Table Of ContentMNRAS000,1–??(2017) Preprint17February2017 CompiledusingMNRASLATEXstylefilev3.0
The statistical challenge of constraining the low-mass IMF
in Local Group dwarf galaxies
Kareem El-Badry,1(cid:63) Daniel R. Weisz,1 and Eliot Quataert1
1Department of Astronomy, University of California, Berkeley, CA, USA.
7 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
1
0
2
ABSTRACT
b WeuseMonteCarlosimulationstoexplorethestatisticalchallengesofconstrainingthe
e characteristic mass (m ) and width (σ) of a lognormal sub-solar initial mass function
c
F (IMF) in Local Group dwarf galaxies using direct star counts. For a typical Milky
6 Way(MW)satellite(MV =−8),jointlyconstrainingmc andσ toaprecisionof(cid:46)20%
1 requires that observations be complete to (cid:46) 0.2M(cid:12), if the IMF is similar to the MW
IMF. A similar statistical precision can be obtained if observations are only complete
] down to 0.4M , but this requires measurement of nearly 100× more stars, and thus,
(cid:12)
A
a significantly more massive satellite (M ∼ −12). In the absence of sufficiently deep
V
G data to constrain the low-mass turnover, it is common practice to fit a single-sloped
. power law to the low-mass IMF, or to fit mc for a lognormal while holding σ fixed.
h
We show that the former approximation leads to best-fit power law slopes that vary
p
withthemassrangeobservedandcanlargelyexplainexistingclaimsoflow-massIMF
-
o variations in MW satellites, even if satellite galaxies have the same IMF as the MW.
r Inaddition,fixingσ duringfittingleadstosubstantiallyunderestimateduncertainties
t
s in the recovered value of mc (by a factor of ∼ 4 for typical observations). If the
a IMFs of nearby dwarf galaxies are lognormal and do vary, observations must reach
[
down to ∼mc in order to robustly detect these variations. The high-sensitivity, near-
2 infraredcapabilitiesofJWSTandWFIRSThavethepotentialtodramaticallyimprove
v constraints on the low-mass IMF. We present an efficient observational strategy for
7 using these facilities to measure the IMFs of Local Group dwarf galaxies.
4
3 Key words: stars: mass function – stars: low-mass – methods: statistical
2
0
.
1
0 1 INTRODUCTION 2010,forareview).Someworkshave foundintegratedlight
7 signatures of low-mass IMF variations in the centers of
1 A persistent uncertainty in extragalactic astronomy is
massive early-type galaxies (van Dokkum 2008; Conroy &
: whether the stellar initial mass function (IMF) varies with
v van Dokkum 2012; Cappellari et al. 2012; Smith 2014; van
environment and/or cosmic time. Many theoretical models
i Dokkum et al. 2016), but these results require further in-
X of star formation hold that the shape of the IMF is set
terpretation (McConnell et al. 2016; Coulter et al. 2017).
by local variables such as the Mach number in molecular
r Most IMF studies carried out with direct star counts have
a clouds (e.g. Padoan et al. 1997; Hopkins 2013), the local
found little evidence of IMF variation in the Local Group
Jeans length (e.g. Larson 1998; Narayanan & Dav´e 2012),
(Bessell & Stringfellow 1993; Wyse et al. 2002; Chabrier
or the gas pressure in star-forming regions (e.g. Krumholz
2003a; Covey et al. 2008; Bastian et al. 2010; Bochanski
2011;Krumholzetal.2016),andthuspredictthattheIMF
et al. 2010).
foraparticularstarformationeventshoulddependtosome
extent on the local gas density and metallicity, and on the Dwarf spheroidal and ultra-faint galaxies in the Lo-
instantaneousstarformationrate(seeOffneretal.2014,for cal Group provide a natural laboratory in which to search
a review). for environmental variations in the IMF. Compared to the
On the other hand, most observational studies have Milky Way (MW), Local Group satellites have extremely
measured mass functions consistent with a universal IMF low stellar densities, low metallicities ([Fe/H] ∼ −2), high
across a wide variety of environments (see Bastian et al. alpha abundances ([α/Fe]∼0.4), and high mass-to-light ra-
tios (M/L ∼100, in solar units) (Tolstoy et al. 2009; Frebel
& Norris 2015). In addition, many (though not all) Local
(cid:63) E-mail:[email protected] Group dwarf galaxies exhibit relatively simple star forma-
©2017TheAuthors
2 El-Badry et al.
tion histories, having formed all their stars within (cid:46)2 Gyr, M
V
before z∼3(Brownetal.2012;Vargasetal.2013;Webster 12.9 10.5 9.7 8.9 7.7 6.45.2
etaetnmdaalt.thi2ce0I1MM5W;FW.veairsizateitonalb.e2t0w1e4e)n. OLonceaml Gigrhotupthduwsaerxfpgeacltasxyiess- =Mmin mc=0.22 power law, α Mobs Mmax
ObservationalsearchesforIMFvariationsinMWsatel- dm mit =1.55
l(i1t9e9s8h)aavnedthWusysfearetyiaelld.e(d20m02ix)efdournedsulItMs:FGsriinllmthaierDetraaclo. dn/ ning li mc=0.4
andUrsaMinordwarfspheroidalgalaxiesthatwereroughly =100 bur
consistent with the IMF measured in the MW and in its m) en
g
( o
globular clusters, and Kalirai et al. (2013) found the IMF ξ dr
y typical
oftheSmallMagellanicCloud(SMC)tobeonlymarginally h
M = all stars observations
shallower than that of the MW. On the other hand, Geha star
et al. (2013) found the IMFs of the Hercules and Leo IV 0.1 0.2 0.3 0.4 0.5 0.60.7
ultra-faintdwarfgalaxiestobesignificantlyshallower(more initial mass (M )
bottom-light) than the canonical MW IMF. fl
All published attempts to measure the IMFs of nearby
dwarf galaxies have been limited at low masses by observa- Figure 1.Schematicillustrationofthechallengeofconstraining
tionalsensitivity.AtadistanceofD=150kpc,HubbleSpace the low-mass IMF. Observations sample the IMF between Mobs,
Telescope(HST)photometricstudiesbecomeincompletebe- the lowest stellar mass which can be reached by observations,
low stellar masses of ∼ 0.5M(cid:12). This is significantly more and Mmax, the highest stellar mass which remains on the main
massive than the characteristic mass at which the canoni- sequence. The total mass Mstar corresponds to all stars between
cal MW IMF turns over and begins to significantly deviate Mmin and Mmax,manyofwhichcannotbeobserved.Whenobser-
vationsdonotreachthelow-massturnover,bothpowerlawIMFs
from a power law (mc ∼ 0.22M(cid:12)). No published observa- and lognormal IMFs with a wide range of characteristic masses
tionalstudieshavedetectedaturnoverinthemassfunctions
mc areconsistentwiththedata.
