Measuring the vertical age structure of the Galactic disc using asteroseismology and SAGA
L. Casagrande, V. Silva Aguirre, K.J. Schlesinger, D. Stello, D. Huber, A.M. Serenelli, R. Schoenrich, S. Cassisi, A. Pietrinferni, S. Hodgkin, A.P. Milone, S. Feltzing, M. Asplund
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 2 April 2018 (MN L A TEX style file v2.2)
Measuring the vertical age structure of the Galactic disc usingasteroseismology and SAGA (cid:63)
L. Casagrande , †‡ , V. Silva Aguirre , , K.J. Schlesinger , D. Stello , , , D. Huber , , , ,A.M. Serenelli , , R. Sch ¨onrich , S. Cassisi , A. Pietrinferni , S. Hodgkin , A.P. Milone ,S. Feltzing , M. Asplund Research School of Astronomy & Astrophysics, Mount Stromlo Observatory, The Australian National University, ACT 2611, Australia Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia SETI Institute, 189 Bernardo Avenue, Mountain View, CA 94043, USA Instituto de Ciencias del Espacio (ICE-CSIC / IEEC) Campus UAB, Carrer de Can Magrans, s / n 08193 Cerdanyola del Valls Rudolf-Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OX1 3NP, Oxford, United Kingdom INAF-Osservatorio Astronomico di Collurania, via Maggini, 64100 Teramo, Italy Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK Lund Observatory, Department of Astronomy & Theoretical Physics, Box 43, SE-22100, Lund, Sweden
Received; accepted
ABSTRACT
The existence of a vertical age gradient in the Milky Way disc has been indirectly knownfor long. Here, we measure it directly for the first time with seismic ages, using red giants ob-served by
Kepler . We use Str¨omgren photometry to gauge the selection function of asteroseis-mic targets, and derive colour and magnitude limits where giants with measured oscillationsare representative of the underlying population in the field. Limits in the 2MASS system arealso derived. We lay out a method to assess and correct for target selection e ff ects independentof Galaxy models. We find that low mass, i.e. old red giants dominate at increasing Galacticheights, whereas closer to the Galactic plane they exhibit a wide range of ages and metallic-ities. Parametrizing this as a vertical gradient returns approximately 4 Gyr kpc − for the discwe probe, although with a large dispersion of ages at all heights. The ages of stars show asmooth distribution over the last (cid:39)
10 Gyr, consistent with a mostly quiescent evolution forthe Milky Way disc since a redshift of about 2. We also find a flat age-metallicity relation fordisc stars. Finally, we show how to use secondary clump stars to estimate the present-day in-trinsic metallicity spread, and suggest using their number count as a new proxy for tracing theaging of the disc. This work highlights the power of asteroseismology for Galactic studies;however, we also emphasize the need for better constraints on stellar mass-loss, which is amajor source of systematic age uncertainties in red giant stars.
Key words:
Asteroseismology – Galaxy: disc – Galaxy: evolution – stars: general – stars:distances – stars: fundamental parameters
A substantial fraction of the baryonic matter of the Milky Wayis contained in its disc, where much of the evolutionary activitytakes place. Thus, knowledge of disc properties is crucial for un-derstanding how galaxies form and evolve. Late-type Milky Way- (cid:63) http: // / saga † Stromlo Fellow ‡ [email protected] like galaxies are common in the local universe. However, we can atbest measure integrated properties for external galaxies, while theMilky Way o ff ers us the unique opportunity to study its individualbaryonic components.Star counts have revealed that the disc of the Milky Wayis best described by two populations, one with shorter and onewith longer scale-heights, dubbed the “thin” and the “thick” disc(e.g., Gilmore & Reid 1983; Juri´c et al. 2008). This double discbehaviour is also inferred from observations of edge-on galaxies,where the thick disc appears as a pu ff ed up component extending c (cid:13) a r X i v : . [ a s t r o - ph . GA ] O c t Casagrande et al. to a larger height above a sharper thin disc (e.g., Burstein 1979; vander Kruit & Searle 1981; Yoachim & Dalcanton 2006). Althoughit is usually possible to fit the vertical density / luminosity profileof late-type galaxies as a double-exponential profile, its interpreta-tion is still a matter of debate. In particular, it is unclear if the thinand thick disc in the Milky Way are real, separated structural en-tities, or not (e.g., Norris 1987; Nemec & Nemec 1991; Sch¨onrich& Binney 2009b; Bovy et al. 2012). These di ff erent interpretationson disc’s decomposition underpin much of the theoretical frame-work for understanding its origin and evolution. Models in whichthe thick disc is formed at some point during the history of theGalaxy via an external mechanism (in particular accretion and / ormergers) best fit the picture in which the thin and the thick discare real separated entities (e.g., Chiappini et al. 1997; Abadi et al.2003; Brook et al. 2004; Villalobos & Helmi 2008; Kazantzidiset al. 2008; Scannapieco et al. 2009). This scenario acquired mo-mentum in the framework of cold dark matter models, where struc-tures (and galaxies) in the universe form hierarchically (e.g., White& Rees 1978; Searle & Zinn 1978). Thus, the growth of a spiralgalaxy over cosmic time would be responsible for pu ffi ng up thedisc, also “heating” the kinematics of its stars. In contrast, internaldynamical evolution (primarily in the form of radial mixing e.g.,Sch¨onrich & Binney 2009a,b; Loebman et al. 2011) favours thescenario in which the thick disc is the evolutionary end point ofan initially pure thin disc, without requiring a heating mechanism.Internal sources of dynamical disc-heating, e.g. from giant molecu-lar clouds or clump-induced stellar scattering, may also contributeto thick disc formation (e.g., H¨anninen & Flynn 2002; Bournaudet al. 2009). Although merger events can happen at early times, inthis picture the formation of the Galaxy is mostly quiescent.As is often the case, the real –yet unsolved– picture of galaxyformation is more complicated than the simplistic sketches drawnabove. The latest numerical simulations indicate that hierarchicalaccretion occurring at early times can imprint a signature of hotkinematic and roundish structures. Other processes then factor intothe following more quiescent phases, characterized by the forma-tion of younger disc stars in a more flattened, rotationally sup-ported configuration (e.g., Genzel et al. 2006; Bournaud et al. 2009;Aumer et al. 2010; House et al. 2011; Forbes et al. 2012; Stin-son et al. 2013; Bird et al. 2013). Importantly, high-redshift ob-servations suggest that, for galaxies in the Milky Way mass range,this might not happen in an inside-out fashion (van Dokkum et al.2013).Historically, the study of chemical and kinematic propertiesof stars in the (rather local) disc, has been used to shed light onthese di ff erent formation scenarios. Thin disc stars are observed tobe on average more metal-rich and less alpha-enhanced than thickdisc stars (e.g., Edvardsson et al. 1993; Chen et al. 2000; Reddyet al. 2003; Fuhrmann 2008; Bensby et al. 2014). Due to stellarevolutionary timescales the enrichment in alpha elements happensrelatively quickly (e.g. Tinsley 1979; Matteucci & Greggio 1986).Thus, for the bulk of local disc stars it is customary to interpretthis chemical distinction into an age di ff erence (but see Chiappiniet al. 2015 and Martig et al. 2015 for the possible existence of lo-cal outliers). In this picture the thick disc would be the result ofsome event in the history of the early Galaxy and thus metal-poorand alpha-enhanced. This interpretation however has been recentlychallenged by the observational evidence that alpha-enhanced thickdisc stars may also extend to super-solar metallicities (e.g., Feltz-ing & Bensby 2008; Casagrande et al. 2011; Trevisan et al. 2011;Bensby et al. 2014). This