The K2-HERMES Survey: Age and Metallicity of the Thick Disc
Sanjib Sharma, Dennis Stello, Joss Bland-Hawthorn, Michael R. Hayden, Joel C. Zinn, Thomas Kallinger, Marc Hon, Martin Asplund, Sven Buder, Gayandhi M. De Silva, Valentina D'Orazi, Ken Freeman, Janez Kos, Geraint F. Lewis, Jane Lin, Karin Lind, Sarah Martell, Jeffrey D. Simpson, Rob A. Wittenmyer, Daniel B. Zucker, Tomaz Zwitter, Timothy R. Bedding, Boquan Chen, Klemen Cotar, James Esdaile, Jonathan Horner, Daniel Huber, Prajwal R. Kafle, Shourya Khanna, Tanda Li, Yuan-Sen Ting, David M. Nataf, Thomas Nordlander, Hafiz Saddon, Gregor Traven, Duncan Wright, Rosemary F. G. Wyse
DD RAFT VERSION A PRIL
30, 2019Typeset using L A TEX twocolumn style in AASTeX62
The K2-HERMES Survey: Age and Metallicity of the Thick Disc S ANJIB S HARMA ,
1, 2 D ENNIS S TELLO ,
3, 1, 4, 2 J OSS B LAND -H AWTHORN ,
1, 2 M ICHAEL
R. H
AYDEN ,
1, 2 J OEL
C. Z
INN , T HOMAS K ALLINGER , M ARC H ON , M ARTIN A SPLUND ,
7, 2 S VEN B UDER ,
8, 9 G AYANDHI
M. D E S ILVA , V ALENTINA
D’O
RAZI , K EN F REEMAN , J ANEZ K OS , G ERAINT
F. L
EWIS , J ANE L IN , K ARIN L IND ,
8, 13 S ARAH M ARTELL , J EFFREY
D. S
IMPSON , R OB A. W
ITTENMYER , D ANIEL
B. Z
UCKER ,
10, 15 T OMAZ Z WITTER , T IMOTHY
R. B
EDDING , B OQUAN C HEN , K LEMEN C OTAR , J AMES E SDAILE , J ONATHAN H ORNER , D ANIEL H UBER ,
16, 17, 18 P RAJWAL
R. K
AFLE , S HOURYA K HANNA , T ANDA L I , Y UAN -S EN T ING ,
20, 21 D AVID
M. N
ATAF , T HOMAS N ORDLANDER ,
7, 2 H AFIZ S ADDON , G REGOR T RAVEN , D UNCAN W RIGHT , AND R OSEMARY
F. G. W
YSE
Sydney Institute for Astronomy, School of Physics, The University of Sydney, NSW 2006, Australia ARC Centre of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D) School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA Institute of Astrophysics, University of Vienna, TÃijrkenschanzstrasse 17, Vienna 1180, Austria Research School of Astronomy & Astrophysics, Australian National University, ACT 2611, Australia Max Planck Institute for Astronomy (MPIA), Koenigstuhl 17, D-69117 Heidelberg Fellow of the International Max Planck Research School for Astronomy & Cosmic Physics at the University of Heidelberg Department of Physics & Astronomy, Macquarie University, Sydney, NSW 2109, Australia INAF -Osservatorio Astronomico di Padova Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden Centre for Astrophysics, University of Southern Queensland, Toowoomba, Queensland 4350, Australia Research Centre in Astronomy, Astrophysics & Astrophotonics, Macquarie University, Sydney, NSW 2109, Australia Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA SETI Institute, 189 Bernardo Avenue, Mountain View, CA 94043, USA Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark International Centre for Radio Astronomy Research (ICRAR), The University of Western Australia, 35 Stirling Highway,Crawley, WA 6009, Australia Institute for Advanced Study, Princeton, NJ 08540, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
ABSTRACTAsteroseismology is a promising tool to study Galactic structure and evolution because it can probe the agesof stars. Earlier attempts comparing seismic data from the
Kepler satellite with predictions from Galaxy modelsfound that the models predicted more low-mass stars compared to the observed distribution of masses. It wasunclear if the mismatch was due to inaccuracies in the Galactic models, or the unknown aspects of the selectionfunction of the stars. Using new data from the K2 mission, which has a well-defined selection function, we findthat an old metal-poor thick disc, as used in previous Galactic models, is incompatible with the asteroseismicinformation. We show that spectroscopic measurements of [Fe/H] and [ α /Fe] elemental abundances from theGALAH survey indicate a mean metallicity of log( Z / Z (cid:12) ) = − .
16 for the thick disc. Here Z is the effectivesolar-scaled metallicity, which is a function of [Fe/H] and [ α /Fe]. With the revised disc metallicities, for thefirst time, the theoretically predicted distribution of seismic masses show excellent agreement with the observeddistribution of masses. This provides an indirect verification of the asteroseismic mass scaling relation is goodto within five percent. Using an importance-sampling framework that takes the selection function into account,we fit a population synthesis model of the Galaxy to the observed seismic and spectroscopic data. Assumingthe asteroseismic scaling relations are correct, we estimate the mean age of the thick disc to be about 10 Gyr, inagreement with the traditional idea of an old α -enhanced thick disc. a r X i v : . [ a s t r o - ph . GA ] A p r S HARMA ET AL . Keywords:
Galaxy: stellar content – structure– methods: data analysis – numerical INTRODUCTIONIn recent years, asteroseismology has emerged as a power-ful tool to study Galactic structure and evolution (Miglioet al. 2009; Chaplin et al. 2011a; Miglio et al. 2013;Casagrande et al. 2014, 2016; Sharma et al. 2016, 2017;Anders et al. 2017; Rodrigues et al. 2017; Silva Aguirre et al.2018) However, previous attempts based on data from theoriginal
Kepler mission (Borucki et al. 2010; Stello et al.2013), which was designed for detecting transiting planets,have struggled to match the predictions of stellar-population-synthesis Galactic models to observations, with the modelsproducing too many low mass stars (Sharma et al. 2016,2017). There are three possible causes for this mismatch: (i)inaccuracies in the selection function of stars in the obser-vational catalog, (ii) an incorrect Galactic model, and (iii)systematics in the scaling relations used to relate asteroseis-mic observables ( ∆ ν, ν max ) to density and surface gravity ofthe stars. Using data from the K2 Galactic Archaeology Pro-gram (K2GAP, Stello et al. 2015, 2017), which is a programto observe oscillating giants with the ’second-life’ Kepler mission (K2, Howell et al. 2014) following a well-definedselection function, illuminates the first cause. In this paperwe therefore, first use the K2GAP data to test if predictionsof the Galactic models that are constrained independentlyof the asteroseismic data match the observed asteroseismicdata. This provides an indirect way to test the scaling re-lations. Having shown that the scaling relations are fairlyaccurate, next, we make use of them and the asteroseismicdata to fit some of the parameters in the Galactic model, anddiscuss the implications for our understanding of the Galaxy.Unlike
Kepler , however, the seismic detection completenessof K2 is not 100%. This is because the time span of K2 lightcurves (typically 80 days) is much shorter than that of
Kepler (typically more than a year). Hence, we carefully study thedetection completeness in K2, and devise ways to take theminto account when comparing observations to models.In a Galactic model, the mass distributions of giants issensitive to the age and the metallicity of the stellar popula-tions in the model. While the role of age was investigated inSharma et al. (2016), the possibility of an inaccurate prescrip-tion of metallicity being responsible for the mismatch be-tween observed and predicted mass distributions has not beeninvestigated so far. Many studies have attempted to character-ize the metallicity distribution of the thin and thick discs. Forthe thin disc, there is a well defined radial metallicity gradient( ≈ -0.07 dex/kpc Robin et al. 2003; Hayden et al. 2014) butthe age-metallicity relation is almost flat (Bensby et al. 2014;Casagrande et al. 2016; Xiang et al. 2017; Silva Aguirre et al.2018). For the thick disc, there is a lack of consensus re- garding its properties. The Besançon model adopted a meanmetallicity value of [Fe/H]=-0.78 based upon spectroscopicmeasurements by Gilmore et al. (1995) (mean thick disc[Fe/H] ∼ − .
6) and photometric (U, B, V bands) measure-ments by Robin et al. (1996). The
Galaxia model (Sharmaet al. 2011) also adopted the same prescription for metallicitydistribution of Galactic components as the Besançon model.However, at least four separate studies have compared pre-dictions of the Besançon Galaxy model with that of spec-troscopic observations and find that away from the midplaneand in regions where the thick disc dominates, the metallic-ity distribution of the model is inconsistent with observationsand that a shift of the thick disc metallicity from -0.78 toabout -0.48 is required to make the model agree with obser-vations. Specifically, Soubiran et al. (2003) concluded thethick disc metallicity to be [Fe/H]= − . ± .
05 by spectro-scopically studying about 400 red clump stars in the directionof the North Galactic Pole at a height of 200 < z / pc < R ∼ < z / kpc < R ∼ ∼ − .
48 for thethick disc. Note, the [M/H] of Kordopatis et al. (2011) isprobably close to [Fe/H] but the exact relationship is notknown. Boeche et al. (2013) and Boeche et al. (2014) used
Galaxia to compare predictions of the Besançon model withstars from RAVE (841-879.5 nm at R ∼ | z | >
800 pc, a better match to observations is obtainedif [Fe/H] of the thick disc is set to − .