ofnearbydwarfgalaxies,andtheyhavethusalloptedtofit
a simple power law IMF. However, measuring the turnover
intheIMF(ifindeeditexists)isparticularlydesirable,asit JWST will improve the constraints on the IMFs of nearby
wouldprovideadirecttestoftheoreticallypredictedscalings dwarfgalaxies,andwelayoutanefficientobservingstrategy
of mc with metallicity and density. for characterizing IMF variations in the Local Group with
Thestatisticalpowerofresolvedstellarpopulationstud- JWST. We summarize our results in Section 6.
iestoconstrainthelow-massIMFwillimprovesignificantly
with the introduction of the James Webb Space Telescope
(JWST), whose larger collecting area and improved near- 2 METHODS
infraredsensitivitywillallowobservationstoprobetheIMFs
2.1 Lognormal IMF
ofLocalGroupgalaxiestolowermassesthanpastHSTob-
servations. In the nearest MW satellites, JWST will effi- We assume that the true IMF from which masses are sam-
cientlyobservedowntothehydrogenburninglimit,making pled is a lognormal of the form advocated by Chabrier
it possible to constrain the characteristic mass and slope of (2003b), which at sub-solar masses is given by
the IMF with unprecedented precision.
(cid:34) (cid:35)
Inthispaper,weuseMonteCarlosimulationstoinves- ξ(m)= dn =ξ 1 exp −(logm−logmc)2 , (1)
tigatehowtheaccuracytowhichtheIMFcanberecovered dm 0m 2σ2
fromstarcountsscaleswiththesizeandlimitingmagnitude
of the observed stellar sample. We demonstrate that HST where the logarithm is base 10. Here mc = 0.22M(cid:12) is the
“characteristic mass”at which the logarithmic slope of the
resolved stellar population studies do not have the statisti-
cal power to robustly detect IMF variations in most Local IMF is dlogξ/dlogm = −1,1 and σ = 0.57M(cid:12).2 Hereafter,
we refer to a lognormal IMF with these parameter values
Group dwarf galaxies if variation occurs primarily at lower
asa“ChabrierIMF.”Thenormalizationfactor ξ isdefined
masses. However, we find that similar studies with JWST 0
will have the potential to definitively quantify (or rule out) such that ∫MMmminaxξ(m)dm=1, where Mmin and Mmax are the
low-mass IMF variations for galaxies within ∼100 kpc. minimum and maximum of the mass range over which the
We organize this paper as follows. In Section 2, we de- IMF is populated. For a single-age stellar population, Mmax
scribe how we fit the IMF from a sample of masses or a is the highest initial mass which has not yet left the main
color-magnitude diagram (CMD). In Section 3, we explore sequence, and Mmin is taken to be the hydrogen burning
howaccuratelytheparametersofaMW-likelognormalIMF limit, Mmin ≈ 0.08M(cid:12) (e.g., Kumar 1963; Grossman 1970;
can be constrained with observations reaching a range of
limiting magnitudes. We then investigate the systematics
introduced by simplifications which are commonly made in 1 Note that mc is not the most probable mass or the peak of
the mass probability distribution function dn/dm. The peak is
the absence of sufficiently deep observations, such as fitting at mcexp{−[ln(10)σ]2},whichisalwayslessthan mc. mc isthe
a single power law or holding one of the parameters of a
peakofdn/dlogm, theprobabilitydensityfunctioninlogspace.
lognormal fixed during fitting. In Section 4, we investigate 2 These parameters are for the“system”IMF, which represents
what observations are required to detect variation in the the observable mass function after the single-star IMF is con-
IMF. In Section 5, we discuss how future observations with volvedwiththedistributionofunresolvedmultiple-starsystems.
MNRAS000,1–??(2017)
Constraining the Low-Mass IMF 3
Liebert & Probst 1987). Throughout this work, we take quantifytheaccuracywithwhichtheobservablesystemIMF
Mmax = 0.77M(cid:12), which is approximately the highest mass can be recovered; additional modeling is required to trans-
in a stellar population with Fe/H=−2 that remains on the late the system IMF into a single-star IMF.
mainsequenceafter12.5Gyr(Paxtonetal.2011;Choietal.
2016).
Figure 1 illustrates the typical situation for observa- 2.3 Fitting the IMF
tional IMF studies of Local Group dwarf spheroidal and
Recovering the low-mass IMF from an observed sample of
ultra-faint galaxies. Since these galaxies are dominated by
stars is straightforward compared to the high-mass IMF,
old stellar populations, they have no surviving stars above
because to first order, the galaxy’s star formation history
Mmax≈0.77M(cid:12),andbecauselow-massstarsarefaint,obser-
(SFH)isirrelevant(Miller&Scalo1979;Scalo1986).Since
vations typically do not reach the low-mass turnover. Pub-
lished observational studies of Local Group dwarf galaxies the main-sequence lifetimes of low-mass ((cid:46) 0.75M(cid:12)) stars
exceedtheageoftheuniverse,thepresent-daymassfunction
thus far have, at best, reliably sampled the IMF down to
is identical to the IMF at low masses, irrespective of the
snteealrlarsamteallsisteess (oDf M=ob6s0∼−700.4Mkp(cid:12)c,; fWorysoebseetrvaalt.io2n0s02o;fKvaerliy- galaxy’s star formation history.4
Weexperimentedwithtwodifferentmethodsforfitting
raietal.2013).Studiesofmoredistantsatellites(especially
theIMFtoobservations.Thefirstmethodisidealized,with
older studies using WFPC2) have only reached down to
Mobs=0.5−0.6M(cid:12) (Grillmairetal.1998;Gehaetal.2013). the assumption that the masses of individual stars can be
measuredexactly.Inthiscase,theprobabilityofmeasuring
If observations do not reach down to the low-mass
a particular mass is simply the value of the IMF at that
turnover, one is faced with the underconstrained problem
mass.Assumingmassesareindependentandidenticallydis-
of inferring the form of the IMF from observations sam-
tributed,thelikelihoodfunctionforasetofstellarmassesis
pling a narrow mass range in which the IMF has a near
simply the product of the probability of each mass. Explic-
constantslope. As Figure 1illustrates,theChabrier IMFis
effectively indistinguishable from a power law (ξ(m)∝m−α, itly, given a sample of masses mi and a functional form of
where α = 2.35 is the Salpeter value) with α ≈ 1.55 in the theIMFξ(m,θj),whereθj arefreeparameters,thelikelihood
function is
mass range (0.4 − 0.77)M(cid:12). Data drawn from a Chabrier
IMFinthismassrangearealsoconsistentwithhavingbeen L=p(cid:0)mi|θj(cid:1) =(cid:214)ξ(cid:0)mi,θj(cid:1). (2)
drawn from a wide range of lognormal IMFs with different
i
characteristic masses mc (and appropriately chosen σ val-
ues; e.g., σ = 0.57M(cid:12) for mc = 0.22M(cid:12), or σ = 0.36M(cid:12) for An advantage of this approach compared to the traditional
mc = 0.4M(cid:12)). In general, if both mc and σ are allowed to method of essentially fitting a line to a histogram is that it
doesnotrequireanybinning.Binningmasseswhenfittinga
vary, observations which reach a lower mass limit M and
obs
distributionintroducesunnecessaryambiguityinthechoice
samplealognormalIMFwillbeconsistentwithanylognor-
of bins and can lead to biases in the inferred distribution,
mal IMF with mc ∈[0,Mobs].
