Counting Low-Mass Stars in Integrated Light

1.3 MB·English

by  Charlie Conroy

DRAFTVERSIONJANUARY30,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
COUNTINGLOW-MASSSTARSININTEGRATEDLIGHT
CHARLIECONROY1 &PIETERVANDOKKUM2
DraftversionJanuary30,2012
ABSTRACT
Low-mass stars (M .0.4M⊙) are thought to comprise the bulk of the stellar mass of galaxies but they
constituteonlyoforderapercentofthebolometricluminosityofanoldstellarpopulation.Directlyestimating
2
the number of low-mass stars from integrated flux measurementsof old stellar systems is therefore possible
1
but very challenging given the numerous variables that can affect the light at the percent level. Here we
0
presentanewpopulationsynthesismodelcreatedspecificallyforthepurposeofmeasuringthelow-massinitial
2
massfunction(IMF)downto∼0.1M⊙ formetal-richstellarpopulationswithagesintherange3- 13.5Gyr.
n Our fiducial model is based on the synthesis of three separate isochrones and a combination of optical and
a near-IR empirical stellar libraries in order to produce integrated light spectra over the wavelength interval
J
0.35µm<λ<2.4µmataresolvingpowerofR≈2000. Newsyntheticstellaratmospheresandspectrahave

7 been computed in order to model the spectral variations due to changes in individualelemental abundances
2 including C, N, Na, Mg, Si, Ca, Ti, Fe, and generic α elements. We demonstrate the power of combining
bluespectralfeatureswith surface gravity-sensitivenear-IRfeaturesin orderto simultaneouslyconstrainthe

] low-massIMF,stellarpopulationage,metallicity,andabundancepatternfromintegratedlightmeasurements.
O
Finally,weshowthattheshapeofthelow-massIMFcanalsobedirectlyconstrainedbyemployingasuiteof
C surfacegravity-sensitivespectralfeatures,eachofwhichismostsensitivetoaparticularmassinterval.
.
h Subjectheadings:galaxies:stellarcontent—galaxies:abundances—galaxies:elliptical—stars:fundamental
p parameters—stars: abundances
-
o
r 1. INTRODUCTION Progress can be made by realizing that the spectra of M
t
s dwarfsandMgiantsatthesameeffectivetemperaturedisplay
Inordertounderstandtheformationandevolutionofgalax-
a subtlebutimportantdifferences.Oneofthefirstdetectionsof
[ ieswemustunderstandtheirstellarcontent.Attentionisoften
directed at constrainingthe history of star formation, metal- suchadifferencewasbyWing&Ford(1969)whodiscovered
3 licity, and dust content of galaxies. In such work a key as- averystrongabsorptionfeatureat0.99µminthespectrumof
v sumptionismade: thatthestellarinitialmassfunction(IMF) an M dwarf that is not seen in M giants. This feature, now
7 isuniversalandequaltotheIMFmeasuredinthesolarneigh- knownasthe Wing-Fordband,isattributedtoabsorptionby
0 FeH. It is very strong in late M dwarfs, but is absent in M
borhood. This assumption, and its influence on the derived
0 giants. Among dwarfs, it is also very sensitive to effective
physicalpropertiesofgalaxies,isarguablythelargestsource
0 temperature.
ofuncertaintyinnearlyallworkrelatedtothestellarpopula-
. There are in fact a number of strong absorption features
9 tionsofgalaxies.
that vary with surface gravity at fixed effective temperature.
0 The IMF cannot be easily measured in extragalactic sys-
1 tems (or even in our own Galaxy) because the stellar These featuresinclude NaI at 0.82µm, 1.14µm and 2.21µm,
1 luminosity-massrelation is steeper than the IMF — the for- CaIat1.98µm,CaIInear0.86µm,COnear2.30µm,andH2O
v: merscalesasM3- 4onthemainsequence(andismuchsteeper lliinneesstahraetsatrroennguminedrowuasrfisnathnednweeaar-kIRor.TabhseeNntaiIn,CgiaaInatsn,dwHh2iOle
i along the giant branches) while the latter scales as M- 2.3 or
X CaIIandCOarestrongingiantsandweakindwarfs(seee.g.,
shallowerinthesolarneighborhood. Barringgrossvariation
Frogeletal.1978;Kleinmann&Hall1986;Diazetal.1989;
r in these scalings in other galaxies, they imply that the light
a Ivanovetal.2004;Rayneretal.2009).
from any populationof stars is dominatedby the most mas-
It was recognized early on that dwarf-sensitive and,
sive stars that are still alive. The faint low-mass stars (i.e.,
to a lesser degree, giant-sensitive features could be
M<0.4M⊙),whichprobablydominatethetotalstellarmass used to count the number of low-mass stars in inte-
budget in galaxies, contribute only of order 1% to the inte-
grated light3 (e.g., Spinrad 1962; Spinrad&Taylor 1971;
gratedlight.
Cohen1978;Frogeletal.1978,1980;Faber&French1980;
In early type galaxies the problem is compounded by the
Carteretal. 1986; Hardy&Couture 1988; Couture&Hardy
fact that the stars that dominate the light, K and M giants,
1993; Vazdekisetal. 1996; Schiavonetal. 1997b,a, 2000;
haveasimilarspectraltypeasthestarsthatdominatethemass
Cenarroetal. 2003). The fundamental difficulty then, as
budget,theKandMdwarfs. Thisimpliesthat,tofirstorder,
now, was the separation of abundance effects from giant-
the light from low mass stars looks similar to the light that
to-dwarf ratio effects. Both effects can change the strength
dominates the flux. This is why it has historically been so
of the gravity-sensitive lines. Since the average metallicity
difficulttomeasurethelow-massendoftheIMFinearlytype
of massive elliptical galaxies is believed to be greater than
galaxies.
ourownGalaxy(e.g.,Wortheyetal.1992;Trageretal.2000;
Thomasetal.2005),theuseofstellarspectragatheredinthe
1Harvard-SmithsonianCenterforAstrophysics,Cambridge,MA,USA
2DepartmentofAstrophysicalSciences,YaleUniversity, NewHaven, 3By“integratedlight”wemeanthecombinedlightfromtheentirestellar
CT,USA population,asopposedtoresolvedphotometryofindividualstars.
