Kirby et al. 2011, ApJ, 727, 79

25 Pages·2011·4.26 MB·English

TheAstrophysicalJournal,727:79(25pp),2011February1 doi:10.1088/0004-637X/727/2/79
(cid:2)C2011.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedintheU.S.A.
MULTI-ELEMENTABUNDANCEMEASUREMENTSFROMMEDIUM-RESOLUTIONSPECTRA.IV.
∗
ALPHAELEMENTDISTRIBUTIONSINMILKYWAYSATELLITEGALAXIES
EvanN.Kirby1,5,Judith.G.Cohen1,GraemeH.Smith2,StevenR.Majewski3,SangmoTonySohn4,
andPuragraGuhathakurta2
1CaliforniaInstituteofTechnology,1200EastCaliforniaBoulevard,MC249-17,Pasadena,CA91125,USA
2UniversityofCaliforniaObservatories/LickObservatory,UniversityofCalifornia,1156HighStreet,SantaCruz,CA95064,USA
3DepartmentofAstronomy,UniversityofVirginia,P.O.Box400325,Charlottesville,VA22904-4325,USA
4SpaceTelescopeScienceInstitute,3700SanMartinDrive,Baltimore,MD21218,USA
Received2010September15;accepted2010November16;published2011January5
ABSTRACT
We derive the star formation histories of eight dwarf spheroidal (dSph) Milky Way satellite galaxies from their
alpha element abundance patterns. Nearly 3000 stars from our previously published catalog comprise our data
set. The average [α/Fe] ratios for all dSphs follow roughly the same path with increasing [Fe/H]. We do not
observethepredictedkneesinthe[α/Fe]versus[Fe/H]diagram,correspondingtothemetallicityatwhichTypeIa
supernovaebegintoexplode.Instead,wefindthatTypeIasupernovaejectacontributetotheabundancesofallbut
themostmetal-poor([Fe/H]<−2.5)stars.Wehavealsodevelopedachemicalevolutionmodelthattracksthestar
formationrate,TypesIIandIasupernovaexplosions,andsupernovafeedback.Withoutmetalenhancementinthe
supernovablowout,massiveamountsofgaslossdefinethehistoryofalldSphsexceptFornax,themostluminousin
oursample.Allsixofthebest-fitmodelparameterscorrelatewithdSphluminositybutnotwithvelocitydispersion,
half-lightradius,orGalactocentricdistance.
Keywords: galaxies:abundances–galaxies:dwarf–galaxies:evolution–LocalGroup
Online-onlymaterial:colorfigures
Photometrically derived SFHs are most sensitive to young
1.INTRODUCTION
stars and metal-rich stars because the separation between
Understanding the origins of galaxies requires understand- isochronesincreaseswithdecreasingageandincreasingmetal-
ing the histories of their dark matter growth, gas flows, and licity.Elementalabundancesobtainedfromspectroscopydonot
star formation. Of these, the dark matter growth is the most giveabsoluteages,buttheycanprovidefinerrelativetimereso-
straightforward to model (e.g., Diemand et al. 2007; Springel lutionforold,metal-poorpopulations.Gilmore&Wyse(1991)
etal.2008).Thegasflowhistorypresentsmoredifficultobsta- showed that star formation bursts of varying duration and fre-
cles, such as collisional dissipation, gas cooling, stellar feed- quencyindwarfgalaxiesengravesignaturesontheratioofoxy-
back, and conversion into stars. Despite the challenges, some gen to iron as a function of metallicity. Because oxygen-rich
models—built on top of dark matter simulations—track all of TypeIIsupernovae(SNe)explodewithintensofMyrofastar-
theseprocessesovercosmictime(e.g.,Governatoetal.2007). burst, the oxygen content of stars forming soon after the burst
The results of these models have observational consequences willbehigh.WithinhundredsofMyr,iron-richTypeIaSNebe-
forthepropertiesofthepresentstellarpopulationsofgalaxies. gintoexplode.Theinjectionofironintotheinterstellarmedium
(ISM)depressestheoxygen-to-ironratioofsubsequentlyform-
1.1.MethodsforDeterminingStarFormationHistories ing stars. These processes are generalizable to other elements.
Theabundancesofthenextseveralelementswithevenatomic
The star formation histories (SFHs) of galaxies may be de- numberbeyondoxygen—thealphaelements(Ne,Mg,Si,S,Ar,
duced from the colors and magnitudes of the population and Ca,andTi)—roughlyscalewithoxygenabundance.Theabun-
fromthespectraofthestarsandgas,ifpresent.Distant,unre- dancesofiron-peakelements(V,Cr,Mn,Co,andNi)roughly
solvedgalaxiesdisplayonlyasingle,compositespectralenergy scalewithironabundance.Thetrendofthealpha-to-iron-peak
distribution, which may be examined through calibrations of ratiowithiron-peakabundance,aproxyforelapsedtimeorinte-
spectrophotometricindices(e.g.,Graves&Schiavon2008)or, gratedstarformation,revealstherelativeSFHwitharesolution
insomecases,spectralsynthesis(McWilliam&Bernstein2008; ofabout10Myr,theapproximatetimescaleforaTypeIISN.
Coluccietal.2009).Nearerstellarsystemsmayberesolvedboth
photometricallyandspectroscopically.TheHubbleSpaceTele- 1.2.ChemicalEvolutionModels
scope (HST) has enabled the characterization of the SFHs of
A glance at a diagram of [Mg/Fe] versus [Fe/H] gives a
manynearbygalaxies(Weiszetal.2008;Dalcantonetal.2009;
qualitative sense of a galaxy’s SFH. Converting quantitative
Bernardetal.2009),includingmostofthedwarfgalaxiesinthe
abundancesintoaquantitativeSFHrequiresachemicalevolu-
LocalGroup(Holtzmanetal.2006;Orbanetal.2008).
tionmodel.Pagel(1997)describedindetailhowtocreatesuch
amodel,andTolstoyetal.(2009)reviewedrecentprogresson
∗ DatahereinwereobtainedattheW.M.KeckObservatory,whichisoperated modeling the SFHs of Local Group dwarf galaxies. Matteucci
asascientificpartnershipamongtheCaliforniaInstituteofTechnology,the
(2008)describedthelevelsofapproximationthatthemodelsas-
UniversityofCalifornia,andNASA.TheObservatorywasmadepossibleby
thegenerousfinancialsupportoftheW.M.KeckFoundation. sume.Ingeneral,moresophisticatedandpresumablymoreac-
5 HubbleFellow. curatemodelsreducethenumberofapproximations.Themost
1
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
basicassumptionsareinstantaneousrecyclingandinstantaneous iesmayreflectthestellarpopulationsofthedissolvedbuilding
mixing. Consideration of stellar lifetimes and SN delay times blocks,andtheymayshowhowthesurvivingsatellitesevolved
removes the first approximation. Three-dimensional hydrody- sincethetimeofrapidaccretionontotheMW.