particularly when there are unequal numbers of samples in
each bin (e.g. Ma´ız Apella´niz & U´beda 2005; Maschberger
2.2 Sampling from the IMF & Kroupa 2009).
Wealsoexperimentedwithamorerealistic(thoughstill
WedrawmassesfromalognormalIMFusinginversetrans- idealized), observationally motivated fitting approach using
formsampling,asdescribedinAppendixA.Inpractice,ob- synthetic CMDs generated for a stellar population with a
servations cannot directly recover the true single-star IMF, realistic completeness and metallicity distribution function.
even if observations are arbitrarily deep and can measure In this case, it is not possible to determine the mass of any
stellar masses with zero uncertainty. This is because a frac- individual star exactly, since stars of different metallicities
tionofstarsareinunresolvedmultiple-starsystems(e.g.,bi- have different magnitudes at fixed initial mass.
naries)andthusappearinobservationsasasinglestarwith We describe this approach in detail and compare the
the combined magnitude of all components, corresponding results of CMD fitting to the idealized approach of fitting
toahigherinferredmass.Thiseffectwillmaketheobserva- massesdirectlyinAppendixC.Wefindthat,ifthemetallic-
tionally inferred IMF slightly more top-heavy (i.e., steeper) itydistributionfunctionandobservationalcompletenessare
thantheunderlyingsingle-starIMF,thoughthiseffectonly known, the two fitting approaches yield effectively identical
becomes significant at m(cid:46)0.3M(cid:12) (Kroupa et al. 1991). best-fit values, covariances, and marginalized uncertainties
Rather than explicitly modeling unresolved multiple- inIMFparameters.Thatis,inthelimitofzerophotometric
starsystems,whichhavepoorlyconstrainedbinaryfractions uncertaintyandperfectstellarmodels,noinformationislost
andmassratiodistributions,wedrawmassesfromthe“sys- by fitting observables as opposed to fitting masses directly
tem”IMFreportedinChabrier(2003b).ThesystemIMFis
=observationally motivated: it represents the IMF derived
from the system luminosity function, which is produced by
isderivedbyapplyingasingle-starmass-magnituderelationship
“merging”thecomponentsofeachmultiple-starsysteminto
toanunresolvedmultiple-starsystem.
asingleunresolvedsource.3Thetestswecarryoutherethus 4 Strictlyspeaking,SFHcan haveaslighteffectontherecovered
IMF, because there is weak evolution in the magnitudes of low-
massstarsevenwhiletheyremainonthemainsequence.However,
3 Note that this is defined differently from e.g. the system IMF thiseffectintroducesnegligibleuncertaintyintherecoveredlow-
of Kroupa et al. (2013), which describes the distribution of the massIMF(Gehaetal.2013),becausethemain-sequencelifetimes
totalmassesofmultiple-starsystems.TheChabriersystemIMF oflow-massstarsaremanytimestheageoftheUniverse.
MNRAS000,1–??(2017)
4 El-Badry et al.
as in Equation 2. Having demonstrated as a proof of con- which completeness falls off steeply because stars are too
cept that both fitting approaches produce the same results faint to observe or observations become crowding limited.
in Appendix C, we use the first approach (directly fitting We translate magnitude thresholds in a given filter to M
obs
masses) exclusively for our primary analysis in the interest by interpolating on a grid of isochrones from the MESA
of computational cost. Isochrones and Stellar Tracks project (Mist; Choi et al.
For both fitting approaches, we assume flat priors for 2016), as explained in Appendix B. For this conversion, we
the free parameters of the IMF. When fitting a lognormal, assumeastellarpopulationofage12.5Gyrwithmetallicity
weusethepriorsmc ∈[0.01,1.5]×M(cid:12) andσ ∈[0.1,2]×M(cid:12); [Fe/H]=−2.
whenfittingapowerlaw,wetakeα∈[0,5].Wehaveverified
that our priors are wide enough to be noninformative; that
is, increasing the range of the priors does not change the 3.2 Fitting results
shape of the recovered posterior.
We use the affine-invariant ensemble sampler emcee Figure 2 shows the constraints on mc and σ obtained by
fitting a lognormal IMF to a range of mass samples drawn
(Foreman-Mackey et al. 2013) to sample from the poste-
from a Chabrier IMF. The white lines and blue shaded re-
rior. To verify that our chains have converged, we use the
gions respectively show the median and middle 68 percent
Gelman-Rubin convergence diagnostic (Gelman & Rubin
1991; Gelman et al. 2013). Formally, we require Rˆ < 1.05 ofthemarginalizedposteriordistributionsfor mc (top)and
forallchains,where Rˆ isthepotentialscalereductionfactor σ(bottom).Largeshadedregionsindicatesignificantuncer-
(PSRF).5 We find that in practice, drawing 20,000 samples tainty in the IMF parameters, due both to the degeneracy
from the posterior is always sufficient to satisfy this diag- betweenmc andσ andtostochasticsamplingeffectsarising
from the finite number of stars being sampled (Elmegreen
nostic threshold.
1999;Hernandez2012).Smallerregionsindicatethatmc and
σ are individually well constrained.
In the left two panels, we generate a stellar popula-
3 RECOVERING THE PARAMETERS OF A tionwithtotalmass Mstar =104M(cid:12) (correspondingto Ntot=
LOGNORMAL IMF 34,287 between 0.08M(cid:12) and 0.77M(cid:12)) and vary Mobs, the
In this section, we investigate how accurately the parame- massdowntowhichstarscanbeobserved.AtMobs=0.4M(cid:12),
ters of a lognormal IMF can be recovered from a sample of this Mstar corresponds to Nobs = 8,686 stars included in the
fit, which is a factor of a few more than the number used
masses. We begin by exploring how the observational mass
inmostpreviousobservationalstudiesofLocalGroupdwarf
lower-limit Mobs and total stellar mass Mstar of a sample af-
galaxies.
dfeecgtetnheerasctyrebnegttwheeonf cthonesttwraoipntasraomnemtecrsa.nWdeσa,ssausmweeltlhaasttthhee For Mobs (cid:38)0.2M(cid:12), the middle 68% ranges for both mc
true IMF is a Chabrier lognormal IMF with mc = 0.22M(cid:12) and σ are large, indicating weak constraints on the indi-
and σ = 0.57M(cid:12). We consider the case in which the true vidual IMF parameters. As we will show, this is largely the
IMF does vary in Section 4. result of the degeneracy between mc and σ. Strong con-
straintsonmc andσ canbeobtainedonlywhen Mobs(cid:46)mc.
