Chemical and photometric evolution models for disk, irregular and low mass galaxies

2 MB·

by  Mercedes Mollá

Mon.Not.R.Astron.Soc.000,1–??(2011) Printed30June2014 (MNLATEXstylefilev2.2)
Chemical and photometric evolution models for disk, irregular and
low mass galaxies
Mercedes Molla´ 1,2 ⋆
4 1DepartamentodeInvestigacio´nBa´sica,CIEMAT,Avda.Complutense40,E-28040Madrid.Spain.
1 2IAG,UniversityofSa˜oPaulo,05508-900,Sa˜oPaulo-SP.Brasil
0
2

n
a AcceptedReceived;inoriginalform
J

4
ABSTRACT
] We summarize the updated set of multiphase chemical evolution models performed with
A
44 theoreticalradial mass initial distributionsand 10 possible valuesof efficienciesto form
G
molecularcloudsandstars.Wepresenttheresultsabouttheinfallratehistories,theformation
. ofthedisk,andtheevolutionoftheradialdistributionsofdiffuseandmoleculargassurface
h
density,stellarprofile,starformationratesurfacedensityandelementalabundancesofC,N,O
p
andFe,findingthattheradialgradientsfortheseelementsbeginverysteeper,andflattenwith
-
o increasingtimeordecreasingredshift,althoughtheouterdisksalwaysshowacertainflatten-
r ingforalltimes.Withtheresultingstarformationandenrichmenthistories,wecalculatethe
t
s spectralenergydistributions(SEDs)foreachradialregionbyusingtheonesforsinglestellar
a populationsresultingfromtheevolutivesynthesismodelpopstar.Withthese SEDswemay
[
computefinallythebroadbandmagnitudesandcolorsradialdistributionsintheJohnsonand

1 intheSLOAN/SDSSsystemswhicharethemainresultofthiswork.Wepresenttheevolution
v ofthesebrightnessandcolorprofileswiththeredshift.
1
6 Keywords: galaxies:abundances–galaxies–evolutiongalaxies–photometry
7
0
.
1
0 1 INTRODUCTION Itwasearlyevidentthatitwasimpossibletoreproducethese
4
observationsbyusingtheclassicalclosedboxmodel(Pagel,1989)
1 Chemicalevolutionmodelstostudytheevolutionofspiralgalaxies
whichrelatesthemetallicityZ ofaregionwithitsfractionofgas
: has been the subject of a high number of works for the last
v overthetotalmass,(stars,s,plusgas,g),µ =g/(g+s).Therefore
decades. FromtheworksbyLynden-Bell (1975);Tinsley(1980); g
i infall or outflows of gas in MWG and other nearby spirals were
X Clayton (1987, 1988); Sommer-Larsen&Yoshii (1989), many
soonconsiderednecessarytofitthedata.Infact,suchasitwases-
r other models have been developed to analyze the evolution of tablished theoretically by Goetz&Koeppen (1992) and Koeppen
a a disk galaxy. The first attempts were performed to interpret
(1994) aradial gradient of abundances may becreated only by 4
the G-dwarf metallicity distribution and the radial gradient of
possible ways: 1) A radial variation of the Initial Mass Function
abundances (Peimbert, 1979; Shaveretal., 1983; Fich&Silkey,
(IMF); 2) A variation of the stellar yields along the galactocen-
1991; Fitzsimmonsetat., 1992; V´ılchez&Esteban, 1996;
tric radius; 3) A star formation rate (SFR) changing with the ra-
Smartt&Rolleston, 1997; Afflerbach,Churchwell,&Werner,
dius; 4) A gasinfall ratevariable, f, withradius. Thefirstpossi-
1997;Estebanetal.,1999;Esteban,Peimbert,&Torres-Peimbert,
bilityisnotusuallyconsideredasprobable, whilethesecondone
1999b; Esteban,Peimbert,Torres-Peimbert,&Garc´ıa-Rojas,
isalready included inmodern models, sincethestellar yieldsare
1999c; Smarttetal., 2001) observed in our Milky Way Galaxy
in fact dependent on metallicity. Thus, from the seminal works
(MWG). A radial decrease of abundances was soon observed
from Lacey&Fall (1985); Gu¨sten&Mezger (1983) and Clayton
in most of external spiral galaxies, too (see Henry&Worthey,
(1987) most of the models in the literature (Diaz&Tosi, 1984;
1999, and references therein), although the shape of the radial
Matteucci&Francois, 1989; Ferrinietal., 1992, 1994; Carigi,
distribution changes from galaxy to galaxy, at least when it is
1994; Prantzos&Aubert, 1995; Chiappini,Matteucci,&Gratton,
measuredindexkpc−11.
1997;Boissier&Prantzos,1999),includingthemultiphasemodel
usedinthiswork, explaintheexistence of thisradial gradient by
thecombinedeffectsofaSFRandaninfallofgaswhichvarywith
⋆ E-mail:[email protected] thegalactocentricradiusintheGalaxy.
1 Recentresultsseemindicatethattheradialgradientmaybeuniversalfor
allgalaxies whenismeasuredindexreff−1 (Sa´nchezetal.,2013),reff
beingtheradiusenclosingthehalfofthetotalluminosityofadiskgalaxy, yearsagobyGarnett(1998)althoughthestatisticalwasnotlargeenoughto
orwithanyothernormalizationradius,somethingalreadysuggestedsome reachaccurateconclusions.
(cid:13)c 2011RAS
2 Molla´
Most of the chemical evolution models of the literature, and10possiblevaluesofefficienciestoformmolecularcloudsand
included some of the recently published, are, however, only stars.Nowwehaveupdatedthesemodelsbyincludingabulgere-
devoted to the MWG, totally or only for a region of it, halo gionandbyusingadifferentrelationmass–life-meantimeforstars
or bulge (Costa,Maciel,&Escudero, 2008; Tumlinson, 2010; nowfollowingthePadovastellartracks.Thesemodelsdonotcon-
Marcon-Uchida,Matteucci,&Costa, 2010; Caimmi, 2012; sider radial flows, nor stars migration since no dynamical model
Tsujimoto&Bekki, 2012; Micali,Matteucci,&Romano, 2013) isincluded.Thepossibleoutflowsbysupernovaexplosionsisnot
or to any other individual local galaxy as M 31, M 33 or other included, too. We check that with the continuous star formation
local dwarf galaxies (Carigi,Hernandez,&Gilmore, 2002; histories resulting of these models, the supernova explosions do
Va´zquez,Carigi,&Gonza´lez, 2003; Carigi,Col´ın,&Peimbert, notappearinsufficientnumberastoproducetheenergyinjection
2006; Magrinietal., 2007; Barker&Sarajedini, 2008; necessarytohaveoutflowsofmass.Withthesechemicalevolution
Magrinietal., 2010; Marcon-Uchida,Matteucci,&Costa, 2010; model results, we calculate the spectro-photometric evolution by
Lanfranchi&Matteucci, 2010; Herna´ndez-Mart´ınezetal., usingeachtime-stepof theevolutionary model asasinglestellar
2011; Kangetal., 2012; Romano&Starkenburg, 2013; populationsatwhichweassignaspectrumtakenfromthepopstar
Robles-Valdez,Carigi,&Peimbert, 2013a,b). These works evolutionary synthesis models (Molla´,Garc´ıa-Vargas,&Bressan,
perform the models inaTailor-MadeModelsway, done by hand 2009).Ourpurposeintogiveasacataloguetheevolutionofeach
for each galaxy or region. There are not models applicable to radialregionofadiskandthiswaytheradialdistributionsofele-
any galaxy, except our grid of models shown in Molla´&D´ıaz mentalabundances,starformationrate,gasandstarswillbeavail-
(2005,hereinafterMD05)andtheseonesfromBoissier&Prantzos ablealongwiththeradialprofilesofbroadbandmagnitudesforany
(1999,2000),whopresentedawidesetofmodelswithtwoparam- timeofthecalculatedevolution.
eters,thetotalmassorrotationvelocityandtheefficiencytoform The work is divided as follows: we summarize the updated
molecular clouds and stars in MD05 and a angular momentum chemical evolution models in Section 2. Results related with the
parameterinthelastauthorsgrid. surface densities and abundances are given in Section 3. We
Besides that, these classical numerical chemical evolution describe our method to calculate the SEDs of these theoretical
modelsonlycomputethemassesinthedifferentphasesofaregion galaxies and the corresponding broad band magnitudes and col-
(gas, stars, elements...) or the different proportions among them. orsinSection4.Thecorrespondingspectro-photometricresultsare
Theydonotusetogivethecorresponding photometricevolution, shown in Section 5 where we give a catalog of the evolution of
preventingthecomparisonofchemicalinformationwiththecorre- thesemagnitudesintherest-frameofthegalaxies.Someimportant
spondingstellarone.Thereexistafewconsistentmodelswhichcal- predictionsarisefromthesemodelswhicharegivenintheConclu-
culateboththingssimultaneouslyinaconsistentway,asthosefrom sions.