2 CONROY&VANDOKKUM
solar neighborhoodforthe interpretationof metal-richellip- dividualelemental abundancevariations. Our principle goal
ticalswasviewedwithskepticism. ThecollectionofMgiant is to show that it is possible to separate the effects of IMF,
spectra in the Galactic bulge, which is more metal-rich than age, metallicity, and abundance pattern, through the consid-
the disk, was recognized to be an important step in under- erationofavarietyofspectralfeaturesacrosstheopticaland
standingthespectraofmassiveellipticals(Carteretal.1986; near-IR wavelength range. These models can be contrasted
Frogel&Whitford1987),butthisapproachhasnotyetbeen withmodelssuchasBruzual&Charlot(2003),whichmodel
asfruitfulasinitiallyhoped. spectralenergydistributionsatsolarabundancepatterns,and
The early work on this topic suffered from additional un- modelsthatconsidertheeffectofage,metallicity,andabun-
certainties. Thelackofaccuratestellarevolutioncalculations dance patterns on the Lick index system (e.g., Trageretal.
across the main sequence and through advanced evolution- 2000;Thomasetal.2003b;Schiavon2007). Themodelpre-
aryphasesimpliedthatmodelconstructionsufferedfromma- sented herein is, to our knowledge, the first to consider the
joruncertaintiesthatwere difficultto quantify. Anotherma- responseof the entireopticaland near-IRspectrumto varia-
jor limitation was the poor near-IR detector technology that tionsinindividualelementalabundancepatterns.
existed in the 1980s and 1990s. This made it very difficult In the following sections we describe the details of the
to measure spectra at the sub-percent level precision neces- model(§2), providea brief comparisonto data (§3), present
sary to measure the low-mass IMF in integrated light. In ourresults(§4), anddiscussseveralimplicationsand setour
theinterveningdecadesbothoftheseuncertaintieshavebeen findings in a broader context (§5). A summary of our main
significantly reduced (though not eliminated). In addition, conclusionsisalsoprovided(§6).
substantial progress has been made in understandingthe re-
2. MODELCONSTRUCTION
sponse of stellar spectra to elemental abundance variations
(e.g., Tripicco&Bell 1995; Kornetal. 2005; Servenetal. 2.1. Overview
2005). All of these factors imply that the time is ripe to re- Ourgoalistoconstructmodelsfortheintegratedlightspec-
considerthepossibilityofmeasuringtheIMFfromintegrated trumofapopulationofstarsasafunctionofpopulationage,
lightspectra. metallicity,abundancepattern,andIMF.Twoingredientsare
Recently we obtained spectra for eight massive galax- necessary to construct such models: stellar evolution calcu-
ies in the Coma and Virgo clusters with σ > 250 kms- 1 lations (i.e., isochrones), and stellar spectral libraries. The
(vanDokkum&Conroy 2010). Thanks both to the new isochronesdictate which stars are to be included in the syn-
CCDs installed on the Low Resolution and Imaging Spec- thesis,dependingontheageandmetallicityofthestellarpop-
trometer (LRIS) mounted on the Keck I telescope, and the ulation. The isochrones also determine the stellar mass of
stackingof these spectrain the restframe(therebyaveraging eachstar, giventhephysicalparametersofthestar(effective
out any residualsky emission and absorptionproblems), we temperature, T ,andbolometricluminosity, L ). Withthe
eff bol
wereabletoobtainaccuratespectraatthe<0.5%level. We appropriate list of stars for a given age and metallicity, and
combined these data with a preliminary version of the pop- thestellarmassofeachstar,theintegratedlightsinglestellar
ulation synthesis modeldescribed herein to concludethat in population (SSP) spectrum can be simply constructed from
thesemassiveellipticalstheratiooflowtosolarmassstarsis theintegral:
muchlargerthanintheGalaxy.TheIMFinthesemassiveel-
lipticalsappearedtobemuchmore‘bottom-heavy’compared mu(t)
f(λ)= s(λ,m)φ(m)dm, (1)
totheIMFintheGalaxy. Z
The primary uncertainty in that work was the importance ml
of abundance effects; the empirical stars used in our model wheretheintegraloverstellarmasses,m,rangesfromthehy-
are of approximately solar metallicity, while the ellipticals drogenburninglimit,ml,tothemostmassivestarstillaliveat
havemetallicitiesinexcessofsolarandareα-enhanced(e.g., timet,mu(t). Throughoutthispaperweadoptml =0.08M⊙.
Wortheyetal. 1992; Trageretal. 2000; Thomasetal. 2005). In the above equation, f is the integrated spectrum, s is the
In a follow-up to our initial results, we obtained spectra for spectrumofanindividualstar,andφistheIMF.Wewillcon-
four globular clusters (GCs) in M31 that have iron abun- siderfourIMFsinthepresentwork: aChabrier(2003)IMF,
dancesandabundancepatterns(i.e.,α- enhancement;[α/Fe]) which is a fit to the IMF of stars in the disk of the Milky
comparabletothemassiveellipticals(vanDokkum&Conroy Way,aSalpeterIMF,whichhasalogarithmicslopeofx=2.3
2011). A directcomparisonof thespectraofthese two pop- (x≡- dlnφ/dlnm),a‘bottom-heavy’IMFthathasalogarith-
ulations revealed significant differences only in the surface mic slope of x=3, and a ‘bottom-light’ IMF advocated by
gravity-sensitive lines. This constituted a critical test of our vanDokkum (2008). The bottom-light IMF has a similar
interpretationoftheellipticaldatabecausetheM31GCshave shape to the Chabrier IMF except that the former rolls over
lowmass-to-lightratios(Straderetal.2011),andsocouldnot at≈2M⊙ whilethelatterrollsoverat≈0.1M⊙.
havebottom-heavyIMFs. Wethereforeconcludedthattheel- Inthefollowingsectionswedescribetheisochrones,stellar
lipticals containedproportionallymanymore low mass stars libraries, and synthesis techniques used to create integrated
than the GCs, again consistentwith a dwarf-richIMF in the spectraasafunctionofpopulationage,metallicity,andIMF.
massiveellipticals.
2.2. Isochrones
Inthepresentworkwepresentdetailedpopulationsynthe-
sismodelswiththegoalofextractingthelow-masscontribu- Accurate isochronesare requiredthat extendfrom the hy-
tion to the integrated flux. The models span a range in age, drogen burning limit to the end of the asymptotic giant
metallicity, abundance patterns, IMFs, and cover the wave- branch(AGB)phase.Unfortunately,nosinglesetofpublicly-
length range 0.35µm<λ<2.4µm at a resolving power of availablestellarevolutioncalculationsexiststhatmeetsthese
R≈2000. Animportantfeatureofourapproachistheuseof requirements. We therefore need to stitch together several
syntheticspectratogaugethesensitivityofthespectrumtoin- separate evolutionarycalculations. For the bulk of the main
sequenceandredgiantbranch(RGB)weusetheDartmouth
COUNTINGLOW-MASSSTARSININTEGRATEDLIGHT 3
4 3.0
MS -> RGB tip
x=3
2.5 Salpeter
2 Chabrier
2.0 bottom-light
])LO • 5.0 Gyr 13.5 Gyr MF)
og(L [bol 0 log(I 11..05
l
-2
0.5
-4 PDLayadortonmvaouth [[[ZZZ///ZZZO O O •••]]]===000...200,,, [[[aaa ///FFFeee]]]===000...040 0.0
1.0
6000 5000 4000 3000 2000 0.2 0.4 0.6 0.8 1.0
T (K) mass (M )
eff O • 0.8
er
FIG.1.— Left panel: HRdiagram showing how stellar evolution calcu- b
m
lations from three groups are stitched together to create an isochrone that
u 0.6
spans all stages of stellar evolution (through the end of the AGB) for all n
masses. Isochrones areshownfortwoagesaslabeled inthepanel. Right ve
panel: Luminosity–mass relation as a function of metallicity. Here only ati 0.4
the Dartmouth-based isochrones areshown forsimplicity. Notice that the ul
Lbol(M)relationisessentiallyinsensitivetometallicityasidefromthechange um
intheturn-offmass. c 0.2
13.5 Gyr Z SSP
sol
isochrones (Dotteretal. 2008), which have been shown to 0.0
reproduce very well the main sequence morphology of old 1.0
star clusters (Anetal. 2009). These isochrones only extend
to core helium ignition at the tip of the RGB. We therefore
0.8
usethePadovaisochrones(Marigoetal.2008)forhorizontal
s
bthreanLcyho(nHiBso)cahnrodnAesG(BChevaborluietrio&n4B. aFroarffMe1<9907.;1B7Mar⊙affweeetuasle. e mas 0.6
1th9e9s8e)a.rTehleowmmaxaismsusmtarasgtehaintetvhoelLveyolinttlmeoodveelrstihse8raGnygre.3Si-n1c4e ulativ 0.4
m
Gyr,weusetheir8Gyrmodelsthroughout.Theprecisevalue
u
of0.17M waschosensothattheDartmouthisochronesjoin c
⊙ 0.2
smoothlywiththeLyonisochronesin T and L .