This makes intuitive sense, since the lognormal IMF has a
3.1 Overview nearly constant slope at m (cid:29) mc and only begins to turn
over at m∼mc. Above the turnover, mc and σ are strongly
Wesimulateobservationsofagalaxywithtotalstellarmass
degenerate, because a wide range of lognormal IMFs with
MMsmtaarx a=nd0.a77CMh(cid:12)abbriyerdrIMawFingbetNwobesenmMasmseins =fro0m.08tMhe(cid:12)IaMnFd 0ur(cid:46)e 1m).c (cid:46) Mobs can all fit the data equally well (see Fig-
between Mobs and Mmax.6 Due to incompleteness at lower In the right panels of Figure 2, we again show con-
masses, only the fraction of stars cobs with masses greater straintsonmc andσ,butnowvaryMstarwhileMobs=0.4M(cid:12)
than M can be observed, so
obs is held fixed. As expected, stronger constraints can also be
obtainedbyincreasingMstar,butverylargenumbersofstars
Nobs=Ntotcobs= Mmstarcobs=Mstar∫∫MMMmombasxaxξξ(m(m))mddmm. (3) asMarsemtarreemqimuuipsrterodbvteeomionebcnrtteaaoinsnemdthebeaynfiiandguffcauicaltlcoorcnoosnftsrt∼arian1itn0st.stFotohorabettxacaiamnnptbhleee,
Mmin
obtained by decreasing Mobs by 0.2M(cid:12).
In practice, the lower-limit for observations is not a
For reference, Figure 2 also shows the absolute magni-
threshold mass, but rather a magnitude threshold below
tudes of individual stars (left) and the total galaxy (right)
thatcorrespondtotheplottedvaluesof Mobs and Mstar.We
5 The PSRF quantifies the ratio of the mean variance within calculate MV for individual stars from the Mist isochrones
individual chains to the variance of the mean across all chains. asexplainedinAppendixBandMV forawholegalaxyusing
In the limit of infinite samples, Rˆ →1; values of Rˆ greater than Equation 4.
∼1.1indicatepoorconvergence(Gelmanetal.2013).
6 Inthisformulation,Mstarrepresentsthetotalstellarmassinthe
region of the galaxy covered by observations. If only a fraction 3.3 Fitting a power law IMF to data drawn from
of the galaxy is observed, the total stellar mass will be greater a lognormal
than Mstar. Note also that Mstar represents only the initial mass
contributionofmain-sequencestars.Wedonotaccountformass Intheabsenceofsufficientlydeepobservationswiththesta-
loss,evolvedstars,orremnants. tistical power to jointly constrain mc and σ of a lognormal
MNRAS000,1–??(2017)
Constraining the Low-Mass IMF 5
M total M
V V
6.4 7.7 8.9 9.7 10.5 12.9 -5.0 -7.5 -10.0 -12.5
0.4
0.3
)
fl
M
0.2
(
c
m
0.1
0.0
0.8
)
fl
0.6
M
(
σ
0.4
fixed M =104M fixed M =0.4M
star obs
fl fl
0.2
0.6 0.5 0.4 0.3 0.2 0.1 103 104 105 106
M (M ) M (M )
obs star
fl fl
Figure2.Best-fitvaluesofmc (top)andσ(bottom)obtainedfromfittingalognormalIMFtosamplesofstarsdrawnfromaChabrier
IMF. White line and shaded regions show the median and middle 68% of the maginalized posterior for each IMF parameter. Left: we
fix the total mass of all stars in the range 0.08−0.77M(cid:12) at 104M(cid:12) and vary Mobs, the mass threshold down to which stars can be
observed. Right: we fix Mobs at 0.4M(cid:12) (corresponding to a magnitude threshold of MV ≈8.9) and vary the total stellar mass. Tight
constraintsonmc andσcanbeobtainedeitherbyobservingsystemswithmorestarsorbypushingobservationstofaintermagnitudes,
but unrealistically large stellar samples are needed to meaningfully constrain the low-mass the IMF at the magnitude limits of current
studieswithHST.
IMF,itiscommontomakesimplifyingassumptions,suchas α≈1.55.Thissuggeststhat,forthischoiceofMstar =104M(cid:12),
fittingasinglepowerlawinplaceofalognormalorholding observationswouldhavetoreach Mobs(cid:46)0.3M(cid:12) beforethey
σ fixed when fitting a lognormal (e.g. Grillmair et al. 1998; could begin to detect deviations from a power law.
Wyse et al. 2002; Geha et al. 2013; Kalirai et al. 2013).
Figure3alsoshowstheresultsofpreviousstudieswhich
This avoids the degeneracy between mc and σ, since either havefitpowerlawIMFstostarcountdatafromLocalGroup
approachessentiallyreducesthefittoaonedimensionalop-
dwarf galaxies. Black stars show results for ultra-faint and
timization problem.
dwarf spheroidal galaxies, which have very low metallici-
Figure3showstheresultsoffittingapowerlaw(ξ(m)∝ ties, no gas, and are dominated by old stellar populations;
m−α) to masses drawn from a Chabrier IMF. As in the left we plot results for Draco (Grillmair et al. 1998), Ursa Mi-
panelofFigure2,weholdMstarfixedat104M(cid:12) whilevarying nor (Wyse et al. 2002), Hercules and Leo IV (Geha et al.
M .Here,the68percentprobabilityrangeinα isnarrow: 2013). We also show the result of Kalirai et al. (2013) for
obs
for Mobs (cid:46) 0.5M(cid:12), α is constrained within (cid:46) 10%. The im- theIMFoftheSMC;weplotthispointinbluetoemphasize
proved constraints on α at lower M primarily reflect the the SMC’s more complicated star formation history. These
obs
fact that deeper observations increase the total number of previous studies have reached down to M values between
obs
stars included in the fit. 0.37M(cid:12) and 0.6M(cid:12). None of them detected a turnover at
lower masses, and they thus all opted to fit a single power
The best-fit logarithmic slope α becomes shallower for
law.
decreasing M , since deeper observations begin to sample
obs
the mass regime near the turnover in the IMF. However, We note that the best-fit logarithmic slope of α ∼1.55
for values of Mobs in the range (0.3−0.6)M(cid:12) (with Mmax = for masses in the range typically probed by observations
0.77M(cid:12)),thebest-fitlogarithmicslopeisalmostconstantat (0.4M(cid:12) (cid:46) m (cid:46) 0.77M(cid:12)) is significantly shallower than the
MNRAS000,1–??(2017)
6 El-Badry et al.
M nificantlydifferentpredictionsforobservationswhichsample
V
12.9 10.5 9.7 8.9 7.7 6.4 only a narrow range of masses near 0.5M(cid:12).
Second,thelogarithmicslopeoftheKroupaIMFsteep-
ens discontinuously from α = 1.3 to α = 2.3 at m = 0.5M(cid:12).