can be explained if –at least some of– thethick disc is composed of stars originating from the inner Galaxy, where the chemical enrichment happened faster. In terms of kine-matics, thin disc stars are cooler (i.e. with smaller vertical velocitywith respect to the Galactic plane) and have higher Galactic rota-tional velocity compared to thick disc stars then referred to as kine-matically hot. Low rotational velocities (due to larger asymmetricdrift) imply higher velocity dispersion for thick disc stars, whichthen point to older ages, either born hot or heated up. In fact, theage-velocity dispersion relation has long been known to indicate theexistence of a vertical age gradient (e.g. von Hoerner 1960; Mayor1974; Wielen 1977; Holmberg et al. 2007): its direct measurementis the subject of the present study.The dissection of disc components based only on chemistryand kinematic is far from trivial (e.g., Sch¨onrich & Binney 2009b).In this context, stellar ages are expected to provide an additionalpowerful criterion. Also, age cohorts are easier to compare with nu-merical simulations than chemistry based investigations, bypassinguncertainties related to the implementation of the chemistry in themodels. From the observational point of view however, even whenaccurate astrometric distances are available to allow comparison ofstars with isochrones, the derived ages are still highly uncertain,and statistical techniques are required to avoid biases (e.g., Pont& Eyer 2004; Jørgensen & Lindegren 2005; Serenelli et al. 2013).Furthermore, isochrone dating is meaningful only for stars in theturno ff and subgiant phase, where stars of di ff erent ages are clearlyseparated on the H-R diagram. This is in contrast, for example, tostars on the red giant branch, where isochrones with vastly di ff er-ent ages can fit observational data such as e ff ective temperatures,metallicities, and surface gravities equally well within their errors.As a result, so far the derivation of stellar ages has been essen-tially limited to main-sequence F and G type stars with Hipparcos parallaxes, i.e. around ∼
100 pc from the Sun (e.g., Feltzing et al.2001; Bensby et al. 2003; Nordstr¨om et al. 2004; Haywood 2008;Casagrande et al. 2011). All these studies agree on the fact that thethick disc is older than the thin disc. Yet, only a minor fraction ofstars in the solar neighbourhood belong to the thick disc.It is now possible to break this impasse thanks to asteroseis-mology. In particular, the latest spaceborne missions such as
CoRoT (Baglin & Fridlund 2006) and
Kepler / K2 (Gilliland et al. 2010;Howell et al. 2014) allow us to robustly measure global oscillationfrequencies in several thousands of stars, in particular red giants,which in turn make it possible to determine fundamental physicalquantities, including radii, distances and masses. Most importantly,once a star has evolved to the red giant phase, its age is determinedto good approximation by the time spent in the core-hydrogen burn-ing phase, and this is predominantly a function of the stellar mass.Although mass-loss can partly clutter this relationship, as we willdiscuss later in the paper, to first approximation the mass of a redgiant is a proxy for its age. In addition, because of the intrinsic lu-minosity of red giants, they can easily be used to probe distancesup to a few kpc (e.g. Miglio et al. 2013b).This has profound impact for Milky Way studies, and in factsynergy with asteroseismology is now sought by all major surveysin stellar and Galactic archaeology (e.g. Pinsonneault et al. 2014;De Silva et al. 2015). With similar motivation, we have started theStr¨omgren survey for Asteroseismology and Galactic Archaeology(SAGA, Casagrande et al. 2014, hereafter Paper I) which so far hasderived classic and asteroseismic stellar parameters for nearly 1000red giants with measured seismic oscillations in the
Kepler field. Inthis paper we derive stellar ages for the entire SAGA asteroseis-mic sample, and use them to study the vertical age structure of theMilky Way disc. Our novel approach uses the power of seismol-ogy to address thorny issues in Galactic evolution, such as the age- c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc metallicity relation, and to provide in situ measurements of stellarages at di ff erent heights above the Galactic plane, at the transitionbetween the thin and the thick disc. The study of the vertical metal-licity structure of the disc with SAGA will be presented in a com-panion paper by Schlesinger et al. (2015).The paper is organized as follows: in Section 2 we review theSAGA survey, and present the derivation of stellar ages for the seis-mic sample. In Section 3 we investigate the selection function ofthe Kepler satellite, and identify the colour and magnitude intervalswithin which the asteroseismic sample is representative of the un-derlying stellar population in the field. This allows us to define clearselection criteria, which are then used in Section 4 to derive verti-cal mass and age gradients. We provide raw gradients, i.e. obtainedby simply fitting all stars that pass the selection criteria in Section4.2. The biases introduced by our target selection criteria are alsoassessed, and gradients corrected for these e ff ects are presented inSection 4.3. The implications of the age-metallicity relation, and ofthe age distribution of red giants to constrain the evolution of theGalactic disc are discussed in Section 5. In Section 6 we suggest us-ing secondary clump stars as age candles for Galactic Archaeology.Finally, we conclude in Section 7. The purpose of the SAGA is to uniformly and homogeneously ob-serve stars in the Str¨omgren uvby system across the
Kepler field,in order to transform it into a new benchmark for Galactic stud-ies, similar to the solar neighbourhood. Details on survey rationale,strategy, observations and data reduction are provided in Paper I,and here we briefly summarize the information relevant for thepresent work. So far, observations of a stripe centred at Galacticlongitude l (cid:39) ◦ and covering latitude 8 ◦ (cid:46) b (cid:46) ◦ have beenreduced and analyzed. This geometry is particularly well suited tostudy vertical gradients in the Galactic disc.The Str¨omgren uvby system (Str¨omgren 1963) was de-signed for the determination of basic stellar parameters (see e.g.,´Arnad´ottir et al. 2010, and references therein). Its y magnitudes aredefined to be essentially the same as the Johnson V (e.g., Bessell2005), and in this work we will refer to the two interchangeably.SAGA observations are conducted with the Wide Field Cam-era on the 2 . all stars in the magnitude range 10 (cid:46) y (cid:46)
14, wheremost targets were selected to measure oscillations with
Kepler . Thisrequirement can be easily achieved with short exposures on a 2 . Kepler measured oscilla-tions are essentially detected in our survey (with a completeness (cid:38)
95 per cent, see Paper I). SAGA is magnitude complete to about y (cid:39)
16 mag, thus providing an unbiased, magnitude-limited cen-sus of stars in the Galactic stripe observed. Stars are still detectedat fainter magnitudes ( y (cid:46) ,
000 ob-jects in the stripe observed so far. Thus, we can build two samplesfrom our observations. First, a magnitude complete and unbiasedphotometric sample down to y (cid:39)
16 mag, which we refer to asthe full photometric catalog. Second, we extract a subset of 989stars which have oscillations measured by
Kepler , dubbed the as-teroseismic catalog. Note that the stars for which
Kepler measuredoscillations were selected in a non trivial way. By comparing the properties of the asteroseismic and full photometric catalogs wecan assess the
Kepler selection function (Section 3).Before addressing the
Kepler selection function, we briefly re-call the salient features of the full photometric and asteroseismiccatalog. For the asteroseismic catalog we also derive stellar ages,and discuss their precision.