namicalsimulationsremovethesecondapproximation. Inaseriesofarticles,Lanfranchi&Matteucci(2003,2004,
A chemical evolution model reflects the history not only of 2007, 2010) and Lanfranchi et al. (2006, 2008) presented nu-
star formation but also of gas flow. A complete explanation merical models that tracked the evolution of several elements
of metallicity and alpha element distributions requires both in dwarf spheroidals (dSphs). The models plausibly explained
inflows and outflows. The metallicity distribution functions theMDFsandtheavailablemulti-elementabundancemeasure-
(MDFs)ofnearbyGalacticGdwarfscannotbeexplainedwith ments in dSphs. However, large samples of published abun-
aclosedboxmodel(vandenBergh1962;Schmidt1963).Pagel dance measurements in any individual dSph have been sparse
(1997)discussedsomeoftheproposedsolutionstotheGdwarf until recently (Shetrone et al. 2009; Kirby et al. 2009, 2010b;
problem, including variable nucleosynthesis yields, bimodal Letarteetal.2010).OtherchemicalevolutionmodelsofdSphs
starformation,andpre-enrichment.Oneofthemostpromising have examined the effects of reionization (Fenner et al. 2006)
solutionsisinfallingmatter(Larson1972).Gasesundoubtedly and star formation stochasticity (Carigi & Hernandez 2008).
flowoutofthegalaxy,eitherfromSNwinds(Mathews&Baker Recchietal.(2001)constructedoneofthefirsthydrodynamical
1971;Larson1974)orstrippingfromtheinfluenceofexternal modelsofdwarfgalaxyevolution.Inparticular,theysimulateda
orhostgalaxies(Tinsley&Larson1979;Lin&Faber1983).For galaxysimilartoIZw18.Marcolinietal.(2006,2008)published
example,interactionswiththeMilkyWay(MW)couldremove hydrodynamical simulations of an isolated, Draco-like dSph.
gasfromthesatellitegalaxiesdiscussedhere.Bothinflowsand Their models relaxed the assumption of instantaneous mixing
outflows affect the star formation rate (SFR) throughout the andallowedinhomogeneouschemicalenrichment.Someofthe
history of the galaxy. Therefore, they shape the MDF and the newesthydrodynamicalmodels(Revazetal.2009;Sawalaetal.
trendof[α/Fe]with[Fe/H]. 2010)trackedboththekinematicsandabundancesofthestars
Chemical evolution models suffer from uncertainties in astheyform.Theyattemptedtoexplainnotonlychemicalabun-
the initial mass function (IMF) of stars and stellar lifetimes dancepatternsbutalsodynamicalpropertiesofdSphs,suchas
(Romano et al. 2005), nucleosynthesis yields (Romano et al. theseeminglyuniversaldynamicalmassmeasuredwithintheir
2010), and the delay time distribution (DTD) for Type Ia SNe optical radii (Mateo 1998; Strigari et al. 2008) and out to the
(Matteucci et al. 2009). However, these limitations have not edgeoftheirlightdistributions(Gilmoreetal.2007).
prevented the models from providing good fits to abundance
data.EvenmodelswithsomeofthefirsttheoreticalSNyields 1.3.HistoryofChemicalAnalysisofMilkyWaySatellites
(Woosley et al. 1993) successfully reproduced the observed
metallicity distribution and abundance patterns in the Galaxy Theearliestindicationsofheavyelementabundancespreads
(Pagel & Tautvaisˇiene˙ 1995). Models with newer SN yields among red giants of the dSph systems in Draco, Ursa Minor,
alsomatchthesolarneighborhoodabundancedistributionsvery Sculptor, and Fornax were first obtained by the multichannel
well (e.g., Romano et al. 2010). Nonetheless, uncertainties in scanner observations of Zinn (1978, 1981), initial efforts at
the model assumptions do complicate the interpretation of the spectroscopy (Norris & Bessell 1978; Kinman & Kraft 1980;
model results. For example, changing the Type Ia DTD, par- Kinman et al. 1981; Stetson 1984; Smith 1984; Lehnert et al.
ticularlytheturn-ontime,affectsthederivedtimescaleforstar 1992), and both broad and narrow band photometry (Demers
formation.Thebestwaytocircumventtheseuncertaintiesisto etal.1979;Smith&Dopita1983).Theglobularclusters(GCs)
applythesamemodelconsistentlytoseveralsystemsandcom- oftheFornaxsystemprovedtodifferintheirmetallicities(Zinn
pare them differentially. Although the absolute ages or SFRs & Persson 1981). The presence of carbon stars (Aaronson &
maybeaffectedbysystematicerrorsinthemodel,therelative Mould1980;Aaronsonetal.1982,1983;Azzopardietal.1985)
quantitiesbetweendifferentgalaxieswillbemeaningful. and so-called anomalous Cepheids (Demarque & Hirshfeld
LocalGroupdwarfgalaxiesmakegoodsubjectsforchemical 1975; Norris & Zinn 1975; Hirshfeld 1980; Smith & Stryker
evolutionmodels.First,theLocalGroupcontainsmanyresolved 1986) further indicated the potential complexity of the stellar
dwarf galaxies (Mateo 1998; Tolstoy et al. 2009) with stars populations in dSphs. Carbon stars are exceedingly rare in
bright enough for medium- or high-resolution spectroscopy. GCs, while the period–luminosity relations of the anomalous
Second, dwarf galaxies span a wide range of properties, in- Cepheidsimpliedthattheyaremoremassivethantypicalcluster
cludingvelocitydispersionandluminosity.Thepopulationsof Cepheids (Zinn & Searle 1976). As a consequence, by the
the lowest luminosity galaxies enable the study of star forma- mid-1980s, circumstantial evidence was building to suggest
tion on small scales (Martin et al. 2008a; Norris et al. 2008). thatdSphshadmorecomplexandpossiblymoreextensivestar
The changes in populations for more luminous or more mas- formationandchemicalevolutionhistoriesthanGCs.
