the left panel of this figure, we show the prediction of our photo-chemical model for the CMD of the Sculptor dSph, whereason the right panel we show for comparison the observed CMD. The synthetic and the observed CMDs are shown as 2-D histograms,with the bin size in both the x - and y -dimensions being 0 .
02 dex and the color-coding representing the number of stars, on a logarithmicscale, residing within each grid element.
The synthetic CMD corrected for the photometric noiseis then convolved with the results of the artificial star testperformed by de Boer et al. (2011). In particular, by follow-ing a standard procedure, de Boer et al. (2011) inserted inthe observed images a large catalog of artificial stars; afterreducing and analyzing the altered images, they could com-pute the fraction of artificial stars recovered in the data as afunction of their input magnitude and color. We use the re-sults of their test to compute the recovered fraction in eachgrid element, so as to throw out from the synthetic CMD theremaining lost fraction. By means of this type of analysis,one can apply to the synthetic CMD the same completenessprofile as is present in the data. After correcting the syn-thetic CMD for the incompleteness, the number of syntheticstars strongly reduces, becoming N tot , syn ≈ In this Section, we present the results of our photo-chemicalmodel for the CMD of the Sculptor dSph. The main resultof our work in shown in Fig. 5, where the synthetic CMD ofSculptor (left diagram) is compared with the observed one(right diagram). In order to better understand where thediscrepancies between the observed and the synthetic CMDreside, in Fig. 6 we plot the residuals, which correspond tothe color-coding in the figure. In particular, to better visu-alize the differences, we define the residual in the ij -th gridelement as: R ij = n ij, obs − n ij, syn √ n ij, syn , (2)with n ij, obs and n ij, syn being the number of stars in theobserved and synthetic CMD, respectively. The regions ofthe observed Sculptor CMD without any synthetic star areshown in Fig. 6 as a greyscale density plot.On the one hand, by a visual inspection of Figs. 5 and6, we can obtain a quite good agreement for the red giant branch (RGB), the horizontal branch (HB) and the asymp-totic giant branch (AGB) of the observed CMD. On theother hand, at fainter magnitudes, particularly in the sub-giant branch (SGB) and at the MSTO, the observed CMDextends towards slightly bluer colors than the synthetic one.Moreover, our model cannot reproduce the observed popula-tion of blue straggler stars which extend the Main Sequencetowards blue colors and could be – in principle – reproducedby including the effect of merging binary stellar systems.We do not include blue straggler stars in our photo-chemicalmodel.We remark on the fact that it is not obvious that amodel reproducing the chemical evolution of Sculptor canalso capture the main features of the observed galaxy CMD.In fact, the final configuration of the synthetic CMD turnsout to be highly affected by the variation of the free param-eters of chemical evolution models. In particular, by increas-ing the SFE, the stellar metallicities accordingly increase atany fixed galactic time, causing the entire synthetic CMD toshift towards redder colors. The IMF acts in a similar wayas the SFE, with the additional effect of filling up the var-ious stellar evolutionary phases in the CMD with differentrelative fractions. Finally, the wind parameter and the infalltime-scale crucially affect the spread of the predicted CMD,since they determine the length of the bulk of the galaxystar formation activity.In Fig. 7a), we compare the predicted stellar ( V − I )-color distribution (black solid line) with the observed one(blue histogram); this quantity turns out to be particularlysensitive to metallicity variations among the galaxy stellarpopulations. In Fig. 7b), the predicted stellar luminosityfunction in the I -band (black solid line) is compared withthe observed one (blue line with error bars, which are shownas a shaded blue area); the trend of this second quantityis more affected by the galaxy SFH and stellar lifetimes.An age indicator for the galaxy is given by the fraction of MNRAS , 1–8 (2016)
F. Vincenzo et al.
Figure 6.
In this Figure, we show the residuals (see eq. 2 ) forthe comparison between the observed and the synthetic CMD.The greyscale density plot represents the regions of the observedSculptor CMD where no synthetic stars are predicted. We find agood agreement for the RGB, HB and AGB stars, whereas theMSTO and the SGB of the observed CMD extend towards bluercolors than the synthetic CMD. We cannot reproduce the popu-lation of blue straggler stars in the observed CMD, since we donot include them in our model. stars on the HB relative to the one on the RGB; we predict N HB /N RGB ≈ . V − I )-color distribution; in particular, the firstpeak in Fig. 7a) (the one at bluer colors) is determined bythe MSTO and SGB stars, whereas the second peak (theone at redder colors) is the signature left by the ascendingRGB and HB stars.Concerning the blue portion of the color distribution,from a visual inspection of Fig. 7a), we cannot reproduce theobserved population of blue straggler stars, which – as afore-mentioned – are not included in our photo-chemical model.Furthermore, a remarkable discrepancy resides in the decay-ing trend of the blue wing of the predicted color distribu-tion, which contains a lower number of stars than the data,and in the predicted ‘saddle’, which turns out to be higherthan the observed one. This can be also appreciated by look-ing at the residual plot in Fig 6, where the observed CMDclearly contains a larger number of MSTO and SGB starswith blue colors than the synthetic CMD. This discrepancyseems likely the signature of metal-poor stellar populationsin the Sculptor dSph, which our one-zone chemical evolutionmodel has not been able to capture. Nevertheless, in princi-ple, it could also indicate a predicted age-metallicity relationwhich is steeper than what seems to be required by observa-tions; in fact, one would obtain a similar discrepancy if alsothe synthetic metal-rich stars are older than the observedones.Interestingly, by looking at Fig. 3, the observed stel-lar MDF suggests the presence of two distinct peaks, corre-sponding to two separated main stellar populations in thegalaxy. The latter feature cannot be resolved by our bestchemical evolution model, which indeed predicts the stellar Figure 7.