eff bol
TheswitchtotheLyonmodelsbelow0.17M⊙ ismadebe- 0.0
causetheDartmouthmodelsapplythesurfaceboundarycon-
1.0
ditionatthephotosphere(T = Teff)ratherthanatthebaseof 1.0
the atmosphere(τ =100) as in the Lyon models. The depth
atwhichthesurfaceboundaryconditionissetisparticularly sity 0.8 10-1
importantforthehighdensity, lowtemperatureenvironment o
n
encounteredinverylowmassstars. Stellarinteriorcodesare mi 0.6
u 10-2
not ideally suited to compute the physical conditionsall the e l
wayto T = T . We notein passingthatthePadovamodels v
areevenlessaecffcurateatlowmassesthantheDartmouthmod- ulati 0.4 10-03.0 0.2 0.4 0.6 0.8 1.0
els,andtheyonlyextendto0.15M⊙,whichiswhyweusethe m
u 0.2
Padovamodelsonlyforadvancedevolutionaryphases. c
IntheleftpanelofFigure1weshowtheresultingcompos-
iteisochroneforasolarmetallicitypopulationwithanageof 0.0
13.5Gyr. Forcomparison,wealsoshowanisochronefora5 0.0 0.2 0.4 0.6 0.8 1.0
Gyrsolarmetallicitypopulation. Therightpanelofthesame mass (MO •)
figure shows the L- M relation for the main sequence and
RGB. InthispanelwealsoshowthesensitivityoftheL- M
relationtometallicity.Themetallicityvariationsincludea0.2 FIG.2.—Cumulativecontributiontothetotalnumberofstars,totalstellar
mass,andbolometricluminosityasafunctionofstellarmass.Contributions
dexincreaseinthetotalmetalabundance(Z),anda0.4dexin- areplotted asafraction ofthetotal. Theinset inthebottom panelshows
creaseintheα-elements([α/Fe]). Noticethatwhilethemain thecumulative luminosityonalogarithmic scale. Stellar remnantsarenot
includedinthisfigure.ResultsareshownforthreeIMFs,formodelswithan
ageof13.5Gyrandsolarmetallicity. Thesmallcontributionfromlow-mass
4 Due to different assumptions in the Dartmouth and Padova models,
starstotheintegratedluminosity,despitetheirlargecontributiontothetotal
isochronesatagivenageandmetallicity donotjoinperfectly smoothlyin numberofstars andthestellar mass, isadirect consequence oftheL- M
massbetweenthetipoftheRGBintheDartmouthmodelsandtheHBin
relationshowninFigure1.
thePadovamodels. Inordertoensureamonotonicincreaseinmassalong
thecombinedisochrone,wehaveshiftedthePadovamassessothattheyjoin
smoothlywiththeendoftheDartmouthisochrones.Theseshiftsrangefrom sequence turn-off mass changes slightly, the main sequence
10- 3M⊙at13Gyrto0.05M⊙at3Gyr,andtheyhavenoeffectonourresults. L- M relation is almost entirely insensitive to these modest
4 CONROY&VANDOKKUM
1.2
K1V M3V M4III
1.0 Teff = 5257 K Teff = 3161 K Teff = 3562 K
0.8
x
u
l 0.6
f
0.4
0.2
0.0
0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5
l (m m) l (m m) l (m m)
FIG.3.—SpectralenergydistributionsforthreestarsfromtheIRTFlibrary.TheIRTFspectra(solidblacklines)arecomparedtoopticalandnear-IRphotometry
(symbols)andoursyntheticspectra(bluelines). Thesyntheticspectrawereselectedtohavethesame Teffandlog(g)astheobservedstarsandwerescaledto
matchtheobservedspectraattheblueandredendsofthedata.ThetheoreticalspectraaredashedwheretheyoverlapinwavelengthwiththeIRTFspectra,and
solidwheretheyareusedtoextrapolatetheIRTFspectra. TheSEDswerenormalizedtounityandsmoothedby100Å fordisplaypurposes.Spectraltypesand
estimatedeffectivetemperaturesareshownineachpanel.
metallicitychanges. calibration is provided by the JHK photometry. The spec-
Inthe followingwe makeuse ofthe isochronesin two re- tralresolutionisR=2000,andthesignal-to-noiseratiovaries
spects.First,theyareusedtodeterminewhichstarstoinclude from∼100- 1000withamedianat1µmof700. We donot
in thesynthesis(e.g.,stars thatfallnearthe isochronein the attempt to correct for dust extinction because the E(B- V)
leftpanelofFigure1). Second,theyareusedtoassignmasses values providedin the IRTF library are consistent with zero
tothestarsbasedon L . Throughoutthispaperwewilluse within the measurementuncertainties. In addition, the aver-
bol
thesamesolarmetallicityisochrones,evenwhensynthesizing agedistancetothestarsinoursampleisonly32pc,suggest-
modelswithdifferentabundancepatterns.Thissimplification ingthatthespectrasufferfromminimalextinction.
is motivated by two facts: first, the L- M relation is almost EffectivetemperaturesareestimatedforeachIRTFstarvia
entirely insensitive to metallicity over the range of metallic- the standard technique of using calibrated effective temper-
ities we will consider (rightpanel of Figure 1). This means ature - color relations. The T - (V - K) relation for gi-
eff
that the masses assigned to stars are robust against changes ants was adopted from Alonsoetal. (1999) for F0-K5 gi-
inmetallicity.Second,althoughchangesinmetallicityinduce ants, Ridgwayetal. (1980) for K5-M6 giants, and from
changesinthetemperaturesofstars,thiseffectisdwarfedby Perrinetal. (1998) for M6-M8 giants. For dwarfs we adopt
themuchlargerchangeinthespectraofstarsatfixedtemper- thenew T scale fromCasagrandeetal.(2008,2010). The
eff
aturewhenmetallicityvaries,especiallyforoldages(>5Gyr; T scale of Casagrande et al. is consistent with earlier re-
eff
seeCoelhoetal.2007;Schiavon2007;Leeetal.2009). lations from Alonsoetal. (1996) and Ramírez&Meléndez
InFigure2weshowtheIMFsconsideredhereinandthecu- (2005),modulo∼100Koffsetsthatcanbeattributedtodif-
mulativecontributiontothetotalnumberofstars,totalstellar ferences in the photometric absolute calibration adopted by
mass,and L asafunctionofstellarmassandIMF.Stellar differentauthors.