sivegalaxiesshowhowstarformationrespondstogalaxysize Since that time, the application of ground-based CCD and
(Mateo 1998; Kirby et al. 2010a). Third, dwarf galaxies host HST imaging has led to greatly improved color–magnitude
some of the most metal-poor stars known (Kirby et al. 2008, diagrams (CMDs) that have clearly shown the presence of
2009; Geha et al. 2009; Cohen & Huang 2009, 2010; Frebel significant internal age spreads within some of the MW’s
et al. 2010a, 2010b; Simon et al. 2010; Norris et al. 2010a, retinue of dSphs, such as Carina, Fornax, Leo I, and Sextans
2010b;Starkenburgetal.2010;Tafelmeyeretal.2010).These (e.g., Mighell 1990, 1997; Smecker-Hane et al. 1996; Hurley-
starsretainthechemicalimprintoftheISMwhentheuniverse Keller et al. 1998; Buonanno et al. 1999; Gallart et al. 1999a,
was less than 1 Gyr old. Therefore, dwarf galaxies permit the 1999b;Savianeetal.2000;Leeetal.2009).Spectroscopywith
study of star formation not only on small scales but also at large ground-based telescopes has demonstrated the presence
earlytimes.Finally,dwarfgalaxiesmaybetheprimarybuilding ofabundanceinhomogeneitiesinthemajorityofthesesystems
blocks for the MW halo (Searle & Zinn 1978; White & Rees (e.g.,Suntzeffetal.1993;Smecker-Haneetal.1999;Shetrone
1978). The stellar populations of the surviving dwarf galax- et al. 2001b, 2003; Tolstoy et al. 2001, 2003, 2004; Winnick
2
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
Table1
ChemicalEvolutionModelVariables
Variable Description Units
t Timesincestartofsimulation Gyr
M Massofasinglestar M(cid:4)
ξj(t) Gasmassinelementj M(cid:4)
Xj(t) Massfractioninelementj Dimensionless
Y Primordialheliummassfraction(XHe(0)) Dimensionless
Mgas(t) Totalgasmass M(cid:4)
Z(t) Metalfraction(allelementsheavierthanHe) Dimensionless
ξ˙j(t) Timederivativeofξj M(cid:4)Gyr−1
ξ˙j,∗(t) SFR,orrateofgaslossinelementjduetostarformation M(cid:4)Gyr−1
ξ˙j,II(t) TypeIISNorHNyieldrateforelementj M(cid:4)Gyr−1
(cid:4)HN FractionofHNeamongstarswithM(cid:2)20M(cid:4) Dimensionless
ζj,II(M,Z) MassofelementjejectedbyoneTypeIISN M(cid:4)
ξ˙j,Ia(t) TypeIaSNyieldrateforelementj M(cid:4)Gyr−1
tdelay TypeIaSNdelaytime Gyr
ΨIa(tdelay) TypeIaSNDTD SNGyr−1M(cid:4)−1
ζj,Ia MassofelementjejectedbyoneTypeIaSN M(cid:4)
ξ˙j,AGB(t) AGByieldrateforelementj M(cid:4)Gyr−1
ζj,AGB(M,Z) MassofelementjejectedbyoneAGBstar M(cid:4)
A∗ NormalizationofSFRlaw(freeparameter) M(cid:4)Gyr−1
α SFRexponentofMgas(freeparameter) Dimensionless
Ain Normalizationofgasinfallrate(freeparameter) M(cid:4)Gyr−1
τin Gasinfalltimeconstant(freeparameter) Gyr
Aout GaslostperSN(freeparameter) M(cid:4)SN−1
Mgas(0) Initialgasmass(freeparameter) M(cid:4)
2003; Pont et al. 2004; Geisler et al. 2005; McWilliam & toprevious photometricandspectroscopicstudies(Section3).
Smecker-Hane2005a,2005b;Battagliaetal.2006;Kochetal. Next, we change some of the model variables to estimate the
2006;Bosleretal.2007;Sbordoneetal.2007;Gullieusziketal. systematic errors in the derived SFHs (Section 4). Then, we
2009;Cohen&Huang2009,2010;Kirbyetal.2009). explore how the abundance distributions, SFHs, and gas flow
histories change with galaxy properties such as luminosity
1.4.ChemicalEvolutionModelsfortheNewCatalog
andvelocitydispersion(Section5).Finally,weenumerateour
conclusions(Section6).
In this article, we interpret the multi-element abundance
distributions in eight dSphs with our own chemical evolution
model.Thedatasetisourcatalogofabundancesbasedonspec- 2.CHEMICALEVOLUTIONMODEL
tralsynthesisofmedium-resolutionspectrafromtheDEIMOS
In order to provide a rough interpretation of the abundance
spectrograph on the Keck II telescope (Kirby et al. 2010b,
trends in Paper II’s catalog, we have developed a rudimentary
hereafterPaperII).Thecatalogcontains2961starswithabun-
model of chemical evolution. Table 1 defines the symbol for
dancemeasurements.ThenumberofstarsineachdSphranges
each variable or constant in the model. The model supposes
from141(Sextans)to827(LeoI).Itisthelargesthomogeneous
thatadwarfgalaxyatanyinstantisachemicallyhomogeneous
chemicalabundancedatasetindwarfgalaxies.Thetypicalareal
systemthatcan accreteorlosegas.The ejectaofType IISNe
coverageisabout300arcmin2atornearthecenterofeachdSph.
enrichthegasaccordingtothetotallifetimeofmassive(10 <
Themedianuncertaintyon[Fe/H]is0.12dex.Thefractionof
M/M(cid:4) <100)stars,whiletheTypeIaSNefollowtheobserved
thesamplewith[Mg/Fe]uncertaintieslessthan0.2(0.3)dexis
DTD (Maoz et al. 2010; see below). Stars form according
42%(53%).Thatfractionincreasesto54%(69%)for[Ti/Fe], to the Kroupa et al. (1993) IMF (dN/dM = 0.31M−2.2 for
whichiseasiertomeasurethan[Mg/Fe].For(cid:5)[α/Fe](cid:6)(theav- 0.5<M/M(cid:4) <1anddN/dM =0.31M−2.7 forM >1M(cid:4)).
erageof[Mg/Fe],[Si/Fe],[Ca/Fe],and[Ti/Fe]),thefraction
ThecalculationtracksthemassofH,He,Mg,Si,Ca,Ti,and
increasesto71%(88%). Feateachtimestep(Δt =1Myr).Thecalculationisterminated
Our one-zone model is simple, but it incorporates some of
whenthesystemreacheszerogasmass.
thenewestSNyieldsandthemostrecentlymeasuredDTDfor
Wedefineξ (t)asthegalaxy’sgasmassofelementjattime
j
TypeIaSNe.Thebiggestadvantageofourdatasetisthatitis
t.Thegalaxy’stotalgasmassattimetis
homogeneous. All of the spectra were obtained with the same
(cid:2)
spectrographconfiguration,andalloftheabundancesweremea- M (t)= ξ (t) (1)
suredwiththesamespectralsynthesiscode.Thus,thederived gas j
starformationandgasflowhistoriesfromourmodel—despite j
its simplicity—will be easy to interpret differentially. In other
≈ξ (t)+ξ (t)+20.4[ξ (t)+ξ (t)+ξ (t)+ξ (t)]
words, the absolute ages and SFRs may be affected by model H He Mg Si Ca Ti
uncertainties,butthetrendswithgalaxyproperties,suchaslu-
minosity,shouldreflectthetrueSFHs. +1.07ξ (t). (2)
Fe
Webeginbydescribingourmodel(Section2).Then,weapply
the model to the eight dSphs by finding the solution that best The summation in Equation (1) is over all elements in the
matches the abundances. We discuss how our results compare periodictable.However,ourmodeltracksonlysevenelements.