The full photometric catalog provides uvby photometry for sev-eral thousands of stellar sources down to a magnitude complete-ness limit of y (cid:39)
16. The asteroseismic catalog is a subset of thetargets in the full photometric catalog, and thus the data reductionand analysis is identical for all stars in SAGA. In this work, forboth the asteroseismic and the full photometric sample (or any sub-sample extracted from them) we use only stars with reliable pho-tometry in all uvby bands (Pflg = flag, see also figure 4 in PaperI). This requirement excludes stars whose errors are larger thanthe ridge-lines defined by the bulk of photometric errors, as cus-tomarily done in photometric analysis, and does not introduce anybias for our purposes. Furthermore, we have verified that the frac-tion of stars excluded as a function of increasing y mag is nearlyidentical for both the asteroseismic and the full photometric sam-ple. This is true for the two samples as a whole, or when restrict-ing them only to giants. A flag also identifies stars with reliable[Fe / H], i.e. those objects for which the Str¨omgren metallicity cali-bration is used within its range of applicability (Mflg = ). This flagautomatically excludes stars with [Fe / H] (cid:62) .
5, where such highvalues could be the result of extrapolations in the metallicity cali-bration and / or stem from photometric and reddening errors. Again,this limit is not expected to introduce any bias given the paucity (oreven the non-existence, see e.g., Taylor 2006) of star more metal-rich than 0 . Kepler selection func-tion and to study the vertical stellar mass and age struture in theGalactic disc.
The SAGA asteroseismic catalog consists of 989 stars identified bycross-matching our Str¨omgren observations with the dwarf sam-ple of Chaplin et al. (2014) and the (cid:39) ,
000 giants from the
Ke-pler
Asteroseismic Science Consortium (KASC, Stello et al. 2013;Huber et al. 2014). Within SAGA, a novel approach is developedto couple classic and asteroseismic stellar parameters: for each tar-get, the photometric e ff ective temperature and metallicity, togetherwith the asteroseismic mass, radius, surface gravity, mean densityand distance are computed (Casagrande et al. 2010, 2014; SilvaAguirre et al. 2011, 2012). A detailed assessment of the uncertain-ties in these parameters is given in Paper I. For a large fraction ofobjects, evolutionary phase classification identifies whether a star isa dwarf (labelled as “Dwarf”, 23 such stars in our sample), is evolv-ing along the red giant branch (“RGB”) or is already in the clumpphase (“RC”). It was possible to robustly distinguish between thelast two evolutionary phases for 427 stars, whereas for the remain-ing giants no classification is available (“NO”). In this paper, werefer to all stars classified as “RGB”, “RC”, or “NO” as red giants. c (cid:13) , 000–000 Casagrande et al.
Figure 1.
Panel a): comparison between ages obtained with mass-loss (vertical axis, η = . ff erent seismic classification, as labelled (see also Paper I). Error bars are formal uncertainties, only. Panel b) to e): same as above, but showing thefractional age di ff erence as function of di ff erent parameters. With the information available for each asteroseismic target it israther straightforward to compute stellar properties. As described inPaper I, we apply a Bayesian scheme to sets of BaSTI isochrones (Pietrinferni et al. 2004, 2006). Flat priors are assumed for agesand metallicities over the entire grid of BaSTI models, meaningthat at all ages, all metallicities are equally possible. A Salpeter(1955) Initial Mass Function (IMF, α = − .
35) is also used (seedetails in Silva Aguirre et al. 2015). The adopted asteroseismicstellar parameters are derived using non-canonical BaSTI modelswith no mass-loss, but we explore the e ff ect of varying some of theBaSTI prescriptions as described further below. As input parame-ters we consider the two global asteroseismic parameters ∆ ν and ν max and the atmospheric observables T e ff and [Fe / H]. The infor-mation on the evolutionary phase (“RGB”, “RC”) is included as aprior when available, otherwise the probability that a star belongs toa given evolutionary status is determined by the input observables,the adopted IMF and the evolutionary timescales. The median and68 per cent confidence levels of the probability distribution functiondetermine the central value and (asymmetric) uncertainty of ages,which following the terminology of Paper I we refer to as formaluncertainties.Overshooting in the main-sequence phase can significantlychange the turn-o ff age of a star and is therefore important for ourpurposes. In order to assess its impact, in the present analysis weexplore the e ff ect of using BaSTI isochrones computed from stel- http: // / BASTI lar models not accounting for core convective overshoot during thecentral H-burning stage (dubbed canonical) as well as isochronesbased on models including this e ff ect (dubbed non-canonical, andadopted as reference). We note that all sets of BaSTI isochronestake into account semiconvection during the core helium-burningphase. The BaSTI theoretical framework for mass-loss implementsthe recipe of Reimers (1975): dMdt = η × − LgR (cid:34) M (cid:12) yr (cid:35) , (1)where η is a (free) e ffi ciency parameter that needs to be constrainedby observations (see e.g. the recent analyses by McDonald & Zijl-stra 2015; Heyl et al. 2015). As we discuss further below, mass-losse ffi ciency can have a considerable impact on the parameters rele-vant for the present analysis. Sets of BaSTI stellar models havebeen computed for di ff erent values of η ; we explore its e ff ects byusing the no mass-loss isochrones ( η =
0) as our reference set andcompare to the stellar properties derived with the η = . ∆ ν and ν max , and thus the mass is rather independent of themodels adopted. Paper I demonstrated that for most stellar param-eters, assuming a very e ffi cient mass-loss ( η = .