2.2 Thus, although the shallower IMF slopes found in Local
Dra
Groupdwarfgalaxiesthusfararesignificantlydifferentfrom
α 2.0
w SMC theKroupavalueatm≥0.5M(cid:12),theyareconsistentwiththe
UMi Kroupavalueatslightlylowermasses.Asthereislittlephys-
a 1.8
l icalreasontoexpectadiscontinuityintheslopeoftheIMF
r
e 1.6 (Miller & Scalo 1979), one might reasonably expect inter-
w mediateαvaluesbetween1.3and2.3atmassesnear0.5M(cid:12).
po 1.4 Kalirai et al. (2013) interpreted their best-fit slope of
LeoIV α=1.90+0.15 as an indication that the SMC’s IMF is some-
1.2 Kroupa (2002) −0.10
Her whatshalloweratlowermassesthantheKroupavalue,and
significantly shallower than the α = 2.35 prediction from
1.0
0.1 0.2 0.3 0.4 0.5 0.6 Salpeter (1955). Figure 3 shows that the measured slope is
actually somewhat steeper than what would be measured
M (M )
obs if masses were drawn from a Chabrier IMF in the range
fl
0.37M(cid:12) ≤ m ≤ 0.77M(cid:12). However, Kalirai et al. (2013) fit
tFaiignuedreby3.fitBtiensgt afitpolowgearriltahwmtico mIMaFssesslodpreaw(nξ(fmro)m∝amC−hαa)broiber- imngasstehseiurpMtmoaxMamnadx =Mm0i.n9,3Mwe(cid:12)firnadthαer=th1a.n580+0..007.5076Mf(cid:12)o.rAmdaospsets-
drawnfromaChabrierIMF.Thisisslopeisshallowerthan
lMogmnaxor=m0a.l77IMM(cid:12)F.F(witittihngtoataplowMesrtalraw=t1o04oMbs(cid:12)e)rvbateitownesenwhMichobsreaancdh the SMC value, but only at the (2−3)σ level.
Mobs ∼ 0.5M(cid:12) yields a logaritmic slope α ∼ 1.55, which is sig- Finally, we note that, with the exception of the data
nificantly shallower than than the canonical value of α = 2.3. forUrsaMinor,theobservationalresultsplottedinFigure3
Observationally-inferredIMFconstraintsareoverplottedforfive wereobtainedinstudiesthatattemptedtoexplicitlymodel
LocalGroupdwarfgalaxies:UrsaMinor(Wyseetal.2002,noer- the effects of binaries and recover the single-star IMF, not
rorbarsincluded),HerculesandLeoIV(Gehaetal.2013),Draco the system IMF. The system IMFs corresponding to these
(Grillmair et al. 1998), and the Small Magellanic Cloud (SMC; resultsareexpectedtobeslightlyshallower(lowerα).How-
Kaliraietal.2013).Notethatthefirstfourgalaxiesaredominated
ever, as demonstrated by Kroupa et al. (1991, their Figure
byoldstellarpopulations,whiletheSMCsamplecontainedsome
3), accounting for binarity has negligible effects on the in-
youngerstars.
ferred IMF except at very low masses (m (cid:46) 0.3). We find
that at m ≥ 0.4M(cid:12), the single-star and system IMFs re-
Kroupa (2002) value of α = 2.3 for m > 0.5M(cid:12) and the portedinChabrier(2003a)differbylessthan10%,sowedo
Salpeter (1955) value of α = 2.35 at all masses. Previous not expect binarity to significantly change our results.
However,futureobservationswithJWSTwillprobethe
studies have suggested that the shallower α values mea-
IMFs of nearby dwarf galaxies to significantly lower masses
suredinLocalGroupdwarfgalaxiesareevidenceofsystemic
(see Section 5), and we expect more significant differences
variation in the the IMF with environment. For example,
Geha et al. (2013) pointed out that the value α = 1.2+0.4 between the system and single-star IMFs in this regime. In
−0.5 particular, because the conversion between the system and
found in Hercules is shallower than the Kroupa value at
single-star IMFs depends on the assumed binary fraction,
the 5.4σ level. However, Figure 3shows that the shallowest
environmental variations in the binary fraction or mass ra-
observationally-inferred IMFs are all approximately consis-
tios in binary systems could lead to variation in the system
tent with having been drawn from a Chabrier lognormal
IMF,evenintheabsenceofvariationinthesingle-starIMF.
IMF.
There are two reasons for this somewhat counterintu-
itive result. First, the Chabrier and Kroupa IMFs, though
3.4 Distinguishing a lognormal IMF from a power
similarinoverallshape,havedifferentinstantaneousslopes:
law
at m=0.5M(cid:12), the Chabrier IMF has an instantaneous log-
arithmic slope dlogξ/dlogm ≈ −1.48; at the same mass, Wenowconsiderhowdeepobservationsmustreachinorder
thelogarithmicslopeoftheKroupaIMFsteepensto−2.3.7 to robustly detect deviations from a power law if the true
Thus, slopes of α ∼ 1.55 at m ≥ 0.5M(cid:12) appear significantly IMF is lognormal. Existing observational studies of Local
shallower than the Kroupa value (the 99% (3σ) confidence Group dwarf galaxies have shown that their IMFs do not
interval on the Kroupa slope is α = 2.3±0.3 in this mass showanydeviationfromapowerlawwithinthemassrange
range (Kroupa 2001)), but are entirely consistent with be- ofobservations,sowewishtodeterminewhetherthisconsti-
ing drawn from a Chabrier IMF. We therefore stress that, tutes evidence that these galaxies have IMFs different from
although many works treat both the Kroupa and Chabrier the MW.
IMFsasidentical“canonical”IMFs,thetwoIMFsmakesig- To this end, we draw masses from a Chabrier lognor-
mal IMF and fit them with both a power law and a log-
normal IMF, for observations reaching a range of values of
7 Note, however, that it is not the instantaneous slope at Mobs, Mobs and Mstar. Then, for each set of model fits, we use the
butrathertheaverageslopebetween Mobs and Mmax,thatdrives BayesianInformationCriterion(BIC;Schwarz1978)toeval-
thefit. uate whether the sample of masses is fit significantly bet-
MNRAS000,1–??(2017)
Constraining the Low-Mass IMF 7
12.9 10.5 9.7 MV 8.9 7.7 6.4 1.50 Mstar=104Mfl, Mmin=0.4Mfl
)C 20 MCMC
BI 104M 105M 106M
w (∆ 15 103M fl fl fl 1.15
ower la 10 fl ∆BIC = "Decisive" )fl 00..999575 ility
st p 5 M 0.80 0.683 0.0 0.2 0.4 ab
gain 0 (σ mc (Mfl) ob
e a 0.57 pr
nc 5 0.45
e
d
vi 10
e 0.1 0.2 0.3 0.4 0.5 0.6
M (M ) 0.10
obs
fl 0.00 0.12 0.25 0.38 0.50
m (M )
c
Figure4.DifferenceinBayesianinformationcriterion(BIC)be- fl
tweenapowerlawIMFandalognormal.Higher∆BICindicates
stronger deviations from a power law and evidence for a lognor-
Figure 5. Likelihood for fitting a lognormal IMF. Orange con-
mal.Inallcases,wedrawmassesfromaChabrierlognormalIMF.