3
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
Therefore,weassumetheratioofthesumofallelementsfromLi mass during its lifetime and because some mass is locked up
toTi,inclusive,tothesumofMg,Si,Ca,andTiisthesameasin foreverinanSNremnant.
theSun.Thisratiois20.4(Anders&Grevesse1989).Similarly, Nomoto et al. modeled both normal core-collapse SNe and
weassumethesolarratioforthesumofallelementsVthrough very energetic hypernovae (HNe). The lowest mass HN they
Ge compared to Fe: 1.07. Elements beyond Ge are neglected. modeled is 20M(cid:4). The fraction of stars at least this massive
Equation(2)reflectstheseapproximations.Forconvenience,we that explode as HNe is (cid:4) . Nomoto et al. adopted (cid:4) = 0.5
HN HN
definethemetallicityofthegasasfollows: for their own model of the solar neighborhood. Romano et al.
(2010) explored the cases of (cid:4) = 0 and 1. In our own
Z = Mgas(t)−ξH(t)−ξHe(t). (3) experimentation, we have foundHtNhat (cid:4)HN = 0 produces good
M (t) matchestothedSphabundancepatternsatthelowestvaluesof
gas
[Fe/H],andweadoptthisvalueforthemodel.InSection4.2,
Wealsodefinethegas-phasemassfractioninanelementj: weexploretheeffectofincreasing(cid:4) onthemodel.
HN
Thefollowingintegralgivestheinstantaneouschangeingas
X (t)= ξj(t) . (4) massfromtheejectaofTypeIISNe(M(cid:4)Gyr−1):
j
M (t) (cid:7)
gas
The following subsections explain the components of the ξ˙j,II =0.31M(cid:4)0.7 100M(cid:4)ζj,II(M,Z(t −τ∗(M)))
models. Each component is expressed as the time change in 10M(cid:4)
ξ (t),whereξ˙ ≡dξ (t)/dt.
j j j
× ξ˙∗(t −τ∗(M))M−2.7dM. (7)
2.1.StarFormationRate
The coefficient 0.31M0.7 is the normalization from the IMF.
Forsimplicity,weassumethattheSFRisapowerlawinthe (cid:4)
This integral depends on the SN yields (ζ ), the recent SFH
gasmassofthegalaxy.Withthisassumption, j,II
(cid:3) (cid:4) (ξ˙∗),andthehigh-massIMFslope(M−2.7).Inpractice,thisin-
ξ˙j,∗ =A∗Xj(t) 1M0g6aMs(t(cid:4)) α. (5) toevgerralainsparerrafyormofed10n0umloegraicriatlhlymwiciatlhlyNsepwatcoend˙-Cmoatessseisntbegetrwateioenn
10M(cid:4) and100M(cid:4).Thevaluesofζj,II andξ∗ areinterpolated
ThevariablesA∗andαarefreeparametersinthemodel.Inthe ontothisarray.Themetallicityusedtolookuptheappropriate
completechemicalevolutionequation(Equation(16)),thesign SN yields is consistent with the metallicity of the gas at the
ofξ˙j,∗ isnegativebecauseξj representsthegasmass,whichis time the exploding star formed. (In other words, at any given
depletedduetostarformation. timestep,themetallicitiesofthelowermassSNearelessthan
Equation(5)isageneralizationofaKennicutt–Schmidtlaw themetallicitiesofhighermassSNefrommorerecentlyformed
(Schmidt 1959; Kennicutt 1998), which connects the SFR to stars.)
the gas surface density, Σ . Surface density is perhaps more TheinstantaneousTypeIISNrate(SNGyr−1)isgivenbya
gas
appropriate for disks than spheroids. Desiring a more three- relatedintegral:
dimensionalproperty,wehaveusedthegasmass,M ,instead (cid:7)
ofΣgas.Thevolumedensity,ρgas,wouldbeabetterdgeasscription, N˙II =0.31M(cid:4)0.7 100M(cid:4)ξ˙∗(t −τ∗(M))M−2.7dM. (8)
but the difference between M and ρ is simply a constant
gas gas 10M(cid:4)
becauseourmodelisone-zoned.
This integral is performed over the same array of massive star
2.2.TypeIISupernovae lifetimesasafunctionofmassasforEquation(7).Thevaluewill
beusedtodeterminethemasslostfromSNwinds(Section2.6).
Inourmodel,starsmoremassivethan10M(cid:4)andlessmassive
than100M(cid:4)explodeaccordingtotheirtotallifetimes(Padovani 2.3.TypeIaSupernovae
&Matteucci1993;Kodama1997):
(cid:5) (cid:6) We adopt the Type Ia SN yields of Iwamoto et al. (1999).
τ∗(M)= 1.2(M/M(cid:4))−1.85+0.003 Gyr. (6) ThemassofelementjejectedperTypeIaSNisζj,Ia.TheSNe
explode according to a function that approximates the DTD
This formula is valid for stars more massive than 6.6M(cid:4), observed by Maoz et al. (2010; see Figure 1). The following
(inclusive of our entire mass range for Type II SNe). Maeder equationdescribestheadoptedDTD:
& Meynet (1989) give slightly different formulas for stars ⎧
less massive than 60M(cid:4), but the differences do not affect the ⎪⎨0(cid:5) (cid:6) tdelay <0.1Gyr
chemicalevolutionmodelappreciably. Ψ = 1×1(cid:12)0−3 S(cid:13)NGyr−1M(cid:4)−1 .
