Stellar population synthesis based modelling of the Milky Way using asteroseismology of dwarfs and subgiants from Kepler

0.78 MB·

by  Sanjib Sharma

DRAFTVERSIONJANUARY11,2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
STELLARPOPULATIONSYNTHESISBASEDMODELLINGOFTHEMILKYWAYUSINGASTEROSEISMOLOGYOF
DWARFSANDSUBGIANTSFROMKEPLER.
SANJIBSHARMA1,DENNISSTELLO1,2,DANIELHUBER1,2,3,JOSSBLAND-HAWTHORN1,TIMOTHYR.BEDDING1,2
DraftversionJanuary11,2017
ABSTRACT
EarlyattemptstoapplyasteroseismologytostudytheGalaxyhavealreadyshownunexpecteddiscrepancies
for the mass distribution of stars between the Galactic models and the data; a result that is still unexplained.
7 Here, werevisittheanalysisoftheasteroseismicsampleofdwarfandsubgiantstarsobservedbyKeplerand
1 investigateindetailthepossiblecausesforthereporteddiscrepancy. WeinvestigatetwomodelsoftheMilky
0 Way based on stellar population synthesis, Galaxia and TRILEGAL. In agreement with previous results, we
2 findthatTRILEGALpredictsmoremassivestarscomparedtoGalaxia,andthatTRILEGALpredictstoomany
bluestarscomparedto2MASSobservations. Bothmodelsfailtomatchthedistributionofthestellarsamplein
n
(logg,T )space,pointingtoinaccuraciesinthemodelsand/ortheassumedselectionfunction.Whencorrected
a eff
J forthismismatchin(logg,Teff)space,themassdistributioncalculatedbyGalaxiaisbroaderandthemeanis
shiftedtowardlowermassescomparedtothatoftheobservedstars. Thisbehaviourissimilartowhathasbeen
0
reportedfortheKeplerredgiantsample. Theshiftbetweenthemassdistributionsisequivalenttoachangeof
1
2%inν ,whichiswithinthecurrentuncertaintyintheν scalingrelation. Applyingcorrectionstothe∆ν
max max
] scaling relation predicted by the stellar models makes the observed mass distribution significantly narrower,
A butthereisnochangetothemean.
G Subject headings: Galaxy: disk – Galaxy: stellar content – Galaxy: structure – asteroseismology – stars:
fundamentalparameters
.
h
p
- 1. INTRODUCTION we can obtain asteroseismic information that is sensitive to,
o
andhencecapableofmeasuring,stellarradiusandmassina
r Our current understanding of the Milky Way has, to a
t large extent, been informed by stellar data from large scale largelymodelindependentway.
s A promising approach to take advantage of the large en-
a photometric,astrometric,andspectroscopicsurveys,suchas,
semblesofseismically-inferredstellarpropertiesistoinvoke
[ 2MASS(Skrutskieetal.2006),SDSS(Juric´etal.2008),Hip-
stellar population synthesis-based models of the Milky Way
parcos(ESA1997),GCS(Nordstro¨metal.2004),RAVE(Ko-
1 (e.g. Miglio et al. 2009; Chaplin et al. 2011a; Sharma et al.
rdopatisetal.2013), SEGUE(Yannyetal.2009), APOGEE
v 2016). This offers a way to link stellar structure and evo-
(Zasowskietal.2013),andGaia-ESO(Gilmoreetal.2012).
4 lution with Galactic structure and evolution by combining
As a result, we have already come a long way from simple
6 isochrones with star-formation history, the initial mass func-
empirical models of the Galaxy that fit star counts in a few
4 tion,andthespatialdistributionofstarsoftheGalaxy.Thisal-
lines of sight (Bahcall & Soneira 1980b,a, 1984), to mod-
2
lowsonetopredictstellarobservablesliketemperature,pho-
0 els that aim to be dynamically self-consistent (Robin et al.
tometry, asteroseismic parameters, as well as fundamental
. 2003; Binney 2010, 2012b; Binney & McMillan 2011; Bin-
1 stellarpropertiessuchasradiusandmass.
ney 2012a; McMillan & Binney 2012; Czekaj et al. 2014;
0 Chaplin et al. (2011a) used the first seven months of Ke-
Scho¨nrich & Binney 2009a,b; Sharma et al. 2014). Some of
7 pler data of dwarfs and subgiants, to compare the distribu-
these models, such as , Besanc¸on (Robin et al. 2003), TRI-
1 tions of seismically-inferred radii and masses of about 400
LEGAL (Girardi et al. 2005), and Galaxia (Sharma et al.
: stars with a synthesis-based Galactic model using TRILE-
v 2011) have been constructed to directly satisfy the observa-
GAL.Theyfoundthattheradiusdistributionofthesynthetic
i tionalconstraintsfromvariouslargescalephotometric,astro-
X populationmatchedthedata,butthemassdistributionsignif-
metricandspectroscopicsurveysoftheMilkyWay.However,
r to understand the Milky Way’s formation history, and hence icantly under-predicted the number of low-mass stars (M <
a 1.15M ),andhenceover-predictedthenumberofmoremas-
tofurtherverifythemodels, itisimportanttoknowthefun-
sive(y(cid:12)ounger)stars. UsingredgiantsfromKepler,wefound
damental properties of the stars, including radius and mass.
theoppositeeffectwhencomparingtheobservedmasseswith
Untilrecently,ithasbeendifficulttoreliablydeterminethese
predictions from the Galactic model Galaxia (Sharma et al.
properties model-independently for large numbers of distant
2016). WeshowedthatintheKeplerregionforamagnitude
stars. Fortunately,thespacemissionsCoRoT(Baglin&Frid-
limitedsample,TRILEGALpredictsmorebluestarsascom-
lund2006)andKepler(Boruckietal.2010),andnowalsoK2
pared to 2MASS, while Galaxia has no such problem. Be-
(Howelletal.2014),providehighlyaccuratetime-seriespho-
cause blue stars are young and massive, this suggests that
tometry of thousands of stars across the Galaxy, from which
TRILEGAL overpredicts the number of young and massive
stars. Hence,itisimportanttocomparetheobservedmasses
1SydneyInstituteforAstronomy,SchoolofPhysics,UniversityofSyd-
ney,NSW2006,Australia ofdwarfsandsubgiantswiththepredictionsfromGalaxia.