tours enclose regions corresponding to 68.3%, 95.4%, and 99.7%
Then, using only masses above a threshold Mobs, we fit both a probability. Inset shows samples from the posterior distribution
powerlawandalognormaltothesampledmassesandcalculate
the BIC. Correctly inferring that the IMF is lognormal, not a formc obtainedfromMCMCfitting.Bluehistogramcorresponds
to holding σ fixed; black histogram shows results of allowing σ
lpaorwgeersalamwp,lree(qhuiigrhesMeisttahre)r. deepobservations(low Mobs)oravery teoravtae,ryhoalnddinmgaσrgfiinxaedlizdinugr.inBgeficattuisnegmsicgnainfidcaσntlayreunhdigehrelystidmegaetnes-
theuncertaintyinmc.
ter by a lognormal IMF than by a power law. The BIC is
a rough diagnostic for comparing how well different mod-
els fit a particular dataset. It is designed to balance higher Inconclusion,wehaveshownthatpublishedstarcount
likelihood against increasing model complexity, since more dataintheLocalGroupdonotreachsufficientlylowstellar
complexmodelswithlargernumbersoffreeparameterscan masses to robustly measure the low-mass turnover in the
often produce higher likelihood through overfitting. IMF: existing studies have only measured the slope of the
Ingeneral,modelswithlowerBICarepreferred.InFig- IMF over a narrow mass range (0.4M(cid:12) (cid:46) m (cid:46) 0.8M(cid:12)) in
ure4,weplotthedifferenceinBICbetweenapowerlawand whichalognormalIMFiseffectivelyindistinguishablefroma
lognormal IMF as a function of M for different values of powerlaw.IftheIMFislognormal,observationsmustreach
obs
Mstar. A common rule of thumb is that a ∆BIC > 10 is con- approximatelydowntomc,whichforaChabrierIMFism≈
sidered “decisive” evidence against the model with higher 0.22 (MV ≈10.2 for an old, metal-poor stellar population).
BIC (Liddle 2007); thus, IMF realizations which lie above In addition, we find that the shallower IMF slopes
the dashed line in Figure 4 strongly favor a lognormal IMF reported in Local Group dwarf galaxies thus far between
over a power law. Points below the line indicate that there masses of ∼ 0.4M(cid:12) and ∼ 0.8M(cid:12) are approximately consis-
isnotastrongpreferenceforonemodelortheother,or(for tent with a Chabrier IMF, despite apparent disagreement
∆BIC < −10), that a power law model is strongly preferred with the Kroupa slope in this mass regime. We emphasize
over a lognormal. thatdeeperobservationsarerequiredforrobustdetectionof
Figure 4 shows that for Mstar = 104M(cid:12), the lognormal IMF variations.
fitisnotdecisivelypreferreduntil M ∼0.2,comparableto
obs
mc. This can be understood intuitively: the lognormal IMF
has a nearly constant slope at m (cid:29) mc, and it is thus not 3.5 Fixing model parameters during fitting
possibletodistinguishbetweenasimplepowerlawandand
anIMFwhichturnsoveratlowermasseswithinsufficiently Instead of fitting a power law IMF, another possible ap-
deep data. However, statistical power to constrain the IMF proach for constraining the IMF when data are only avail-
increases with increasing Mstar, so that atfixed Mobs, devia- able over a narrow range of masses is to hold one of the
tions from a power law IMF can be detected more robustly IMF parameters fixed during fitting. If either mc or σ is
for larger Mstar. held fixed (and mc (cid:28) Mobs, so the IMF resembles a power
We note that none of the published IMF constraints law), then varying the parameter which is not held fixed
shown in Figure 3 reach sufficiently low masses to robustly effectively varies the power law slope at m(cid:29)mc.
ruleoutaturnoverattheChabriercharacteristicmass.The Figure5showsisoprobabilitycontoursforfittingalog-
constraints from Kalirai et al. (2013) come closest, but we normal IMF to a sample of masses and compares the con-
find that for their sample size (∼ 5000 stars with masses straintsonmc obtainedwhenσ isheldfixedatσ=0.57M(cid:12)
below 0.93M(cid:12)), observations would have to reach down to (theChabriervalue)tothoseobtainedwhenbothσ andmc
∼0.25M(cid:12) beforetheBICwoulddecisivelyfavoralognormal are allowed to vary. In the inset, we show draws from the
to a power law IMF. marginalizedposteriorformc whenbothparametersareal-
MNRAS000,1–??(2017)
8 El-Badry et al.
lowed to vary (black) and when σ is held fixed at 0.57M(cid:12) bilityregionextendsalmosttomc =0.22M(cid:12),thiswouldnot
(blue). represent strong evidence of variation in the IMF.8
Whenbothparametersareleftfree,mcandσarehighly Figure 6 shows that variation in the IMF can only be
covariant. Although essentially any mc (cid:46) 0.4M(cid:12) provides robustly detected if Mobs ∼ mc; that is, if observations are
a good fit when paired with an appropriate choice of σ, deepenoughtodetectthebeginningoftheturnover.Inthe
all acceptable fits reside in a nearly one-dimensional region rightpanel,where Mobs=0.5M(cid:12),awiderangeofIMFswith
of parameter space. Thus, when σ is held fixed, fitting a 0.1M(cid:12) (cid:46) mc (cid:46) 0.5M(cid:12) are all consistent with the Chabrier
lognormal yields much stronger constraints on mc. value. In other words, if observations only reach down to
It is important to note, however, that holding σ fixed 0.5M(cid:12),onecouldobservegalaxieswithlognormalIMFsand
essentially amounts to putting a delta function prior on σ, characteristic masses anywhere in the range 0.1M(cid:12) (cid:46)mc (cid:46)
whichisgenerallynotjustified.Becauseofthestrongdegen- 0.5M(cid:12), and the data would not provide sufficiently strong
eracy between mc and σ, holding σ fixed will dramatically constraints to rule out a Chabrier IMF with mc =0.22M(cid:12).
underestimatethetrueuncertaintyinmc:naivelyinterpret- However, it becomes much easier to accurately con-
ingtheblueposteriorformc inFigure5,onemightconclude strain IMF parameters, and thus, to detect deviations from
that the characteristic mass mc is was well-constrained to a Chabrier IMF, for lower Mobs. For Mobs = 0.3M(cid:12) (mid-
0.2M(cid:12) < mc < 0.3M(cid:12) when in fact, a much larger range of dle),onlytheIMFswith0.2M(cid:12) (cid:46)mc (cid:46)0.4M(cid:12) areapproxi-
values can provide equally good fits. This is not necessarily matelyconsistentwiththeChabriervalue,andtypical68%
a problem, if the constraints from fitting are compared to error bars are smaller than for Mobs = 0.5M(cid:12). Finally, for
otherstudiesinthesamemassrange,sinceallacceptablefits Mobs =0.1M(cid:12), mc can be constrained with percent-level ac-
have basically the same shape across the mass range where curacy inall cases, andallthe IMFswith mc differentfrom
data are available (see Figure 1). the Chabrier value can be ruled out. The dramatically im-
On the other hand, attempts to extrapolate the fitted proved constraints on mc for lower Mobs in this (admittedly
IMFtolowermassesinordertoconstraintheturnovermass idealized)experimentaretheresultoftwofactors:first,ob-
will yield erroneously strong constraints: clearly, if observa- servations reach low enough that the turnover can be de-
tionsonlysampleasmallrangeofmassesatm(cid:29)mc,where tecteddirectlyforlower Mobs,andsecond,thetotalnumber
theIMFhasnotyetbeguntoturnover,theycannotreliably of stars contributing to the fit increases for deeper observa-
constrain mc or the turnover mass. We therefore caution tions.