Starsmoremassivethan100M(cid:4) donotforminthismodel. Ia ⎪⎩ × tdelay −1.1 tdelay (cid:2)0.1Gyr
TheTypeIISNejectaaremixedhomogeneouslyandinstanta- Gyr
neouslyintotheISMoftheentiredSph. (9)
We adopt the Type II SN nucleosynthetic yields of Nomoto
et al. (2006). The symbol ζj,II(M,Z) represents the mass in Thevariabletdelay isusedinsteadofttoindicatethattheDTD
elementjejectedfromtheTypeIISNexplosionofastarwith willbeintegratedfromtimetintothepast.
aninitialmassM.Itisafunctionofbothinitialstellarmassand Unfortunately, the abundance distributions derived from the
metallicity. Nomoto et al. tabulated the yields for seven initial chemicalevolutionmodeldependsensitivelyonthenormaliza-
masses ranging from 13M(cid:4) to 40M(cid:4) and four metallicities tionandturn-ontimeofΨIa.Bothofthesequantities—partic-
fromZ =0toZ =0.02.Thetotalmassoftheejectaisalways ularlytheturn-ontime—havelargeuncertainties.Thenormal-
lessthanthebirthmassofthestarbecausethestarlosessome izationaffects[Fe/H]andtheslopeof[α/Fe]with[Fe/H].We
4
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
10-1 heavierthan10M(cid:4) explodeasTypeIISNe(Section2.2).The
clusters lowermasslimitweconsiderforAGBstarsis0.865M(cid:4),which
nearby galaxies
isthestellarlifetimecorresponding totheageoftheuniverse,
Magellanic SNRs
-1) 10-2 0.4 < z < 1.2 ellipticals 13.6Gyr,accordingtoEquation(12).TheAGBmassreturnrate
M inM(cid:4)Gyr−1isgivenby
SN Gyr-1 10-3 ξ˙j,AGB =0.31M(cid:4)0.2 (cid:7) 1M(cid:4) ζj,AGB(M,Z(t −τ∗(M)))
Ψ (Ia 10-4 0.865M(cid:4)
10-5 × ξ˙∗(t −τ∗(M))M−2.2dM
0.1 1.0 10.0
t (Gyr)
delay (cid:7)
Figure 1. Type Ia SN DTD, as measured by Maoz et al. (2010). The data 10M(cid:4)
comefromavarietyofstarformationenvironments,giveninthefigurelegend. + 0.31M(cid:4)0.7 ζj,AGB(M,Z(t −τ∗(M)))
Equation(9)givestheexpressionforthisfunction.ComparethisfiguretoMaoz 1M(cid:4)
etal.’sFigure2.
havechosen1×10−3SNGyr−1M−1forthenormalizationbe- × ξ˙∗(t −τ∗(M))M−2.7dM. (13)
(cid:4)
cause that is the value that Maoz et al. (2010) reported. Even
Compared to SN ejecta, AGB ejecta affect the chemical
thoughthedatainFigure1areeasilyconsistentwithhalfthat
evolution of the elements considered here to a small degree.
value,thelargervaluebetterreproducestheslopeof[α/Fe]with
AGB ejecta are more important for other elements, such as C,
[Fe/H]formanyofthedSphs.Theturn-ontimedeterminesthe
N,andO.
time or [Fe/H] at which [α/Fe] begins to drop. We have cho-
sen0.1Gyrbecausethatisapproximatelythemaximumvalue 2.5.GasInfall
acceptable for the DTD data (Figure 1). See Section 4.1 for a
discussionoftheeffectofincreasingthisminimumdelaytime Infall of gas during the star formation lifetime of a dSph is
to0.3Gyr. requiredtoexplainitsMDF(Kirbyetal.2010a,hereafterPaper
TheinstantaneousTypeIaSNrateisgivenbycombiningΨ III). Therefore, our model allows pristine gas to fall into the
withthepastSFH: Ia dSph.ThegashasaheliumfractionofY = XHe(0) = 0.2486,
which is the value obtained when the WMAP7 (Larson et al.
(cid:7)
0 2010) baryon-to-photon ratio is applied to the formula of
N˙Ia = ξ˙∗(tdelay)ΨIa(t −tdelay)dtdelay. (10) Steigman(2007).Therestoftheinfallinggasishydrogen.
t The MDFs of the dSphs are generally more peaked than a
closed box model predicts. One scenario that explain such a
ThemassreturnedtotheISMistheproductoftheSNIayields
distribution is gas infall that first increases and then decreases
(ζ )andtheSNIarate:
j,Ia
(Lynden-Bell1975;Pagel1997).Wefindthataquickincrease
ξ˙ =ζ N˙ . (11) of the rate of gas falling into the galaxy followed by a slower
j,Ia j,Ia Ia decreaseintheinfallratedoeswellatreproducingthedata.We
parametrizethegasinfallrateasfollows:
2.4.AsymptoticGiantBranchStars
(cid:3) (cid:4)
butWsiingdnsififcraonmt alomwo-uanntdofinmtearsmsetdoiathtee-mISaMss.sTtahresrsetatursrnloasesmleaslsl ξ˙j,in =AinXj(t =0) Gtyr e−t/τin. (14)
than1%ofthismassbeforereachingtheasymptoticgiantbranch
(AGB; van den Hoek & Groenewegen 1997). Therefore, we The term X (t = 0) means that the infalling gas is primordial
j
considermasslossontheAGBonly. (metal-free).ThevariablesA andτ arefreeparametersinthe
in in
We adopt the AGB yields of Karakas (2010), who tracked model.
all of the elements we consider here except Ca and Ti. (We
assume that the fraction of Ca and Ti in AGB ejecta is the 2.6.SupernovaWinds
sameasinthematerialthatformedthestar.)Weassumeallof
TheMDFsofdSphsrequiregasoutflow.Ifthatwerenotthe
the mass is ejected in the final time step of the star’s lifetime.
case,themetallicitieswouldapproachtheSNyields,whichare
ThisassumptionisappropriatebecauseanAGBstar’sthermal
much larger than observed in even the most metal-rich star in
pulsationperiod,duringwhichitlosesmostofitsmass,lastson
any dSph. Gas may be lost through SN winds, stellar winds,
theorderof1Myr(Marigo&Girardi2007),whichisthelength
or gas stripping from an external source. All of these sources
ofonetimestepinourmodel.Equation(6)givesthelifetimes
undoubtedlyoccuroveradSph’slifetime,butSNwindsarethe
of stars more massive than 6.6M(cid:4). Less massive stars obey moststraightforwardtoincludeinachemicalevolutionmodel.
Padovani&Matteucci’s(1993)andKodama’s(1997)equation:
Weignoreothersourcesofgasloss.