2StellarAstrophysicsCentre,DepartmentofPhysicsandAstronomy, Besides inaccuracies in the Galactic model, there are a
AarhusUniversity,DK-8000AarhusC,Denmark numberofotherfactorsthatcouldcontributetothemismatch
3SETI Institute, 189 Bernardo Avenue, Mountain View, CA 94043, in the mass distributions seen by Chaplin et al. (2011a). (i)
USA
2
The observed asteroseismic information is in the form of
25
the average seismic parameters ∆ν (average frequency spac-
20 (a) ingbetweenovertoneoscillationmodes)andν (frequency
max
at maximum oscillation power) that are extracted from time
15 mean=1.0006
series photometry using a specific algorithm/method. Prior
stddev=0.02
p
10 tothelaunchofKepler, about2000starswithKp<12mag
were selected as potential asteroseismic dwarf and subgiant
5 targets based on their parameters in the Kepler Input Cata-
log (KIC) (Brown et al. 2011). They were all observed with
0
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 shortcadenceforonemontheachduringaninitial10-month
∆νC14/∆νC11 seismic survey phase. Four hundred stars showed detectable
25 oscillationsafterthefirstsevenmonthsandwerepresentedby
Chaplinetal.(2011a)(hereafterdenotedasChaplin-11sam-
20 (b) ple).Followingthecompletionofthe10-monthsurvey,anup-
dateddwarf/subgiantsample(518stars)showingoscillations
15 mean=1.0051
stddev=0.03 was presented by Chaplin et al. (2014) (here after denoted
p
10 as Chaplin-14 sample), where a method different from that
ofChaplinetal.(2011a)wasusedforextractingseismicpa-
5 rameters. IntheChaplin-14sample467starshavemeasured
values of ∆ν, ν and T . Of these, 290 stars are in com-
0 max eff
0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 mon with the previous Chaplin-11 sample. In Figure 1, we
νC14/νC11
max max comparethe∆νandν valuesofChaplin-2014withthose
max
ofChaplin-2011,forstarscommontobothsamples. Itcanbe
FIG.1.—(a)Ratioof∆νforthe290starsincommonbetweentheChaplin-
seen that there are no systematic shift between the two data
2014andChaplin-2011samples. (b)Sameaspanel(a)butforνmax. The
dashedlineinpanel(b)showstheratioofνmax, adoptedinChaplin-2014to sets. This means there is no method-specific differences in
thatadoptedinChaplin-2011. Thereisnosyst(cid:12)ematicshiftbetweenthetwo the two data sets. In this paper we mainly make use of the
methods. Thestandarddeviationisoftheorderoftheuncertaintiesonthe Chaplin-2014sample,becauseithasmorestars.
estimatedvaluesofνmax(4%)and∆ν(2%).
Inaccuraciesintheselectionfunctioncanleadtoamismatch 2.2. Scalingrelationsandsolarreferencevalues
and need to be checked. There could be systematics associ- Thestellarmassandradiuscanbeestimatedfromtheseis-
atedwiththealgorithmusedtoestimateaverageseismicpa- mic parameters ∆ν and ν , and the effective temperature
max
rameters. (ii) The probability to detect oscillations can dif- T usingthefollowingscalingrelations:
eff
ferfromonealgorithmtoanotherandthiscanleadtodiffer-
encesintheselectionfunction. SincetheanalysisbyChaplin M (cid:18) νmax (cid:19)3(cid:18) ∆ν (cid:19)−4(cid:18) Teff (cid:19)1.5
etal.(2011a),anewdatasetofdwarfsandsubgiantshasbeen = (1)
M f ν f ∆ν T
publishedbyChaplinetal.(2014),whichusedadifferental- (cid:12) νmax max,(cid:12) ∆ν (cid:12) eff,(cid:12)
gorithm to estimate the seismic parameters. Additionally, it R (cid:18) νmax (cid:19)(cid:18) ∆ν (cid:19)−2(cid:18) Teff (cid:19)0.5
= . (2)
contains more stars and extends to slightly lower gravities. R f ν f ∆ν T
Hence, it is necessary to investigate the mass distributions (cid:12) νmax max,(cid:12) ∆ν (cid:12) eff,(cid:12)
with the new data set as well. (iii) Theoretical modelling of These relations are based on the relations ∆ν ∝ ρ1/2, and
thestellaroscillationspredictdeparturesfromthe∆νscaling ν ∝g/T1/2(Brownetal.1991;Kjeldsen&Bedding1995;
relation (Stello et al. 2009; White et al. 2011; Miglio et al. max eff
Belkacemetal.2011),whichinturnarebasedontheassump-
2013) and the effect of this needs to be taken into account.
tion that the structure of any given star is homologous with
(iv) To estimate mass from average seismic parameters, one
respecttotheSun. Thisassumptionisnotstrictlycorrectand
has to adopt certain solar reference values. Currently, there
canleadtodeparturesfromthescalingrelations. Toaccom-
is no consensus on the choice of these with systematic dif-
modatethesedepartureswehaveintroducedthefactors f
ferences ranging from 1% to 2%. Hence, one needs to in- νmax
and f . Thereisalsoconsiderableuncertaintyregardingthe
vestigate whether the discrepancy between observations and ∆ν
choiceofsolarreferencevaluesandthisleadstouncertainties
predictions is less or greater than the current diversity in the
in f and f , when we adopt a specific set of canonical
solarreferencevalues. νmax ∆ν
solarreferencevalues. Below,wediscussthisindetail.