4) or neglectingcore overshooting during the H-core burning phase a ff ects the re-sults significantly less than the formal uncertainty of the propertyunder consideration. The same, however, does not hold for ages. c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 2.
Panel a): final global age uncertainties (cid:104) σ (cid:105) (as defined in the text) for stars with di ff erent seismic classification plotted as function of their ages.Squares highlight stars with [Fe / H] < −
1, whereas continuous ridge-lines mark uncertainties between 10 and 100 per cent. Dashed line is the maximumuncertainty formally possible at old ages because of the sharp cut imposed at 15 Gyr.
Panel b) to d): fractional age uncertainties as function of di ff erentparameters. Although the mass of a red giant is a useful proxy for age, it is im-portant to distinguish between the initial mass which sets the evo-lutionary lifetime of a star, and the present-day (i.e. actual) stellarmass, which is derived from seismology. Thus, when inferring anage using the actual stellar mass, the value derived will depend onthe past history of the star, whether or not significant mass-loss hasoccurred during its evolution.There are very few observational constraints on mass-loss.Open clusters in the
Kepler field suggest a value of η between 0 . . ∼ (cid:12) ; observations of globular clusters reveal that mass-loss seems to be episodic and increasingly important when ascend-ing the red giant branch (see e.g., Origlia et al. 2014), with recentstudies suggesting a very ine ffi cient mass-loss during this phase(Heyl et al. 2015). For the asteroseismic sample we also derive agesusing BaSTI isochrones with η = .
4. This value corresponds to ane ffi cient mass-loss process, and it is often used e.g., to reproducethe mean colours of horizontal branch stars in Galactic globularclusters, although this morphological feature is a ff ected by other,also poorly constrained, parameters (e.g., Catelan 2009; Grattonet al. 2010; Milone et al. 2014). By deriving ages with both η = .
4, we can compare two extreme cases and derive a conserva-tive estimate of the age uncertainty introduced by mass-loss.The comparison of ages derived with and without mass-lossis done in Figure 1. Ages of dwarf stars are obviously unaf-fected by mass-loss, and the same conclusion holds for stars with“RGB” classification. It must be noticed that the distinction be-tween “RGB” and “RC” is based on the average spacing between mixed dipole modes, and this measurement largely depends on thefrequency resolution which smoothes over the spacing (e.g., Bed-ding et al. 2011). A clear identification of “RGB” stars is thus pos-sible for log g (cid:38) . ff ect of mass-loss increases when moving to lowergravities, and it is most dramatic for stars in the clump phase.Isochrones including mass-loss return younger ages than thosewithout mass-loss; this can be easily understood since a given mass–seismically inferred– will correspond to a higher initial mass incase of mass-loss, and thus evolve faster to its presently observedvalue. From Equation 1 it can be seen that the rate of mass-losshas an inverse dependence on mass. This implies a decreasing im-portance of mass-loss for increasing stellar mass. This is evidentin Figure 1e, where only masses below about 1 . M (cid:12) are signifi-cantly a ff ected by mass-loss. The fractional di ff erences shown inFigure 1 deserve an obvious -yet important- word of caution. Wedefine the reference ages as those without mass-loss. Within thiscontext, a fractional di ff erence of e.g., 50 per cent means that ageestimates decrease by half when we factor in mass-loss. Should thesame di ff erence be computed using η = . ff ect ofcanonical and non-canonical models (for a given η ). The di ff erenceis negligible above 3 Gyr, with di ff erences of a few per cent or less,while at younger ages (i.e. for masses above (cid:39) . M (cid:12) ) the e ff ect c (cid:13) , 000–000 Casagrande et al. can amount to a few hundreds Myr, thus translating in age di ff er-ences of few tens of per cent for the youngest stars. The reasonfor this is that in this mass range, the inclusion of overshooting inthe main-sequence phase plays a significant role in the turn-o ff age.Although part of this di ff erence is compensated by a quicker evo-lution in the subgiant phase for stars with smaller helium cores (i.e.with no overshooting, see Maeder 1974), the e ff ect remains in moreadvanced stages.To determine our final and global uncertainties on ages weadopt the same procedure used for other seismic parameters, butalso account for the uncertainties related to mass-loss and the use of(non)-canonical models (see Paper I for details on the GARSTECgrid and Monte-Carlo approach discussed below). Briefly, we addquadratically to the formal asymmetric uncertainties obtained fromour η = i) theGARSTEC grid, ii) the Monte-Carlo approach, iii) implementingmass-loss with η = . iv) using BaSTI canonical models. Inmost cases the uncertainties listed in i) to iv) dominate over theasymmetric formal uncertainty. For plotting purposes we use (cid:104) σ (cid:105) defined as the average of the (absolute) value of the upper and lowerage uncertainty. Figure 2 shows both the absolute and relative ageuncertainty of each star in our sample, along with their dependenceon log g , [Fe / H] and mass.For most of the stars, the age uncertainty is between 10 and30 per cent. When restricting to gravities higher than log g (cid:39) . ff ect of mass-loss is weak forstars ascending it, and where seismic classification is able to sep-arate “RGB” from “RC” stars. There is only a handful of “RGB”stars with uncertainties larger than about 30 per cent: those are lo-cated at the base of the red giant branch and have ∆ ν , but not ν max measurements, explaining their larger errors. For dwarfs, our ageuncertainties are also consistent with the results of Chaplin et al.(2014), who found a median age uncertainty of 25 per cent whenhaving good constraints on T e ff and [Fe / H]. For the 20 dwarfs wehave in common with that work, which span an interval of about10 Gyr, the mean age di ff erence is 1 Gyr with a scatter of 3 Gyr.The largest di ff erences occour for the most metal-poor stars, andthe stars having Pflg and Mflg di ff erent from zero. These discrep-ancies likely arise from Chaplin et al. (2014) assuming a constant[Fe / H] = − . (cid:104) σ (cid:105) is a ff ected, see Fig-ure 2). Notice that our global uncertainties (which include the ef-fect of di ff erent models and mass-loss assumptions) partly blur thislimit. We also remark that the accuracy of asteroseismic masses(and thus ages) obtained from scaling relations is still largely unex-plored, especially in giants (see e.g., Miglio et al. 2013a). There arealso indications that in the metal-poor regime ([Fe / H] (cid:46) −
1) scal-ing relations might overestimate stellar masses by 15 −
20 per cent(Epstein et al. 2014), thus returning ages systematically youngerby more than 60 per cent (see e.g., Jendreieck et al. 2012). In ab-sence of a more definitive assessments on the limits of the scalingrelations, this source of uncertainty has not been included in ourerror budget. Our metal-poor stars are highlighted in Figure 2, andthey cover the entire age range (i.e. we do also have metal-poor oldstars) with formal uncertainties between 20 and 30 per cent (shouldscaling relations for metal-poor stars be trusted).