√ Ourcomputationofgaslossisfairlysimple.Thegalaxyloses
τ∗(M)=100.334− 1.790−0.202.13121[67.764−log(M/M(cid:4))] Gyr. (12) a fixed amount of gas for every SN that explodes. The blown-
outgasmassdoesnotvarywithSNtypebecausetheexplosion
EachAGBstarejectsζ solarmassesofelementj.Stars energies for Types II and Ia SNe are similar. See Recchi et al.
j,AGB
lighterthan10M(cid:4) participateinAGBmasslosswhereasstars (2001), Romano et al. (2006), and Marcolini et al. (2008) for
5
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
examplesofchemicalevolutionmodelsthattreatedtheenergy progenitors form depends sensitively on the turn-on time.
inputfromthetwoSNetypesdifferently.Therateofgaslossis The uncertainty in the turn-on time translates to a large
uncertaintyintheFeabundancedistribution.Withallother
ξ˙j,out =AoutXj(N˙II+N˙Ia). (15) modelparametersheldfixed,anearlierturn-ontimewould
cause the metallicity of the MDF peak to increase and
TheparameterAout isafreeparameterinthemodel.Anenergy [α/Fe] at a given metallicity to decrease. See Matteucci
argument shows that the ejected gas mass is of the order of et al. (2009) for a detailed discussion of the effect of
104M(cid:4)SN−1.OneSNexplodeswithatypicalenergyof1051erg adjustingtheratioofprompttodelayedTypeIaSNe.
(Woosley&Weaver1995).Inthelatestagesofexpansion,the
2. TheSNyieldsareimperfect.AswementioninSection3,
kineticenergyoftheejectaisE ∼ 8.5×1049 erg(Thornton
ej we needed to increase the [Mg/H] output of the model
et al. 1998). A typical line-of-sight velocity dispersion for a
by 0.2 dex (see Franc¸ois et al. 2004). Furthermore, Ti is
dwarf galaxy is σ ∼ 10kms−1. Given the virial theorem
los severelyunderproducedinourmodel.Therefore,wedonot
(GM/R =3σ2 )andtheescapevelocity(v2 =2GM/R),the
los esc considerTiabundancesatall.
gasmassejectedasaresultofSNblowoutisMej =Eej/ve2sc = 3. Our model assumes instantaneous mixing. Relaxing this
Eej/(6σl2os)∼7×103M(cid:4)SN−1. approximationwouldrequiremultiplezones,whichwedo
A metal-enhanced wind can prevent the galaxy from be- notconsider forthesakeofcomputational simplicity.See
coming too metal-rich without such a large gas loss (Vader Mori et al. (2002), Marcolini et al. (2006, 2008), Revaz
1986). For simplicity, we assume that the SN winds have the etal.(2009),andSawalaetal.(2010)forthree-dimensional
samechemicalcontentasthegasremaininginthegalaxy.See chemicalmodelsofdwarfgalaxies.
Section4.3forafurtherdiscussionofincludingmetal-enhanced 4. We also assume instantaneous gas cooling. The cooling
windsinthemodel. time for gas to become available for star formation (after
accretion or ejection from SNe and AGB stars) may be
2.7.CompleteChemicalEvolutionEquation longerthanthemodeltimestep,Δt =1Myr.Amoreproper
Thecompleteequationthatdescribesthechemicalevolution treatmentofthecoolingtime,suchasinahydrodynamical
ofthegalaxy’sgasis model,mightresultinslightlylongerSFdurationsthatwe
derivewithinstantaneouscooling.
(cid:7)
t 5. Onarelatednote,wealsoignoredynamicalprocesses.Our
ξj(t)=Mgas(0)+ (−ξ˙j,∗+ξ˙j,II+ξ˙j,Ia (16) adoption of a single value of A , the gas ejected from
out
0
thegalaxy inthewindofoneSN,implicitlyassumesthat
+ξ˙ +ξ˙ −ξ˙ )dt. thepotentialofthegalaxyishomogeneousandstatic.This
j,AGB j,in j,out
assumption is inconsistent with our allowance of gas to
Theinitialgasmass,M (0),isafreeparameter.Anon-zeroini- flow into the galaxy. Although dark matter dominates the
gas
tialgasmassmayseeminconsistentwithEquation(5)because dynamical mass of dSphs, they undoubtedly change their
the gas should form stars as it falls into the galaxy. However, dark matter masses during their star formation lifetimes
the galaxy could acquire gas available for star formation—via (Robertsonetal.2005;Bullock&Johnston2005;Johnston
gravitationalcollapseorcooling,forexample—onatimescale etal.2008).Furthermore,baryonic(adiabatic)contraction
fasterthanthestarformation(SF)timescale.Wewillshowthat canaffectstarformationandfeedbackinthedensecenters
the non-zero initial gas mass is more important for the more ofthedSphs(Napolitanoetal.2010).
luminousdSphs. 6. Weconsideronlyoneparametrizationofthegasinfallrate.
BecausetheSFRisproportionaltothegasmass,thegasin-
2.8.ShortcomingsoftheModel fallrateessentiallyshapesthedifferentialMDF.Differently
shapedgasinfallhistoriesmightbetterreproducethedSph
Ourmodelincorporatesrealisticconditionsindwarfgalaxies.
MDFs.Externalinfluencesonthegasflow(oralternatively,
We model chemical evolution using an observed Type Ia SN
availabilityofgascoolenoughtoformstars)thatwedonot
DTD(Maozetal.2010).Wealsotakeintoaccountthelifetimes
considerincludereionization(Bullocketal.2000)andtidal
ofTypeIISNprogenitors,ratherthanassuminginstantaneous
andrampressurestripping(Lin&Faber1983).
recycling. The delay helps to shape the metal-poor abundance
7. Wemodelonlyoneepisodeofstarformation.CMDshave
distributionsbecauseitaffectstherapidriseinmetallicityafter
revealed extended and possibly bursty SFHs in several
theonsetofstarformation.
dSphs in our sample (Fornax and Leo I and II). These
However, our model is not as sophisticated as some other
bursts will not be included in our model. In these cases,
chemical evolution models of dwarf galaxies (e.g., Marcolini
wedefertothephotometricallyderivedSFHs.Infact,we
et al. 2008; Revaz et al. 2009; Sawala et al. 2010). In the
suggestforfuturestudyamoresophisticatedanalysisthat
next section, we show the best model fits to eight different
modelsboththeCMDandabundancedistributions.
MWsatellitegalaxies.Thesimplicityofourmodelreducesthe
8. The infalling gas is assumed to be metal-free at all times.
computationaldemandoffindingthebestsolution.Nonetheless,
In reality, the metallicity may have increased over time
weenumeratesomeshortcomingswhichaffecttheinterpretation
becausethesourceofthenewgasmayhavebeenblowout
oftheabundancedistributions.
from prior SF episodes in the galaxy in question or other
1. The turn-on time for Type Ia SNe is poorly constrained. galaxies. This gas would have been enriched by SNe and
Maoz et al. (2010) showed that it is almost certainly othernucleosyntheticsources.