5. The former studymakes use of dwarfs while the latter uses giants. More re-cently, results of Hayden et al. (2015) using giants from theAPOGEE survey (1.51-1.70 µ m at R ∼ − .
36) for the thick disc, byconsidering stars between 1 < z / kpc < < R / kpc < α enhancement into account. This translatesto [Fe/H] ∼ − . Z (cid:12) = 0 . α /Fe]=0.24),which compared to the Besançon model is more in line withrecent spectroscopic measurements but is still lower than theAPOGEE measurements.We now have a large sample of stars with very precisemetallicity measurements from spectroscopic surveys likethe Galactic Archaeology for HERMES (GALAH De Silvaet al. 2015) , K2-HERMES (a GALAH-like survey dedicatedto K2 follow-up, Wittenmyer et al. 2018) and Apache PointObservatory Galactic Evolution Experiment (APOGEE Ma- EW CONSTRAINTS FROM
K2 3jewski et al. 2017) surveys. In this paper, we use data fromthe GALAH survey to determine the metallicity distributionof the stellar populations in the Besançon-based Galacticmodel that we later use for asteroseismic analysis. Obser-vationally, it is difficult to measure the metallicity of the stel-lar populations like the thin and the thick discs that are usedin Galactic models. This is because the thick and thin discsoverlap considerably such that it is difficult to identify indi-vidual stars belonging to each of the discs. Hence, we adopta forward modeling approach where we fit a Galactic modelto the observed data and try to answer the following ques-tion. What is the metallicity of the thick and thin discs thatbest describes the spectroscopic data from GALAH? Next,we use data from the APOGEE survey to verify our best fitmodel. Unlike spectroscopic studies of
Kepler seismic tar-gets, the selection function of the K2-HERMES stars is thesame as for the K2 seismic targets. We take advantage of thisfact to then directly check if the metallicity distribution of theasterosesimic data in K2, whose mass distributions we wishto compare with Galactic models, is in agreement with themodels.Finally, unlike the original
Kepler survey, which was con-fined to one direction of the sky, the K2 targets span alongthe ecliptic allowing us to test our Galactic models in variousregions of the Galaxy. Of particular importance is the abilityof K2 to investigate the thick disc of the Milky Way. Thethick disc is one of the most intriguing components of theGalaxy and its origin is not well understood. Compared tothe thin disc, it is old, alpha-enhanced, metal poor, has highervelocity dispersion, and has a larger scale height. A compli-cation with the thin disc vs. thick disc nomenclature, is thatthe scale-length for the thick disc is shorter than for the thindisc (Bovy et al. 2012; Xiang et al. 2017; Mackereth et al.2017); the thick disc truncates near the solar circle wherethe thin disc dominates and beyond the solar circle the thindisc flares with increasing Galactic radius (see discussion andFig. 1 in Bland-Hawthorn et al. 2018). Although numerousspectroscopic surveys have targeted thick disc stars, a char-acterization of the thick disc using asteroseismic data has notbeen carried out. It was not possible to do using the
Kepler data because its field of view was close to the Galactic plane.To move beyond this limitation, a number of K2GAP cam-paigns were selected at high Galactic latitudes, which meansthat a significant fraction of stars in the K2GAP are expectedto be thick disc stars. Here, we use the K2 data to answerthe following question. What is the thick disc age that bestdescribes the seismic masses and spectroscopic data?The paper is structured as follows. In Section 2, we de-scribe the asteroseismic and spectroscopic data used for thestudy. In Section 3, we discuss the methods that we use.Here we describe the selection function of the sample anddiscuss how we take it into account when forward modeling the simulated Galactic data. In Section 4, we present our re-sults where we compare model predictions with observationsand also tune the metallicity and the age distributions in ourmodel to fit the data. In Section 5 we discuss and concludeour findings. DATA2.1.
Target selection
The stars in this study were observed by K2 as part of theK2GAP Guest Observer program (Stello et al. 2015, 2017).The stars that we use span four K2 campaigns − C1, C4,C6, and C7 − whose sky distributions are shown in Figure 1.These K2 campaigns cover different regions of the Galaxyand sample a wide variety of Galactic stellar populations in-cluding old, young, thin disc, thick disc, inner disc, and outerdisc. C1 and C6 are at high galactic latitudes and hence arelikely to have more old thin disc and thick disc stars owingto the larger scale height of such stars. C4 and C7 are atlower latitudes and are likely to be dominated by young thindisc stars. C4 is towards the Galactic anti-center and samplesthe outer disc, whereas C7 is towards the Galactic center andsamples the inner disc.The stars follow a simple color magnitude selection basedon the 2MASS photometry, which is given in Table 1. Thefollowing equation was used to convert 2MASS magnitudeto an approximate V band magnitude (Sharma et al. 2018). V JK = K + . J − K s + . + .
382 exp[( J − K − . / . < V JK < . c denotes the pointing iden-tifier of these 1 degree radius zones.In the K2GAP survey, the proposed stars were ranked inpriority by V magnitude. For the dense field of C7, they wereadditionally restricted to only three circular zones, to makethe spectroscopic follow-up more efficient. During the finalK2 mission-level target selection process for each campaign,the K2GAP target list was truncated at an arbitrary point ( V -magnitude) based on target allocation. Hence those selectedstars will follow the K2GAP selection function. However,targets from other successful Guest Observer programs thatoverlap with lower ranked (fainter) K2GAP targets could stillend up being observed. These stars would not satisfy the S HARMA ET AL . G a l a c t i c l a t i t u d e [ d e g r ee ] Figure 1.
Field of view of K2 campaigns 1 to 19 in Galactic coordinates. Orange fields (campaigns C1, C4, C6 and C7) are studied in thispaper. Campaigns C1 and C6 are at high Galactic latitude pointing away from the Galactic midplane, while campaigns C4 and C7 are at lowGalactic latitude close to the Galactic plane. Campaign C7 is towards the Galactic center at (0,0) and C4 is towards the Galactic anti-center.
Figure 2.
Angular distribution of the seismic sample on sky (orange dots). Each field of view comprises of 21 CCD modules with a smallspacing between them. Two CCD modules were broken and hence no observations fall within them. Black dots are stars with spectroscopicinformation from the K2-HERMES survey. The black dots lie within a circle of 1 degree radius centered on the CCD modules. CampaignsC1, C4, and C6 have no angular selection for the seismic sample. In C7, stars are confined to three circular regions. Panel (d) shows theK2-HERMES sky pointings and their identifier.
Table 1.
Number of K2 GAP targets for each campaignCampaign Proposed Observed Following N giants N giants N giants Selection functionselection with ν max with ∆ ν ν max +spec.1 9108 8630 8598 1104 583 455 (( J − K s ) > . < H < . J − K s ) > . < V JK < . J − K s ) > . < V JK < . J − K s ) > . < V JK < . c ∈ { , } ) OR((14 . < V JK < . c ∈ { } )))N OTE —Circular pointing identifier c is shown in Figure 2d. K2GAP selection function. It is straightforward to locate thetruncation point from the lists of proposed and observed tar-gets by plotting the fraction of proposed to observed stars as a function of row number. A sharp fall in this ratio identifiesthe location of the truncation point. The K2GAP-proposedstars, the K2GAP-observed stars, and the K2GAP-observed
EW CONSTRAINTS FROM
K2 5
Table 2. ≤ ’BBB’ = ’AAA’ J,H,K photometric qualityBflag = ’111’ = ’111’ blend flagCflag = ’000’ = ’000’ contamination flagXflag = 0 = 0Aflag = 0 = 0prox > > stars following the K2GAP selection function are listed inTable 1. 2.2. Asteroseismic data
Our primary asteroseimic sample comes from stars ob-served by K2. The K2 time-series photometry is sampledroughly every 30 minutes, and span about 80 days per cam-paign. This allows us to measure the seismic signal in giantsbrighter than
Kepler magnitude,
K p , of ∼
15 in the range10 (cid:46) ν max /µ Hz (cid:46)
270 (1 . (cid:46) log g (cid:46) . g stars due to theirhigher noise levels and lower oscillation amplitudes (Stelloet al. 2017). We adopt the sesimic results from Stello et al.(2017) (C1) and Zinn et al. (2019,in prep) (C4, C6, andC7). Throughout the paper we focus on stars with 10 <ν max /µ Hz < ν max , and the frequency sepa-ration between overtone oscillation modes, ∆ ν . Here, we usethe results of the two pipelines called BAM (Zinn et al. 2019in prep) and CAN (Kallinger et al. 2010), which are bothbased on Bayesian MCMC schemes for accessing whetheroscillations are detected in a given dataset, and to obtain sta-tistically robust uncertainties on each measurement. Typi-cally, only 50-70% of stars for which oscillation are detected(meaning ν max is determined) do the pipelines also measurea robust ∆ ν (Stello et al. 2017, Zinn et al. 2019 accepted).In this paper, in addition to comparing the predictions ofour new Galactic model against results from K2, we alsocompare against the results from the Kepler mission. For thiswe use the catalog of oscillating giants by Stello et al. (2013),in which the global seismic parameters were estimated usingthe Huber et al. (2009) pipeline (SYD). The exact selectionfunction of oscillating giants in
Kepler is not known. How-ever, an approximate formula3 . R (cid:12) < R KIC < . R Earth (cid:112) . σ LC / . , (2) was derived by Sharma et al. (2016) and we use this to sub-select targets from the above catalog. Here, R KIC is thephotometry-based stellar radius as given in the
Kepler inputcatalog of Brown et al. (2011), σ LC = (1 / c Kepler ) (cid:113) c Kepler + × max(1 , Kp / (3)is the long cadence (LC) noise to signal ratio, and c Kepler =3 . × . − Kp ) + is the number of detected e − per LCsample (Jenkins et al. 2010). For comparing the Kepler re-sults with Galactic models, the synthetic g band SDSS pho-tometry was corrected using Equation 4 from (Sharma et al.2016) and then R KIC was estimated from synthetic photome-tery using the procedure outlined in Brown et al. (2011).2.3.