Va´zquez,Carigi,&Gonza´lez (2003); Boissier&Prantzos (2000)
or those from Fritze-vonAlvensleben,Weilbacher,&Bicker
(2003);Bickeretal.(2004);Kotullaetal.(2009,hereaftergalev).
2 THECHEMICALEVOLUTIONMODEL
The latter, galev evolutionary synthesis models, describe the
DESCRIPTION
evolutionofstellarpopulationsincludingasimultaneoustreatment
ofthechemicalevolutionof thegasandof thespectral evolution The model shown here are the ones from MD05 and therefore a
of the stellar content. These authors, however, treat each galaxy more detailed explanation about the computation is given in that
as a whole for only some typical galaxies along the Hubble work.Westartedwithamassof primordialgasinasphericalre-
sequence and does not perform the study of radial profiles of gionrepresentingaprotogalaxyorhalo.Theinitialmasswithinthe
mass, abundances and light simultaneously. The series of works protogalaxyisthedynamicalmasscalculatedbymeansoftherota-
byBoissier&Prantzos(1999,2000);Prantzos&Boissier(2000); tionvelocity,V(Radius),throughtheexpression(Lequeux,1983):
Boissieretal.(2001)seemsoneoffewthatgivemodelsallowing
M(Radius)=M =2.32105RadiusV2(Radius) (1)
an analysis of the chemical and photometric evolution of disk H,0
galaxies. with M inM ,RadiusinkpcandV inkms−1.WeusedtheUni-
⊙
Giventheadvancesintheinstrumentation,itisnowpossible versalRotationCurvefrom(URC)fromPersicetal.(1996)tocal-
tostudyhighredshiftgalaxiesasthelocaloneswithspatialresolu- culateaset of rotationvelocity curvesV(Radius) and fromthese
tionenoughgoodtoobtainradialdistributionsofabundancesand velocity distributions we obtained the mass radial distributions
ofcolorsormagnitudesandthustoperformcarefulstudiesofthe M(Radius)(seeMD fordetailsandFig.2showing thesedistribu-
possibleevolutionofthedifferentregionsofdiskgalaxies.Forin- tions).Itwasalsopossibletousethoseequationstoobtainthescale
stancetochecktheexistenceofradialgradientsatotherevolution- lengthofthediskR ,theopticalradius,definedastheonewhere
D
arytimesdifferentthanthepresent(Crescietal.,2010;Yuanetal., thesurfacebrightnessprofileis25magarcsec−2,which,ifdisksfol-
2011; Queyreletal., 2012; Jonesetal., 2013) and their evolution lowtheFreeman’s(Freeman,1970)law,isRopt =3.2R ,andthe
D
withtimeorredshiftisnowpossible.Itisalsopossibletocompare virial radius, which we take as the galaxy radius Rgal. The total
thesegradientswiththeradialdistributionsfromthestellarpopula- massofthegalaxy Mgalistakenasthemassenclosedinthisra-
tionstostudypossiblemigrationeffects.Itisthereforeimportantto diusRgal.TheexpressionfortheURCwasgivenbymeansofthe
haveagridofconsistentchemo-spectro-photometricmodelswhich parameter λ = L/L , the ratio between the galaxy luminosity, L,
∗
allowstheanalysisofbothtypesofdatasimultaneously. andtheoneoftheMWG,L ,inbandI.Thisparameterdefinesthe
∗
The main objective of this work is to give the spectro- maximum rotation velocity, Vmax and the radii described above.
photometric evolution of the theoretical galaxies presented in Thus, weobtained the values of themaximum rotation velocities
MD05, for which we have updated the chemical evolution mod- andthecorrespondingparametersandmassradialdistributionsfor
els.Inthatworkwepresentedagridofchemicalevolutionmodels asetofλvaluessuchasitmaybeseeninTable1fromMD05.
for440theoreticalgalaxies,with44differenttotalmass,asdefined To the radial distributions of disks calculated by means of
by its maximum rotation velocity, and radial mass distributions, Eq.1describedabove,wehaveaddedaregionlocatedatthecenter
(cid:13)c 2011RAS,MNRAS000,1–??
Chemical andphotometricmodels forlatetypegalaxies 3
(R=0)torepresentabulge.Thetotalmassofthebulgeisassumed
Table1.TheoreticalgalaxymodelsselectedtorepresentasimulatedHub-
asa10%ofthetotalmassofthedisk.Theradiusofthisbulgeis blesequence
takenasR /5.Bothquantitiesareestimatedfromthecorrelations
D
foundbyBalcells,Graham,&Peletier(2007)amongthediskand dis Vmax Mgal Ropt τc nt ǫν ǫδ
thebulgesdata. kms−1 1011M⊙ kpc Gyr
3 48 0.3 2.3 31.6 8 0.037 2.610−4
10 78 1.3 4.1 15.5 7 0.075 1.510−3
21 122 4.3 7.1 8.1 6 0.15 1.010−2
2.1 Theinfallrate:itsdependenceonthedynamicalmass
24 163 9.8 10.1 5.4 5 0.30 5.010−2
andonthegalactocentricradius 28 200 17.9 13.0 4.0 4 0.45 1.410−1
We assume that the gas falls from the halo to the equa- 35 250 33.5 16.9 2.9 3 0.65 3.410−1
39 290 52.7 20.6 2.3 1 0.95 8.810−1
torial plane forming out the disk in a scenario ELS
(Eggen,Lynden-Bell,&Sandage, 1962). The time-scale of
this process, or collapse-time scale τ , characteristic of every
gal,c
galaxy,changeswithitstotaldynamicalmassMgal,followingthe giveninTable1fromMDtoo,withthecharacteristicτ 3obtained
c
expressionfromGallagheretal(1984): foreachgalaxytotalmassMgal.
By taking into account that the collapse time scale depends
τ ∝Mgal−1/2T, (2)
gal,c on thedynamical mass, and that spiral disks show a clear profile
where Mgal isthe total massof the galaxy, and T is itsage. We ofdensitywithhighervaluesinsidethanintheoutsideregions,we
assumeallgalaxiesbegintoformatthesametimeandevolvesfor mayassumethattheinfallrate,andthereforethecollapsetimescale
atimeof T = 13.2Gyr. Weuse thevalue of 13.8 Gyr, given by τcoll, has a radial dependence, too. Since the mass density seems
the Planck experiment (PlanckCollaborationetal., 2013) for the tobeanexponential inmost ofcases, wethenassumedasimilar
ageof theUniverseand thereforegalaxiesstarttoformat atime expression:
tstart =0.6Gyr. (Radius−Rc)
NormalizingtoMWG,weobtain: τcoll(Radius)=τcexp λ (4)
D
M where λ is the scale-length of the collapse time-scale, taken as
τ =τ MWG, (3) D
gal,c MWG,cs Mgal around the half of the scale-length of surface density brightness
distribution,R ,thatisλ =0.15Ropt∼0.5R .