0.1 Gyr or less (at least in the Magellanic Clouds and 9. Themodelingresultforagivengalaxyrepresentsonlypart
higher redshift elliptical galaxies), but the DTD slope ofthatgalaxy’sstellarpopulation.Ourspectroscopicsam-
(td−e1la.y1)isdivergentastdelayapproacheszero.Therefore,the pleswerecentrallyconcentratedtomaximizethenumberof
number of Type Ia SNe that explode shortly after their memberstarsonaDEIMOSslitmask,butmostdSphshave
6
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
radial population gradients (e.g., Sculptor, Battaglia et al. theestimateofmeasurementuncertainty(δ(cid:4) )inthatquantity.
i,j
2008). As a result, we preferentially probe the younger, Starswithlargeruncertaintieshavelessweightinthelikelihood
more metal-rich populations. MW satellite galaxies also calculation than stars with smaller uncertainties. (Although
shedstarsastheyinteractwiththeGalaxy.Majewskietal. Figures2–9showonlystarswithuncertaintieslessthan0.3dex,
(2000b)identifiedstarsfromtheCarinadSphbeyondCa- there is no error cut in the likelihood calculation. Instead,
rina’stidalradius.Majewskietal.(2002)andMun˜ozetal. we downweight stars with large uncertainties.) The chemical
(2006)discussedtheimplicationsforCarina’spresentstel- evolution model traces a path (cid:4) (t) in the four-dimensional
j
larpopulation.Inparticular,theremainingstarsareonaver- space. The probability that a star formed at a point t on the
ageyoungerandmoremetal-richthantheloststars.Conse- path is dP/dt = M˙∗(t)/M∗, where M∗ is the galaxy’s final
quently,thespectroscopicsamplefavorstheyounger,more stellarmass.Thelikelihoodthatonestarconformstothemodel
metal-richstars. is the line integral of dP/dt along the path (cid:4) (t). The total
j
likelihood L is the product of the individual likelihoods of the
Someoftheseshortcomingsareobservationalortheoretical
Nstars:
uncertainties (1–2), which can only be resolved with a more ⎛ ⎞
tfihcoartoioungsh(i3n–v8e)s,tiwgahtiicohncoafnSbNerraetseoslvoerdyiwelidths.mOotrheerssoparheisstiimcaptelid- L= (cid:14)N (cid:7) t⎝(cid:14)√ 1 exp−((cid:4)i,j −(cid:4)j(t))2⎠M˙∗(t)dt
models. The last shortcoming (9) could be resolved by an in- i=1 0 j 2πδ(cid:4)i,j 2(δ(cid:4)i,j)2 M∗
tensive,wide-fieldcampaignwiththeintenttorecoverspectra
for a magnitude-limited sample of red giants in a dSph. This (cid:3)
projectwouldrequireagreatdealoftelescopetime,butitcould × √ 1 exp−(M∗,obs−M∗,model)2
beaccomplishedinprincipleforoneortwodSphs.Foreground 2πδM∗,obs 2(δM∗,obs)2
contaminationcouldbeminimizedbyselectingadSphathigh
Galactic latitude or photometrically pre-selecting likely mem- (cid:4)
1 −(M )2 0.1N
bers(e.g.,Majewskietal.2000a). × √ exp gas,obs . (17)
2πδMgas,obs 2(δMgas,obs)2
3.GASFLOWANDSTARFORMATIONHISTORIES
The second line of the equation requires that the final stellar
We apply our chemical evolution model to eight dSphs: massofthemodel(M∗,model)matchestheobservedstellarmass
Fornax,LeoI,Sculptor,LeoII,Sextans,Draco,CanesVenatici (M∗,obs) within the observational uncertainties. We adopt the
I, and Ursa Minor. We use the abundance measurements from stellar masses of Woo et al. (2008). They did not study Canes
PaperII.Foreachgalaxy,weattempttomatchsimultaneously VenaticiI.Weassumethatthegalaxyhasaboutthesamestellar
the distribution of [Fe/H] and the trends of [Mg/Fe], [Si/Fe], massasUrsaMinorbecauseithasthesameluminositywithin
and [Ca/Fe] with [Fe/H] by adjusting the six free parameters the observational uncertainties. The third line of the equation
listedatthebottomofTable1. ensures that the dSph ends up gas free. We fairly arbitrarily
Unfortunately, some elemental abundances could not be assume an uncertainty of δMgas,obs = 103M(cid:4) because even
matched for any combination of parameter values. In particu- lower values of δMgas,obs cause the chemical evolution model
lar,themodelunderpredicts[Mg/H]and[Ti/H].Franc¸oisetal. toconvergeonspurioussolutions.Theexponent0.1N setsthe
(2004)constructedachemicalevolutionmodelfortheMWand relativeinfluenceofthefinalstellarandgasmasscomparedto
also encountered trouble in reproducing the yields. They con- theabundancedistributions.Thisvaluewaschosensothatthese
cludedthattheSNyieldsshouldbemodified.Theyspecifically quantities did not dominate the likelihood but also so that the
singled outMgfor being underproduced byboth Type IaSNe modeledgalaxiesendedupgas-freeandwithaboutthecorrect
andlow-massTypeIISNe.Wefeelcomfortablemodifyingthe stellarmass.
model results for [Mg/H] because chemical evolution models For computational simplicity, we minimize the quantity
bydifferentauthorsoverawiderangeofgalaxymassesandages Lˆ =−lnL:
indicatethatsuchmodificationisnecessary.Weadd0.2dexto ⎛ ⎞
(cid:7)
[HMowg/eHve]rt,othberiNngomthoetomeotdaell.i(n2t0o0b6e)tTteyrpaegIrIeSemNeynitewlditfhorth[Teid/aFtea]. Lˆ = −(cid:2)N ln t⎝(cid:14)√ 1 exp−((cid:4)i,j −(cid:4)j(t))2⎠
is about −0.1 dex, which is far below the value observed for i=1 0 j 2πδ(cid:4)i2,j 2δ(cid:4)i2,j
metal-poor stars in dSphs or in the MW halo. Rather than at-
treemsuplttifnogrtToi.cNororemcottsouecthaal.laalrsgoeidgenfiocreit,thweeiriTginoyrieeldthseinmtohdeeirl × M˙∗(t)dt + 0.1N(cid:3)(M∗,obs−M∗,model)2
ownchemicalevolutionmodelofthesolarneighborhood. M∗ 2(δM∗,obs)2
In Paper III, we found the best-fit analytical chemical evo-
(cid:4)
lution models for the same eight dSphs based on their MDFs (M )2
alone. We repeat the process here for our more sophisticated + 2(δMgas,obs )2 +ln(2π)+ln(δM∗,obs)+ln(δMgas,obs) . (18)
model.AsinPaperIII,weusemaximumlikelihoodestimation gas,obs
tofindthebest-fitmodelparameters.