In the top panel of this figure, we compare the pre-dicted stellar ( V − I )-color distribution (black solid line) withthe observed one (blue histogram with errorbars). In the bottompanel, we compare the predicted stellar luminosity function in the I -band (back solid line) with the observed one (blue line, withthe shaded blue area representing the 1- σ errors). MSTO starsare predicted to reside in the synthetic CMD at m I > ∼ . . < ∼ m I < ∼ . I -band stellar luminosity function at m I ≈ . MDF to have a single peak, lying between the two of theobserved distribution. We remark on the fact that this fea-ture is peculiar to the Romano & Starkenburg (2013) MDF,which combines the DART sample – determining the lowmetallicity portion of the observed MDF – with the oneby Kirby et al. (2009, 2010), concentrated towards slightlyhigher metallicity. The regions where the observed CMDcontains a larger number of blue (metal-poor) stars than thesynthetic one likely correspond to the low-metallicity peak inthe observed MDF, which is also the most pronounced one.Accordingly, the higher ‘saddle’ in the predicted color distri-bution (see the top panel in Fig. 7) confirms that the modelpredicts galaxy stellar populations which are intermediatebetween the observed metal-poor and metal-rich ones.The observed stellar MDF, as derived by Romano &Starkenburg (2013), includes stars with r ell ≥ . MNRAS000
In the top panel of this figure, we compare the pre-dicted stellar ( V − I )-color distribution (black solid line) withthe observed one (blue histogram with errorbars). In the bottompanel, we compare the predicted stellar luminosity function in the I -band (back solid line) with the observed one (blue line, withthe shaded blue area representing the 1- σ errors). MSTO starsare predicted to reside in the synthetic CMD at m I > ∼ . . < ∼ m I < ∼ . I -band stellar luminosity function at m I ≈ . MDF to have a single peak, lying between the two of theobserved distribution. We remark on the fact that this fea-ture is peculiar to the Romano & Starkenburg (2013) MDF,which combines the DART sample – determining the lowmetallicity portion of the observed MDF – with the oneby Kirby et al. (2009, 2010), concentrated towards slightlyhigher metallicity. The regions where the observed CMDcontains a larger number of blue (metal-poor) stars than thesynthetic one likely correspond to the low-metallicity peak inthe observed MDF, which is also the most pronounced one.Accordingly, the higher ‘saddle’ in the predicted color distri-bution (see the top panel in Fig. 7) confirms that the modelpredicts galaxy stellar populations which are intermediatebetween the observed metal-poor and metal-rich ones.The observed stellar MDF, as derived by Romano &Starkenburg (2013), includes stars with r ell ≥ . MNRAS000 , 1–8 (2016) he photo-chemical evolution of the Sculptor dSph Starkenburg (2013) sample being contributed by stars with r ell ≤ . r ell = 0 . / H] abun-dances than the predicted MDF of the best model. In thisway, the metal-poor stellar population could be reproducedby assuming a lower SFE than the metal-rich one. Aftersuperimposing the two stellar populations in the syntheticCMD with appropriate weights, one would extend the syn-thetic CMD towards slightly bluer color, hence obtaininga better agreement with the observed CMD. Finally, wecannot exclude that the lack of binary stars in our photo-chemical model might also contribute to the discrepancy be-tween the observed and the synthetic CMD, since their in-clusion would cause a broadening of the MS and therefore aredistribution of the colors of the synthetic stars in Fig. 7a).In Fig. 7b), the observed stellar I -band luminosity func-tion is compared with the predicted one. For the RGBand AGB stars, there is a good agreement between themodel and data. Furthermore, the model predicts a peakat m I ≈ . I -band stellar luminosity function because of the large fore-ground contamination in the redder part of the observedCMD, both at fainter and at brighter I -band magnitudesthan the ones of the observed HB. If we had considered onlySculptor stars in the innermost regions of the galaxy (e.g.with r ell ≤ . In this work, we have presented a new approach to drawthe synthetic CMD of galaxies. In particular, we have devel-oped ab-initio a new photo-chemical model, which we haveapplied to reproduce the observed CMD of Sculptor dSph.Our numerical code starts from the predictions of chemicalevolution models about the galaxy SFH and age-metallicityrelation. Then, by assuming the PARSEC stellar evolution-ary tracks, we can ‘light up’ the stars with different age,mass and metallicity of chemical evolution models, in orderto draw a synthetic CMD. We have defined the best chemi-cal evolution model for Sculptor as the one reproducing theobserved galaxy stellar MDF.Several improvements could be done in our photo- chemical model, by considering for example an underlyingcosmological framework, whose primary effect would be toinfluence the evolution of the galaxy gas mass assembly withtime. Interestingly, a very first attempt to draw the CMD ofa dSph galaxy within a full cosmological framework by mak-ing use of a semi-analytical model for the galaxy formationand evolution is represented by the work of Salvadori et al.