D D D
where MMWG ∼ 1.81012M⊙ is the total mass of MWG and Obviouslythecollapsetimescaleforthebulgeregionisob-
τMWG,c = 4 Gyr(seedetailsinthenextparagraph)istheassumed tained naturally from the above equation with R = 0. We show
characteristiccollapse-timescaleforourGalaxy. inthe upper panel of Fig. 1 thecollapse timescale τ , in natu-
coll
Theaboveexpressionimpliesthatgalaxiesmoremassivethan rallogarithmicscale,asafunctionofthegalactocentricradius,for
MWG form in a shorter time-scale, that is more rapidly, than seven radial distributions of total mass, as defined by their maxi-
the least massive galaxies which will need more time to form mumrotationvelocity, Vmax, andplottedwithdifferent color, as
their disks. This assumption is in agreement with the observa- labeled.Theseseventheoreticalgalaxiesareusedasexamplesand
tionsfromJimenezetal.(2004);Heavensetal.(2004);Pe´rezetal. theircharacteristicsaresummarizedinTable1,wherewehavethe
(2013)whichfindthatthemostmassivegalaxieshavetheirstellar numberofthedistribution,dis,correspondingtocolumn2fromTa-
populationsinplaceatveryearlytimeswhilethelessmassiveones ble1inMDincolumn1,themaximumrotationvelocity,Vmax,in
form most of their stars at z < 1. This isalso in agreement with kms−1,incolumn2,thetotalmass,Mgal,in1011M units,incol-
⊙
self-consistent cosmological simulations which show that a large umn 3, the theoretical optical radius Ropt, following Persicetal.
proportionofmassiveobjectsareformedatearlytimes(highred- (1996) equations, in kpc, in column 4, the collapse time scale in
shift)whiletheformationoflessmassiveonesismoreextendedin thecharacteristicradius,τ ,inGyrincolumn5,thevaluentwhich
c
time,thussimulatingamodernversionofthemonolithiccollapse definestheefficiencies(seeEq.22and23insection2.2)incolumn
scenarioELS. 6,andthevaluesfortheseefficienciesincolumns7and8.
Thecalculated collapse-timescaleτgal,c isassumed that cor- The red line corresponds to a MWG-like radial distribution.
respondstoaradialregionlocatedinacharacteristicradius,which The long-dashed black line shows the time corresponding to 2
isRc=Ropt/2∼6.5kpcfortheMWGmodelwhichusesthedis- timestheageoftheUniverse.Suchaswemaysee,themostmas-
tributionwithλ = 1.00andnumber28,withamaximumrotation sive galaxies would have the most extended disks, since the col-
velocityVmax = 200kms−1.ThevalueτMWG,c = 4 Gyrwasde- lapse timescale issmaller than the age of the universe for longer
terminedbyadetailedstudyofmodelsforMWG.Weperformed radii,thusallowingtheformationofthediskuntilradiiaslargeras
alargenumber ofchemicalevolutionmodelschangingtheinputs 20kpc,whiletheleastmassiveoneswouldonlyhavetimetoform
free-parametersandcomparingtheresultswithmanyobservational thecentral region, smaller than 1-2kpc, asobserved. The dashed
data(Ferrinietal.,1992,1994)toestimatethebestvalue(seesec- (gray)linesshowthetimecorrespondingto2timestheageofthe
tion2.1.2).Similarcharacteristicradii2forourgridofmodelswere Universe,T =13.8 Gyr.Thedottedblacklinedefinesthe
UNIVERSE
collapsetimescaleforwhichthemaximumradiusforthediskof
theMWGmodelwouldbe13kpc,theopticalradiusanditcorre-
2 Alltheseradiiandvaluesarerelatedwiththestellarlightandnowiththe
spondstoacollapsetimescaleof5timestheageoftheUniverse.
mass,butweclearthatwedonotusetheminourmodelsexcepttodefine
Other authors have also included a radial depen-
thecharacteristicradiusRcforeachtheoreticalmassradialdistribution.The
dence for the infall rate in their models (Lacey&Fall,
free parameters are selected forthe region defined bythis Rcbuttaking
intoaccountthatwenormalizethevaluesafteracalibrationwiththeSolar
Neighborhoodmodel,achangeofthisradiuswouldnotmodifyourmodel
results. 3 Fromnowwewillusetheexpressionτcforτgal,cinsakeofsimplicity.
(cid:13)c 2011RAS,MNRAS000,1–??
4 Molla´
Carigi&Peimbert (2008); Marcon-Uchida,Matteucci,&Costa
(2010), aslabeled. Since theyuse straight linesthecollapse time
scale for our model results shorter for the inner disk regions
(except for the bulge region R < 3− 4kpc) and longer for the
outerones,thantheonesusedbytheotherworks.Thiswillhave
consequences in the radial distributions of stars and elemental
abundancesaswewillsee.
2.2 Thestarformationlawintwo-steps:theformationof
moleculargasphase
Thestarformationisassumeddifferentinthehalothaninthedisk.
In the halo the star formation follows a Kennicutt-Schmidt law.
In the disk, however, weassume a more complicated star forma-
tionlaw,bycreating molecular gasfromthediffusegasinafirst
step,againbyaKennicutt-Schmidtlaw.Andthenstarsfromfrom
the cloud-cloud collisions. There is a second way to create stars
fromtheinteractionofmassivestarswiththesurroundingmolecu-
larclouds.
Thereforetheequationsystemofourmodelis:
dg
H = −(κ +κ )gn − fg +W (5)
dt 1 2 H H H
ds
1,H = κ gn −D (6)
dt 1 H 1,H
ds
2,H = κ gn −D (7)
dt 2 H 2,H
dg
D = −ηgn +α′c s +δ′c2 + fg +W (8)
dt D D 2,D D H D
dc
D = ηgn −(α +α +α′)c s −(δ +δ +δ′)c2 (9)
dt D 1 2 D 2D 1 2 D
ds
1,D = δ c2 +α c s −D (10)
Figure 1. a) Dependence of the collapse time scale, τcoll, in natu- dt 1 D 1 D 2D 1,D
ral logarithmic scale, (in Gyr) on the galactocentric radius, Radius, in ds
kpc. Each line represents a given maximum rotation velocity, Vmax d2t,D = δ2c2D+α2cDs2D−D2,D (11)
or radial mass distribution as labeled. The dashed (gray) and dotted
dr
(black) lines show the time corresponding to 2 and 5 times, respec- H = D +D −W (12)
dt 1,H 2,H H
tively, the age of the Universe, TUNIVERSE = 13.8 Gyr. b) Com-
dr
parison of the radial dependence of the collapse time scale, τcoll, D = D +D −W (13)
in natural logarithmic scale, for our MWG model, corresponding to dt 1,D 2,D D
Vmax = 200kms−1, shown by the solid red line and labeled MD, Xi,H = (Wi,H−Xi,HWH) (14)
withtheradialfunctionsusedbyChangetal.(1999);Boissier&Prantzos dt g
H
(1999); Chiappini,Matteucci,&Romano (2001); Rendaetal. (2005); X [W −X W + fg (X −X )]
Carigi&Peimbert(2008);Marcon-Uchida,Matteucci,&Costa(2010),la- i,D = i,D i,D D H i,H i,D (15)
dt g +c
beledCHA99,BP99,CHIA01,REN05,CAR08andMAR10,respectively. D D
These equations predict the time evolution of the different
phases of the model: diffuse gas, g, molecular gas, c, low mass
1985; Matteucci&Francois, 1989; Ferrinietal., 1992;
stars, s , and intermediate massand massive stars, s , and stellar
1 2
Portinari,Chiosi,&Bressan, 1998; Boissier&Prantzos, 2000) remnants, r,(whereletters Dand H correspond todiskandhalo,
with different expressions. In fact this dependence, which pro-
respectively).Starsaredividedin2ranges, s beingthelowmass
1
duces an in-out formation of the disk, is essential to obtain
stars, and s the intermediate and massive ones, considering the
2
the observed density profiles and the radial gradient of abun- limitbetweenbothrangesstellaramassm=4M .X arethemass
⊙ i
dances, such as it has been stated before (Matteucci&Francois, fractionsofthe15elementsconsideredbythemodel:1H,D,3He,
1989; Ferrinietal., 1994; Boissier&Prantzos, 2000). In the 4He,12C,16O,14N,13C,20Ne,24Mg,28Si,32S,40Ca,56Fe,andthe
bottom panel of the same Fig. 1 we show the collapse time richneutronisotopescreatedfrom12C,16O,14Nandfrom13C.