against fixing model parameters during fitting. If observa- We note that, even for the cases with higher Mobs, it is
tionsdonotreachsufficientlylow-massstarstomeasurethe alwayspossibletoobtainreasonablytightconstraintsonmc
turnover in the IMF, it is better to fit a simple power law (and to detect deviations compared to the Chabrier value)
thantofitamorecomplicatedIMFwhilefixingoneormore when Mobs (cid:46)mc. This is because when Mobs (cid:46)mc, the low-
of the parameters. mass turnover is directly probed by observations, and the
IMFisnolongerwell-fitbyapowerlawoverthemassrange
where data are available. Thus, there is no longer a signif-
icant degeneracy between mc and σ, and both parameters
can be constrained strongly. Existing observations can thus
already rule out lognormal IMFs with mc (cid:38)0.4M(cid:12).
4 WHAT IS NEEDED TO DETECT
VARIATION IN THE IMF?
5 LOOKING FORWARD TO JWST
One of the goals of studying the IMFs of dwarf spheroidal
and ultra-faint galaxies is to detect or rule out systematic Having quantified the sample size and limiting magnitude
IMFvariationswithenvironment.Inthissection,weinvesti- required to constrain generic lognormal IMFs using star
gate how deep observations must reach in order to robustly counts,wenowturntotheprospectofmeasuringtheIMFs
detect variation in the IMF (if it exists). We assume that of Local Group dwarf galaxies with JWST. How accurately
the IMF is lognormal, but that the characteristic mass mc theIMFofanindividualgalaxycanbeconstraineddepends
varies across galaxies. Since observations of nearby dwarfs onMobs(whichisafunctionofthemagnitudedowntowhich
have found only marginal evidence of IMF variations at observationsarecomplete)andonthetotalnumberofstars
m (cid:38) 0.5M(cid:12), we intentionally choose mc and σ so that all sampled(whichisafunctionoftheangularsizeofthegalaxy
IMFs have approximately the same slope at m∼0.5M(cid:12). and the number of pointings).
Figure 6showshowaccuratelytheformoftheIMFcan We begin by calculating a realistic limiting magnitude
berecoveredfor different Mobs,the minimummassdownto and corresponding Mobs for observations with JWST. Be-
whichobservationsarecomplete.ForeachIMFsamplereal- causecool,low-massstarsaremostluminousintheNIR(for
ization,weplotthemedianandmiddle68%percentrangeof
themarginalizedposteriorformc againstthetruemc ofthe
8 WealsoexperimentedwithusingaKStestasanon-parametric
IMF from which the sample was drawn. If the middle 68%
region contains mc = 0.22M(cid:12), we label the IMF realization mbeeiansgudreraowfnwfhreotmhearCahgaivberniesraImMpFle,fionfdminagssreesswulatssctohnastiswteernetswimitih-
“consistent with Chabrier”; i.e., observing such a sample of
larwiththemetricdescribedabove.Weoptedtousetheparamet-
masseswouldnotraiseanysuspicionofvariationintheIMF.
ricversionforourprimaryanalysisbecausewefoundsignificant
IftheChabriervalueofmc =0.22M(cid:12)fallsoutsidethemiddle variations in the KS-test p values across different realizations of
68% credibility region, we label the realization inconsistent draws from identical IMFs. This is a generic shortcoming of the
withtheChabriervalue.Ofcourse,ifthemiddle68%credi- KStest;seeIvezi´cetal.(2014).
MNRAS000,1–??(2017)
Constraining the Low-Mass IMF 9
M =0.1M M =0.3M M =0.5M
obs obs obs
fl fl fl
consistent with Chabrier
0.5
inconsistent
)
fl0.4
M
( 0.3
c
m
0.2
t
i
f
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
true m (M ) true m (M ) true m (M )
c c c
fl fl fl
Figure 6. Accuracy to which the characteristic mass mc of a lognormal IMF can be recovered for three different observational stellar
masslowerlimits, Mobs.Wedrawmassesfrom14differentlognormalIMFswithcharacteristicmassesmc intherange0.01M(cid:12)−0.6M(cid:12),
always for a total Mstar =104M(cid:12). For each mc, we choose σ such that the IMF at masses above 0.5M(cid:12) resembles a Chabrier IMF (see
Figure1).WethenfitalognormalIMFtothedrawnmasses.Pointsanderrorbarsshowthemedianandmiddle68%ofthemarginalized
posterior draws for mc. Gray points indicate that the data cannot rule out a Chabrier IMF with mc =0.22M(cid:12); black points indicate
that mc =0.22M(cid:12) fallsoutsidethe68%constraintson mc,andthus,thatthedataappeartobeinconsistentwithbeingdrawnfroma
ChabrierIMF.
0.8 F200W filtersareoptimalforobservationsoflow-massstars,
HST, F814W < 27.5 becausealthoughthepeakin fλissomewhatbluer,thesefil-
0.7 ters are significantly wider than the F090W and F115W fil-
JWST, F150W < 28.5
tersandthusadmitmoretotalphotons.Wefindthatforan
0.6 exposuretimeofapproximately20,000secondsintheF150W
filter,JWSTcanobtainasignal-to-noiseratioSNR=5−10
)
fl0.5 or better for stars brighter than 28.5 Vega mag.
M
Figure 7 shows the expected lowest mass M which
s obs
( 0.4 n JWST will reach as a function of distance. We compute
Mobs0.3 ST gai tMeropbsolfartoimngaopnpaargernitdmofaigsoncithurdoeneastuasinggivtehnedMisitsatnccoedbey.Fionr-
W
J comparison, we show the approximate Mobs limit of current
0.2 optical observations with HST.10 This HST curve is com-
puted assuming a magnitude limit of 27.5 Vega mag in the
0.1
F814W filter, which can be obtained for an exposure time
anddesired S/N ratiosimilartoourassumptionsforJWST
0.0
101 102 103 (Geha et al. 2013; Kalirai et al. 2013).