Inthispaperwerevisitthedwarf/subgiantanalysisofChap-
It is clear from the scaling relations that to estimate mass
lin et al. (2011a) and analyze each of the above mentioned
and radius one has to adopt some solar reference values,
factors. In Section 2, we discuss the observational data and
∆ν andν . Unfortunately,thereisnoconsensusonthe
theGalacticmodels.Systematicsassociatedwiththedifferent max,
cho(cid:12)ice of sola(cid:12)r reference values. Ideally, to estimate them
datasetsandGalacticmodelsarediscussedhere. InSection
we would require high quality data of the Sun in the Kepler
3,weanalyzetheasteroseismicinformationfordifferentdata
bandpass,which,unfortunatelyisnotavailable. Thedatafor
setsanddifferentGalacticmodelsanddiscusstheroleofthe
theSunisavailableinotherbandpassesandthishasbeenan-
selectionfunction. Wealsodoaquantitativestudyofthedif-
alyzed. The results using the SOHO/VIRGO green channel
ference between observed and predicted mass distributions.
data (Frohlich et al. 1997) are shown in Table 1 for various
Finally,inSection4wediscussandconcludeourfindings.
methods. Whiletheestimatesof∆ν agree(differenceabout
0.2%),theestimatesofν differ(cid:12)significantly(difference
2. DATA,SCALINGRELATIONSANDGALACTICMODELS max,
about2.5%)andsofarthisdi(cid:12)sagreementhasnotbeenexplic-
2.1. Observationaldata
itlyexplained. Themostlikelycauseforthedifferencesisthe
3
ThemainGalacticstellar-population-synthesismodelused
TABLE1
SOLARREFERENCEVALUESFORDIFFERENTMETHODSOFCOMPUTING inthispaperisfromtheGalaxia4 code(Sharmaetal.2011).
AVERAGESEISMICPARAMETERS. ItusesaGalacticmodelthatisbasedontheBesanc¸onmodel
byRobinetal.(2003)butwithsomemodifications. Galaxia
Method ∆ν (µHz) νmax, (µHz)
SYDa 135(cid:12).10 0.01 3090(cid:12) 3e has a 3D extinction scheme that is based on Schlegel et al.
CANb 134.88±0.04 3120±5 (1998)dustmaps. Wealsoapplyalowlatitudecorrectionto
CORc 134.90±0.1 3060±10 thedustmapsasinSharmaetal.(2014). Theisochronesused
OCTd 135.03±0.1 3140±10 topredictthestellarpropertiesarefromthePadovadatabase
aHuberetal.(2009±,2011,2013)± (Bertellietal.1994;Marigoetal.2008). Theuniquefeature
bKallingeretal.(2010) of Galaxia is its novel star-spawning scheme which, unlike
cMosseretal.(2012);Hekkeretal.(2013) previouscodes,doesnotdiscretizethespatialdimensionsinto
dHekkeretal.(2013) multiplelinesofsight;insteaditgeneratesacontinuousthree-
eHuberetal.(2011)reportanuncertaintyof30µHzforνmax, and10µHz dimensionaldistributionofstars.
for∆ν . However,thisisforone30daysolartimeseriessubs(cid:12)etoutof111
analyz(cid:12)edbythemintotal. For comparison, we also used the TRILEGAL5 Galac-
tic stellar-population-synthesis model (Girardi et al. 2005).
TRILEAGLasacodeisveryflexibleandoffersmultipleop-
tions to the user to change various aspects of the Galactic
3.0 KIC
model,i.e.,IMF,SFR,agescaleheight,localsurfacedensity
2.5 Galaxia
ofstarsandsoon. However,thereisadefaultversionofthe
TRILEGAL
2.0 Galactic model that is advocated in the (Girardi et al. 2005)
p1.5 paperandiscommonlyused. Weherewishtoinvestigatethe
discrepanciesreportedbyChaplinetal.(2011a),whodonot
1.0
mention any specific changes to the default settings. So we
0.5 choosetousethedefaultmodelofTRILEGALforouranaly-
0.0 sis.
0.5 0.0 0.5 1.0
− Unlike Galaxia, TRILEGAL cannot generate stars over a
J-Ks(mag)
wide angular area, so we generated stars along 21 lines of
FIG.2.—(a)J Kscolordistributionofstarswithr<14intheKIC(black) sight pointing towards the centers of the 21 Kepler CCD-
comparedwithp−redictionsfromGalaxia(red)andTRILEGAL(green). (b) modules. We used the default settings but with binary stars
aspanel(a),butforg rcolor. Theintegratedprobabilitydistributionsare turned off. To model the dust, we used the 3D extinction
scaledtounity. −
modelofGalaxia.OnenoticeabledifferencebetweenGalaxia
andTRILEGAListhatGalaxiausesaconstantstar-formation
rate, whilethedefaultsettinginTRILEGALusesatwo-step
method-specificsystematicsassociatedwiththeestimationof
star-formation rate, which is twice as large between 1-4 Gyr
ν . This would argue for the use of “method-specific” so-
max asatanyothertime.