scale τcoll, in natural logarithmic scale, assumed in different Thereforewehavedifferentprocessesdefinedinthegalaxy:
chemical evolution models of MWG, as a function of the galac-
tocentric radius. The red solid line corresponds to our MWG (i) Star formation by spontaneous fragmentation of gas in the
model (λ = 1.00 and Number 28 of the mass distributions halo:∝κ gn,whereweusen=1.5
1,2 D
of Table 1) from MD. The other functions, straight lines, are (ii) Cloudsformationbydiffusegas:∝ηgn withn=1.5,too
D
those used by Changetal. (1999); Boissier&Prantzos (1999); (iii) Starformationduetocloudcollision:∝δ c2
1,2 D
Chiappini,Matteucci,&Romano (2001); Rendaetal. (2005); (iv) Diffusegasrestitutionduetocloudcollision:∝δ′c2
D
(cid:13)c 2011RAS,MNRAS000,1–??
Chemical andphotometricmodels forlatetypegalaxies 5
(v) Inducedstarformationduetotheinteractionbetweenclouds whichdefinestheabsolutelevelofabundancesforagivenmodel,
andmassivestars:∝α c s depends on both ingredients. Inthiswork weused theIMF from
1,2 D 2,D
(vi) Diffuse gas restitution due to the induced star formation: Ferrini,Palla&Penco (1990) withlimitsm = 0.15 and m =
low up
α′c s 100M . The stellar yields are from Woosley&Weaver (1995)
D 2,D ⊙
(vii) Galaxyformationbygasaccretionfromthehaloorproto- for massive stars (m ≥ 8M ) and from Gavila´n,Buell,&Molla´
⊙
galaxy: fg (2005); Gavila´n,Molla´,&Buell (2006) for low and intermediate
H
mass stars (0.8M < m ≤ 8M ). Stars in the range 0.15M <
whereα,δ,ηandκaretheproportionalityfactorsofthestars ⊙ ⊙ ⊙
m < 0.8M have no time to die, so they still live today and
andcloudformationandarefreeinputparameters.4. ⊙
do not eject any element to the interstellar medium. The mean
Thus,thestarformationlawinhaloanddiskis:
stellar lifetimes are taken from the isochrones from the Padova
Ψ (t) = (κ +κ )gn (16) group (Bressan,Chiosi,&Fagotto, 1994; Fagottoetal., 1994a,b;
H 1 2 H
Ψ (t) = (η +η )c2 +(α +α )c s (17) Girardietal., 1996), instead using those from the Geneva group
D 1 2 D 1 2 D 2D Schalleretal.(1992).Thischangeisdoneforconsistencysincewe
Althoughthenumberofparametersseemstobelarge,actually usethePadovaisochronesonthepopstarcodethatwewillusefor
notallofthemarefree.Forexample,theinfallrate, f,istheinverse thespectro-photometricmodels.ThesupernovaIayieldsaretaken
of the collapse time τcoll, as we described in the above section; fromIwamotoetal.(1999).Thecombinationofthesestellaryields
Proportionalityfactorsκ,η,δandαhavearadial dependence, as with this IMF produces the adequate level of CNO abundances,
weshowinthestudyofMWGFerrinietal.(1994),whichmaybe abletoreproduce mostofobservational dataintheMWGgalaxy
usedinalldisksgalaxiesthroughthevolumeofeachradialregion (Gavila´n,Buell,&Molla´, 2005; Gavila´n,Molla´,&Buell, 2006),
andsomeproportionalityfactorscalledefficiencies.Theseefficien- inparticulartherelativeabundancesofC/O,N/O,andC/Fe,O/Fe,
ciesorproportionalityfactorsoftheseequationshaveaprobability N/Fe. The study of other combinations of IMF and stellar yields
meaningandthereforetheirvaluesareintherange[0,1].Theeffi- willbeanalyzeinMolla´,etal.(2013).ThesupernovatypeIarates
cienciesarethen:theprobabilityofstarformationinthehalo,ǫκ, are calculated by using prescriptions from Ruiz-Lapuenteetal.
theprobabilityofcloudformation,ǫη;cloudcollision,ǫδ;andthe (2000).
interactionbetweenmassivestarsǫ .Thislastonehasaconstant
α
valuesinceitcorrespondstoalocalprocess.Theefficiencytoform
starsinthehaloisalsoassumedconstantforallofthem.Thus,the
3 RESULTS:EVOLUTIONOFDISKSWITHREDSHIFT
numberoffreeparametersisreducedtoǫ andǫ .
η δ
Theefficiencytoformstarsinthehalo,ǫ ,isobtainedthrough The chemical evolution models are given in Tables 2 and 3. We
κ
the selection of the best value κ to reproduce the SFR and abun- showasanexamplesomelinescorrespondingtothemodelnt=4
dances of the Galactic halo (see Ferrinietal., 1994,for details). anddis = 28, thewholeset of resultswillbegiven inelectronic
Weassumed thatitisapproximately constant forallhaloswitha format asacatalogue. InTable2wegive thetypeof efficiencies
valueǫ ∼1.5−610−3.Thevalueforǫ isalsoobtainedfromthe ntandthedistributionnumberdisincolumns1and2,thetimein
κ α
best value α for MWG and assumed as constant for all galaxies Gyrincolumn3,thecorrespondingredshiftincolumn4,theradius
since these interactions massive stars-clouds are local processes. ofeachdiskregioninkpcincolumn5,andthestarformationrate
The other efficiencies ǫη and ǫδ may take any value in the range in the halo and in the disk, in M⊙yr−1 , in columns 6 and 7. In
[0-1].Fromourpreviousmodelscalculatedforexternalgalaxiesof nextcolumnswehavethetotalmassinthehaloandinthedisk.In
differenttypes(Molla´etal.,1996),wefoundthatbothefficiencies columns8to16weshowthemassineachphaseofthehalo,diffuse
mustchangesimultaneouslyinordertoreproducetheobservations, gas,low-massstars,massivestars,andmassinremnants(columns
withhighervaluesfortheearliermorphologicaltypesandsmaller 8to11)andofthedisk,diffuseandmoleculargas,low-massstars,
forthethelaterones.InMD05thereisacleardescriptionaboutthe massivestars,andmassinremnants(columns12to16).
selectionofvaluesandtherelationǫ −ǫ .Asasummary,wehave We will give the results obtained for our grid of models by
η δ
calculatedtheseefficiencieswiththeexpressions: showingthecorrespondingonestothegalaxiesfromTable1asa
function of the redshift or of the galactocentric radius. When the
nt
ǫη = exp20 (18) radialdistributionsareplotted,wedothatforseveraltimesorval-
ues of redshifts. We assume that each time in the evolution of a
nt
ǫδ = exp 8 (19) galaxycorrespondstoaredshift.Tocalculatethisredshift,weuse
therelationredshift-evolutionarytimegivenbyMacDonald(2006)
selecting 10 values nt between 1 and 10 (we suggest to select a
withthecosmological parametersfromthesamePLANKexperi-
value nt similartothe Hubble typeindex to obtain model results
mentPlanckCollaborationetal.(2013)(Ω = 0.685,H = 67.3),
fittingtheobservations). Theefficienciesvaluescomputedforthe λ 0
andthiswaythetimeassumedforthebeginningofthegalaxyfor-
gridfromMD05areshowninTable2fromthatwork.
mation,t =0.6Gyr,correspondstoaredshiftz=7.
start
2.3 Stellaryields,InitialMassfunctionandSupernovaIa
3.1 Theformationofthedisk
rates
Theprocessofinfallofgasfromtheprotogalaxytotheequatorial
The selection of the stellar yields and the IMF needs to be done
planedependsontimesincethemassremainingintheprotogalaxy
simultaneously since the integrated stellar yield for any element,
orhaloisdecreasingwithtime.InFig.2werepresenttheresulting
totalinfallofmassforour44radialdistributionsofmassasafunc-
4 Sincestarsaredividedintwogroups:thosewiths1,ands2,theparam- tionoftheredshiftz.Sincethisprocessisdefinedbythecollapse
eters are divided in the twogroups too, thus κ = κ1 +κ2, δ = δ1 +δ2, time-scale,thetotalinfallrateforthewholegalaxyonlydepends
α=α1+α2. on the total mass, and therefore there would be only 44 possible
(cid:13)c 2011RAS,MNRAS000,1–??