Spectroscopic data
The spectroscopic data come from the K2-HERMES (forseismic K2 targets) and the GALAH surveys (non seismictargets) being conducted at the 3.9-m AAT located at Sid-ing Spring observatory in Australia. The spectra were col-lected using the multi object High Efficiency and Resolu-tion Multi-Element Spectrograph (HERMES) spectrograph(Sheinis et al. 2015). The K2-HERMES survey uses the sameinstrument setup as the GALAH survey (Martell et al. 2017)and the TESS-HERMES survey (Sharma et al. 2018). Thereduction is done using a custom IRAF based pipeline (Koset al. 2017). The spectroscopic analysis is done using theGALAH pipeline and is described in (Buder et al. 2018a). Ituses Spectroscopy Made Easy (SME) to first build a trainingset by means of a model driven scheme (Piskunov & Valenti2017). Next,
The Cannon (Ness et al. 2015) is used to es-timate the stellar parameters and abundances by means of adata driven scheme. METHODS3.1.
Galactic models
In this paper we perform two kinds of analysis, one is tocompare the predictions of theory with observations and theother is to fit Galactic models to the observed data. For this,we use population synthesis based Galactic models. Themodels consist of four different Galactic components, thethin disc, the thick disc, the bulge, and the stellar halo. Thefull distribution of stars in space, age, and metallicity Z , isgiven by p ( R , z , Z , τ | θ ) = (cid:88) k p ( k ) p ( R , z , Z , τ | θ, k ) , (4)with k denoting a Galactic component, θ the parameters gov-erning the Galactic model, R the Galactocentric cylindricalradius, z the Galactic height, τ the age, and Z the metallicityof the stars in the Galactic component. To sample data from S HARMA ET AL .a prescribed population synthesis model we use the
Galaxia code (Sharma et al. 2011). It uses a Galactic model that is ini-tially based on the Besançon model by Robin et al. (2003) butwith some crucial modifications. The density laws and theinitial mass functions for the various components are givenin Table 1 of Sharma et al. (2011) and are same as in Robinet al. (2003). The density normalizations for various compo-nents are given in Table 4, these differ slightly from Robinet al. (2003) and a discussion of the changes is given in Sec-tion 3.6 of Sharma et al. (2011). The thin disc spans an agerange of 0 to 10 Gyr and has a star formation rate which isalmost constant. The thin disc has a scale height which in-creases with age according to Equation 18 in Sharma et al.(2011). In this paper, we leave the thick disc normalizationas a free parameter and solve for it using data from Gaia DR2.Other differences between
Galaxia and the
Besançon model are as follows.
Galaxia is a robust statistical sam-pler, it provides continuous sampling over any arbitraryvolume of the Galaxy. This enables rigorous compar-isons with observed stellar surveys for an arbitrary selec-tion function.
Galaxia has a 3D extinction scheme that isbased on Schlegel et al. (1998) dust maps. We also ap-ply a low-latitude correction to the dust maps as describedin Sharma et al. (2014). The isochrones used to predictthe stellar properties are from the Padova database usingCMD 3.0 (http://stev.oapd.inaf.it/cmd), with PARSEC-v1.2Sisochrones (Bressan et al. 2012; Tang et al. 2014; Chenet al. 2014, 2015), the NBC version of bolometric cor-rections (Chen et al. 2014), and assuming Reimers massloss with efficiency η = 0 . Y = 0 . + . Z relation and their solar metal content is Z (cid:12) = 0 . Galaxia modeldenoted by MP (metal poor) is from Sharma et al. (2011), ithas an old metal poor thick disc and a thin disc whose meanmetallicity decreases with age as in Robin et al. (2003). Themodel denoted by MR (metal rich) has metal rich thick andthin discs. The FL (flat) model also has a metal rich thick andthin disc, but unlike other models its thick disc spans an agerange from 6 to 13 Gyr with a uniform star formation rateand no variation of metallicity with age. For each Galacticcomponent k , the IMF, the formula for spatial distributionof stars, and the density normalizations are given in Sharmaet al. (2011).To compare predictions of Galactic models to asteroseis-mic data, we need to estimate the observed seismic quantities ν max and ∆ ν for the synthetic stars. The seismic quantities http://galaxia.sourceforge.net Table 3.
Galactic models with different age and metallicity distri-bution functions.Model Thick Thin a MP b (cid:104) [M / H] (cid:105) -0.78 [0.01, 0.03, 0.03, 0.01,-0.07, -0.14, -0.37] σ [M / H] c ) 11 [0, 0.15, 1, 2, 3, 5, 7]Max(Age) 11 [0.15, 1, 2, 3, 5, 7, 10]MR (cid:104) [M / H] (cid:105) -0.16 [0.01, 0.03, 0.03, 0.01, 0, 0, 0] σ [M / H] (cid:104) [M / H] (cid:105) -0.14 0.0 σ [M / H] a Thin disc consists of 7 distinct populations with different ageranges and d[M / H] / d R = − .
07 dex/kpc b The [M/H] values correspond to [Fe/H] values used by Robin et al.(2003), ignoring α enhancement. c In units of Gyr
Table 4.
The IMFs and the density normalizations of Galactic com-ponents. The parameters α and t α are used to specify the IMF(number density of stars as a function of mass stellar mass M ),which is of the following form, ∝ M α for M / M (cid:12) < ∝ M α for M / M (cid:12) > α α Thin (0 < Age / Gyr < a (cid:12) yr − -1.6 -3.0Thin (7 < Age / Gyr < a (cid:12) yr − -1.6 -3.0Thick b ρ (cid:12) , thick -0.5 -0.5Stellar Halo b . × M (cid:12) pc − -0.5 -0.5Bulge c − -2.35 -2.35 a Star formation rate for an IMF spanning a mass range of 0.07 to100 M (cid:12) . b Local mass density of visible stars c Central density are estimated from effective temperature T eff , surface gravity g , and density ρ using the following asteroseismic scaling re-lations (Brown et al. 1991; Kjeldsen & Bedding 1995; Ulrich1986). ν max ν max , (cid:12) = gg (cid:12) (cid:18) T eff T eff , (cid:12) (cid:19) − . (5) ∆ ν ∆ ν (cid:12) = f ∆ ν (cid:18) ρρ (cid:12) (cid:19) . (6)Here, f ∆ ν = (cid:18) ∆ ν . µ Hz (cid:19) (cid:18) ρρ (cid:12) (cid:19) − . (7) EW CONSTRAINTS FROM
K2 7is the correction factor derived by Sharma et al. (2016)by analyzing theoretical oscillation frequencies with GYRE(Townsend & Teitler 2013) for stellar models generated withMESA (Paxton et al. 2011, 2013). We used the code AS-FGRID (Sharma et al. 2016) that computes the correctionfactor as a function of metallicity Z , initial mass M , evolu-tionary state E state (pre or post helium ignition), T eff , and g .3.2. Importance-sampling framework
To constrain the parameters of a Galactic model fromthe observed data we developed and used an importance-sampling framework, which we now describe. Suppose wehave collected some data regarding some variable x , suchas metallicity Z or seismic mass, subject to some selectionfunction S . Then suppose that we have a Galactic model pa-rameterized by θ from which we can draw samples subjectto the same selection function S . To constrain the model, westart with a base model parameterized by some θ , then tochange the model to one parameterized by a new θ , we sim-ply reweight the samples from the simulation parametrizedby θ instead of drawing from a new simulation. When themodel changes from θ to θ , the new weights for a star i be-longing to a Galactic component k are given by w i = p ( R i , z i , Z i , τ i | θ, k ) / p ( R i , z i , Z i , τ i | θ , k ) . (8)In general, such a change can alter the number of visiblestars of your synthetic Galaxy, but as long as the parametersgoverning the density distribution of the stars are unaltered,the changes are minimal. In this paper, we are mainly con-cerned with only altering the thick disc parameters like meanage and metallicity. We also alter the metallicity of the oldthin disc, but this change is minor and can be ignored forthe present discussion related to the number of visible stars.The base model that we use is based on the Besançon model,which was constructed by Robin et al. (2003) to satisfy theobserved star counts in the Galaxy. When the thick discparameters, like mean age and/or the metallicity are modi-fied, we adopt the following procedure to address the slightchange that is expected in the number of visible thick discstars. We measure f SGP , the ratio of stars that lie between2 < | z | / kpc < ◦ radius conearound the south Galactic pole ( b = − . ◦ ) and have Gaiamagnitudes 0 < G <