bol
remnants are not included in the figure (though they are in- BolometricluminositieswereestimatedforeachIRTFstar
cludedinthetotalstellarmassbudgetintherestofthepaper). by extrapolating the observed spectrum both blueward and
Thisfiguresummarizesthewell-knownfactthatforstandard redwardwithaspectrumfromoursyntheticlibrary(described
IMFslow-massstarsdominatethetotalstellarmassandnum- inthenextsection)atthesame T . Figure3illustratesthis
eff
ber of stars, but contribute only a small fraction of the total procedure. IneachpanelweshowanIRTFspectrum,theac-
bolometricluminosity. companyingBVJHK photometry,andthesyntheticspectrum
usedtoextrapolatetheIRTFspectrumbothbluewardandred-
2.3. Empiricalspectrallibraries ward. Integratingoverthe fullwavelengthrangethen yields
Empirical spectral libraries constitute the core feature of anestimateof Lbol. ThewidewavelengthrangeoftheIRTF
ourmodel.Wemakeuseoftwoseparatelibraries,theMILES spectraensuresthatthebulkoftheluminosityisdirectlymea-
library, which covers the wavelength range 0.35µm< λ < suredforlate-typestars. Forexample,theIRTFspectrasam-
0.74µm(Sánchez-Blázquezetal.2006),andtheIRTFlibrary ples70- 85%ofthebolometricluminosityforM-typestars,
of cool stars, which covers the wavelength range 0.81µm< 40- 70% for G and K-type stars, and 20- 40% for F-type
λ<2.4µm(Cushingetal.2005;Rayneretal.2009). stars. TheBandV photometryprovideanimportantconsis-
We select stars from the main IRTF library that are not tencycheckforourbluewardextrapolation,asisevidentfrom
classified as chemically peculiar, have measured parallaxes Figure3.
(asobtainedthroughSIMBAD)andthatbelongtoluminosity Forthesereasons Lbolisaveryrobustparameterestimated
classIIIorV.Thisleavesuswith91starsranginginspectral fortheIRTFstars.Wethereforeuse Lboltoestimatethestellar
typefromF0toM9.TheIRTFlibraryincludesfluxcalibrated massforeachIRTFstarbasedonthe Lbol- Mrelationshown
spectraandBVJHK photometryforeachstar. Absoluteflux in Figure 1 and described in §2.2. The resulting mass—K-
COUNTINGLOW-MASSSTARSININTEGRATEDLIGHT 5
1.0
Dynamically-based 0
Isochrone-based
0.8
1
)
)MO • 0.6 −2m s] 2 MILES
ass ( g [c 3 IRTF
m 0.4 g(
o
l
4
0.2
5
0.0
4 6 8 10 6000 5000 4000 3000 2000
M T (K)
K eff
FIG.4.—Stellarmass–K-bandabsolutemagnituderelationderivedforthe FIG.5.—LocationintheHRdiagramofstarsfromtheempiricalIRTFand
IRTF stars (symbols) compared to the relation derived from dynamically- MILESstellarlibraries. OnlythesubsetofIRTFstarsusedinourfiducial
basedmasses.Dynamicalmassesarederivedfromanalysisofstarsinbinary modelareshown. Alsoshownisthefiducial isochrone (Z=Z⊙,t=13.5
orbits(Delfosseetal.2000).IRTFstellarmassesarederivedfromatheoret- Gyr).
icalLbol(M)relation;seethetextfordetails.
band luminosity relation for main sequence stars is shown fortheir Teff andlog(g)sincethosevaluesaremoreaccurate
in Figure 4. These isochrone-based masses are compared (beingbased onhighresolutionspectra), and resultin better
to the mass-luminosity relation derived for stars in binary agreementwiththelocationof13.5Gyrisochrones.Adoption
orbits by Delfosseetal. (2000). The dynamicaly-estimated oftheseparametershasessentiallynoimpactontheresulting
mass-luminosityrelationiscompletelyindependentfromour spectral synthesis, since the Teff and log(g) values are only
isochrone-based method, and so the impressive agreement usedtopairIRTFandMILESstars.
provides strong support to our approach. Note that only The MILES library consists of 985 stars covering a wide
twostarspindownthedynamically-basedM - Mrelationat range in atmospheric parameters (Sánchez-Blázquezetal.
K
0.45M⊙<M<0.6M⊙, so the mild divergencebetween the 2006). The spectra have a FWHM of 2.5Å and are relative
isochroneanddynamically-basedmassesinthisrangeisnota fluxcalibrated,butnotabsolutefluxcalibrated. Atmospheric
significantsourceofconcern.Onceamassisassignedtoeach parameters([Fe/H],log(g),and T )werederivedforMILES
eff
star,wecanthenestimateitslog(g)basedontheisochrones. starsinCenarroetal.(2007). WeusetheMILESlibraryv9.1
We have surveyedthe literature for independently-derived (Falcón-Barrosoetal.2011),whichcorrectsseveralsmaller-
physicalparametersforoursampleof91IRTFstars. Wehave rorsin previousreleases. We removedhotvariablestarsand
found parametersfor 35 of the stars (38% of the sample) in binaries(asidentifiedinSIMBAD),andstarswhose T ap-
eff
one or more of the MILES, ELODIE (Prugniel&Soubiran pearedtobeinerrorbasedonvisualinspection.Thisresulted
2001), SPOCS (Valenti&Fischer 2005), and Indo-US in28starsbeingremovedfromthelibrary.
(Valdesetal. 2004; Wuetal. 2011) libraries. The mean and ThetwocoolestMdwarfsintheMILESlibraryhavespec-
standarddeviation of the differencesin T are ≈20 K and tral types M6 and M7. Both stars are magnetically active,
eff
≈100 K, while for log(g) the mean and standard deviation asevidencedbystrongBalmeremissionlinesintheiroptical
of the offsets are ≈0.05 and ≈0.2. These differences are spectra.Activityoccursin≈60%ofsuchstars,andincreases
comfortablywithin the typicalerrorson T and log(g)and to ≈75% for M8V stars (Westetal. 2004). In addition to
eff
giveconfidencetoourindependently-derivedparameters.The Balmeremission,thesestarsalsodisplaystrongCaIIH&K
differencesaresmallestbetweenourparametersandthosede- linesinemission. ThethirdcoolestMdwarf,ofspectraltype
rivedbytheSPOCSsurvey,whichisencouragingbecausethe M3,alsodisplaysweakH&Kemissionlines. Wewillreturn
SPOCSsurveyconsistsofveryhighresolution(R∼70,000) to the importance of magnetic activity and associated chro-
spectra and very careful consideration of the available line mosphericemissionlinesincoolMdwarfslaterinthepaper.
data in their analysis. These surveys also provide estimates The MILES and IRTF libraries are combined in the fol-
ofthemetallicityofeachstar. Theaveragemetallicityofthe lowing way. First, we select stars from MILES that have a
IRTFstarsincommonwithMILESis- 0.07,whileforstarsin metallicityof- 0.1<[Fe/H]<0.1.Thisleaves197starsinthe
commonwithSPOCS the averagemetallicityis +0.05. This MILESlibrary.WethentakeeachIRTFstarandpairittothe
supports our assumption that the IRTF stars are of approxi- MILES library via interpolation in T - log(g) space. Fig-
eff
matelysolarmetallicity. ure5showshowMILESandIRTFstarspopulatethisspace.