larvalues,meaningthevaluesreturnedfromsolardatawhen
using the same method (pipeline) as used for the rest of the
stellarsample. However, thereisnostrongevidencetoback 2.3.1. ComparingGalacticmodelswiththeKeplerInputCatalog
uptheuseofmethod-specificvalues.Onthecontrary,Hekker Before comparing the Galactic models with the data, it is
et al. (2013) found that for red giants, method-specific solar important to check that they reproduce the stellar photom-
reference values introduce biases. In other words, method- etry in the KIC. This is because the KIC formed the basis
specific systematics in νmax for the Sun are not necessarily fortheselectionoftheobservedsampleofstars. Hence,any
representativeofthesystematicsforotherstars. significantmismatchbetweenthemodelsandtheKICwould
Similar to giants, for dwarfs and subgiants it is not clear indicate a fundamental problem with the models, making it
whetheroneshouldadoptmethodspecificsolarreferenceval- difficulttoperformmeaningfulcomparisonswiththeseismic
ues. The Chaplin-14 sample adopted the SYD method (Hu- data.
beretal.2009,2011)forcomputingtheseismicparameters. InSharmaetal.(2016),weanalyzedthe(J K )colordis-
s
The solar reference values corresponding to this method are tributionofamagnitudelimitedsample(r<1−4mag)ofstars
∆ν = 135.1 µHz, νmax, = 3090 µHz. For the Chaplin- in the Kepler field and below we summarize the results for
11(cid:12)sample, however, a di(cid:12)fferent method was used, namely the completeness of this paper. We generated two synthetic
the OCT pipeline (Hekker et al. 2013) available at that time catalogsoftheMilkyWay,usingeachofGalaxiaandTRILE-
whose method specific solar reference values were ∆ν = GAL.StarsfromthesyntheticcatalogsandfromtheKICthat
134.9 µHz and νmax, =3150 µHz. These were adopted(cid:12)by laywithin8degreesofthecenteroftheKeplerfieldandwith
Chaplinetal.(2011a)(cid:12)forcomputingthestellarmassesintheir magnituder<14wereselectedforcomparison. IntheKIC,
sample. As discussed earlier (Figure 1), there is no system- therepeatabilityofphotometryforstarsbrighterthanmagni-
atic shift in seismic parameters between the Chaplin-11 and tude 14 is about 2%, so it is very likely to be complete for
Chaplin-14datasets.Hence,νmax, =3090µHzisanequally r<14. WeshowtheresultingdistributionsofJ Ks colorin
validchoiceforestimatingmasses.(cid:12)Forclaritythroughout,we Figure2. ThepredictionofGalaxiaisinexcelle−ntagreement
adopt∆ν =135.1µHz,andνmax, =3090µHz. withthestarsfromtheKIC,butthatfromTRILEGALshows
Basedo(cid:12)nthediscussioninthepr(cid:12)evioustwoparagraphs,we asignificantdifference. TRILEGALcorrectlyreproducesthe
conclude that fνmax is uncertain by at least 2%, and we will region around the red peak of the color distribution, but not
usethisfactlater.
4http://galaxia.sourceforge.net
2.3. Galacticmodels 5http://stev.oapd.inaf.it/cgi-bin/trilegal
4
0.35 12
(a) (b)
0.30 TRILEGAL 10
0.25 Galaxia
8
Galaxia-matchedtoTRILEGAL
0.20
6
0.15
4
0.10
0.05 2
0.00 0
0 2 4 6 8 10 12 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Age(Gyr) J-Ks
0.0018 2.5
0.0016 (c) (d)
0.0014 2.0
0.0012
1.5
0.0010
0.0008
1.0
0.0006
0.0004 0.5
0.0002
0.0000 0.0
7000 6500 6000 5500 5000 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Teff (K) M(M )
(cid:12)
FIG.3.—ComparisonofstellarpropertiesofTRILEGAL(black)andGalaxia(blue). Shownaredistributionsofage(panela),color(panelb),temperature
(panelc),andmass(paneld). ThecurvesinredshowthepredictionsofGalaxiawhenitsstarsarereweightedtomatchtheagedistributionoftheTRILEGAL
stars.Inpanela,theredandblacklinesareontopofeachother.
aroundthebluepeak. Specifically,itoverestimatesthenum- encesbetweenGalaxiaandTRILEGALaremainlyrelatedto
ber of stars to the left of the blue peak and underestimates ageand/ormass,andnotduetodifferencesinisochrones. If
the number of stars on the right side of the blue peak. Stars the difference were due to isochrones, then even after forc-
leftwardofthebluepeakaretypicallyyoungmain-sequence ing a match on mass, age and metallicity, the two models
stars,suggestingthatTRILEGALoverpredictsthenumberof would have shown differences in the color distribution. The
youngstarsintheKeplerfield. two main factors that control the age distribution of a stellar
Now, to better understand the above mentioned difference samplearethestar-formationrateandtheagescaleheightre-
betweenTRILEGALandGalaxia,weselectedstarsfromboth lation.BothofthesefactorsaredifferentbetweenGalaxiaand
modelssatisfying4000K<T <6600K,3.7<logg<4.2 TRILEGALandcouldberesponsibleforthemismatchinthe
eff
and r<12 mag (we call this selection criteria S ). This J K colordistribution.