14 from Gaia DR2 (Gaia Collaborationet al. 2018). Using f SGP estimated from Gaia DR2, we solvefor the normalization factor ρ (cid:12) , thick and reweight the thickdisc of the model such that f SGP in the selection-function-matched mock sample matches with that of the Gaia DR2data. Following this global normalization, the stars are fur-ther reweighted to satisfy the color magnitude selection func- tion of the observational data to which the model is beingfitted.To fit the model to the data we need to compute the likeli-hood of the data given the model and this is done as follows.Let x q be the q th percentile of the distribution of some vari-able x . For this variable, suppose we have observed samples X o and samples from some model X m , with the model beingparameterized by θ and S being the selection function. Theprobability of the observed data given the model can then bewritten as p ( X o | θ, S ) = (cid:89) q √ πσ x exp (cid:18) − ( x o , q − x m , q ) σ x (cid:19) , with (9) σ x = ( σ x , o / n eff , o + σ x , m / n eff , m )Here, n eff is the effective number of stars, which for stars withdifferent weights is given by (cid:0)(cid:80) w i (cid:1) / (cid:80) w i according toKish’s formula. We make use of 16, 50 and 84 percentiles tocompute the likelihood of the data given the model. The x o , q and x m , q denote the q th percentile obtained from samples X o and X m respectively. For multiple data sets, X = { X o , ..., X nso } with each of them having their own selection function S = { S , ..., S ns } , the full likelihood is p ( X | θ, S ) = (cid:89) i p ( X io | θ, S i ) . (10)In this paper, the importance-sampling framework is usedfor estimating the metallicity of the thick disc using spectro-scopic data from the GALAH survey and to estimate the ageof the thick disc from the asteroseismic data from K2. Forthe former (metallicity estimation), we bin up the stars lyingin 5 < R / kpc <
11 and 1 < | z | / kpc < R and 0.33 kpc in | z | . We use log Z / Z (cid:12) as the ob-served variable x and fit for the mean and the dispersion ofthe thick disc metallicity and the mean metallicity of the oldthin disc (age greater than 3 Gyr) in the Galactic model. Forthis we use the MR model from Table 3. For the latter (ageestimation), we bin up the stars into different K2 campaignsand 3 different giant classes. We follow Sharma et al. (2016)by using the temperature-independent seismic mass proxy κ M = (cid:18) ν max ν max , (cid:12) (cid:19) (cid:18) ∆ ν ∆ ν (cid:12) (cid:19) − (11)(12)as the variable x and fit for the age (mean) and metallicity(mean and dispersion) of the thick disc. For this we use theFL model from Table 3. For each selection of stars the likeli-hood is computed using Equation 9. The κ M is closely relatedto the stellar mass M , which is given by M M (cid:12) = κ M (cid:18) T eff T eff , (cid:12) (cid:19) . . (13) S HARMA ET AL .Given that temperatures are not always readily available forthe observed stars, we use κ M instead of mass when compar-ing theoretical predictions to observations. This also removesany ambiguity in temperature scale differences between themodels and the data. For simplicity we will in the followingrefer to κ M as mass.3.3. Detection completeness
Before we can compare the mass distributions, we have tomake sure that the observed and simulated data satisfy thesame selection function. In other words, we have to prop-erly forward-model the simulated data and make it satisfythe same observational constraints that the observed data sat-isfies.The duration of the K2 campaigns sets a lower limit onthe detectable ν max of about 10 µ Hz below which the seismicdetection efficientcy drops. The observational cadence setsan upper limit of about ν max = 270 µ Hz (Stello et al. 2015).The amplitude of oscillations decreases with increasing ν max (less luminous stars) and the photometric noise increases to-wards fainter stars. This makes it harder to detect oscillationsfor stars that have higher ν max and/or are faint. This bias isclearly visible as missing stars in the top right corner of Fig-ure 3(a,c,e,g), which shows the distribution of observed starsin the ( ν max , V JK ) plane.To model the seismic detection probability we followedthe scheme presented by Chaplin et al. (2011b) and Cam-pante et al. (2016). For this, we used the mass, radius,and effective temperature of each synthetic star to predictits total mean oscillation power and granulation noise in thepower spectrum. The oscillation amplitude was estimatedas A = 2 . (cid:0) L / L (cid:12) (cid:1) . (cid:0) M / M (cid:12) (cid:1) − . (cid:0) T eff / T eff , (cid:12) (cid:1) − followingStello et al. (2011). The granulation power was estimatedusing the Kallinger et al. (2014) model. The apparent mag-nitude was used to compute the instrumental photon-limitednoise in the power spectrum, which combined with granu-lation noise gave the total noise. For the instrumental noisewe use Equation 3 (formula given by Jenkins et al. 2010).For K2, we scaled the noise by a factor of three to takeinto account the higher noise in the K2 data compared tothe Kepler data and also applied a minimum threshold of 80ppm. The mean oscillation power and the total noise werethen used to derive the probability of detecting oscillations, p detect , with less than 1% possibility of false alarm. Stars with p detect > . Galaxia simulated stars are shown in Figure 3(b,d,f,h) as theratio between the number of observed to predicted stars. The ν max was estimated using Equation 5. The figure shows thatthe fraction of predicted to observed stars is close to one overmost of the regions where we have observed stars. However,C4, C6, and C7 show a slight tendency of having a lower l o g [ m a x / H z ] (a) C1, 1053 (b) C1, 0.771.01.52.02.5 l o g [ m a x / H z ] (c) C6, 1910 (d) C6, 0.731.01.52.02.5 l o g [ m a x / H z ] (e) C4, 1801 (f) C4, 0.668 10 12 14 V JK [mag]1.01.52.02.5 l o g [ m a x / H z ] (g) C7, 1488 8 10 12 14 V JK [mag](h) C7, 0.7010 N obs N obs / N pred Figure 3.
Distribution of observed (left panels) stars in the( ν max , V JK ) plane for four K2 campaigns and the Kepler field. Theright panels plot the ratio of observed to predicted oscillating giantsin each bin. The predictions are based on simulations using
Galaxia .The dashed line represents the equation ν max = − V JK − than predicted number of stars towards the top right cornerof each panel (higher V JK and higher ν max ), where the signal-to-noise ratio of the oscillations is low. This is probably be-cause for these campaigns, the detections are based solely onautomated pipelines, with no additional visual inspection asfor C1 (Stello et al. 2017). The mean detection fraction is0.72, and the cause for fewer detections is not clear. Usingthe deep-learning-based pipeline of Hon et al. (2018) resultedin slightly more ν max detections, raising the mean detectionfraction to 0.78, but the fraction still remained significantlyless than one.3.4. The distribution of ν max and apparent magnitude We now check the distribution of apparent magnitudes and ν max in more detail. In Figure 4, the ν max distributions show apeak, which corresponds to the red clump stars. For the sim-ulated data (orange line) the peak is close to 30 . µ Hz. Thelocation of the peak varies very little across different cam-paigns, it is about 1 µ Hz higher for the low latitude cam-paigns C4 and
Kepler . For the observed data analyzed with
EW CONSTRAINTS FROM
K2 9 p (a)C10.000.040.080.12 p (b)C60.000.030.060.09 p (c)C40.000.030.060.09 p (d)C716 20 24 28 32 36 40 44 max [ Hz]0.000.030.060.09 p (e) Kepler
K2-CANGalaxia(MP) K2-BAMGalaxia(MR) Kepler-SYD
Figure 4.
The probability distribution of ν max for observed and pre-dicted oscillating giants. The dashed line, ν max = 30 . µ Hz, showsthe approximate location of the peak in the distribution of the pre-dicted stars. The peak corresponds to the location of the red clumpgiants. The location of the peak is not sensitive to the choice of theGalactic model, but the distribution is sharper for the MR model.The peak for the CAN pipeline is systematically lower as comparedto the BAM pipeline. For C4 the location of the peak for both theCAN and the BAM pipelines is higher as compared to predictions. the BAM pipeline, except for C4, the location of the peakdoes not show any obvious shift with respect to the pre-dicted peak. For C4 the BAM peak is about 3 µ Hz higher.For all campaigns, the location of the peak for the CANpipeline is systematically lower by 2 µ Hz compared to theBAM pipeline. This suggests that the CAN pipeline system-atically underestimates ν max for stars around the red clumpregion compared to the scaling relation prediction. The peakfor the Kepler data obtained using the SYD pipeline also didnot show any shift with respect to the predicted peak. To con-clude, we see systematic differences between campaigns andbetween pipelines, they are small but could be important forcertain applications and hence should be investigated furtherin future. p (a)105314111367C10.000.150.30 p (b)191028192627C60.000.150.30 p (c)180128952716C49 10 11 12 13 14 15 16 17 V JK [mag]0.00.30.6 p (d)148820732123C7K2-CAN Galaxia(MP) Galaxia(MR) Figure 5.