6 Molla´
Table2.Evolutionofdifferentphasesalongthetime/redshiftforthegridofmodels.WeshowasexampletheresultsforthepresenttimeofaMWG-like
model(nt=4,dis=28).Thecompletetablewillbeavailableinelectronicformat.
nt dis t z Radius ΨH ΨD MH MD
Gyr kpc M⊙yr−1 M⊙yr−1 109M⊙ 109M⊙
4 28 1.3201e+01 0.00 0 1.5431e-05 4.9526e-02 1.0337e+00 8.4818e+00
4 28 1.3201e+01 0.00 2 2.5875e-10 2.1748e-02 9.7257e-03 4.2363e+00
4 28 1.3201e+01 0.00 4 2.9111e-08 5.7276e-02 6.4913e-02 9.7481e+00
4 28 1.3201e+01 0.00 6 2.1827e-04 1.8737e-01 3.7799e-01 1.2042e+01
4 28 1.3201e+01 0.00 8 9.6978e-03 4.7546e-01 2.8732e+00 9.6868e+00
4 28 1.3201e+01 0.00 10 3.4321e-02 3.8606e-01 6.6828e+00 5.0472e+00
4 28 1.3201e+01 0.00 12 4.9261e-02 1.8346e-01 8.7949e+00 2.0751e+00
4 28 1.3201e+01 0.00 14 5.2833e-02 7.0495e-02 9.5228e+00 7.8723e-01
4 28 1.3201e+01 0.00 16 5.2396e-02 2.2987e-02 9.7360e+00 2.9398e-01
4 28 1.3201e+01 0.00 18 5.1446e-02 5.7661e-03 9.8290e+00 1.1004e-01
4 28 1.3201e+01 0.00 20 5.0964e-02 9.0888e-04 9.9186e+00 4.1360e-02
4 28 1.3201e+01 0.00 22 5.1061e-02 7.9634e-05 1.0014e+01 1.5583e-02
4 28 1.3201e+01 0.00 24 5.1747e-02 4.7257e-06 1.0104e+01 5.8713e-03
Table2.Cont.Evolutionofthedifferentphasesalongthetime/redshiftforthegridofmodels
gH s1,H s2,H remH gD cD s1,D s2,D remD
109M⊙ 109M⊙ 109M⊙ 109M⊙ 109M⊙ 109M⊙ 109M⊙ 109M⊙ 109M⊙
2.9016e-03 8.6815e-01 1.2048e-05 1.6259e-01 2.7416e-02 1.4705e+00 7.1689e+00 3.3978e-04 1.2371e+00
1.3879e-05 8.1686e-03 1.7090e-07 1.5430e-03 1.4016e-02 1.4705e+00 3.5667e+00 1.5015e-04 6.2852e-01
4.0674e-04 5.4436e-02 4.7128e-07 1.0070e-02 4.1804e-02 1.4705e+00 8.2575e+00 3.9145e-04 1.3868e+00
1.7779e-01 1.7061e-01 2.0974e-06 2.9597e-02 1.3955e-01 1.4705e+00 1.0242e+01 1.2864e-03 1.5234e+00
2.4434e+00 3.7340e-01 6.7054e-05 5.6303e-02 3.5938e-01 1.4705e+00 8.0530e+00 3.2457e-03 1.0218e+00
6.0752e+00 5.3545e-01 2.3428e-04 7.1939e-02 3.9304e-01 1.4705e+00 3.9522e+00 2.6251e-03 4.4801e-01
8.1503e+00 5.7161e-01 3.3513e-04 7.2706e-02 2.8478e-01 1.4705e+00 1.4482e+00 1.2453e-03 1.5079e-01
8.9024e+00 5.5139e-01 3.5900e-04 6.8586e-02 1.7444e-01 1.4705e+00 4.4266e-01 4.7796e-04 4.2150e-02
9.1454e+00 5.2546e-01 3.5588e-04 6.4789e-02 9.5622e-02 1.4705e+00 1.1078e-01 1.5556e-04 9.4763e-03
9.2582e+00 5.0801e-01 3.4936e-04 6.2419e-02 4.6570e-02 1.4705e+00 2.0439e-02 3.8890e-05 1.5411e-03
9.3567e+00 5.0019e-01 3.4607e-04 6.1374e-02 2.1461e-02 1.4705e+00 2.3765e-03 6.0946e-06 1.5895e-04
9.4528e+00 4.9999e-01 3.4672e-04 6.1317e-02 1.0009e-02 1.4705e+00 1.7210e-04 5.3111e-07 1.0686e-05
9.5352e+00 5.0643e-01 3.5137e-04 6.2099e-02 4.4897e-03 1.4705e+00 9.3097e-06 3.1422e-08 5.5691e-07
results, one for each maximum rotation velocity, or mass of the Vmax > 120kms−1 –the green line– (that is all lines except the
theoreticalgalaxy.However,sincewehaveassumedaninfallrate orangeandmagentaones)coincideinasamelocusforthepresent
variablewiththeradius,eachradialregionofagalaxyhasadiffer- time,andreproducewelltheobservedvaluegivenbySancisietal.
entinfallrate.Weshowasashadedregionthelocuswhereourre- (2008)andrepresentedbytheredfullhexagon.
sultsforallradialregionsofthewholesetofmodelsfall.Overthis
As a consequence of this infall of gas scenario, the disk is
regionweshowassolidlinestheresultsforthewholeinfallofthe
formed.Theproportionofmassinthediskcomparedwiththeto-
same7theoreticalgalaxiesasinthepreviousFig.1,andwiththe
taldynamicalmassofthegalaxyis,asexpected,dependentonthis
samecolorcoding,aslabeled.Thedashedpurpleandyellowlines
totalmass.InFig.3weshowthefractionMgal/M ,whereM is
correspond to the prescriptions given by Dekel,Sari,&Ceverino D D
themassinthediskresultingfromtheappliedcollapsetimescale
(2009)andFaucher-Gigue`re,Keresˇ,&Ma(2011)fortheinfallof
prescriptions,asafunctionofthefinalmassinthedisk.Thesere-
gasasobtainedbytheircosmologicalsimulationstoformmassive
sults,shownasredpoints,arecomparedwiththelineobtainedby
galaxies. In both cases these expressions depend on the dynami-
Mateo(1998)forgalaxiesintheLocalGroup,solidcyanline,and
calhalomass,sowehaveshownbothlinesfor Mdyn = 1012M
⊙
withtheratiobyShankaretal.(2006)calculatedthroughthehalo
which will be compared to our most massive model which is on
andthestellarmassdistributionsingalaxies,solidblackline.We
thetop(blackline)andwhichhaswithasimilartotalmass.There
alsoplottheresultsobtainedbyLeauthaudetal.(2010)fromcos-
are some differences with the results from cosmological simula-
mologicalsimulationsforthreedifferentrangesinredshift,suchas
tions for the lowest redshifts, for which our models have lower
labeled in the figure. Our results have a similar slope to the one
infall rates than these cosmological simulations. However we re-
fromMateo(1998),buttheabsolutevaluegivenbythisauthoris
mind that these simulations prescriptions are valid for spheroidal
slightlylower,whichisprobablyduetoadifferentvalue M /Lto
galaxies. We show as dotted purple lines 2 other lines follow- ∗
transformtheobservations(luminosities)instellarmasses.Ourre-
ing Dekel,Sari,&Ceverino (2009) prescriptions but for masses
sultsare close to the ones predicted by Shankaretal. (2006) and
21012M and31011M ,aboveandbelowthestandarddashedline,
⊙ ⊙ Leauthaudetal. (2010) obtained with different techniques. These
checkingthatthislastoneisverysimilartoourcyanline.There-
authorsfindinbothcasesanincreaseforhighdiskmasseswhich,
fore to decrease the infall rate for the most recent times is prob-
obviously,isnotapparentinourmodels,sincewehaveassumeda
ably a good solution to obtain disks. In fact all our models for
continuousdependenceofthecollapsetimescalewiththedynami-
(cid:13)c 2011RAS,MNRAS000,1–??