ˆ
The likelihood that a particular model matches the data is We find the values of the six parameters that minimize L
theproductofprobabilitydistributions.Eachstarisrepresented usingPowell’smethod.Wecalculateuncertaintiesonthemodel
by a probability distribution in a four-dimensional space. The parameters via a Monte Carlo Markov chain. We perform at
four dimensions are [Fe/H], [Mg/Fe], [Si/Fe], and [Ca/Fe]. least104trialsforeachdSphafteraburn-inperiodof103trials.
Wedenotethesequantitiesas(cid:4) ,whereirepresentstheithstar The dSphs with shorter SF durations require less computation
andjidentifiesoneofthefourie,jlementratios.TheGaussianis time, and we were able to perform up to 5 × 104 trials for
centeredonthestar’sobservedvalues.Thewidthineachaxisis someofthedSphs.AsinPaperIII,themodeluncertaintiesare
7
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
H] 100 Fornax )yr-1 235000
d[Fe/ 6800 M G 125000
N / 2400 106 15000
e]d 01..050 f (M)in 12050 Final mass: 1g.a9s × 107M
Mg/F 0.0 106 10 stotatrasl
[ −0.5 M ( 5
1.0 yr)-1 800
G 60
Fe] 0.5 M 40
Si/ 0.0 06
[ 1 20
−0.5 R ( 0
F
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...055
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure2.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforFornax.Left:thetoppanelshowstheobservedMDFastheblackhistogramandthe
modeledMDFinred.Themodelisconvolvedwithanuncertaintyfunctiontomimicthebroadeningofthehistograminducedbyobservationalerror.Acrossmarks
each1Myrtimestep,butthesearetoocloselyspacedtodiscernformostofthemetallicityrange.Veryfewstarsareexpectedtohaveformedatthelowmetallicities
wherethecrossesaredistinguishable.Theotherpanelsshowtheobserved[Mg/Fe],[Si/Fe],[Ca/Fe],and[Ti/Fe]ratiosasblackpointswhosesizesareinversely
proportionaltomeasurementuncertainties.Onlypointswithuncertaintieslessthan0.3dexareshown.Theredlinesshowtheabundanceratiosofthestarsandgasat
eachtimestep.Wedonotshowthemodelresultsfor[Ti/Fe]becausetheSNyieldsareinaccurate.Right:thegasflowandSFHforthebest-fitmodel.Fromtopto
bottom,thepanelsshowthegasinflowrate;thestellar,gas-phase,andtotalbaryonicmass;theSFR;theironandmagnesiumabundances;andthe[Mg/Fe]ratio,all
asafunctionoftime.Thesecondpanelalsogivesthefinalstellarmassinthemodel.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
d[Fe/H] 1168020000 Leo I M Gyr)-1 112050000
N / 2400 106 50
d 1.00 f (in 205 Final mass: 4.6 × 106M
) 20 gas
Fe] 0.5 M 15 stars
Mg/ 0.0 106 10 total
[ −0.5 M ( 5
1.0 yr)-1 80
G 6
Fe] 0.5 M 4
Si/ 0.0 06
[ 1 2
−0.5 R ( 0
F
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...505
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure3.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforLeoI.SeeFigure2foradetailedexplanation.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
8
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
H] 40 Sculptor yr)-1 6800
d[Fe/ 2300 M G 40
N / 10 106 20
d1.00 f (in 100 Final mass: 1.2 × 106M
) 8 gas
Fe] 0.5 M 6 stars
Mg/ 0.0 106 4 total
[ −0.5 M ( 2
1.0 Gyr)-1 23..050
Fe] 0.5 M 12..50
Si/ 0.0 06 1.0
[ 1
−0.5 FR ( 00..05
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...505
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure4.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforSculptor.SeeFigure2foradetailedexplanation.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
d[Fe/H] 234000 Leo II M Gyr)-1 468000
N / 10 106 20
d1.00 f (in 102 Final mass: 1.4 × 106M
) 10 gas
e] 0.5 M 8 stars
F
[Mg/ −00..50 M (106 246 total
1.0 Gyr)-1 22..005
Fe] 0.5 M 1.5
Si/ 0.0 06 1.0
[ 1
−0.5 R ( 00..05
F
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...505
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure5.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforLeoII.SeeFigure2foradetailedexplanation.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
the two-sided 68.3% confidence intervals. These uncertainties Table 3 lists the total star formation durations for the most
incorporate only observational uncertainty and not systematic likelymodels.Thedurationisnotafreeparameterbutaresult
modelerrors.Table2liststhesolutionsforeachdSphinorder ofthemodel.Thetablealsolistssometimescalesderivedfrom
ofdecreasingluminosity. HST CMDs (Dolphin et al. 2005; Orban et al. 2008). It is not
9
TheAstrophysicalJournal,727:79(25pp),2011February1 Kirbyetal.
H] 15 Sextans yr)-1 10800
Fe/ 10 G 60
d[ M 40
N / 5 106 20
d1.00 f (in 102 Final mass: 8.5 × 105M
) 10 gas
e] 0.5 M 8 stars
F
[Mg/ −00..50 M (106 246 total
1.0 yr)-1 2.00
G 1.5
Fe] 0.5 M 1.0
Si/ 0.0 06
[ 1 0.5
−0.5 R ( 0.0
F
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...505
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure6.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforSextans.SeeFigure2foradetailedexplanation.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
H] 30 Draco yr)-1 110200
d[Fe/ 20 M G 6800
N / 10 106 2400
d1.00 f (in 102 Final mass: 9.2 × 105M
) 10 gas
e] 0.5 M 8 stars
F
[Mg/ −00..50 M (106 246 total
1.0 yr)-1 2.00
G 1.5
Fe] 0.5 M 1.0
Si/ 0.0 06
[ 1 0.5
−0.5 R ( 0.0
F
1.0 S
Fe] 0.5 H] -1
[Ca/ 0.0 [X/ -2 [Fe/H]
−0.5 -3 [Mg/H]
1.0 1.0
Ti/Fe] 00..05 Mg/Fe] -000...505
[ −0.5 [ -1.0
-1.5
−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.0 0.5 1.0 1.5
[Fe/H] t (Gyr)
Figure7.Observedabundanceratiosandthebest-fitgasflowandSFHmodelforDraco.SeeFigure2foradetailedexplanation.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
possible to measure photometrically the total star formation morerecentlythan10Gyr.Forsmallorzerovaluesoff ,the
10G
durationforpredominantlyancientstellarpopulationsbecause CMDshowsthatthepopulationisancient,butthereisnotime
10Gyrisochronesareextremelysimilarto13Gyrisochrones. resolution.Wealsoshowthestellarmass-weightedmeanageτ
Therefore, we have quoted f , the fraction of stars formed (Orbanetal.2008).ForthethreedSphswithintermediate-aged
10G
10