Fortwo IRTFstars onthe subgiantbranch(HD10697and The MILES library is considerably larger than the IRTF li-
HD25975)we haveadoptedtheSPOCSandIndo-USvalues brary, so this pairing is straightforward. Once the pairing is
6 CONROY&VANDOKKUM
made,theinterpolatedMILESspectrumisnormalizedagainst
theV- bandmagnitudeoftheIRTFstar. Withthisprocedure
weobtainanempiricalspectrumforeachstarin T - log(g)
eff 1.1
spacethatspansthewavelengthrange0.35µm<λ<2.4µm
withonlyamodestgapat0.74µm<λ<0.81µm. 1.0
The astute reader will notice in Figure 5 that there are no 0.9
MILESdwarfstarscoolerthan T =2750K,corresponding x
to M =0.15M⊙. Because of this,effthe five IRTF stars cooler flu 0.8
than this are paired to the coolest MILESstar available. As 0.7
MILESspectraarealwaysnormalizedtotheV- bandphotom-
etryoftheIRTFstars,theoverallchangeinV- bandluminos- 0.6 CaII CO
0.5
ity with mass is correctly modeled with our approach. It is
0.845 0.850 0.855 0.860 0.865 2.28 2.30 2.32 2.34
only the surface gravity and temperature-sensitive lines that l (m m) l (m m)
areaffectedbythelackofcoolerdwarfsintheMILESlibrary.
1.1
Finally,wepointoutthat<0.15M⊙starscontributetentimes 1.0
less flux at 0.5µm compared to >1µm, so the approximate
treatmentofthese stars in the opticalshouldnotbe a source 0.9
ofconcern.
x 0.8
u
fl
2.4. Syntheticspectrallibraries 0.7
M7V
Our empirical library, by construction, is very well suited 0.6 M4V
K2III
for generating integrated light spectra of stellar populations 0.5 NaI FeH M4III
withapproximatelySolarabundanceratios. However,itdoes
0.816 0.818 0.820 0.822 0.824 0.990 0.995 1.000
notallow us to investigate changesin the overallmetallicity l (m m) l (m m)
orchangesintheabundancesofindividualelements. Weturn
tosyntheticspectratoincorporatethisoptioninourmodeling. FIG.6.— IRTF spectra for stars of various spectral types and luminos-
Thesesyntheticspectraareusedtocalculaterelativechanges ityclasses. Theplotsfeaturefoursurfacegravity-sensitiveandtemperature-
sensitivespectralfeatures. Thetoppanelshighlightfeaturesthatarestrong
with respectto a fiducialsynthetic model, and these relative
ingiantscomparedtodwarfs,whilethebottompanelshighlightfeaturesthat
changesarethenappliedtoindicesthatwemeasuredfromour arestrongindwarfscomparedtogiants. Spectraarenormalizedinorderto
empiricalspectra(asine.g.,Trageretal.2000;Thomasetal. highlightthegravity-sensitivelines. Noticeinparticularthetwospectrathat
2003b;Schiavon2007). Asyntheticspectrallibrarydoesnot havethesamespectraltype(M4)butdifferentluminosityclass(Vvs.III).
existthatmeetsourneeds,sowe havegeneratedourownli-
The line data are critical for construction of realistic stel-
braryspecificallyforthisproject.
lar atmospheres and spectra. We have adopted the atomic
We have made use of the ATLAS model atmosphere and
and molecular linelists available on Kurucz’s webpage5, in-
spectrum synthesis package (Kurucz 1970, 1993), ported to
cluding the molecules H O, TiO, FeH, C , CH, CN, NH,
Linux by Sbordoneetal. (2004). In particular, we used the 2 2
SiO, SiH, OH, MgH, CO, and H . In particular, we use the
ATLAS12modelatmospherecode,whichutilizestheopacity 2
TiOlinelistfromSchwenke(1998)andtheH Olinelistfrom
sampling technique. The use of ATLAS12 for our purpose 2
Partridge&Schwenke (1997). The FeH linelist is from the
isrelativelynovelbecausewe allowtheopacitytovarywith
workofDulicketal.(2003),butwithapartitionfunctiontab-
eachelementalabundancevariation.Inpreviousworkrelated
ulatedbyKuruczspecificallyforthisproject.
to the response of spectral indices to individual abundance
Wehaveidentified20pointsalongthe13.5Gyrisochrone
variations (e.g., Tripicco&Bell 1995; Kornetal. 2005), a
shownin Figure1 rangingfromthe tip of the RGB downto
singleopacitydistributionfunctionwasusedforallindividual
T =3170K andlog(g)=5.05,correspondingtoamainse-
abundancevariations.Theatmospheresareplane-paralleland eff
stationary. Thestandardmixing-lengththeoryisadoptedfor quencemass of 0.15M⊙. A synthetic spectrum is computed
at each of these points for later use in the spectral synthe-
the treatment of convection, and local thermodynamicequi-
sis. Modelatmospheresforcoolerdwarfsfailedtoconverge.
librium(LTE)isassumed.
Wethereforeextrapolateourspectrallibraryin T tocooler
Oncetheatmospheresaregenerated,theSYNTHEprogram eff
dwarfsdownto0.08M . Thisapproximationwillhavelittle
(Kurucz&Avrett1981)isusedtogenerateaveryhighreso- ⊙
impact on our results because we only use the model spec-
lution (R=100,000) spectrum for each star over the wave-
tratocreatedintegratedlightspectraforaChabrierIMF,for
lengthrange0.3µm<λ<3.0µm. Amicroturbulentvelocity
ofV =2.0kms- 1isadopted,independentofspectraltypeand whichstarswith0.08M⊙<M<0.15M⊙contribute≈10- 3of
t thetotalflux.Noticethatpreviouswork(e.g.,Tripicco&Bell
luminosity. Wehaverecomputedasubsetofthemodelswith
1995; Kornetal. 2005; Servenetal. 2005) only considered
V =1 kms- 1 in order to explore the sensitivity of the spec-
t two or three points along the isochrone in order to gauge
tra to this parameter. This change in Vt results in a sizable therelativeresponseofspectralfeaturesto changesinabun-
changeintheequivalentwidthsofmanyfeatures(≈0.2- 2Å; dances.
seealsoTripicco&Bell1995). Wecareonlyabouttherela- In the present work we adopt the latest solar abundances
tiveresponseofthemodels(seebelow),andthisismuchless from Asplundetal. (2009). These new abundances for C,
effectedby a changeinVt. For example,consideringthe re- N,O,andNearesignificantlylowerthantheoldsolarabun-
sponse of the model spectra to a factor of two enhancement dancesofAnders&Grevesse(1989) andGrevesse&Sauval
in iron abundance, the difference betweenV =2 kms- 1 and (1998). In particular, the new [O/H] abundance is 0.24 dex
t
1kms- 1 islessthan0.1Å inequivalentwidthforalllinesof
interest. 5http://kurucz.harvard.edu/
COUNTINGLOW-MASSSTARSININTEGRATEDLIGHT 7
C ,CH,Ca,Cr,Fe,Mg,MgH,Na,andTiO(Tripicco&Bell
2
TABLE1 1995;Kornetal.2005).Inthefollowingwork,wewillutilize
DEFINITIONOFSPECTRALINDICES the Lick index wavelength intervals for the index bandpass
andpseudocontinuaasdefinedinTrageretal.(1998),butwe
Indexa Feature Blue Red Notes
Continuum Continuum makenoefforttoplace ourindicesonthezero-pointsystem
(Å) (Å) (Å) oftheLicksystem. TheseindicesaredefinedinTable1. All
indices in the present work (except for TiO0.89) are quoted
CaII0.39 3899.5- 4003.5 3806.5- 3833.8 4020.7- 4052.4 1
CN0.41 4143.3- 4178.3 4085.0- 4097.6 4245.5- 4285.5 2 as equivalent widths (EWs) with units of Å (the procedure
CaI0.42 4223.5- 4236.0 4212.2- 4221.0 4242.2- 4252.2 2 formeasuringEWsisdescribedinWortheyetal.1994). All
CH0.43 4282.6- 4317.6 4267.6- 4283.8 4320.1- 4336.5 2 wavelengthsinthispaperarequotedinvacuum.