Promisingly, JWST is forecasted to reach significantly
distance (kpc)
lower mass stars at fixed distance. For example, at d =150
kpc,theapproximatedistanceofthemost-distantMWsatel-
lites for which previous studies have attempted to measure
Figure7.Lowestobservablestellarmassasafunctionofdistance
theIMF(Gehaetal.2013),HSTobservationscanonlyreach
for HST ACS and JWST observations, assuming 20,000 seconds
of integration time. At a fixed distance, JWST can reach stars downtoMobs≈0.5M(cid:12) withtypicalintegrationtimes,which,
as we showed in Section 3.2, is not deep enough to jointly
nearly 0.3M(cid:12) less massive than HST. JWST observations will
reachthehydrogenburninglimitfordistances(cid:46)60kpc. constrain σ and mc for a Chabrier IMF. At the same dis-
tance, JWST is projected to reach down to Mobs ∼0.22M(cid:12),
a typical low-metallicity M-dwarf, fλ peaks near 0.9µm), it
isbesttousefiltersnearthiswavelength.WeusetheSTScI
JWST exposure time calculator9 to determine the optimal 10 Previous studies attempting to constrain the IMFs of nearby
dwarf galaxies have used the HST F606W and F814W filters
filters for observing low-mass stars and the projected limit-
rather than NIR filters, both because HST is significantly less
ing magnitude in these filters. We find that the F150W and
sensitive(andhaspoorerangularresolution)intheNIRthanat
optical wavelengths, and because the low-mass stars reached by
previous studies (m ∼ 0.5M(cid:12)) are somewhat less red than the
9 https://jwst.etc.stsci.edu/ lowest-massstarsthatcanbereachedwithJWST.
MNRAS000,1–??(2017)
10 El-Badry et al.
-12 Here ΥV is theV−band stellar mass-to-light ratio:11
Sculptor
-9 UMiDSreaxCIarina CLVenoIII 10-1y ()Mfl ΥV = ∫MM∫mmMinMaxmmiξnax(mξ)(mLV)m(md)mdm, (5)
al MV -6 0.02 BooIBIIooIUMaHIeLreoIV taint wmhaesrsemL,Vn(omrm)iaslitzheedVa−tbthaendmlaugmniitnuodsietyzeorfopaositnatr.Iwnitphraicntiitciael,
ot 0.05 er wecalculateLV(m)forapopulationofagivenageandmetal-
t -3 0.10 UMaIICBer 10-2 unc ltihcaittyobbyseinrvteartpioonlastiwniglloonnalygroibdsoerfvseteallafrramctoidoenls.foWbse=as0s.u1moef
mc all stars above M , due to the limited number of point-
obs
ingsandthedifficultyofdistinguishingbetweenthesatellite
[Fe/H] < - 1.5 andforegroundMWstarsinthegalaxy’soutskirts.Thatis,
0
f represents the fraction of a galaxy’s angular area that
1.0 1.5 2.0 2.5 3.0 obs
log(distance (kpc)) is imaged, and the observable Mstar = fobsMstar,tot.
IneachpixelofFigure8,wecalculatethevalueof Mstar
corresponding to MV using Equation 4, and the value of
Figure8.ProjectedaccuracytowhichtheIMFcanberecovered Mobs byinterpolatingtheJWSTcurveinFigure7.Wethen
from JWST observations of Local Group dwarf galaxies. Gray draw masses between Mobs and Mmax from a Chabrier IMF
pixelsshowthestrengthofconstraintsonmc obtainedbyfitting of total mass Mstar between Mmin and Mmax, following the
alognormalIMFtosamplesofstarsdrawnforpopulationswith procedure outlined in Section 3. Finally, we fit a lognor-
differentMobsandtotalMV;lighterpixelcolorsindicatestronger mal IMF to the drawn masses using our standard MCMC
constraints. Local Group dwarf galaxies with [Fe/H] < −1.5 are
procedure outlinedinSection2.3. Wemeasure the 68% un-
overplotted,withthemostpromisingtargetslabeled.
certaintyintherecoveredvalueofmc andcoloreachpixelin
Figure8accordingly.Lightlyshadedpixelsindicatethatmc
could be recovered with little uncertainty, while dark pixels
correspond to large uncertainties and poor constraints on
mc.
which is sufficiently deep to detect the low-mass turnover
InFigure8,wealsooverplotLocalGroupdwarfgalax-
andconstrain mc and σ within∼20%,iftheIMFissimilar ies that represent possible targets for observational studies
toaChabrierIMF.Inaddition,forthenearestMWsatellites
to measure the IMF from direct star counts. Data for these
at d (cid:46)60 kpc, JWST observations will reach stars down to
galaxies are taken from the Nearby Dwarfs Database pre-
thehydrogenburninglimit.Thesesignificantimprovements
sentedinMcConnachie(2012).Weonlyplotlow-metallicity
overHSTstembothfromJWST’slargercollectingareaand
objects with measured [Fe/H] < −1.5. These represent the
from its sensitivity to longer wavelengths, where low-mass
most“extreme”environments, in which one might most ex-
stars are more luminous.
pect systematic IMF variations compared to the MW. In
addition, these galaxies typically have primarily old stellar
populations with simple SFHs, which makes it easier to de-
termine their IMFs from a CMD.“Promising”objects (i.e.,
those in regions of parameter space where our simulations
5.1 Optimal targets in the Local Group suggestthattheIMFcouldbeconstrainedwithlittleuncer-
tainty)arelabeled.Pointsintheupperrightcornerarepri-
WenowcombinetheprojectedsensitivityofJWSTwiththe marilythesatellitesofM31,whicharetoofaintandcrowded
dependenceofIMFconstraintson Mstar and Mobs derivedin to be promising targets.
theprevioussectionstodeviseanobservationalstrategyfor Sofar,wehaveonlyconsideredthe Mobs and Mstar that
constraining the low-mass IMF in the Local Group. Nearer canbeachievedfortargetsofagivenMV atagivendistance,
galaxiescanbeobserveddowntofainter Mobs,butthereare without taking into account possible complications due to
more massive galaxies, which offer better statistical power crowding or the number of pointings required for a given
(i.e., larger Mstar), at larger distances. An additional con- source. We now consider the effects of these issues, in order
sideration is that more distant objects subtend a smaller to refine the list of promising targets.
angular size on the sky, and thus require fewer pointings, In Figure 9, we plot the half-light radius,r , and max-
h
but are also more likely to be affected by crowding. imum central surface number density of stars, Σ , of the
0
We investigate the tradeoff between distance and the same galaxies shown in Figure 8. We roughly estimate Σ
0
number of observable stars in Figure 8, which shows the
constraints on mc expected for a galaxy at a given distance
(which determines M ) with a given absolute magnitude
obs
(which determines Mstar). For a given total MV, the total 11 Note that unlike in the previous sections, the integral is over
all stars, including those that have left the main sequence. The
stellar mass is
resultingΥV representsatime-averagedquantity,whichcanfluc-
tuatesignificantlyduetostochasticsamplinginthefaintestgalax-
ies,whoselightisdominatedbyasmallnumberofevolvedstars
Mstar = fobsΥV ×10−MV/2.5. (4) (Hernandez2012).
MNRAS000,1–??(2017)