Figure 3.
Age histogram for radial-velocity single members of the opencluster NGC 6819, selected according to the seismic membership of Stelloet al. 2011. Thin black dotted line indicates all stars belonging to the cluster,without any further pruning. Thin continuous black line is when restrictingthe sample to stars with good Mflg and Pflg. Thick blue lines (dotted andcontinuous) are when further restricting to “RGB” stars. Circles in the upperpart of the plot identify the mean value of each histogram, together with itsstandard deviation (outer bar) and the standard deviation of the mean (innerbar).
As for the other seismic parameters in Paper I, the solar-metallicityopen cluster NGC 6819 o ff ers an important benchmark to checkour results. In Figure 3 we show the age distribution of its clus-ter members, from using all seismic members (Stello et al. 2011)to only a subset of them with the best Str¨omgren photometry andseismic evolutionary phase classification. We recall that for eachstar belonging to the cluster, we use its own metallicity rather thanimposing the mean cluster [Fe / H] for all its members. Requiringgood Mflg and Pflg does not seem to reduce the scatter, and thusimprove the quality of the ages. This is partly expected: althoughour Bayesian scheme fits a number of observables, the main factorin determining ages is the stellar mass, which is mostly constrainedby the asteroseismic observables. More crucial in improving theage precision is to select “RGB” stars only, from which we derive amean (and median) age of 2 . ± . ff ries et al. 2013). Values in the literature range from 2 . . ff erences depends on the models used ineach study, as well as on the reddening and metallicity adopted forthe cluster. We remark though the nearly perfect agreement withthe age of 2 . ± .
20 Gyr from the white dwarf cooling sequenceand 2 . ± . ff match whenusing the same BaSTI models (Bedin et al. 2015).Moving to the entire asteroseismic sample, Figure 4 showsthe age distribution of all stars, which peaks between 2 and 4 Gyr.While this distribution is not a proof of the reliability of the agesin itself, the ability to single out a population of “known” ages is. c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 4.
Age distribution of the entire asteroseismic sample ( centralpanel ), and when splitting stars according to seismic classification ( right-hand side panels ). The lowest right panel is the age distribution of starshaving certain “RC” classification and sorted into primary (blue) and sec-ondary (pink) clump according to their log g (see description in the text).Dotted lines are same distributions once members of the cluster NGC 6819are excluded. Such a population is provided by secondary clump stars, which arebound to be young ( (cid:46) g = .
5. Thus, there is a certain level of contami-nation between the two phases, which surely broads the age distri-bution of plausibly secondary clump stars. In addition there is alsocontamination from members of NGC 6819 which peaks around2 Gyr. Once the seismic members of the cluster are excluded, thetypical age of secondary clump stars shifts to younger values, inaccordance with expectations (e.g., Girardi 1999), providing futherconfidence on our asteroseismic ages. Should the same figure bedone using ages derived with mass-loss, the overall distributionswould remain quite similar, but the tail at older ages would be re-duced, in particular for “RC” stars.The above comparisons tell us that despite the various uncer-tainties associated with age determinations, our results are mean-ingful. On an absolute scale, the age we derive for the open clusterNGC 6819 is in agreement with the values reported in literatureusing a number of di ff erent methods. This holds at the metallicityof this cluster, which nevertheless is representative of the typicalmetallicity of most stars in the Kepler field. On a di ff erential scale,once “RC” stars are identified as primary or secondary, they showdi ff erent age distributions. Despite our rough log g criterium mightpartly blur this di ff erence, the ability to recover the presence ofyoung secondary clump stars gives us further trust on our ages. In order to use our sample for investigating age and metallicity gra-dients in the Galactic disc, we need to know how stars with di ff erentproperties are preferentially, or not, observed by the Kepler satel-lite. In other words, we need to know the
Kepler selection function.The selection criteria of the satellite were designed to optimizethe scientific yield of the mission with regard to the detection ofEarth-size planets in the habitable zone of cool main-sequence stars(Batalha et al. 2010). Even so, deriving the selection function forexoplanetary studies is far from trival (Petigura et al. 2013; Chris-tiansen et al. 2014). For the sake of asteroseismic studies, entries inthe KASC sample of giants (c.f. with Section 2.2) are based on anumber of heterogeneous criteria (Huber et al. 2010; Pinsonneaultet al. 2014). Fortunately, the full Str¨omgren catalog o ff ers a way ofassessing whether seismic giants with particular stellar propertiesare more likely (or not) to be observed by the Kepler satellite.
Kepler selection function
Stellar oscillations cover a large range of timescales; for solar-likeoscillations –as we are interested here– these range from a few min-utes in dwarfs (cf e.g., with 5 minutes in the Sun, Leighton et al.1962) to several days or more for the most luminous red giants (e.g.,De Ridder et al. 2009; Dupret et al. 2009). The
Kepler satellite hastwo observing modes: short-cadence (one minute), for dwarfs andsubgiants (a little over 500 objects with measured oscillations inthe
Kepler field, see Chaplin et al. 2011, 2014) and long-cadence(thirty minutes) well suited for detecting oscillations in red giants.With the exception of a few hundreds of dwarfs, most of thestars for which
Kepler measured oscillations are giants. In order toassess how well these stars represent the underlying stellar popula-tion of giants, we use the full photometric catalog to build an unbi-ased sample of giants with well-defined magnitude and colour cuts.This task is facilitated both by the relatively bright magnitude limitwe are probing, meaning that within a colour range most late-typestars are indeed giants, as well as by the fact that Str¨omgren colourso ff er a very powerful way to discriminate between cool dwarfs andgiants. We use the ( b − y ) vs. c plane, which due to its sensitivityto T e ff and log g (in the relevant regions), can be regarded as theobservational counterpart of an H-R diagram (e.g., Crawford 1975;Olsen 1984; Schuster et al. 2004). Working in the ( b − y ) vs. c planealso avoids any metallicity selection on our sample. In fact, as wediscuss below, we build our unbiased sample using cuts in b − y colour, whereas metallicity acts primarily in a direction perpendic-ular to this index, by broadening the distribution of stars along c .Figure 5 shows the ( b − y ) vs. c plane for the full photomet-ric SAGA sample when restricted to y (cid:54)
14 mag, approximatelythe magnitude limit of the asteroseismic sample (a more precisemagnitude limit will be derived in the next Section). This diagramis uncorrected for reddening, which is relatively low in the SAGAGalactic stripe studied here . In particular, in the following we fo-cus on giants, all located across the same stripe and having similarcolours and magnitudes, meaning that reddening a ff ects both theasteroseismic and the photometric sample in the same way.In the left-hand panel of Figure 5, gray dots nicely map thesequence of hot and turn-o ff stars for b − y (cid:46) .