FeI0.44 4370.3- 4421.6 4360.3- 4371.6 4444.1- 4456.6 2
C20.47 4635.3- 4721.6 4612.8- 4631.6 4744.1- 4757.8 2 WesupplementtheLick-basedblueindiceswithadditional
FeI0.50 4979.2- 5055.4 4947.9- 4979.2 5055.4- 5066.7 2 indicesasdefinedinTable1. Theseindiceswerechosenfor
MgH0.51 5070.5- 5135.6 4896.5- 4959.0 5302.6- 5367.6 2 their promise in constraining the number of low mass stars
FeI0.52 5247.2- 5287.2 5234.7- 5249.7 5287.2- 5319.7 2
MgI0.52a 5161.6- 5194.1 5144.1- 5162.8 5192.8- 5207.8 2 from integrated light (in practice, we selected features that
MgI0.52b 5165.0- 5220.0 5125.0- 5165.0 5220.0- 5260.0 3 changedbymorethan1%betweenaChabrierandx=3IMF;
FeI0.53 5313.6- 5353.6 5306.1- 5317.4 5354.9- 5364.9 2 see Figure 10 below). The majority of the new indices are
NaI0.59 5878.5- 5911.0 5862.2- 5877.2 5923.7- 5949.7 2 defined and measured in a manner analogous to the Lick-
NaI0.82 8177.0- 8205.0 8170.0- 8177.0 8205.0- 8215.0 3
CaII0.86 8484.0- 8513.0 8474.0- 8484.0 8563.0- 8577.0 4,5 based indices. Exceptions include the calcium triplet (CaT)
8522.0- 8562.0 8474.0- 8484.0 8563.0- 8577.0 at0.86µm, which is comprisedof threedistinct featuresand
8642.0- 8682.0 8619.0- 8642.0 8700.0- 8725.0 sothe totalEWisthesum oftheEWsoftheindividualfea-
MgI0.88 8801.9- 8816.9 8777.4- 8789.4 8847.4- 8857.4 6 tures(seeCenarroetal.2001,fordetails). Likewise,theCaI
TiO0.89 8835.0- 8855.0 8870.0- 8890.0 3,7
FeH0.99 9905.0- 9935.0 9855.0- 9880.0 9940.0- 9970.0 3 featureat1.98µmisadoubletevenwhendopplerbroadened
NaI1.14 11372- 11415 11360- 11370 11417- 11427 3 toσ=300kms- 1,andsothisfeatureisdefinedasthesumof
KI1.17 11680- 11705 11667- 11680 11710- 11750 3,5 twoEWs. TheCOindexat2.3µmisdefinedbythefirsttwo
11765- 11793 11710- 11750 11793- 11810
AlI1.31 13115- 13165 13090- 13113 13165- 13175 3 bandheadsofthe12COmolecule. Finally,theTiO0.89index
CaI1.98 19770- 19795 19740- 19765 19800- 19840 3,5 istheonlyindexdefinedasaratiooffluxesbetweenblueand
19845- 19880 19800- 19840 19885- 19895 redpseudocontinua,ratherthananequivalentwidth.
NaI2.21 22047- 22105 22035- 22045 22107- 22120 3
CO2.30 22932- 22982 22860- 22910 23020- 23070 3,5 ThenamesgiventotheindicesinTable1highlightthepri-
23220- 23270 23150- 23200 23300- 23350 maryspeciescontrollingthestrengthofeachindex.Inreality,
eachindexis sensitive to a numberof atomicand molecular
species(see e.g.,Tripicco&Bell 1995;Kornetal. 2005, for
NOTE. — (1) Servenetal. (2005); (2) Lick-based index (Trageretal.
details).
1998);(3)newindex; (4)Cenarroetal.(2001);(5)Indexwithmultiplefea-
tures. In such cases the total EW is the sum of the individual EWs; (6) The use of indices suffers from several well-known prob-
Diazetal.(1989);(7)Indexdefinedasafluxratiobetweenblueandredpseu- lems. TheEWofanabsorptionfeatureismeasuredwith re-
docontinua. specttoapseudocontinuum,whichitselfiscomposedofab-
a Index names indicate the primary species and central wavelength in µm.
sorptionfeatures. Thevariationofanindexduetochangesin
Wavelengthsareinvacuum.
metallicity or age may therefore be due to a combinationof
lower than the Anders&Grevesse (1989) abundances. This changesinthefeatureitselfandchangesinthepseudocontin-
resultsinsubstantiallybetteragreementbetweenmodelspec- uum.Moreover,eachindexisinrealityablendofseveralab-
traanddataforlate-typeMdwarfsduetotheresultingweaker sorptionlinesarisingfrommorethanoneelement. Thespec-
H Oopacity. tralshapeofthefeatureoftencontainsadditionalinformation,
2
We have run models with individualabundancevariations beyond its EW, but this information is lost in an index. We
forC,N,Na,Mg,Si,Ca,Ti,Fe,andgenericα- elements6,al- willhighlightthesecaveatsinourresultswhereapplicable.
thoughinthepresentworkwefocusprimarilyontheelements We emphasize that our use of indices in this work is re-
C,Na,Ca,Fe,andα. Wehavealsorunmodelswithafactor stricted to “intuition-building” in the form of index-index
oftwovariationinthetotalmetallicity,Z,withabundancera- plots— we do notadvocateusing indiceswhenactually fit-
tiosfixedattheirsolarvalues. Wevaryeachabundanceboth ting data because of the limitations of indices highlighted
higherandlowerthanitscanonicalvalue. Thisisincontrast above.Themodelspectrawillbemadeavailableuponrequest
topreviouswork,whichonlyincreasedtheabundancesofel- so that others can measure their own indices and/or directly
ementsinordertogaugetheireffectonthespectrum. comparemodelstodataattheappropriateresolution.