Chemical andphotometricmodels forlatetypegalaxies 7
dFiiaglurreegi2o.nTshoefeovuorlu4t4iognaolafxtyhemiansfsalvlaglausesrarteeparleosnegnttehdebryedtshheifsthzadfoerdazlolnrae-. mFiogduerles3re.sTulhtseararetiotheMMsgDoallidasreadfudnoctst.ioTnheofcythaenmanadssbilnactkhelindeisskarMeDth.eOruer-
Thesolidlinesshowtheevolution ofthewholeinfallrateofthesame7 sultsobtainedbyMateo(1998)andShankaretal.(2006)fromobservations
theoreticalgalaxiesshowninFig.1withthesamecolorcoding.Thehigh- oftheLocalGroupofgalaxiesandfromhalodata,respectively.Magenta,
est the total mass, the highest the infall rate. Dashed purple and yellow blueandgreendashedlinesareresultsfromLeauthaudetal.(2010)fordif-
linesrepresenttheprescriptions fromDekel,Sari,&Ceverino(2009)and ferentranges ofredshift aslabeled. TheMWGpointisrepresented bya
Faucher-Gigue`re,Keresˇ,&Ma(2011),respectively.Thedottedpurplelines bluefulltriangleandthedottedblacklinemarksthelimitbetweendiskand
aretheDekel,Sari,&Ceverino(2009)prescriptionsformasses21012M⊙ spheroidalgalaxiesfromGonza´lezDelgadoetal.(2013).
and31011M⊙,whiletheredhexagoninthepresenttimeistheestimated
valuegivenbySancisietal.(2008).
the local Universe data obtained by Leroyetal. (2008). The red
starmarkstheaveragegivenbySancisietal.(2008).
calmass.Shankaretal.(2006)analyzedtheluminosityfunctionof
halosanddeterminetherelationwiththemassformedthere,while
Leauthaudetal.(2010)computecosmologicalsimulationsandob- 3.3 Evolutionwithredshiftoftheradialdistributionsindisks
tain the relation between the dynamical mass and the final mass
Innextfiguresweshowtheresultscorrespondingtotheseventhe-
intheirdisks.Aswesaybefore,disksobtainedinsimulationsare
oretical galaxies used as examples and whose characteristics are
smallerthanobservedwhatincreasestheratioMgal/M .Moreover
D
giveninTable1.Wehaveselected7galaxieswhichsimulategalax-
thechange ofslopeinthesecurvesdefinesthelimitinwhichthe
ies along the Hubble Sequence. They have have different masses
ellipticalgalaxiesbegintoappear,shownbythedottedblackline
and sizes and wehave also selected different efficiencies toform
asgivenbyGonza´lezDelgadoetal.(2013).Thereforeitispossible
starsinordertocomparewithrealgalaxies.Innextfigureswewill
thattheseauthorsincludesomespheroidsandgalaxiesSOintheir
show our results for 7 values of evolutionary times or redshifts:
calculationswhicharenotcomputedinourmodels.Itisnecessary
z = 5,4,3,2,1,0.4and0,withcolorspurple,blue,cyan,green,
say, however, that the MWG value in this plot is slightly above
magenta, orange and red,respectively. Theresultingpresent time
thislimit,andjustwheretheShankaretal.(2006)linechangethe
radial distributions in disks for diffuse and molecular gas, stellar
slope.TakingintoaccountthatmoremassivethanMWGthereex-
mass, and star formation rate for the galaxies from Table 1 are
ist,maybethislimitsisnottotallycorrect,andthat,insteadasharp
showninFig.5.Inthisfigureweshowagalaxyineachcolumnand
cut,thereisamixofgalaxiesinthiszoneoftheplot.
adifferentquantityineachrow.Thustop,top-middleandbottom-
middlepanelsshowthediffusegas,themoleculargasandthestel-
larsurfacedensities,respectively,allinM pc−2 unitsandinlog-
3.2 TherelationoftheSFRwiththemoleculargas ⊙
arithmicscale,whilethebottom panel corresponds tothesurface
Sincetheformationofmoleculargasisacharacteristicswhichdif- densityofthestarformationrateinM Gyr−1pc−2.
⊙
ferentiatesourmodelfromotherchemicalevolutionmodelsinthe TheradialdistributionsofdiffusegasdensityΣ showamax-
HI
literature,wewouldliketocheckifourresultingstarformationis imuminthedisk.Theradiusofthemaximumisnearthecenterof
inagreementwithobservations.Wecomparetheefficiencytoform thegalaxyforz=5andmoveoutwardswiththeevolution,reach-
starsfromthegasinphaseH ,measuredasSFR/MH ,withdata ingaradius∼2/3Roptkpcforthepresent,thatis1.3timesRc,and
2 2
inFig.4.Intheupperpanelweshowtheresultsforagalaxylike ∼ 2theeffectiveradiusinmassorradiusenclosedthehalfofthe
MWG,whereeachcoloredlinerepresentsadifferentradialregion: stellarmassofthedisk(seenextsection).ThedensityΣ inthis
HI
Solidred,yellow,magenta,blue,greenandcyanlines,correspond maximumreachesvalues∼50M pc−2 inearlytimesorhighred-
⊙
toradialregionslocatedat2,4,6,8,10and12kpcoftheGalactic shift.Forthepresent,themaximumdensityis∼ 10M pc−2,very
⊙
centerintheMWGmodel.Inthebottompanelsolidblack,cyan, similar in most of theoretical galaxies. These radial distributions
red,blue,green,orange,andmagentalinescorrespondtothegalax- reproduceverywelltheobservationsofHI.Theradialdistribution
iesofdifferentdynamicalmassesandefficienciesgiveninTable1 forregionsbeyondthispointshowsanexponentialdecreasingwith
andtakenasexamples.Observationsatintermediate-highredshift slopesflatternowthatinthehigh-redshiftdistributions.
byDaddietal.(2010);Genzeletal.(2010),areshownascyandots TheradialdistributionsofmoleculargasdensityΣ showba-
H2
and blue triangles, respectively, while the green squares refers to sicallythesamebehavior,withamaximuminthediskstoo.Ineach
(cid:13)c 2011RAS,MNRAS000,1–??