Magnitude distribution of observed oscillating giantsfrom K2 along with predictions from
Galaxia corresponding tomodel MP and MR. The number of stars with ν max detections inthe observed sample and those predicted by model MP, and MR arealso listed in each panel. The corresponding distributions of V JK for K2 are shownin Figure 5. Overall the observed distributions match wellwith the model predictions. For C6, the model predicts morestars for V JK > .
5, the cause of which is not yet clear, butwe found that this has no impact on our conclusions relatedto the mass distribution of stars that we present in this paper.3.5.
Classifying giants into different classes
In the seismic analysis, ν max is easier to detect as comparedto ∆ ν . Hence, there are stars with a ν max measurement butno ∆ ν measurement and this needs to be taken into accountwhen comparing model predictions with observations. To ac-complish this, we first study the ∆ ν -detection completenessof our sample and then devise ways to account for it whencomparing model predictions with observations.The probability, p ∆ ν , of having a ∆ ν measurement giventhat we have a measurement of ν max is shown in Figure 6, asa function of ν max . This was derived by binning the stars in ν max and then computing in each bin the ratio of the num-ber of stars with a ∆ ν measurement ( N ∆ ν ) to those with a ν max measurement ( N ν max ). We see three distinct phases. Thefirst is for ν max < µ Hz, where p ∆ ν is constant but low.The second is for 25 < ν max < µ Hz, where p ∆ ν increaseswith ν max . And the third is for ν max > µ Hz, where p ∆ ν HARMA ET AL . [ Hz]0.00.20.40.60.81.01.2 N / N m a x C1 C6 C4 C7
Figure 6.
The ratio of the number of stars with and without ∆ ν measurements for various K2 campaigns. The black dot marks thefrequency of the peak, ν max , RC , in the ν max distribution of the ob-served stars and is due to RC stars. The ratio shows a sharp increasefor stars with ν max > ν max , RC . is again constant and close to 1 (except for C7 where p ∆ ν is lower for ν max > µ Hz). The drop in p ∆ ν as ν max de-creases from 50 to 30 µ Hz, coincides with the increase infraction of red-clump stars as predicted by a
Galaxia simula-tion (see orange dots in Figure 7). This drop could be becausethe power spectra of red-clump stars are more complex thanRGB stars (Chaplin & Miglio 2013) and this makes the ∆ ν measurement harder to obtain. For ν max < µ Hz, we mainlyhave RGB stars, but the p ∆ ν is still low, and this could be dueto the limited frequency resolution of the K2 data starting toaffect our ability to obtain a clear ∆ ν measurement towardsthe low ν max stars. Although all four campaigns show similar p ∆ ν for high ν max stars, we note that for ν max < µ Hz, p ∆ ν is about a factor of two lower for C1 and C4 compared to C6and C7. The cause for this different behavior is not clear.We have seen in Figure 6 that ∆ ν detections are incom-plete with a completeness that depends on stellar type (evo-lution stage). This suggests that we should study the differ-ent types of giants separately. Below we describe a schemeto segregate stars into three giant classes, the high luminos-ity RGB stars, the RC stars, and the low luminosity RGBstars. The segregation is done in the ( κ M , ν max ) plane. Byconstruction the high luminosity RGB class will have somecontamination from AGB stars and the RC class will havecontamination from RGB stars.Figure 7 shows the distribution of stars in the ( κ M , ν max )plane both for the observed and Galaxia -simulated data.Although RC stars typically have ν max ∼ µ Hz, Figure 7shows that the high κ M stars can have ν max reaching up to ∼ µ Hz. This is the main reason why we decided not toisolate RGB stars solely from their ν max . Based on simula-tions by Galaxia we instead fit and obtain two curves ν lowermax = 6 . κ M − . κ M + .
914 (14) ν uppermax = 33 . κ M − . κ M + .
647 (15)
Figure 7.
Distribution of stars in the ( κ M , ν max ) plane. The bluelines split the plane into three distinct regions, the predominantlyhigh-luminous RGB stars (left), the predominantly red-clump stars(middle) and the low-luminosity RGB stars (right). Left panels(a,c,e,g) show results from K2 based on the CAN pipeline. Rightpanels (b,d,f,h) show predictions from Galaxia . The overplotted or-ange points denote the red clump stars. that enclose about 92% of the RC stars (blue lines). In Fig-ure 7, it can be seen that the red clump stars are nicely en-closed by the blue lines.These curves are then used to classify stars into the threecategories; a) ν max < ν lowermax (high-luminosity RGB starsor hRGB), b) ν lowermax < ν max < ν uppermax ) (RC stars), and c) ν max > ν uppermax (low-luminosity RGB stars or lRGB). Basedon Galalxia simulations, the fraction of RGB stars in thethree categories averaged across all campaigns was found tobe a) 0.87, b) 0.18, and c) 0.97, suggesting that each categoryis dominated by the desired stellar type in that category, i.e.,RGB, RC and RGB respectively. RESULTS4.1.
Constraints from spectroscopic surveys
Large scale surveys of the Milky Way were not availableat the time the
Besançon model was constructed as imple-mented in
Galaxia . The situation has changed now, with sur-veys like APOGEE and GALAH providing high-resolutionspectra for hundreds of thousands of stars, which sample the
EW CONSTRAINTS FROM
K2 11 [ / F e ] (a)(log g < 3.5)2.0 1.5 1.0 0.5 0.0 0.5[Fe/H]0.150.000.150.300.450.60 [ / F e ] (b)(log g < 3.5)&(5 < R /kpc < 7)&(1 < | z |/kpc < 2)10 Figure 8.
Distribution of GALAH giants in the ([Fe/H],[ α /Fe])plane. Giants were selected using log g < .
5. (a) Distribution of allgiants. (b) Giants restricted to 5 < R / kpc < < | z | / kpc < Galaxy well beyond the solar neighborhood. Hence it is pos-sible to characterize the the thick disc better than before.To study the elemental composition of the thick disc weneed to identify stars belonging to the thick disc. This can bedone using height above the Galactic plane or rotational ve-locity. We choose the former approach as the overlap of thethin and thick disc is quite strong in rotational velocity. Toisolate thick disc stars, we select stars with (5 < R / kpc < < | z | / kpc <
2. This region provides the largest num-ber of thick disc stars with the least amount of contamina-tion from the thin disc, as can be seen in Figure 4 fromHayden et al. (2015). In Figure 8, we further illustrate thisusing data from the GALAH survey (Buder et al. 2018b).Three populations are visible in Figure 8a; the stellar halo at[Fe/H] ∼ − .
75, the thin disc at [Fe/H] ∼
0, and the thick discat [Fe/H] ∼ − .
39. After selecting stars by location, the thindisc sequence almost vanishes and the halo can be identifiedas a separate over-density in Figure 8b. For this particularspatial selection, the median and the spread of the distribu-tion of [Fe/H] and [ α /Fe] are listed in Table 5 and comparedwith that of APOGEE. Also given are metallicity estimates[M/H] constructed using the formula[M / H] = log (cid:18) ZZ (cid:12) (cid:19) = [Fe / H] + log(10 [ α / Fe] . + . . (16) Table 5.
Abundance of iron and alpha elements for thick disc stars.Median and standard deviation based on 16-th and and 84-th per-centile values are listed. The first four rows show the abundancesfor stars with 5 < R / kpc < < | z | / kpc < α /Fe] log( Z / Z (cid:12) )med sdev med sdev med sdevAPOGEE -0.294 0.28 0.186 0.08 -0.160 0.24GALAH DR2 -0.367 0.24 0.218 0.08 -0.196 0.21GALAH DR2c a -0.316 0.21 0.239 0.07 -0.131 0.18GALAH mock b -0.170 0.25GALAH DR2c c -0.162 0.17 a Calibrated b Mock catalog generated by
Galaxia with log( Z / Z (cid:12) ) ∼N ( − . , . ) for the thick disc c Fitting a Galactic model to GALAH stars with 5 < R / kpc < < | z | / kpc <
3, 12 < V JK < field_id <6546 and positiverotation. by Salaris & Cassisi (2005). Given an isochrone grid con-structed for metallicities Z using solar-scaled compositionwith a specified Z (cid:12) , the above formula provides an approx-imate estimate of metallicity Z or [M/H] for a given [Fe/H]and [ α /Fe]. In Table 5, although we choose to show the me-dian, the mean values were also very similar with the differ-ence between the two being less than 0.01 dex (after discard-ing stars with [Fe/H] < − .
25, which most likely belong tothe stellar halo).
Table 6.
Polynomial coefficients of calibration equation y calib = y + c + c [Fe / H] + c [Fe / H] to correct for systematics in the Cannonbased estimates against the SME based estimates. The equation wasderived using giants having ν max estimates from asteroseismologyand with − . < [Fe / H] < .