2.5. Spectralindices 2.6. Behaviorofthestellarlibraries
Inlatersectionswewillrelyonspectralindicesinorderto Figure6showssurfacegravity-sensitivespectralregionsfor
investigate the sensitivity of various spectral features to the selectedIRTFstars.Thisfigurehighlightsthepotentialpower
parameters age, metallicity, and IMF. Spectral indices have ofcertain spectralfeaturesatdiscriminatingbetweendwarfs
beenutilizedfordecadestocompresstheamountofinforma- andgiants. InthisfigureanM4dwarfiscomparedtoanM4
tioninaspectrum.Themostwell-knownaretheLick/IDSin- giant. These two stars have the same spectral type and thus
dices (Bursteinetal. 1984; Wortheyetal. 1994; Trageretal. broadlythesame SED shape, butclearlyhaveverydifferent
1998).Theseindicesresideintheblueregionofthespectrum strengths of these spectral features. The M4 giant displays
(4000Å<λ<6400Å), and include indices sensitive to CN, noFeHlinesbutinsteadshowsTiOlinesatalmostthesame
wavelengthas FeH. Also notice the dramatic increase in the
6WeincludetheelementsO,Ne,Mg,Si,S,Ca,andTias“α”elementsin FeH strength toward the latest M dwarfs. These stars also
thepresentwork. show verydifferentfeaturesaroundthe CaII linescompared
8 CONROY&VANDOKKUM
Mass (for dwarfs) Mass (for dwarfs) Mass (for dwarfs)
0.07 0.12 0.58 0.76 0.89 0.07 0.12 0.58 0.76 0.89 0.07 0.12 0.58 0.76 0.89
8 12
empirical giants 10 model giants
6 empirical dwarfs 10 model dwarfs
4 8
0.82 2 0.86 5 0.99 6
NaI 0 CaII FeH 4
0
−2
2
−4
−5 0
−6
2000 3000 4000 5000 6000 2000 3000 4000 5000 6000 2000 3000 4000 5000 6000
T (K) T (K) T (K)
eff eff eff
Mass (for dwarfs) Mass (for dwarfs) Mass (for dwarfs)
0.07 0.12 0.58 0.76 0.89 0.07 0.12 0.58 0.76 0.89 0.07 0.12 0.58 0.76 0.89
14 15 35
12 30
10 10 25
KI1.17 68 CaI1.98 5 CO2.30 1250
4
10
0
2
5
0
−5 0
2000 3000 4000 5000 6000 2000 3000 4000 5000 6000 2000 3000 4000 5000 6000
T (K) T (K) T (K)
eff eff eff
FIG.7.—Dependenceofselectedspectralindicesoneffectivetemperaturefordwarfs(redpointsandlines)andgiants(blackpointsandlines).Theempirical
IRTFspectra (symbols)arecompared tothemodelsynthetic spectra (lines). Atthetopofeachpanel Teff isconverted intostellar massfordwarfs viathe
isochronesdiscussedin§2.2.
tobothslightlywarmerdwarfsandgiants. Thedifferentrela- dwarfs, which is probably due to problems with the FeH
tiveresponseoftheNaIandFeHlinestolater-typeMdwarfs linelist and/or partition function, and a CaI1.98 index for
impliesthattheselinesaresensitivetoIMFvariationsindif- dwarfs that is weaker than observed, which may be due to
ferentways. thefactthatthepseudocontinuumisstronglyaffectedbyH O
2
InFigure7weshowselectedspectralindicesasafunction inthiswavelengthregion.
of effective temperature for giants and dwarfs. This figure The trends in this figure set the stage for the results to be
comparestheempiricalIRTFspectratothesyntheticspectra. discussed in §4 and in particular §4.4 where we discuss the
As expected, the indices NaI0.82 and FeH0.99 are stronger abilityofcombinationsoftheseindicestoconstraintheshape
in dwarfs than in giants, and FeH0.99 becomes very strong ofthelow-massIMFintheintegratedlightofoldstellarpop-
forthecoolestdwarfs. TheCO2.30andCaII0.86indicesalso ulations.
behave as expected in that they are stronger in giants than
dwarfs. Notice though that the CO2.30 index becomes pro- 2.7. Synthesis
gressivelystrongerfordwarfswith T <3000K(seeFigure
eff With the isochrones and stellar libraries described in pre-
6).
vious sections we are now able to construct integrated light
Two new spectral regions are explored in this figure: the
SSP spectra via Equation 1. We emphasize that the synthe-
KIdoubletat1.17µmandtheCaI doubletat1.98µm. These
sisincludesnotonlytheRGBbutalsotheHBandAGB.For
features are both stronger in dwarfs than in giants, but they
a13.5GyrsolarmetallicitypopulationwithaChabrierIMF,
behaveinverydifferentways.TheKI1.17indexisevenmore
starsontheHBandAGBcomprise≈25%oftheintegrated
sensitivetotheverycooleststarsthanFeH0.99,whileCaI1.98
light at 1µm, so their inclusion in the synthesis is essential.
peaks in strength at ≈3700 K (correspondingto ∼0.55M⊙ When adding the HB and AGB we frequently used an em-
fordwarfs).
pirical star that was already counted on the RGB7. In such
It is noteworthy that the synthetic spectra reproduce the
casesonlythestellarmassischangedinaccordancewiththe
qualitative behavior of all the indices shown in Figure 7 L - MrelationontheHBandAGB.
and reproduce in detail the NaI0.82, CaII0.86, KI1.17, and bol
Abundanceeffectsareimplementedinadifferentialsense.
CO2.30 indices. The models extend only to 3170K, so the
This is the standard method of implementing abundance ra-
models cannot be compared to the coolest M dwars. The
tio effects in population synthesis (e.g., Trageretal. 2000;
synthetic spectra predict a CaII0.86 index stronger than ob-
servedforthecoolestgiants, whichmightbe relatedto non-
7Whilethisiscommonpractice,someauthorsuseonlygenuineAGBstars
LTEeffects,aFeH0.99indexstrongerthanobservedformost
inthesynthesis(e.g.,Mármol-Queraltóetal.2008).
COUNTINGLOW-MASSSTARSININTEGRATEDLIGHT 9
2.0 1.4
best model
1.5 M31 GC stack 1.2
x x
u 1.0 u 1.0
fl fl
0.5 0.8
0.0 0.6
%) 20 %) 10
( 10 BC03 ( 5
a a
dat 0 dat 0
odel/ −10 best model solar model odel/ −5
m −20 m −10
0.375 0.380 0.385 0.390 0.395 0.400 0.40 0.41 0.42 0.43 0.44 0.45 0.46
l (m m) l (m m)
1.2 1.2
1.0 1.0
x x
u u
fl fl
0.8 0.8
0.6 0.6
%) 4 %) 4
a ( 2 a ( 2
dat 0 dat 0
el/ −2 el/ −2
d d
o −4 o −4
m m
0.48 0.50 0.52 0.54 0.56 0.58 0.58 0.60 0.62 0.64 0.66 0.68 0.70
l (m m) l (m m)
1.2 1.10
1.05
1.0
x x
u u 1.00
fl fl
0.8
0.95
0.6 0.90
%) 4 %) 4
a ( 2 a ( 2
dat 0 dat 0
el/ −2 el/ −2
d d
o −4 o −4
m m
0.82 0.84 0.86 0.88 0.90 0.92 0.98 0.99 1.00 1.01 1.02
l (m m) l (m m)
FIG.8.—Comparisonofastackedspectrumoffourmetal-richM31GCstomodels. Abest-fitmodelwithvariableabundanceratios(the“bestmodel”),is
comparedtooursolarmetallicitymodelandthehigh-resolutionsolarmetallicitymodelfromBruzual&Charlot(2003,BC03).Bothmodelanddataspectrawere
continuumnormalizedbyathird-orderpolynomialwithineachwavelengthinterval.Thebottomportionofeachpanelshowstheratiobetweenmodelanddatain
percentunits. Thedottedlinesdemarcatea±2%deviation. Forthebest-fitmodel,thedeviationsarelessthan2%redwardof0.4µmandtheroot-mean-square
deviationis1.2%,testifyingtothehighfidelityofourmodels.