8 Molla´
Figure5.Evolutionwithredshiftofthesurfacedensityofdiffusegas,ΣHI,moleculargas,ΣH2,andstellarmass,Σ∗,inM⊙pc−2,intop,middle-top,and
middle-bottom rows,andthestarformation rate, Ψ,inM⊙Gyr−1pc−2,inthe bottom one.Allinlogarithmic scale. Eachcolumnshowstheresults fora
theoreticalgalaxyoftheTable1,fromthegalaxy,withVmax=78kms−1,tothemostmassiveonethebottomforVmax=290kms−1.Eachlinecorresponds
toadifferentredshift:purple,blue,cyan,green,magenta,orangeandredforz=5,4,3,2,1,0.4and0,respectively
galaxy,however,thismaximumislocatedslightlyclosertothecen- in a large number of spiral disks for Σ and SFR distri-
H2
terthattheonefromtheHIdistribution.Theyshowanexponential butions (Martin&Kennicutt, 2001; Nishiyama&Nakai, 2001;
functiontoo,afterthismaximum.Theseradialdistributionsseems Reganetal.,2001),wethink,however,thattheyarestrongerthan
moreanexponential shapewithaflatteningatthecenter that the observed.
onesfromHIwhichshowaclearermaximum,mainlyfortheleast
From all panels we may say that spiral disks would have a
massivegalaxies.
morecompactappearanceathighredshiftwithhighervaluesmax-
Thestellarprofiles,Σ ,showtheclassicalexponential disks. imumandsmallerphysicalsizes,ascorrespondtoainside-outdisk
∗
Thesizeofthesestellardisks,andthescalelengthoftheseexpo- formationasassumed.Thepresentradialdistributionsshowflatter
nentialfunctionsareinagreementwithobservations.Howeverthey shapes,withsmallervaluesinthemaximumandintheinnerdisks
presentadecreasingorflatteningintheinnerregionsinmostcases, andhigherdensitiesintheouterregions.
althoughsomeofthemhaveaabruptincreasejustinthecenter.It Thestarformationisaround2orderofmagnitudelargeratz=
isnecessaryremainthatthesemodelsarecalculatedtomodelspi-
5thannowforthemassivegalaxies,inagreementwithestimatedof
ral disks, and the bulges are added by hand without any density
thestarformationintheUniverse(Glazebrooketal.,1999)while
radialprofile.Probablythisproducesabehaviornottotallyconsis-
forthesmallergalaxiesthemaximumvalueislowandverysimilar
tentwithobservationsatthecenterofgalaxies.
thenthannow.
In these three panels we show the results only for densities
InFig.8,weshow thesame four radial distributionsfor the
higherthan0.1M pc−2,whichcorrespondstoaatomicdensityn∼
⊙ lowestmassgalaxyinourgridwithamaximumrotationvelocity
0.1cm−2whichweconsideralowerlimitforobservations. ofVmax∼50kms−1.Inthiscase,andtakingintoaccountthelimit
The star formation radial distributions show similar shapes oftheobservationaltechniques,weseethatthegalaxywillbeonly
than Σ and Σ with similar decreasing in the inner re- detectedintheregionaround1kpcandonlyforredshiftz<2since
H2 ∗
gions of disks. Although these decreases have been observed forearliertimesthanthisthegalaxywillbeundetected.
(cid:13)c 2011RAS,MNRAS000,1–??
Chemical andphotometricmodels forlatetypegalaxies 9
Figure6.Theevolutionofthehalfmassradii,Reffmass,asafunctionof
theredshiftzformodelswithefficienciessetsfromnt=1to9andformass
distribution numbers from 10 to 44 (see Tables 1 and 2 from MD) cor-
responding tomaximum rotation velocities intherange [80-400]kms−1.
Black,gray,green,red,orange,magenta,purple,cyanandbluedotscorre-
spondtont=1,2,3,4,5,6,7,8,and9respectively.
Figure4.Theefficiencytoformstarsfrommoleculargas,SFR/MH2,in galaxies,asmagentaandpurpledots,whichcorrespondtoirregu-
logarithmic scale, asafunction ofredshift zforourgridofmodels.The largalaxiesshowthischangeoftheslopeatz<2.
evolutionwithredshiftforallradialregionsandgalaxiesareshownbythe
shadedzonewhilesolidblackpointsrepresentthegridresultsforz = 0.
Upperpanel:Solidblack,red,yellow,magenta,blue,greenandcyanlines,
3.5 Evolutionofelementalabundances
correspondtoradialregionslocatedat0,2,4,6,8,10and12kpcofthe
GalacticcenterintheMWGmodel.Bottompanel:Solidblack,cyan,blue, TheresultingelementalabundancesaregiveninTable3.Foreach
red,green,orange,andmagentalinescorrespondtogalaxiesofdifferentdy- typeofefficienciesntandmassdistributiondis,columns1and2,
namicalmassesandefficiencies(seeTable1).Inbothpanelsobservations
wegivethetimetinGyrincolumn3,thecorrespondingredshift
at intermediate-high redshift are from Daddietal. (2010); Genzeletal.
zincolumn4,andtheRadiusinKpcincolumn5.Theelemental
(2010),shownascyandotsandbluetriangles,respectively,whilethegreen
abundancesforH,D,3He,4He,12C,13C,N,O,Ne,Mg,Si,S,Ca
squares refers tothelocal Universe data obtained byLeroyetal. (2008).
andFeareincolumns6to19asfractioninmass.
TheredstarmarkstheaveragegivenbySancisietal.(2008).
Theradialdistributionsofelementalabundancesareshownin
Fig.7.Therewehaveforthesame6galaxiesofFig.5fromtheleft
totherightcolumn,theabundanceevolutionforC,N,OandFe,as
3.4 Thehalfmassradii
12+log(X/H)fromtoptobottompanels.Ineachpanel,asbefore,
Sinceweknowthemassofthediskandthecorrespondingoneto werepresenttheradialdistributionsfor7differentredshifts,z=5,
eachphasewemayfollowtheincreaseofthestellarmassineach 4,3,2,1,0.4and0,withthesamecolorcoding.Thewellknown
radialregionandcalculateforeachevolutionarytimeoredshiftthe decreasingfromtheinnertotheouterregions,calledtheradialgra-
radiusforwhichthestellarmassisthehalfofthetotalofthemass dientsofabundances,appearinthethefinalradialdistributionsfor
instarsinthegalaxy,Reff .Thisradiusobviouslywillevolve thepresenttimeinagreementwithdata.Howevertheslopeisnot
mass
withtimeandsincewehaveassumeanin-outscenarioofdiskfor- unique in most of cases. The radial distribution for any element,
mation,itmustincreasewithit.WeshowthisevolutioninFig.6, 12+log(X/H),isnotastraightlinebutacurvewhichisflatterin
where we plot with different color the evolution of different effi- thecentralregionandalsointheouterdiskinagreementwiththe
ciencies models: black, gray, green, red, orange,magenta, purple, mostrecentobservations(Sa´nchezetal.,2013).
cyanandbluefor setsnt =1to9,respectively, formassdistribu- Thedistributionsofabundanceschangewiththelevelofevo-
tionsfromnumber10to44,whichcorrespondtomaximumrota- lution of a galaxy. A very evolved galaxy, that is, as the one
tionvelocitiesintherangeVmax ∼ 80to400kms−1.Thelowest for the most massive ones, and/or those with the highest effi-
massgalaxies,withVmax<78kms−1(orthoseforwhichefficien- ciencies to form stars, show flatter radial gradients that those
ciescorrespondtont=10)arenotshownsince,aswewillshowin which evolve slowly, which have steeper distributions. There ex-
Fig.8,theyonlywouldhaveavisiblecentralregion.Itisevident ists a saturation level of abundances, defined by the true yield,
that the effective radius increases, being smaller to high redshift, and given by the combination of stars of a certain range of
thisincreasebeginslaterforthelatertypesofsmallerefficiencies mass and the production of elements of these stars. For our
than for the earlier ones or with high efficiencies. Blue and cyan combination of stellar yields from Woosley&Weaver (1995);
pointsarebelow1 kpcforredshiftshigherthan1,whiletheothers Gavila´n,Buell,&Molla´ (2005); Gavila´n,Molla´,&Buell (2006)
begin to increase already at z =5-6. Moreover a change of slope and IMF from Ferrini,Palla&Penco (1990), this level is 12 +
withaabrupt increaseinthesizeoccursatz ∼ 3untilz ∼ 2,for log(X/H) ∼ 8.8,8.0,9.0and 8.2 dex for C,N, O and Ferespec-
nt<5,whenthestarformationsuffersitsmaximumvalueinmost tively.Asthegalaxyevolvesthesaturationlevelisreachedinthe
of these galaxies which is agreement with the data. Intermediate outerregionsofgalaxy,thusflatteningthegradient.