3. The calibration is applied to giantswith [Fe / H] > − .
5, the giants are identified using the Ciardi et al.(2011) definition. y c c c log g +3.4987e-01 +7.4591e-01 +1.5727e-01 T eff K +1.5658e+02 +2.1861e+02 -3.9895e+00[Fe / H] +1.9087e-01 +2.7875e-01 +2.4761e-02[ α / Fe] -2.5775e-02 -4.1510e-02 -3.2592e-02
In Table 5, for GALAH two different estimates are given,the first is based on the GALAH DR2 pipeline, the secondnamed GALAH DR2c is based on a calibration correctionthat we derive and apply to the GALAH DR2 estimates.GALAH DR2 estimates are based on
The Cannon method(Ness et al. 2015), which was trained on results from theSME (Piskunov & Valenti 2017) pipeline . However, asshown in Figure 9, for giants we find subtle systematics in theGALAH DR2 stellar parameters compared to that of SMEestimates, where ν max estimated from asteroseismology was2 S HARMA ET AL . l o g g (a)1500150 T e ff [ K ] (b)0.20.10.00.10.2 [ F e / H ] (c)1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.25[Fe/H] Cannon0.20.10.00.10.2 [ / F e ] (d)Cannon - SME, with max Cannon - SME, logg>3.5
Figure 9.
Comparison of GALAH-DR2 Cannon-based (data-driven) estimates to that of SME-based (model-driven) estimates.The plots shows systematic trends as a function of Cannon-basediron abundance [Fe/H]. The giants (blue) and dwarfs (orange) areshown separately. The giants shown are seismic giants from K2,and for them SME was run using ν max estimated from asteroseis-mology as a prior. The seismic giants show strong systematictrends while dwarfs have negligible systematics. The dotted lineis a two degree polynomial fit to the trends for the seismic giantswith − . < [Fe / H] < . used as a prior. The systematics are particularly significantfor stars with [Fe/H]> 0.0. We use the seismic giants in theSME training set to recalibrate the GALAH results. The co-efficients of the calibration equation are given in Table 6. Acomparison of GALAH stellar parameters (both calibratedand uncalibrated) with those from APOGEE for commonstars is shown in Figure 10. The calibrated GALAH grav-ities and temperatures match better with APOGEE. In therange − . < [Fe / H] < .
0, where the majority of the sam-ple is found, the calibrated [Fe/H] also matches better withAPOGGE. Outside this range some systematics exist. The[ α /Fe] shows slight offsets in zero points but no significanttrend is seen. l o g g (a) GALAH-APOGEE
GALAH DR2GALAH DR2c (calibrated)1500150 T e ff (b)0.20.10.00.10.2 [ F e / H ] (c)1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75[Fe/H] GALAH [ / F e ] (d) Figure 10.
Comparison of GALAH DR2 stellar parameters withAPOGEE stellar parameters. Results corresponding to both uncali-brated and calibrated GALAH DR2 data are shown.
We see from Table 5 that the estimates for the meanmetallicity of stars above the midplane from APOGEE, theGALAH DR2, and the GALAH DR2c agree to within 0.07dex. The lowest values are for GALAH DR2c and the highestare for GALAH DR2. We now investigate if the metallicityof stars in 5 < R / kpc <
11 and 1 < | z | / kpc < Galaxia sample matched to the GALAHsurvey, with a thick disc having a mean [M/H] metallicity of-0.18, for stars lying in the same spatial selection. The esti-mated metallicity is higher by only 0.01 dex compared to themetallicity of the thick disc that was used in the model. Thissuggests that the metallicity of stars with 5 < R / kpc <
11 and1 < | z | / kpc < EW CONSTRAINTS FROM
K2 13 l o g Z / Z [ d e x ] (a)0.0<|z|<0.25 6 9R [kpc] (b)0.25<|z|<0.5 6 9R [kpc] (c)0.5<|z|<0.75 6 9R [kpc] (d)0.75<|z|<1.0 6 9R [kpc] (e)1.0<|z|<2.0 6 9R [kpc] (f)2.0<|z|<3.0GALAH APOGEE Galaxia(MP) Galaxia(MR) Figure 11.
Mean metallicity as function of Galactocentric radius R for different slices in height | z | . The observed results are from GALAH(calibrated) and APOGEE. Selection-function-matched Galaxia predictions based on two different Milky Way models (the old MP model andthe new MR model) are also shown. The metallicity profile has a gradient close to the plane but is flat above the plane. The dashed line forreference denotes [M/H] with a radial gradient of -0.07 dex/kpc. this, we fitted a Galactic model to the GALAH DR2c data ly-ing within 5 < R kpc <
11, 0 . < | z | / kpc <
3, 12 < V JK < field_id < − . < R / kpc <
11 and 1 < | z | / kpc <
2, is due to theinclusion of stars lying between 2 < | z | / kpc < < | z | / kpc < R for different slices in height | z | .Results from APOGEE and GALAH DR2c are shown sep-arately. We also plot Galaxia predictions from the MP andMR models. To eliminate stars belonging to the halo we re-strict the analysis to stars having positive rotation about theGalaxy. The MP model clearly has a thick disc, which is toometal poor to fit the observed data for | z | > < | z | / kpc < Kepler (Kep)) and dif-ferent seismic classes (hRGB, RC, lRGB). Predictions from4 S
HARMA ET AL . p (a)C1( 58.5), hRGB (b)C1( 58.5), RC (c)C1( 58.5), lRGB0.01.53.0 p (d)C6( 49.8), hRGB (e)C6( 49.8), RC (f)C6( 49.8), lRGB0.01.53.0 p (g)C4(-25.9), hRGB (h)C4(-25.9), RC (i)C4(-25.9), lRGB0.01.53.0 p (j)C7(-14.6), hRGB (k)C7(-14.6), RC (l)C7(-14.6), lRGB2.0 1.5 1.0 0.5 0.0 0.5[M/H]0.01.53.0 p (m)Kep( 13.5), hRGB 2.0 1.5 1.0 0.5 0.0 0.5[M/H] (n)Kep( 13.5), RC 2.0 1.5 1.0 0.5 0.0 0.5[M/H] (o)Kep( 13.5), lRGBObserved Galaxia(MP) Galaxia(MR) Figure 12.
The distribution of metallicity[M/H] for RGB and red clump stars with seismic detections from K2 campaigns C1, C4, C6, C7, and
Kepler . The left panels (a,d,g,j,m) show high luminosity RGB stars, middle panels (b,e,h,k,n) show red clump stars, and right panels (c,f,i,l,o)show low luminosity RGB stars. Observed data is compared against predictions from theoretical models, the default model of
Galaxia -MP,which has a metal poor thick disc and the new model MR, which has a more metal rich thick disc. The metallicity for the K2 stars is fromthe K2-HERMES survey while for the
Kepler stars we adopt APOGEE-DR14 metallicities. The Galactic latitude of each field is enclosed inparenthesis.
Galaxia -MP (orange) and the new
Galaxia -MR (green) areshown alongside the observed data (blue). The
Galaxia -MPsamples have many more metal poor stars with [M/H] < − . Galaxia -MR samples, which has ametal rich thick disc ( (cid:104) [M / H] (cid:105) ∼ − . Kepler field the
Galaxia -MR samples are still too metal poor.For hRGB in C7 the
Galaxia -MR samples are too metal rich.4.2.
Constraints from asteroseismology
In this section we present results making use of the aster-oseismic data. We first compare the observed distribution ofseismic masses against the predictions of fiducial Galactic models. Next, we restrict our analysis to thick disc stars andassuming reasonable priors on the thick disc parameters, wedemonstrate that the asteroseismic scaling relations are fairlyaccurate. Finally, assuming the scaling relations to be correctwe estimate the age of the thick disc.4.2.1.
Comparing observed distribution of seismic masses againstpredictions from Galactic models
In Figure 13, we study the distribution of κ M . The orderof the panels is the same as in Figure 12. For hRGB stars,the overall sample size is too small to assess the quality ofhow well the models match the data. Both models seemto perform equally well. However, for the large hRGB Ke-pler sample we do see that the new model provides a visiblybetter match. Now turning to the RC stars, we see acrossthe board that the new MR model performs better than theold MP model, which predicts too many stars with κ M < EW CONSTRAINTS FROM
K2 15 p (a)28146129C1( 58.5), hRGB (b)132529673C1( 58.5), RC (c)420745561C1( 58.5), lRGB012 p (d)131334278C6( 49.8), hRGB (e)53711881427C6( 49.8), RC (f)7721281920C6( 49.8), lRGB012 p (g)55322310C4(-25.9), hRGB (h)43616561677C4(-25.9), RC (i)452916733C4(-25.9), lRGB012 p (j)157349329C7(-14.6), hRGB (k)63813101489C7(-14.6), RC (l)235413305C7(-14.6), lRGB0.5 1.0 1.5 2.0 2.5 3.0 M p (m)89921331496Kep( 13.5), hRGB 0.5 1.0 1.5 2.0 2.5 3.0 M (n)8104961210323Kep( 13.5), RC 0.5 1.0 1.5 2.0 2.5 3.0 M (o)370924242881Kep( 13.5), lRGBObserved Galaxia(MP) Galaxia(MR) Figure 13.