dwarf s
−
was designed to select dwarfs and subgiants, which are the To conclude, we find that TRILEGAL cannot fit the color
main focus of this paper. The distributions of stellar prop- distributionofthestars,probablybecauseitpredictstoomany
erties are shown in Figure 3. The same color difference as youngandmassivestars.Unlesswecanexplainthemismatch
seeninFigure2canbeseenhere. Adifferenceinage, tem- in the color distribution by some other means (e.g., system-
peratureandmassdistributionscanalsobeseen. Mass, age, atics in the isochrones), a color or temperature-limited sam-
andmetallicityarethethreeintrinsicpropertiesofastarthat pleofdwarfsandsubgiantsselectedfromTRILEGALisex-
largely define its observable properties. The metallicity dis- pectedtobebiasedtowardshighermasses.
tributions (not shown here) did not show any significant dif-
ference. Hence,thecolordifferenceismostlikelyduetodif- 3. ANALYSISOFASTEROSEISMICINFORMATION
ferences in mass and age distributions. Next, we therefore TocomparethepredictionsoftheGalacticmodelwiththe
investigate if the color, temperature, and mass distributions asteroseismic information from Kepler, we generated a new
would match (between Galaxia and TRILEGAL) if we were syntheticpopulationofstarsusingbothGalaxiaandTRILE-
toaltertheGalacticmodelofGalaxiainsuchawaythatthe GAL.Thesyntheticstarswerethenselectedtomatchthese-
age distribution of stars obeying the subgiant and dwarf se- lectioncriteriaoftheobservedstars. Themainselectioncri-
lectioncriteriamatchestheTRILEGALprediction. However teria was based on apparent magnitude and a lower limit on
we cannot easily alter the model in Galaxia, instead we use ν . However, not alltargeted starsshowedoscillations. A
max
theideaofimportancesampling. Weassignaweighttoeach scheme to compute the detection probability was presented
Galaxia star such that the weighted age distribution matches inChaplinetal.(2011b)andthiswasusedbyChaplinetal.
thatofTRILEGAL.Usingtheseweightswethencomputethe (2011a)toselectstarsfromaGalacticmodel. Inthisscheme,
weighted distriutions of other quantities like color, tempera- mass,radius,andeffectivetemperatureofeachsyntheticstar
ture,andmass,andcomparethemwiththoseofTRILEGAL. wereusedtopredictitstotalmeanoscillationpowerandthe
We found that the color, temperature and mass distributions granulationnoiseinthepowerspectrum.Theapparentmagni-
of the reweighted Galaxia sample now matched the TRILE- tudewasusedtocomputetheinstrumentalnoiseinthepower
GAL sample. Reweighting the Galaxia sample to match the spectrum which, combined with granulation noise, gave the
massdistributionoftheTRILEGALsampleproducedsimilar total noise. The mean oscillation power and the total noise
results. Thisisexpectedbecausemassandagearecorrelated were then used to derive the probability of detecting oscil-
forthetypesofstarsthatweanalyze. Thisshowsthatdiffer- lations, p , with less than 1% possibility of false alarm.
detect
5
1.0
∆ν =135.1µHz Kepler-C14 ∆ν =135.1µHz Kepler-C14
2.0 νma(cid:12)x,(cid:12)=3090µHz Galaxia 1.0νma(cid:12)x,(cid:12)=3090µHz Galaxia
0.8
1.5
0.8
p p0.6 3.5
1.0
0.4
0.5 (a) 0.2 (b) 0.6
02..00 ∆νmνa(cid:12)x,(cid:12)==13350.910µµHHzz KTRepILleErG-CA1L4 01..00∆νmνa(cid:12)x,(cid:12)==13350.910µµHHzz KTRepILleErG-CA1L4 logg4.0 0.4pMin()detect
0.8
1.5
p p0.6
1.0 4.5
0.4 0.2
0.5 0.2
(c) (d)
0.0 0.0
2.0 ∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L4 1.0∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L4 7000 6500 T6eff00(0K) 5500 5000 0.0
0.8
1.5 FIG.6.—Minimumprobabilityofdetectingoscillationsasafunctionof
p1.0 p00..46 lboagsegdaonndpTreeffd.icTtihoensmfarpomistfhoercstoadreswGiatlhaxaipap.aDreenttecmtiaognniptruodbeabri=lity11isacnodmis-
putedusingtheschemeofChaplinetal.(2011b). Asharptransitionfrom
0.5 (e) 0.2 (f) hightolowdetectionprobabilitycanbeseen.Therectangularboxmarksthe
0.0 0.0 regionwherethesampleisapproximatelycomplete.
0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5
M/M R/R the TRILEGAL model, which was used by Chaplin et al.
(cid:12) (cid:12)
(2011a). Here, the predicted mass distribution is shifted
FIG.4.— MassandradiusdistributionsforChaplin-14sampleobserved slightly to the right. Figure 4e,f also show the predictions
byKepler(black)andpredictedbyGalaxia(red)andTRILEGAL(green).
of TRILEGAL alongside observed stars, but with masses
(a-b): ResultusingGalaxiaandourchoiceofsolarreferencevaluesfor∆ν
andνmax. (c-d): ResultusingTRILEGALandourchoiceofsolarreference of observed stars computed using the solar reference values
values. (e-f): ResultusingTRILEGALandthesolarreferencevaluesused adopted by Chaplin et al. (2011a) in their analysis. As ex-
byChaplinetal.(2011a)(ourreproductionoftheChaplin-11result). The pected, this result is the same as that presented by Chaplin
integratedprobabilitydistributionsarescaledtounity.
etal.(2011a),withasignificantshiftofthemassdistribution
from TRILEGAL compared to the observed distribution. In
Figure 5, we show the same analysis but using the Chaplin-
∆ν =135.1µHz Kepler-C11 ∆ν =135.1µHz Kepler-C11
2.0 νma(cid:12)x,(cid:12)=3090µHz Galaxia 1.0νma(cid:12)x,(cid:12)=3090µHz Galaxia 11sample,whosesizeisslightlysmallerthantheChaplin-14
0.8 sample. Asexpected,weseethesametrendsasseeninFig-
1.5
p p0.6 ure4.