5, whereas the giant Further, E ( b − y ) ∼ . E ( B − V ) and E ( c ) ∼ . E ( b − y ), reddeningthus having limited impact on these indices (Crawford & Barnes 1970).c (cid:13) , 000–000 Casagrande et al.
Figure 5.
Left panel: b − y vs. c plane for the entire SAGA photometric catalog with y (cid:54)
14 and Pflg = . Dotted red line is the main-sequence fiducial of Olsen1984, while the continuous red line is the + .
06 mag shift we use to separate dwarfs from giants. Open squares are cool M giants, also from Olsen 1984. Forreference, the metallicity dependent dwarf sequences of ´Arnad´ottir et al. 2010 are also shown (in blue, for − . (cid:54) [Fe / H] (cid:54) . / H] = − .
0) red giant branch sequence of Anthony-Twarog & Twarog 1994.
Right panel: dark gray circles identify all dwarf / subgiant stars inthe Geneva-Copenhangen Survey, most of the late-type ones being succesfully delimited by our shifted fiducial (in red). Pale gray circles are all asteroseismicgiants having good photometric and metallicity flags and y (cid:54) sequence starts at redder colours, then upturning into the M super-giants at b − y (cid:38) .
0. At the beginning of the giant sequence there isalso an under-density of stars, consequence of the quick timescalesin this phase and mass regime (Hertzsprung gap). Below the giantsis the dwarf sequence, here poorly populated because of our brightmagnitude limit. To exclude late-type dwarfs from the full pho-tometric catalog, we start from the Olsen (1984) fiducial (dottedred line), which is representative of solar metallicity dwarfs. Sincemetallicity spreads the dwarf sequence, we shift Olsen’s fiducial byincreasing its c by + .
06 mag, as shown in Figure 5 (continuousred line). For this shifted fiducial, the linear shape at b − y (cid:39) .
55 ismore appropriate to exclude metal-rich dwarfs (c.f. with ´Arnad´ottiret al. 2010), and it fits well the upper locus of dwarfs in the GCS(shown in the right-hand panel, dark-gray dots). For b − y > . >
97 per cent).Also shown for comparison is an empirical sequence for metal-poorgiants (green dashed line, from Anthony-Twarog & Twarog 1994).Indeed, almost all of the targets with b − y (cid:38) .
5, including theasteroseismic giants, lie on the right-hand side of this metal-poor sequence (as expected, given the typical metallicites encounteredin the disc) thus indicating that an unbiased selection of giants ispossible in the b − y vs. c plane.To summarize, any unbiased, magnitude-complete sample ofgiants used in this investigation will be built by selecting giantsfrom the full photometric catalog in the b − y vs. c plane, withthe colour and magnitude cuts we will derive further below. Asidefrom being shown for comparison purposes, the giant metal-poorsequence discussed above is not used in our selection, while theshifted Olsen’s fiducial derived above is employed to avoid con-tamination from dwarfs. We remark again that at the bright mag-nitudes studied here, contamination from dwarfs is expected to beminimal, most stars with late-type colours being in fact giants.To derive the appropriate magnitude and colour cuts, we firstexplore how the asteroseismic sample of giants compares with theunbiased sample of giants built with the same magnitude limit( y = .
4) and colour range (0 . (cid:54) b − y (cid:54) .
97) comprisingthe asteroseismic one. Should the latter be representative of the un-derlying population of giants within the same colour and magni-tude limits, we would expect the relative contribution of giants at c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 6.
Top panels: cumulative distribution in b − y and y for the theunbiased photometric sample of giants (in black, with gray shaded area in-dicating 1 σ Poisson errors) and the uncut asteroseismic sample of giants (inred, with orange line indicating 1 σ Poisson errors) having the same colourand magnitude limits (0 . (cid:54) b − y (cid:54) .
97 and y (cid:54) . Low panels: same as above, but restrictingboth the asteroseismic and the unbiased photometric sample of giants to0 . (cid:54) b − y (cid:54) . y (cid:54) . each colour and magnitude be the same for both the asteroseismicand the unbiased sample. This comparison is performed in the twoupper panels of Figure 6, for the unbiased photometric sample ofgiants (black line, with gray dashed area representing 1 σ Poissonerrors) and the asteroseismic giants (red line, with orange contourlines representing 1 σ Poisson errors). Since the total number ofstars is di ff erent in the two samples, all curves are normalized toequal area. It is clear from both panels that the asteroseismic and theunbiased sample of giants have di ff erent properties: in fact the as-teroseismic sample has considerably fewer stars towards the bluest(hottest) and reddest (coolest) colours (e ff ective temperatures). Inaddition, the asteroseismic sample begins to lose stars at the faintestmagnitudes.Although the selection of seismic targets by Kepler was het-erogeneous, and not intended for studying stellar populations inthe Galaxy, the observed selection e ff ects are understandable: starswith bluer colours (hotter T e ff ) are at the base of the red giantbranch, where stars oscillate with intrinsically smaller amplitudes,and the Kepler long-cadence mode (thirty minutes) also becomesinsu ffi cient to sample the shorter oscillation periods of these stars.Conversely, on the red side, moving along the red giant branch to-wards cooler T e ff and brighter intrinsic luminosities, the timescaleof oscillations increases, until the characteristic frequency separa-tion can no longer be resolved robustly with the length of our Ke-pler observations (up to Quarter 15, i.e. typically well over 3 years).In order to have an unbiased asteroseismic sample, we mustavoid the incompleteness towards the bluest and reddest colours aswell as at the faintest magnitudes. We explore di ff erent cuts in b − y and y , finding that for 0 . (cid:54) b − y (cid:54) . y (cid:54) . Figure 7.
Normalized metallicity distribution for the unbiased photomet-ric sample of giants (black line, with shaded gray area indicating 1 σ Pois-son errors) and the asteroseismic sample (red line, with shaded orangearea indicating 1 σ Poisson errors) in the same magnitude and colour range0 . (cid:54) b − y (cid:54) . y (cid:54) .