Thomasetal. 2003b; Schiavon 2007; Walcheretal. 2009). spectrumofarbitraryabundancepatternRvia:
Theempiricalstellarlibrariesconstitutethebasesetofmod-
f (R)
els,andthesyntheticstellarlibraries,whichexistforarbitrary syn
f(R)= f , (2)
base
abundancepatterns,areusedtodifferentiallychangethebase fsyn(Z⊙)
models. Inpractice,themodelmakesapredictionforanSSP
where f isthespectrumandwehaveimplicitlyassumedthat
10 CONROY&VANDOKKUM
thebaseempiricalmodelshavesolarmetallicity. Indicesare
thenmeasuredfromthesemodelspectra.
OurmodelthusprovidesSSPsforarbitraryIMFs,forstel-
laragesintherange3- 13.5Gyr,andforarbitraryelemental 1.0
abundancepatterns. Wehaveavoidedyoungeragesbothbe-
causetheIRTFlibrarydoesnotcontainhotstarsandbecause 0.8
atyoungeragesthermally-pulsatingAGBstars,whoseevolu-
tionishighlyuncertain,contributesignificantlytothenear-IR x 0.6
u
flux. fl
Wehaveperformedanumberofteststoensurethatourre- 0.4
sultingbasemodelisnotaffectedbyoneormore‘pathologi-
Chabrier
cal’empiricalstellarspectra.Inadditiontoinvestigatingeach
0.2 x=3
empiricalspectrumbyeye,wehaveconstructedasetofmod-
elsbyremovingonestellarspectrumatatimefromthelibrary
andconstructinganewSSPbasedontheseN- 1spectra.The 1.10 V10 mode ls
resulting indices never changed by more than 2% from our
1.05
fiducial model. As will be shown in later sections, the in-
dexvariationsinducedbymodestvariationsinage,IMF,and
o 1.00
abundances are always much larger than 2%. We conclude ati
thatourresultsarenotundulysensitivetoanyparticularstar x r
intheempiricallibrary. flu 0.95
3. ABRIEFCOMPARISONTODATA 0.90 x=3/Chabrier
Salpeter/Chabrier
In this section we briefly explore the quality of the mod-
els constructed in the previous section by comparing them 0.85
to spectra of globular clusters (GCs). There are a number 0.5 1.0 1.5 2.0 2.5
of massive metal-rich ([Fe/H]&- 0.1) ancient GCs in M31 l (m m)
(Caldwelletal. 2011). Such objects are ideal to compareto
our models because our models are applicable to old metal- FIG.9.—Modelspectraforsolarmetallicityandanageof13.5Gyr.Spec-
traareshownforthreeIMFsandhavebeensmoothedbyaGaussianwith
richsystemsandbecausemassiveGCswillhaveafullypop-
FWHM=40Å inordertohighlightthebroadbanddifferencesbetweenmod-
ulated RGB. Red and near-IR spectra were obtained for the elswithdifferentIMFs.Thegapat0.74- 0.81µmisduetothegapinwave-
GCs B143, B147, B163, and B193 from the LRIS spectro- lengthcoveragebetweentheempiricalMILESandIRTFspectrallibraries.
graphontheKeckItelescope. Thesedataweredescribedin Toppanel: ComparisonbetweenmodelswithaChabrierandanx=3IMF.
Spectraarenormalizedtounityat1.6µm.Bottompanel:Fluxratiobetween
vanDokkum&Conroy (2011). Blue spectra were obtained
modelswithdifferentIMFs,asindicatedinthelegend.Alsoshownareratios
from the Hectospec spectrograph on the MMT by Nelson ofSSPswithdifferentIMFsfromVazdekisetal.(2010,greenlines).
Caldwell (unpublished). The spectra were stacked to create
bandsintheblue,theMgbdoubletat0.52µm,theFeIlinesat
an averageGC spectrum with typicalsignal-to-noise>100.
0.48- 0.54µm,theNaDlineat0.59µm,andtheCaIItripletat
Therearesignificantskysubtractionissuesinthewavelength
∼0.86µm.
regions 0.7µm<λ<0.8µm and 0.93µm<λ<0.98µm so
This figure demonstrates that the models we have con-
thoseregionswillbeomittedinthefollowingcomparison.
structed are capable of reproducing observed spectra at the
Figure8showsthecomparisonbetweenourmodelsandthe
percent level over the entire wavelength range of 0.37µm<
stackedM31GCspectrum. Withineachwavelengthinterval
λ<1.02µm. This comparison also places strong limits on
thespectrahavebeencontinuumnormalizedbyathird-order
thepresenceoftelluriclinesinourmodelspectrabecausethe
polynomial. Our best-fit model was obtained via a Markov
M31 GC spectra are at different observed wavelengths than
ChainMonteCarlofittingtechniquewithvariationineachof
ourempiricalstellarlibrary(TherecessionvelocityofM31is
theelementsC,N,Na,Mg,Si,Ca,Ti,Fe,andwithlock-step
variationintheelementsO,S,andNe.TheIMFisalsovaried, - 300 kms- 1, which means that the GC spectra were shifted
andthe age is keptfixedat13.5Gyr. The data providevery onaverage5- 10Å betweentheobservedandrestframe).Ev-
strongconstraintsontheabundancesoftheseelementsandset idently, sky subtraction and sky absorption corrections have
astrongupperboundontheslopeoftheIMF.Discussionof not introduced errors at the level &1% in either the synthe-
the derivedparametersis beyondthe scope of thispaper but sizedspectraortheM31spectra.
isthefocusofongoingwork.
The purpose of Figure 8 is to demonstrate the high qual- 4. BEHAVIOROFTHEMODELS
ity of our models. The rms differencebetween the data and Inthissectionwediscusshowthemodelsbehaveasafunc-
our best-fit model is 1%, and the percent difference at any tion of the IMF, stellar age, and elementalabundances. The
wavelength point rarely exceeds 2% at λ>0.4µm. In this goalofthissectionistodemonstratethattheintegratedlight
figure we also include comparison to our solar metallicity spectrumofanoldpopulationissufficienttojointlyconstrain
model, and the high-resolutionsolar metallicity model from themeanstellarage, metallicity,abundancepattern,andlow
Bruzual&Charlot (2003). Both of these models assume a mass IMF (0.08M⊙ < M < 1M⊙). All of the models pre-
Chabrier IMF. The solar metallicity models provide a rea- sentedinthissectionareSSPs—thatis,theyarecoevaland
sonable fit to the data over much of the optical wavelength mono-metallic. Throughoutthis section we frequentlyhigh-
range, but at essentially every wavelength the best-fit model light the IMF-dependence of the integrated spectra by con-
performs better. Regions where the best-fit model substan- trasting a Chabrier IMF with a relatively extreme bottom-
tially outperforms the solar models include the CH and CN heavyx=3IMF.Also,inthissectionwerelyonspectralin-