1.0 Thevalueofν usedbyChaplinetal.(2011a)isabout
0.4 max,
2%higherthanthev(cid:12)alueusedbyus(Figure4a-d)andthisre-
0.5 0.2
(a) (b) duces the masses of the observed stars by about 8%, which
0.0 0.0 exacerbates the mass discrepancy between the TRILEGAL
∆ν =135.1µHz Kepler-C11 ∆ν =135.1µHz Kepler-C11
2.0 νma(cid:12)x,(cid:12)=3090µHz TRILEGAL 01..80νma(cid:12)x,(cid:12)=3090µHz TRILEGAL phraesdnicotisoingnainfidctahnetoebffseecrtv.atTioonsc.oTnchleudsme,atlhledimffeisrmenactechinin∆νth(cid:12)e,
1.5
stellar mass distribution found by Chaplin et al. (2011a) can
p p0.6
1.0 0.4 bealleviatedif:(i)oneadoptsavalueofνmax, thatisslightly
smallerand(ii)weuseGalaxia(withdefaults(cid:12)ettings)instead
0.5 (c) 0.2 (d) ofTRILEGAL(withdefaultsettings)astheGalacticmodel.
0.0 0.0 ThisdoesnotnecessarilymeanthattheGalaxiamodeliscor-
2.0 ∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L1 01..80∆νmνa(cid:12)x,(cid:12)==13341.590µµHHzz KTRepILleErG-CA1L1 rleecctti.oTnhfeunmcatisosndaisntdribaubtiioansiinssmenasssitidvueetotothaencinhcooicrereocfttchheosicee-
1.5 oftheselectionfunctioncancanceloutabiasduetoanincor-
p p0.6
1.0 rectgalacticmodel. Hence,inthenextsubsection,weinves-
0.4
tigatetheaccuracyoftheselectionfunction.
0.5 0.2
(e) (f)
0.0 0.0 3.1. Theroleoftheselectionfunction
0.5 1.0 1.5 2.0 2.5 0 1 2 3 4 5
M/M R/R The accuracy of a selection function that is based on de-
(cid:12) (cid:12)
tectabilityofoscillationshingesuponourabilitytoaccurately
FIG.5.—SameasFigure4,butfortheChaplin-11sample. predict the mean oscillation power. The mean oscillation
power is computed from the maximum mode amplitude and
Starswith p >0.9wereassumedtobedetectable. Here- forthisanempiricalrelationisused, whichispronetoinac-
detect
afterwerefertothisselectionfunctionasS0. curaciesandcanpotentiallybiastheselectionfunction.
We applied the S0 selection function to the synthetic stars Wenowstudywhetheralteringtheassumptionsbehindthe
and calculated the mass and radius distributions (Figure 4). selectionfunctionhaveanyeffectonthedistributionofstars
Figures 4a,b show the predictions of Galaxia, which match in the (logg,T ) space. In Figure 6, we show p as a
eff detect
well with the observations. Figure 4c,d show predictions of function of logg and T for stars with r=11 mag, selected
eff
6
ingtheassumednoiseintheselectionfunctionorincreasing
2.5
r would have a similar effect. On average, the stellar mass
increaseasonemovesdiagonallyfromthelowerrighttothe
2.0 upperleft(Figure7). Hence, abiasin(logg,Teff)spacewill
3.5
alsoleadtoabiasinthemassesofthestars.
In Figure 8, we show the distribution of the Kepler sam-
1.5 ple in (T ,logg) space alongside predictions from Galactic
eff
logg4.0 M(M)(cid:12) manoddeplrsedbiacsteedd doinstrtihbeutSio0nssemleacttciohnwfeulnlcitniotnh.e cTehnetraolbsreegrvioedn
1.0
butdifferencescanbeseenalongtheboundary. Bothmodels
overpredictthenumberofstarsataround(6400,3.7). Addi-
4.5 tionally,TRILEGALpredictsfewerstarsaround(5600,4.1),
0.5
whileGalaxiapredictsmorecoolerstars(T<5000K).Atthe
low-temperatureendforevolvedstarsweexpectthedetection
0.0 probability to be high. Hence, the differences seen here are
7000 6500 6000 5500 5000
most likely due to inaccuracies in the models. At high tem-
Teff(K)
perature, the detection probability is low and is sensitive to
FIG.7.—MeanmassofastarasafunctionofloggandTeff,aspredictedby the assumptions made in the selection function. Here, stars
Galaxia,forstarswithapparentmagnituder=11.Themeanmassincreases are close to the instability strip where convection zones are
asonemovesdiagonallyfromthelowerrighttotheupperleft.
thin, whichmakesitdifficulttomodelthemodedrivingand
damping mechanisms. Hence, the differences seen here are
3.2 mostlikelyduetoinaccuraciesintherelationusedtopredict
TRILEGAL modeamplitudes.