(cid:13)c 2011RAS,MNRAS000,1–??
10 Molla´
Table3.Evolutionofelementalabundancesalongthetime/redshiftforthegridofmodels.WeshowasexampletheresultsforthepresenttimeofaMWG-like
model(nt=4,dis=28).Thecompletetablewillbeavailableinelectronicformat.
nt dis t z Radius H D 3He 4He
Gyr kpc
4 28 1.3201e+01 0.00 0 7.1000e-01 7.4918e-08 5.2929e-04 2.7149e-01
4 28 1.3201e+01 0.00 2 7.1020e-01 3.7168e-08 5.2559e-04 2.7085e-01
4 28 1.3201e+01 0.00 4 7.0751e-01 1.4105e-06 4.9819e-04 2.7373e-01
4 28 1.3201e+01 0.00 6 7.2710e-01 2.9584e-05 2.3378e-04 2.5935e-01
4 28 1.3201e+01 0.00 8 7.4243e-01 4.6899e-05 1.0063e-04 2.4818e-01
4 28 1.3201e+01 0.00 10 7.4664e-01 5.1180e-05 7.1091e-05 2.4520e-01
4 28 1.3201e+01 0.00 12 7.4958e-01 5.3626e-05 5.6599e-05 2.4321e-01
4 28 1.3201e+01 0.00 14 7.5423e-01 5.7213e-05 4.2096e-05 2.4017e-01
4 28 1.3201e+01 0.00 16 7.6037e-01 6.2123e-05 2.7653e-05 2.3619e-01
4 28 1.3201e+01 0.00 18 7.6558e-01 6.6386e-05 1.7397e-05 2.3283e-01
4 28 1.3201e+01 0.00 20 7.6841e-01 6.8681e-05 1.2615e-05 2.3100e-01
4 28 1.3201e+01 0.00 22 7.6931e-01 6.9386e-05 1.1281e-05 2.3043e-01
4 28 1.3201e+01 0.00 24 7.6947e-01 6.9516e-05 1.1048e-05 2.3032e-01
Table3.Cont.Evolutionofelementalabundancesalongthetime/redshiftforthegridofmodels
12C 13C N O Ne Mg Si S Ca Fe
3.1109e-03 5.6767e-05 1.3390e-03 6.9834e-03 1.1736e-03 3.2207e-04 1.2311e-03 6.2757e-04 8.8136e-05 2.8875e-03
3.1888e-03 5.8285e-05 1.3916e-03 7.1115e-03 1.1987e-03 3.3036e-04 1.2594e-03 6.4239e-04 9.0264e-05 2.9750e-03
3.1561e-03 5.3970e-05 1.2723e-03 7.0513e-03 1.1641e-03 3.1662e-04 1.2872e-03 6.5643e-04 9.1892e-05 3.0407e-03
2.4417e-03 2.8054e-05 7.2097e-04 5.4124e-03 8.6307e-04 2.2721e-04 9.2775e-04 4.7042e-04 6.5617e-05 2.0596e-03
1.7736e-03 1.3776e-05 4.0653e-04 4.0741e-03 6.4323e-04 1.6486e-04 6.0974e-04 3.0600e-04 4.2665e-05 1.2016e-03
1.5606e-03 1.0075e-05 3.1545e-04 3.6797e-03 5.7863e-04 1.4650e-04 5.1733e-04 2.5795e-04 3.5927e-05 9.5008e-04
1.3936e-03 7.4918e-06 2.3852e-04 3.3407e-03 5.2112e-04 1.3070e-04 4.5084e-04 2.2362e-04 3.1087e-05 7.8546e-04
1.0958e-03 4.5570e-06 1.4171e-04 2.6790e-03 4.1288e-04 1.0266e-04 3.4491e-04 1.7020e-04 2.3619e-05 5.6743e-04
6.7707e-04 2.1410e-06 6.2137e-05 1.6978e-03 2.5908e-04 6.4119e-05 2.0782e-04 1.0209e-04 1.4164e-05 3.2295e-04
3.1234e-04 8.0794e-07 2.2182e-05 8.0723e-04 1.2299e-04 3.0589e-05 9.4069e-05 4.6086e-05 6.4103e-06 1.4037e-04
1.1295e-04 2.7003e-07 8.2285e-06 2.9966e-04 4.6226e-05 1.1834e-05 3.3587e-05 1.6495e-05 2.3115e-06 5.1772e-05
5.0752e-05 1.2185e-07 4.7315e-06 1.3420e-04 2.1235e-05 5.7565e-06 1.4834e-05 7.3585e-06 1.0446e-06 2.6329e-05
3.9167e-05 9.4973e-08 4.1239e-06 1.0244e-04 1.6438e-05 4.5963e-06 1.1325e-05 5.6530e-06 8.0808e-07 2.1822e-05
Theradialgradientflattenswiththeevolutionarytimeorred- 4 THEPHOTOMETRICMODELDESCRIPTION
shift, mainly in the most massive galaxies. However it maintains
ItiswellknownthatgalaxieshaveSEDsdependingontheirmor-
a similar value for the smallest galaxies. The extension in which
phological type(Coleman,Wu,&Weedman,1980).TheseSEDs,
theradialgradientappears,however,changeswithtime.Attheear-
and other data related to the stellar phase, are usually analyzed
liesttimetheradial gradient appears only for thecentral regions,
through (evolutionary) synthesis models(seeConroy,2013, fora
until1kpcintheleftcolumngalaxy,whileitdoesuntil22kpcin
recentandupdatedreviewaboutthesemodels),basedonSSPscre-
theright one, withachange of slopeat around 8kpc. For thera-
ated by an instantaneous burst of star formation (SF). The syn-
dial regions out of this limit, it shows a flat distribution. At the
thesis models began calculating the luminosity (in a broad band
present,thisgradientappearsuntilamoreextendedradius,28kpc
filter or as a SED, F (λ), or by using the spectral absorption
inourmostmassivegalaxy,whileitisonlyintheinner4kpcinthe λ
indices) for a generation of stars created simultaneously, (there-
smallestone.Ifweanalyzetheresultsforthelowestmassgalaxy
fore with a same age, τ and with a same metallicity Z), that is
in Fig. 8, we see that abundances show a very uniform distribu-
the so-called Single Stellar Population (SSP). The evolutionary
tionalongthegalactocentricradiusforallelements,withaslight
codescomputethecorrespondingcolors,surfacebrightnessand/or
increase at the center for all redshifts that may be considered as
spectral absorption indices emitted by a SSP from the sum of
nonexistentwithintheusualerrorsbars.
spectra of all stars created and distributed along a Hertzsprung
Theflatradialgradientinthelowestmassgalaxies,asshown Russell diagram, weighted with an IMF. This SED, given τ and
inFig.8,asthisoneofthemostdistantregionsofthediskinthe Z, is characteristic of each SSP. This way it is possible to ex-
massivegalaxiesatthehighestredshifts,mustbeconsideredasa tract some information of the evolution of galaxies is by us-
productoftheinfallofagasmorerichthatthisoneofthedisk.It ing evolutionary synthesis models in comparison with spectro-
isnecessarytotakeintoaccountthatthehaloisformingstarstoo. photometric observations. This method has been very useful for
Whenthecollapsetimeislonger,asoccursintheouterregionsof the study of elliptical galaxies, for which was developed, with
disks,thereismoretimeandmoregasinthehalotoformstarsand thehypothesis that they are practicallySSPs,allowed toadvance
increase itsabundances. Thus, thegas infallingis, even at avery very much in the knowledge of these objects, determining their
lowlevel(12+log(X/H)∼4−5),moreenrichedthanthegasof ageandmetallicitywithgoodaccurate(Charlot&Bruzual,1991;
thedisk. BruzualA.&Charlot, 1993; Bressan,Chiosi,&Fagotto, 1994;
(cid:13)c 2011RAS,MNRAS000,1–??