5. Only stars with good photometric andmetallicity flags are used. the unbiased photometric sample in b − y and y pass from ∼ − and ∼ − to about 67 per cent and 98 per cent respectively, whenwe use the cuts listed above. This implies that the null assumptionthat the two samples are drawn from the same population can notbe rejected to a very high significance. Equally high levels of sig-nificance are obtained for the other Str¨omgren indices m (73 percent) and c (99 per cent), as well as when the two samples arecompared as function of Galactic latitude b (94 per cent). For allthe above parameters, significance levels of (cid:39)
20 to 80 per centare also obtained using di ff erent statistical indicators such as theWilcoxon Rank-Sum test, and the F-statistic. We remark howeverthat the unbiased photometric sample (641 targets) also includesall the seismic targets (408) within the same magnitude and colourlimits. To relax this condition, we bootstrap resample the datasets10 ,
000 times and find that significance levels for all of the abovetests vary between 30 −
60 per cent when bootstrapping either of thetwo samples, to 20 −
40 per cent when bootstrapping both. Sincefor all these tests significance levels below 5 per cent are generallyused to discriminate whether two samples originate from di ff erentpopulations, we thus conclude that the asteroseismic sample is rep-resentative of the underlying population of giants to a very highconfidence level.Our photometry is significantly a ff ected by binarity only in thecase of near equal luminosity companions (or equal mass, dealingwith giants at the same evolutionary stage). These binaries imprintan easily recognizable signature in the seismic frequency spectrumand are very rare (5 such cases in the full SAGA asteroseismic cat-alog, see discussion in Paper I). When restricting to the unbiasedasteroseismic catalog three such cases survive, implying an occur-rence of near equal-mass binaries of 0 . ± . c (cid:13) , 000–000 Casagrande et al. photometric catalog of giants, suggesting that indeed they have anegligible e ff ect on our results.From the above comparisons, we have already concluded thatthe asteroseismic sample of giants is representative of the underly-ing populations of giants in both colour and magnitude distribution.We expect this to be true for all other properties we are interestedin as well. Whilst this is impossible to verify for masses and ages,Str¨omgren photometry o ff ers a convenient way of checking this inmetallicity space.Before deriving photometric metallicities we must correct forreddening also the unbiased photometric sample (in fact, metal-licities for the asteroseismic sample were derived after correctingfor reddening). Interstellar extinction is rather low and well con-strained in the magnitude range of our targets; we fit the E ( B − V )values of the asteroseismic sample as an exponential function ofGalactic latitude; this functional form reflects the exponential discused to model the spatial distribution of dust in the reddening mapadopted for the asteroseismic sample (see Paper I for a discussion).The fit has a scatter σ = .
015 mag, which is well within the uncer-tainties at which we are able to estimate reddening. More impor-tantly, despite this fit is based upon the asteroseismic sample (theone we want to estimate biases upon), it also reproduces (within theabove scatter) the values of E ( B − V ) obtained from 2MASS, us-ing an independent sample of several thousand stars (see details inPaper I). Such a fit obviously misses any three-dimensional infor-mation on the distribution of dust, but the purpose here is to derivea good description of reddening for the population as a whole, inthe range of magnitudes, colours, and Galactic coordinates coveredin the present study.After correcting for reddening, we apply the same giant metal-licity calibration used for the asteroseismic sample (Paper I) to thephotometric unbiased sample of giants, and compare the two (Fig-ure 7). In both cases we only use stars with good photometric andmetallicity flags (i.e. when the calibration is applied within its rangeof validity). We run the same statistical tests discussed above alsofor the distributions in metallicity, and significance levels variesbetween 15 and 50 per cent depending on the test and / or whetherbootstrap resampling is implemented or not. Based on the abovetests, we can thus conclude that for y (cid:54) . . (cid:54) b − y (cid:54) . / H] distribution of the asteroseismic sample represents thatof the giants in the field within the same colour and magnituderanges.Although we have already constrained the
Kepler selectionfunction using our Str¨omgren photometry, we also explore whether2MASS photometry o ff ers an alternative approach of assessing it,for the sake of other dataset where Str¨omgren is not available (e.g.,such as APOKASC, Pinsonneault et al. 2014). In Figure 8a), dark-gray dots show the K S vs. J − K S colour-magnitude diagram forstars approximately in the same stripe of the asteroseismic sample(73 . ◦ (cid:54) l (cid:54) . ◦ and 7 . ◦ (cid:54) b (cid:54) . ◦ ). In this plot, three mainfeatures are obvious: the overdensity of stars around J − K S (cid:39) . ff stars; the over-density at J − K S (cid:39) .
65 comprising primarily giants, and the blobat J − K S (cid:39) .
85 and faint magnitudes ( K S (cid:38) K S magnitude limit of Kepler is clearly a func-tion of spectral type, or J − K S colour. Using seismic giants only,we derive the following relation between Str¨omgren and 2MASSmagnitudes: K S = . y − . J − K S ) − .
35, with a scatter σ K S = .
09 mag. The inclined black-dashed line in Figure 8a) cor-
Figure 8.
Left panel: K S vs. J − K S diagram (gray dots) for stars inapproximately the same Galactic stripe of the asteroseismic sample (blackopen circles). Filled red circles identify stars belonging to the unbiasedasteroseismic sample built as described in the text, i.e. having good pho-tometric and metallicity flags, and with 0 . (cid:54) J − K S (cid:54) .
782 and K S (cid:54) . − .
28 ( J − K S ). Right panels: cumulative distributions be-tween the unbiased 2MASS photometic sample and seismic giants withsame colour and magnitude cuts (colour code same as of Figure 6). responds to a constant y = .
4, which, as we previously saw, isroughly the limit of the faintest stars selected to measure oscilla-tions in
Kepler . At bright magnitudes, we introduce a similar cut,corresponding to y = . . (cid:54) K S + . + . J − K S )0 . (cid:54) .
4) and colourrange (0 . (cid:54) J − K S (cid:54) . (cid:54) K S (cid:54)
12 (such toencompass our sample, see Fig. 8a) instead of the colour dependentone done above.From the Str¨omgren analysis we already know the magnitudeand colour range where the asteroseismic targets are expected, onaverage, to unbiasedly sample the underlying population of giants.Thus, we can see how these limits convert in the 2MASS system.Using all seismic targets we derive the following relation J − K S = .
977 ( b − y ) with σ J − K S = .
04, which converts 0 . (cid:54) b − y (cid:54) . . (cid:54) J − K S (cid:54) . K S magnitude cut corresponding to 9 . (cid:54) y (cid:54) . The
Kepler field encompasses stars located in the direction of theOrion arm, edging toward the Perseus, and rising above the Galacticplane. The stripe observed so far by SAGA has Galactic longitude l (cid:39) ◦ and covers latitude 8 ◦ (cid:46) b (cid:46) ◦ . Its location in the Galactic c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 9.