3.4
Kepler-C14 The fact that the observed and the predicted distributions
3.6 of stars do not match in (T ,logg) space means the mass
eff
distributions will also not match. To eliminate this bias, we
3.8
g creatednewselectionfunctionsbyresamplingthemodelstars
g4.0 to satisfy the observed distribution of stars in (T ,logg,r)
o eff
l4.2 space. Thedisadvantageofsuchresamplingisthatwereduce
our sensitivity to model-based differences, because the first
4.4
(a) order differences are already taken out. However, they are
4.6 stillusefultounderstandsystematicsrelatedtoasteroseismic
4.8 analysis. Belowaretwowaysfordevisingsuchnewselection
7500 7000 6500 6000 5500 5000 4500
functions.
T
eff
S1: Hereweresamplethemodelstarstomatchthedis-
3.2 • tributionofobservedstarsin(T ,logg,r)space. This
Galaxia eff
3.4 isdonebydividingthe(T ,logg,r)spaceintobinsand
Kepler-C14 eff
thenmakingsurethateachbinhasthesamenumberof
3.6
model and observed stars. Because the number of ob-
3.8
g servedstarsislow,onehastoadoptlargebinsandthis
g4.0 canaffecttheaccuracyoftheselectionfunction.
o
l4.2 S2: Inthisweselectaboxin(T ,logg,r)spacewhere
eff
•
4.4 weexpectthedetectionprobabilitytobecloseto1and
(b)
wherethedistributionoftheobserveddatamatchesthat
4.6
of the Galactic model. Compared to the case S1, the
4.8 case S2 leads to fewer stars, but has a more accurate
7500 7000 6500 6000 5500 5000 4500
T selectionfunction. Theselectedbox,
eff

1 if(5800<T <6600)&
shFoIwGn.8a.s—blSuetarpsoiinnt(slowghgi,leTesffy)nsthpeatciec.sTtahresKweipthlerpddewtecatr>fsa0n.9dasruebgsihaonwtsnaares p(ST ,logg,r)= (3.8<logge<ff4.1)& (3)
rdeedxpinoilnotgs.g,TThheeosbimseurlvaetdeddsattaarshwaseraencuonncveorltvaeindtywiothf1u5n0ceKrtaiinntTieeseftfyapnicdal0o.2f | eff  (r<11)
thatintheobserveddata,150KinTeff and0.2dexinlogg. BothGalactic 0 otherwise,
modelsoverpredictthenumberofstarsataround(6500,3.7). Inaddition
TRILEGALunderpredictsthenumberofstarsataround(5700,4.1)while isshowninFigure6,anditcanbeseenthattheboxis
Galaxiapredictsmorestarsatthelowtemperatureend. mainly inside the region of high detection probability
(red).
from Galaxia. Stars with high p are confined to the re-
detect
gion in (logg,T ) space shown in red. For the red region, In Figure 9, we compare the predicted mass distributions
eff
therightboundaryisbecausetherearenostarstotherightof with observations, for the three different selection functions,
thatboundary. Theleftandlowerboundariesarebecausethe S0, S1 and S2. Compared to the observed distributions, the
oscillationamplitudedecreaseswithincreasingloggandT . predicted distributions are shifted toward lower masses and
eff
Ifwelowertheassumedamplitudesintheselectionfunction, are slightly broader. For S0 the shift is minimal, but for S1
theredregionwouldshiftupwardsandtotheright. Increas- and S2 it is significant. We note that Figure 9a (S0) is the
7
available code ASFGrid6. ASFGrid uses the code MESA
TABLE2
(v6950)(Paxtonetal.2011,2013,2015)forstellarevolution
VAMLAUTECOHFESfνBmaExSFTOWRIWTHHITCHHETMHAESDSIESSTPRRIBEUDTICIOTNEDOBFYOBTHSEERGVAEDLAMCATSICSES andthecodeGYRE(Townsend&Teitler2013)forderiving
MODELS.THEVALUESINSQUAREBRACKETAREFORTHECASEWHEN oscillation frequencies. The correction factors are computed
THE f∆νCORRECTIONFACTORISAPPLIEDTOTHESYNTHETICSTARS. asafunctionofmetallicity, mass, andage, seeSharmaetal.
Selection fνmaxGalaxia fνmaxTRILEGAL (2016) for further details. Applying these corrections makes
Function the observed mass distribution narrower (blue lines in Fig-
S0 1.006[1.004] 0.002 0.982[0.981] 0.002 ure 9). This is expected because for the stars that we study,
S1 1.018[1.015]±0.002 1.002[1.001]±0.002 the mean mass of a star increases with temperature and the
± ±
S2 1.019[1.011] 0.004 1.004[0.994] 0.004 correction factor decreases with temperature. The combined
± ±
effectisthatforhigh-massstars,themassdecreases,andfor
low-mass stars, the mass increases. This leads to narrowing
3.0 of the overall distribution. Computing corrections requires
∆ν=135.1µHz Kepler-C14 metallicity,andweadoptedthespectroscopicmetallicitiesre-
2.5
νmax, =3090µHz Kepler-C14,with∆νcorr ported by Buchhave & Latham (2015). Instead of applying
2.0 (cid:12) Galaxia thecorrectiontotheobservedstars,wecanalsoapplythere-
p1.5 ciprocalcorrectiontothesyntheticstarswhosemetallicityis
known exactly. Doing so had negligible effect on the values
1.0
quotedinTable2(seeresultsinsquarebrackets).