The distribution of κ M for RGB and red clump stars for campaigns C1, C4, C6, C7 and Kepler . The annotation and order of thepanels is the same as in Figure 12. For each panel, the number of stars in each sample are listed on the right hand side. ter than the MP model, which predicts too many stars with κ M < . Galaxia . Specifically, we want to answerthe following questions: a) does the MR model match theK2 data better than the MP model does, b) and does the MRmodel also provide a better match to the
Kepler data, whichhad issues with the selection function.The κ M distributions are in general unimodal. At the mostbasic level a unimodal distribution over a finite domain canbe characterized by a median. We first estimate the mediansand then compute the ratio of medians between the observedand predicted distributions, which we show in Table 7. Ide-ally, we expect the ratio to be close to one, but in previouswork based on Kepler data, we found the median ratio to belarger than one (1.06).The new MR model, anchored on GALAH metilicities ofthe thick disc, is undoubtedly better than the old MP model.For almost all giant classes and campaigns, the median ra-tio for the new MR model is closer to unity than for the oldMP model. The only two exceptions are hRGB for C6 and C7, where the ratio is about 0.85, i.e., the model overpredictsthe masses. However, these samples suffer from low num-ber statistics. Additionally for the hRGB stars in C7, we alsonoticed that the MR model overpredicts the metallicity Fig-ure 12, and this will lead to overestimation of masses in theMR model.4.2.2.
Testing the accuracy of the asteroseismic mass scalingrelation
The fact that the mass distribution of the new model MRmatches the observed seismic masses so well, suggests thatthe asteroseismic scaling relations are fairly accurate. In thefollowing we will explore this more quantitatively by limit-ing the analysis to a single Galactic component and impos-ing reasonable non-seismic priors on its parameters. To dothis, we study the mass distribution of stars lying between1 < | z | / kpc <
3. The Galactic model predicts that about 90%of these stars should be thick disc stars, so we can modelthem as a simple stellar population characterized by someage distribution and metallicity distribution. We have alreadyshown that the metallicity distribution of this population canbe represented by N ( − . , . ). In the following we6 S HARMA ET AL . Table 7.
Ratio of observed (CAN pipeline) median κ M to that predicted by Galaxia for different giant classes. Results for two different Galacticmodels MP (metal poor) and MR (metal rich) are shown. Uncertainties on the computed ratio are also listed.hRGB RC lRGBCampaign
Galaxia (MP)
Galaxia (MR)
Galaxia (MP)
Galaxia (MR)
Galaxia (MP)
Galaxia (MR)1 1 . ± .
05 1 . ± .
04 1 . ± .
03 1 . ± .
03 1 . ± .
009 0 . ± . . ± .
03 0 . ± .
02 1 . ± .
02 0 . ± .
01 1 . ± .
007 1 . ± . . ± .
04 0 . ± .
04 1 . ± .
02 1 . ± .
01 1 . ± .
01 1 . ± . . ± .
03 0 . ± .
03 1 . ± .
02 0 . ± .
01 1 . ± .
01 1 ± . Kepler . ± .
01 0 . ± .
01 1 . ± .
003 1 . ± .
003 1 . ± .
003 1 . ± . M p C1 and C6, 1<|z|/kpc<3, lRGB
K2-CAN(-0.16 dex, 7 Gyr)(-0.16 dex, 10 Gyr)
Figure 14.
The distribution of κ M for lRGB stars in K2 campaignsC1 and C6 that lie between 1 < | z | / kpc <
3. Shown alongside aremass distributions corresponding to simple stellar populations witha Gaussian metallicity distribution and a uniform age distribution(with a width of 2 Gyr). The mean metallicity and the mean age ofeach stellar population is given in the legend. present several pieces of evidence suggesting that the meanage of this high | z | population should be between 8 to 12 Gyr.Firstly, Figure 8 shows that stars between 1 < | z | / kpc < α element abundances and form a distinctsequence in the abundance space. Using dwarf and subgiantsin the solar neighborhood, it has been shown that the starsin the α -enhanced sequence are typically older than 10 Gyr(Bensby et al. (2014) Figure 22 and Hayden et al. (2017) Fig-ure 3). Secondly, chemical evolution models predict that α -enhanced stars must have formed within the first 1 Gyr of thestar formation history of the Milky Way, or else the contri-bution from Type-Ia supernovae would have introduced toomuch iron and hence brought the value of [ α /Fe] down (Pagel2009). When the above fact is combined with Figure 3 fromHayden et al. (2017), which suggest that the oldest thin discstars (stars not enhanced in [ α /Fe]) are around 8 to 10 Gyrold, we reach the conclusion that the α -enhanced populationmust be older than 8 to 10 Gyr. Finally, Kilic et al. (2017)provide one of the most precise and accurate estimates onthe mean age of the thick disc using nearby white dwarfs.They estimate the mean thick disc age to be between 9.5 to9.9 Gyr, with a random uncertainty of about 0.2 Gyr. Hence, f M C1 and C6, 1<|z|/kpc<2, lRGBlog( Z / Z ) ( 0.162, 0.17 )Age Uniform( /2, + /2)0.0 0.2 0.4 0.6 0.8 1.0 p ( f M , | D )/Max( p ( f M , | D )) Figure 15.
The posterior distribution of f M and mean age of thethick disc τ obtained using lRGB stars in K2 campaigns C1 andC6 that lie between 1 < | z | / kpc <
3. The width ∆ τ of the agedistribution was assumed to be 2 Gyr. based on these observational evidence, a reasonable prior forthe mean age of the thick disc is 8 to 12 Gyr.To test the asteroseismic mass scaling relation we selectthe lRGB stars in K2 campaigns C1 and C6 that lie between1 < | z | / kpc <
3. We avoid campaigns C4 and C7 becausethey point into the Galactic plane and hence lack high | z | stars. We restrict our test to lRGB stars because for thesestars there is almost 100% probability both to detect ν max andto detect ∆ ν when a ν max has been measured. The distribu-tion of κ M for the lRGB stars is shown in Figure 14. The dis-tributions of κ M for a stellar population with a metallicity dis-tribution of N ( − . , . ) and a mean age of τ = 10 Gyr isalso shown alongside, showing a good match to the observeddistribution. However, the distribution for the stellar popula-tion with τ = 7 Gyr but the same metallicity distribution asbefore, is shifted too far to the right. Now, to quantify theaccuracy of the asteroseismic mass scaling relation (Eq. 13), EW CONSTRAINTS FROM
K2 17we introduce a factor f M , that is multiplied to κ M for stars inthe model to get a ‘corrected’ mass, and then we investigatehow close to unity this correction factor is when enforcingthat the observed and model mass distributions match.The posterior distribution of f M and the age, τ , conditionalon our data D is given in Figure 15. For the mean age ofthe high | z | population we assume a flat prior in the range 8to 12 Gyr. The analysis was done using the importance sam-pling framework discussed in Section 3.2 and taking the pho-tometric selection function into account. The figure showsthat f M depends upon τ and varies between 0.97 and 1.05 forthe adopted range of τ . This would translate into a maxi-mum deviation of the ν max scaling relation (Eq. 5) of 1-2% ifthe ∆ ν scaling relation (Eq. 6) is true. Or alternatively, thatthe maximum deviation of the ∆ ν scaling relation would beabout 1% if the ν max scaling relation is true. Now, if boththe ∆ ν and the ν max relations are incorrect but conspire tocancel out their inaccuracy when using the mass scaling re-lation (Eq. 13), one could in principle have a scenario wherelarge deviations of the ∆ ν and ν max relations could be hid-den in our mass test. However, this seems not to be the casebecause when testing the radius scaling relation R R (cid:12) = (cid:18) ν max ν max , (cid:12) (cid:19) (cid:18) ∆ ν (cid:12) ∆ ν (cid:12) (cid:19) − (cid:18) T eff T eff , (cid:12) (cid:19) . , (17)which is based on different powers of ∆ ν and ν max , Zinn et al(submitted) finds agreement between seismic and Gaia radiiat the 1% level. Hence, in combination these mass and radiusscaling relation tests show strong evidence that the individual ∆ ν and ν max scaling relations that go into the mass and radiusscaling relations are in fact astonishingly accurate.4.2.3. Constraining the age of the thick disc
Having established that the asteroseismic scaling relationsare good to a high degree of accuracy, it would seem reason-able to now turn the problem around. Hence, in the followingwe assume the relations to be true and use the observed val-ues of κ M to estimate the age and metallicity of the thickdisc. We do this using the importance sampling frameworkdiscussed in Section 3.2. Here, we use the FL Galactic modelfrom Table 3 as the base model and reweight it to simulatesamples corresponding to different values of the parametersof the model. We compute the likelihood of the observed κ M values given the model for different values of the meanmetallicity, log Z / Z (cid:12) , and mean age for the thick disc. Giventhe unknown selection function of the Kepler data, only datafrom the K2 campaigns were used. The results are shown inFigure 16. We adopted a duration of 2 Gyr for the star forma-tion episode of the thick disc. We also investigated shorter (1Gyr) and longer (3 Gyr) star formation durations and foundthat the results were not too sensitive to the exact choice ofthe duration. log Z / Z [dex]789101112 A g e [ G y r ] (a)all Age [Gyr]0.00.20.40.60.81.0 li k e li h oo d (b)10.0 ± 0.25[M/H] -0.162all0.3 0.2 0.1 0.0log Z / Z [dex]789101112 A g e [ G y r ] (c)lRGB 5 6 7 8 9 10 11 12Age [Gyr]0.00.20.40.60.81.0 li k e li h oo d (d)9.2 ± 0.26[M/H] -0.162lRGB3.0 2.5 2.0 1.5 1.0 0.5 0.0(log likelihood)/4.0 Figure 16. (a,c) Likelihood of age and metallicity of the thick discusing asteroseismic information from K2 campaigns C1, C4, C6,and C7. (b-d) The likelihood of thick disc age assuming the thickdisc metallicity to be log( Z / Z (cid:12) ) = − . Figure 16a shows the likelihood when considering all gi-ants. Figure 16b shows the likelihood when only lRGB gi-ants are used. It can be seen that when we only considerthe asteroseismic information, age is degenerate with metal-licity. A decrease in the adopted metallicity by 0.1 dex candecrease the inferred age by about 2 Gyr. Figure 16a showsthat a metal poor thick disc cannot be old. For example, athick disc with log Z / Z (cid:12) = − . DISCUSSION AND CONCLUSIONS8 S