(2008), which adopted the freely available IAC-STAR code(Aparicio & Gallart 2004), however, they did not providea detailed discussion of their findings about the predictedgalaxy CMD. A further improvement in our photo-chemicalmodel would be to include the effects of unresolved binarystars and foreground contamination in the synthetic CMD.The strength of our approach resides in the statisticalsampling of the galaxy predicted SFH and assumed IMF,as well as in the assumption of a very fine grid of stellarisochrones, both in metallicity and in age. The main short-coming is related to the PARSEC stellar isochrones, whichare computed only for Z ≥ − .The SFH can be regulated in our models for dSphsby suitably varying the main parameters triggering the on-set of the galactic wind and determining its intensity. Infact, once the galactic wind has started, the SFR rapidlydrops to zero, since most of the infall mass has been ac-cumulated fast within short typical time-scales. Our bestmodel for Sculptor predicts that ∼
99 per cent of the starsobservable at the present time are formed within the first2 .
16 Gyr of the galaxy evolution. We predict at the presenttime a total stellar mass M (cid:63), best = 1 . × M (cid:12) , whichis of the same order of magnitude as other estimates like M (cid:63) = (1 . ± . × M (cid:12) by de Boer et al. (2012). Alsothe predicted evolution of the SFR as a function of time isin agreement with the findings of de Boer et al. (2012).Stellar systems or interstellar regions with low gas den-sity, such as low-mass dwarf galaxies or the outer partsof spiral galaxies, likely follow a star formation law whichdeviates from the usually assumed Schmidt-Kennicut law,SFR( t ) = νM gas ( t ); for this reason, we have done some nu-merical experiments by assuming the same expression forthe star formation rate as in the original work of Kennicutt(1998) (see also Gatto et al. 2015 for a detailed study inthe context of hydrodynamical simulations). By assumingthe Kennicutt (1998) law, we predict the SFH to be moreconcentrated in the earliest epoch of the galaxy evolutionand the metallicity Z to initially evolve more rapidly thanour best-fitting model; then, at later times, Z remains quiteconstant when the Kennicutt (1998) law is assumed, while itincreases in our best-fitting model. Finally, we find that thefinal total gas and stellar mass are almost the same whenthe two different expressions for the star formation rate areassumed.We have shown that our photo-chemical model is able tocapture the main features of the observed CMD of the Sculp-tor dSph, with the best agreement being obtained for theRGB, HB and AGB stars. The discrepancy has been foundat fainter luminosity in the MSTO and SGB, where the ob-served CMD extends towards bluer colors than the syntheticone. That may be caused by underlying metal-poor stellarpopulations which our photo-chemical model has not beenable to capture as well as to the lack of binary stars in ourmodel, which would also broaden the synthetic CMD at faintmagnitudes. In fact, the predicted stellar MDF is character- MNRAS , 1–8 (2016)
F. Vincenzo et al. ized by a single peak, whereas the observed one suggests thepresence of two peaks, residing at slightly lower and higher[Fe / H] abundances than the model peak. In particular, themore pronounced peak in the observed MDF corresponds tothe one at lower [Fe / H] abundances. Therefore, our photo-chemical model contains stellar populations which are inter-mediate between the metal-poor and the metal-rich ones inthe observed stellar MDF.In order to reduce the discrepancy, one could super-impose the results of multiple one-zone chemical evolutionmodels and find the linear combination which provides thebest agreement with the observed stellar MDF. This will bethe subject of a future work, in which we will also show theeffect of varying the free parameters of chemical evolutionmodels on the synthetic CMD.Although the uncertainties in the assumed complete-ness profile can be important at the MSTO and SGB, thediscrepancy between model and data in the I -band stellarluminosity function for m I > ∼ . ACKNOWLEDGEMENTS
FV thanks E. Brocato for insightful discussions during thevisit at the Astronomical Observatory of Rome in 2014 De-cember, and S. Recchi for his precious suggestions. FM andMT acknowledge financial support from PRIN-MIUR 2010-2011 project ‘The Chemical and Dynamical Evolution of theMilky Way and Local Group Galaxies’, prot. 2010LY5N2T.We thank an anonymous referee for his/her constructivecomments.
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