S0 (a)
0.5
4. DISCUSSIONANDCONCLUSIONS
03..00
We have compared the asteroseismic properties of dwarfs
∆ν=135.1µHz Kepler-C14
2.5 and subgiants observed by Kepler against predictions of two
2.0 νmax,(cid:12)=3090µHz KGeaplalexria-C14,with∆νcorr population synthesis models of the Galaxy, TRILEGAL and
Galaxia. The previous study by Chaplin et al. (2011a) us-
p1.5
ingTRILEGALfoundthatstellarpopulationsynthesisbased
1.0 models overestimated the number of high-mass stars, which
S1 (b) we are able to reproduce. We identified three potential fac-
0.5
torsthatcanshiftthemodelmassdistributionstowardhigher
03..00 masses relative to the observed masses. First, TRILEGAL
2.5 ∆ν=135.1µHz Kepler-C14 most likely overpredicts the number of young massive stars
νmax, =3090µHz Kepler-C14,with∆νcorr as it fails to match the J Ks color distribution of stars in
2.0 (cid:12) Galaxia −
2MASS(Sharmaetal.2016). Second,wefoundthatachoice
p1.5 ofν thatis2%lowerthanthatadoptedbyChaplinetal.
max,
(2011a)(cid:12),whichisequallyvalidgiventheuncertaintyintheac-
1.0
tualvalue,canincreasetheobservedmassesbyabout6%. Fi-
S2 (c)
0.5 nally,wefoundthatifaselectionfunctionbasedonoscillation
0.0 amplitudesisused,theGalacticmodelscannotreproducethe
0.5 1.0 1.5 2.0
distributionoftheobservedsamplein(logg,T )space. This
M/M eff
mightbeduetoinaccuraciesinthemodel, butcouldalsobe
(cid:12)
duetoinaccuraciesintheassumedselectionfunction. Select-
FIG.9.—Massdistributionofobservedstarsalongsidepredictionsfrom ingthesyntheticstarstosatisfythedistributionin(logg,Teff)
Galaxia,usingthreedifferentselectionfunctions.Weusethesamesolarref- spaceremovedthisbiasbutshifted themodelmassdistribu-
erencevaluesforallthreecases.Thebluelineisforthecasewheretheoreti-
tiontolowermasses(Figure9b,c).
callypredictedcorrectionsto∆νscalingrelationareapplied.Thecorrections
werecomputedassumingsolarmetallicity(Z=0.019)foralltheobserved The bias due to the mismatch of the color distribution can
stars. be corrected by using a model such as Galaxia, which does
not show such a mismatch. The bias due to inaccuracies in
theselectionfunctionbasedonoscillationamplitudescanbe
same as Figure 4a. To quantify the shift between the mass reducedbyusingaselectionfunctionbasedonloggandT
eff
distributions,wedeterminedthevalueof f thatminimizes oftheobservedsample. Doingso,wefindthatthemassdis-
νmax
the Kolmogorov-Smirnov statistic between the observed and tributionofGalaxiaisshiftedtowardlowermassesandisalso
predicted distributions. The uncertainty on the estimate was slightly broader compared to the observed distribution. A
computed using bootstrapping. The results are shown in Ta- similar effect was also seen for the Kepler red giant sample
ble2. Thetwoselectionfunctions(S1andS2)thatarebased Sharma et al. (2016), so the underlying cause might be the
onobservableslikegravityandtemperature,givesimilarval- same. Applying corrections to the ∆ν scaling relation pre-
uesfor f ,butthevalueisdifferentforS0. Itisclearthat dictedbystellarmodelsmakestheobservedmassdistribution
νmax
the selection function plays a crucial role and can bias f narrower than observed but does not change the mean. The
νmax
by 2%. The difference between TRILEGAL and Galaxia is disagreement in the mass distributions reported here, trans-
alsoabout2%. latestoabout2%changeinν ,whichiscomparabletothe
max
Detailed theoretical modelling of oscillations shows that currentuncertaintyintheν scalingrelation. Infuture,we
max
therearedeparturesfromthe∆νscalingrelationthatdepend need to verify the scaling relations to better than 2% to put
uponmetallicity,mass,andage. AsmentionedinSection2.2, betterconstraintsontheGalacticmodels.
weaccommodatedthesedeparturesusingthecorrectionfac-
tor f . We computed these corrections using the publicly 6http://www.physics.usyd.edu.au/k2gap/Asfgrid
∆ν
8
However, Galaxia failed to match the distribution of ob- Institute for Theoretical Physics (National Science Founda-
servedstarsinloggandT space. Thisalsoneedstobein- tion Grant No. NSF PHY11-25915) for facilitating helpful
eff
vestigated in future. The mismatch at high T could be due discussionsofresultsinthispaper.WethankWilliamChaplin
eff
to inaccuracies in predicting oscillation amplitudes because forallowingustousehiscodeforcomputingtheprobability
thedetectionprobabilityofastarinthisregionissensitiveto ofdetectingoscillations.
itsassumedamplitude. However,themismatchatlowT is S.S. is funded through ARC DP grant 120104562 (PI
eff
mostlikelyduetoinaccuraciesinthemodel,becauseherewe Bland-Hawthorn)whichsupportstheHERMESproject. D.S.
expect the detection probability to be close to 1. Parallaxes, isfundedthroughFutureFellowshipfromtheAustralianRe-
andhenceluminosities,fromGaiawillhelpresolvethisissue search Council (ARC). J.B.H. is funded through Laureate
becauseluminositycorrelateswithgravity. Fellowship from the Australian Research Council (ARC).
D.H.acknowledgessupportbytheAustralianResearchCoun-
cil’s Discovery Projects funding scheme (project number
ACKNOWLEDGEMENTS DE140101364)andsupportbytheNationalAeronauticsand
WeacknowledgethesupportofGalacticArchaeologyand Space Administration under Grant NNX14AB92G issued
Precision Stellar Astrophysics program organized by Kavli throughtheKeplerParticipatingScientistProgram.
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