HARMA ET AL .Asteroseismology can provide ages for giant stars andhence is a promising tool for studying Galactic structure andevolution. However, it has proven to be difficult to check theaccuracy of the ages and masses estimated by asteroseismol-ogy, due to the shortage of independent estimates of mass andage. Population synthesis based Galactic models, provide anindirect way to validate the asteroseismic estimates. How-ever, previous studies using the
Kepler mission revealed thatthe models predict too many low mass stars as compared toobserved mass distributions, raising doubts on the accuracyof the asteroseismic estimates, the Galactic models, and/orthe selection function. In this paper, we revisit this importantproblem by analyzing asteroseismic data from the K2 mis-sion, which has a well defined selection function. For the firsttime, we show that if the metallicity distribution in the Galac-tic models is updated to measurements from recent spectro-scopic surveys, the distribution of asteroseismic masses is ingood agreement with the model predictions. Using thick discstars we show that the asteroseismic mass scaling relation forlow luminosity red giants should be accurate to 5%. This is inagreement with findings of Brogaard et al. (2018) who testedthe seismic relations using three eclipsing binary systems.We identify three main factors, which, if not taken intoaccount, can lead to discrepancies between observed astero-seismic masses and model predictions. First, in addition toage, the mass distribution giant stars in a stellar population isvery sensitive to its metallicity, hence it is important to get themetallicity distribution of the various Galactic components ina model to agree with observations. Second, certain Galacticcomponents are significantly enhanced in abundance of α el-ements and this should be taken into account, either directlyby using α enhanced isochrones, or indirectly by increasingthe effective metallicity of the solar scaled isochrones. Third,the ∆ ν scaling relation is not strictly valid and there existstheoretically motivated corrections, which should be applied.It was already shown in a previous study (Sharma et al. 2016)that the correction is such that it helps to reduce the mass dis-crepancy.Using a forward modelling approach, where we take theBesançon Galactic model as a prior, we fit for the effectivemetallicity Z (taking α enhancement into account) of the thinand the thick disc using the GALAH data. We find the meanlog Z / Z (cid:12) of the thin disc to be 0 . − .
78 as used in the Besançon model. An increaseof about 0.14 dex in log Z / Z (cid:12) is due to taking the α enhance-ment into account, but about 0.5 dex is due to revision of[Fe/H]. For example, if we consider stars in 5 < R / kpc < < | z | / kpc <
2, which mostly come from the thick disc, both GALAH and APOGEE suggest a mean [Fe/H] ∼ − . . − ± .
25 Gyr (redshift ofabout 1.6), which is broadly consistent with the idea of thethick disc being old and formed early on in the history ofthe Galaxy. What exactly do we mean by thick disc? Tradi-tionally the thick disc was identified as the component withhigher scale height in the solar annulus. Observations alsosuggest the thick disc to be distinct in elemental abundancesfrom the thin disc. Two sequences α + and α o can be seenin the ([ α /Fe],[Fe/H]) plane, with the former (having higher[ α /Fe]) being the thick disc and the later the thin disc. Newresults (Bensby et al. 2011; Bovy et al. 2012; Xiang et al.2017; Mackereth et al. 2017) suggest that the scale lengthof the α + sequence is shorter than that of the α o sequence.Chemical evolution models require the α + sequence to beold. In our forward modelling we do not identify the thickdisc using elemental abundances. Instead the thick disc is in-directly identified by our prior for the spatial distribution ofthin and thick disc stars. In the model, stars with | z | > | z | / kpc > < R / kpc < . ± . | z | > α enhanced. It is consis-tent with results by Mackereth et al. (2017) from APOGEEusing giants, where they show that α enhanced stars have sig-nificantly larger scale height and their mean age is close to orlarger than 10 Gyr. However, the age estimates in Mackerethet al. (2017) are anchored on the asteroseismic age scale. Fi-nally, our estimate (9 . − ± .
25 Gyr) is in excellent agree-ment with estimates of Kilic et al. (2017) of 9 . − . ± . Kepler are higher by about 3%. We also seedifferences in metallicity distributions for these samples and
EW CONSTRAINTS FROM
K2 19this could potentially be responsible for the mass differences.For hRGB and red clumps, the mean predicted mass is lowerthan observed, for campaigns C6 and C7. This could be dueto imperfections in the model, but could also be related to thefact that the detection of ∆ ν is not complete for these stars.We present the selection function for four K2 campaignsand discuss detection biases associated with the K2 data,which should be taken into account when using the K2 data.Probability to detect ν max varies with both ν max and apparentmagnitude. Low-luminosity stars have lower oscillation am-plitudes and cannot be detected at fainter magnitudes. Evenafter we account for the effect of oscillation amplitude andapparent magnitude, comparison with Galactic models showthat the overall detection rate for ν max is about 72%. Usinga deep-learning-based pipeline improves the detection rate to78%, which is still quite low. It is not yet clear as to why thedetection rate is low. It could be that certain specific typeof stars (e.g., red clumps or metal poor stars) have lowerthan expected oscillation amplitudes, or it could be an un-known instrumental effect, or even a problem with the Galac-tic model. There are also biases related to detecting ∆ ν inthe K2 data. The probability to detect ∆ ν has a strong de-pendence on ν max , it is less than 1 for ν max < µ Hz, butis otherwise close to 1. Significant campaign to campaigndifferences are also seen, which needs further investigation.To take the detection biases into account, we propose to splitup the stars into different giant classes based on their detec-tion probabilities. Asteroseismic pipelines also show smallsystematic offsets in estimation of ν max which need furtherinvestigation.Using the seismic sample, we find that the stellar parame-ters for giants in GALAH DR2, which are based on the data-driven The Cannon scheme, have systematic differences withrespect to estimates based on the model-driven SME schemethat is anchored to seismic ν max values. Differences are mostsignificant for stars with [Fe/H]>0. We provide analyticalfunctions to correct for them. The reason for the systematicoffsets is because the giants in the training set used by TheCannon were dominated by non seismic giants. In the ab-sence of seismic ν max , the SME gives biased results. SMEwith Gaia DR2 parallaxes as prior alleviates this problem,however, Gaia DR2 parallaxes were not available at the timeof publication of GALAH DR2. In near future, we will have a much larger sample of starswith asteroseismology from both the K2 and the TESS (Sul-livan et al. 2015) missions. This will allow us to fit moredetailed models of our Galaxy than done here. Specifically,we can study the properties of the stellar populations as afunction of age with much finer age resolution. Future, spec-troscopic surveys, such as the, second phase of GALAH,4MOST (de Jong et al. 2016), WEAVE (Dalton et al. 2018),and SDSSV (Kollmeier et al. 2017), will also produce largesamples of stars with age estimates purely from spectroscopy,based on main sequence turnoff and subgiant stars or basedon giants making use of the age information encoded in car-bon and nitrogen abundances. Asteroseismology in this re-gard is going to play a crucial role by providing independentage estimates.S.S. is funded by University of Sydney Senior Fellowshipmade possible by the office of the Deputy Vice Chancel-lor of Research, and partial funding from Bland-Hawthorn’sLaureate Fellowship from the Australian Research Coun-cil. The GALAH Survey is supported by the AustralianResearch Council Centre of Excellence for All Sky As-trophysics in 3 Dimensions (ASTRO 3D),through projectnumber CE170100013. D.S. is the recipient of an Aus-tralian Research Council Future Fellowship (project numberFT1400147). JBH is supported by an ARC Australian Lau-reate Fel- lowship (FL140100278). MJH is sup- ported by anASTRO-3D Fellowship. S.B. and K.L. acknowledge fundsfrom the Alexander von Humboldt Foundation in the frame-work of the Sofja Kovalevskaja Award endowed by the Fed-eral Ministry of Education and Research. K.L. acknowledgesfunds from the Swedish Research Council (Grant nr. 2015-00415_3) and Marie Sklodowska Curie Actions (CofundProject INCA 600398). T.Z. acknowledges financial supportfrom the Slovenian Research Agency (research core fundingNo. P1-0188). DMN was supported by the Allan C. andDorothy H. Davis Fellowship. J. Z. acknowledges supportfrom NASA grants 80NSSC18K0391 and NNX17AJ40G. Facilities:
AAT
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