Location of the SAGA targets in Galactic coordinates. Stars with di ff erent colours have di ff erent seismic evolutionary classification “Dwarf”, “RGB”,“RC” and “NO” as indicated. Filled circles identify the 373 stars which satisfy the constraints described in Section 4. Panel a): target distribution over theGalactic plane, where the distance of each seismic target from the Sun ( D ) is projected along the line of sight D cos( b ) having direction l (cid:39) ◦ and Galacticlatitude b . The distance between the Galactic Centre (GC) and the location of the Sun ( (cid:12) ) is marked by the gray circle. Galactic longitudes ( l ) at four di ff erentangles are indicated. Panel b): same as above, but as function of Galactic height Z = D sin( b ) and Galactocentric Radius (R GC , computed assuming a solardistance of 8 kpc from the Galactic Centre). Panel c) and d): Z distribution of targets across two orthogonal directions. The multiple beams structure in panelsb) to d) arises from the projection of the gaps in the CCD modules on
Kepler . context is shown in Figure 9; it can be immediately appreciatedfrom panel a) and b) that the geometry of the SAGA survey allowsus to probe distances of several kpc from the Sun at nearly the sameGalactocentric radius, thus minimizing radial variations and greatlysimplifying studies of the vertical structure of the Milky Way disc.At the same time, SAGA spans a vertical distance Z (altitude orheight, hereafter) of about 1 . (cid:39) . (cid:39) ff ects (stemming from the colourand magnitude cuts derived in the previous Section), as well as to c (cid:13) , 000–000 Casagrande et al. account for the fact that in the most general case, the ages of redgiants might not be representative of those of an underlying stellarpopulation. These steps are described further below.
To estimate gradients, we limit our sample to 0 . (cid:54) b − y (cid:54) . y (cid:54) . / or metallicity es-timates (see Section 2). The latter requirement automatically ex-cludes stars with [Fe / H] (cid:62) .
5, but we also limit the metallicityrange to [Fe / H] > − .
0, to remove any halo object, which wouldcontaminate our study of disc gradients . By excluding metal-poorstars we also avoid problems related to the potential inaccuracyof seismic scaling relations in this regime (Epstein et al. 2014).Because we are interested in studying properties of the Galacticdisc via field stars, we also exclude members of the open clusterNGC 6819, based on their seismic membership. Furthermore, weremove all targets classified as “Dwarf”, obtaining a final sampleof 373 giants (i.e. with seismic evolutionary classification “RGB”,“RC” or “NO”, see Section 2.2). They cover heights from ≈ . . ff ect age estimates of unclas-sified and clump stars, whereas “RGB” stars are essentially im-mune to such uncertainty. These stars provide more robust ages,though at the price of a greatly reduced sample size. We also referto Paper I for the uncertainties associated to masses and distances,which are of order 6 and 4 per cent, respectively. In the follow-ing we will determine vertical gradients using both samples when-ever possible: the 373 “Giants” and the 48 best pedigreed “RGB”stars. The bulk of gravities for the “Giants” sample covers the range2 . < log g < .
5, while for “RGB” stars covers 2 . < log g < . We adopt two methodologies to estimate the vertical gradient of ageand mass. First i) we use a boxcar-smoothing technique describedin Schlesinger et al. (2015). Sorting the stars by height above theplane, we calculate the median age (mass) and altitude Z of a frac-tion of the sample at the lowest height. We then step through thesample in altitude, as we want to quantify the age (mass) variationwith height above the plane. Each bin contains the same number ofstars and overlaps by a small fraction with the previous bin. For the“Giants” sample, we explore the range between 18 and 30 stars perbin with overlaps ranging from 8 to 15. The “RGB” sample is muchsmaller and we explore the range between 8 and 10 stars per binwith overlaps ranging from 2 to 4 stars. The binsizes and overlapsexplored contain enough targets so that the overall trend is not dom-inated by outliers, and the median points well reflect the overall be-haviour of the underlying sample. We then perform a least-squaresfit on these median points; the change in slope (i.e. gradient) due todi ff erent choices of binsize and overlap is typically below half theuncertainty of the fit parameter itself. We perform a Monte-Carlo to Note that our colour cut alone already removes many of the metal-poorobjects. explore the sensitivity of the boxcar-smoothing on the uncertaintyof the input ages (masses), and add this uncertainty in quadrature tothose estimated above. We obtain the following raw age and massgradients for the “Giants” 3 . ± . − , − . ± .
10 M (cid:12) kpc − . Similarly, for the “RGB” stars we have − . ± . − , − . ± .
35 M (cid:12) kpc − .Our second estimate of the gradient ii) consists of a simpleleast-squares fit to all of the stars that meet our criteria. Again ouruncertainties include those from the fitting coe ffi cients and from aMonte-Carlo. In this case we obtain for the “Giants” 4 . ± . − , − . ± .
08 M (cid:12) kpc − and for the “RGB” stars 0 ± . − , 0 . ± .
20 M (cid:12) kpc − .With both methods, the gradients for the “RGB” stars haveconsiderably larger uncertainties, which make them consistent withno slope and limit their usability to derive meaningful conclusions.This is due to the small sample size and scatter of the points. Be-cause of this, the χ of the “RGB” fits have the same statisticalsignificance whether we let the slope and intercept be free, or wefix the latter on the “Giants” sample (roughly 3 Gyr and 1 . (cid:12) on the plane). With this caveat in mind (i.e. fixing the intercept),and including in the error budget the uncertainty in the interceptderived from the “Giants”, the raw “RGB” slopes become 3 . ± . − and − . ± .
17 M (cid:12) kpc − for method i) , and 1 . ± . − and − . ± .
14 M (cid:12) kpc − for method ii) .Technique i) and ii) have di ff erent strengths; as the samplesize is small, the least-squares fit takes full advantage of every staravailable. However, the boxcar-smoothing technique avoids beingskewed by any outliers. Additionally, we can see how the uncer-tainties vary with respect to height above the plane by examiningthe variation in each median point.We stress that both methods still need to be corrected for tar-get selection e ff ects, i.e. the gradients above should not be quotedas the values obtained for the Galactic disc. Also, the use of stel-lar masses as proxy for stellar ages is applicable only to red giants.Thus, while it is meaningful to derive a Galactic age gradient byassessing how well our sample of red giants (with known selec-tion function) will convey the age structure of the larger underlyingstellar population (done in the next Section), the stellar mass gra-dient will reflect the mass structure of the underlying populationof red-giants only. For red giants, the relation between mass andage is Age ∝ M − α , with α (cid:39) .
5. Thus, we expect that the agegradient traced by red giants translates into a variety of masses atthe youngest ages, whereas low-mass (i.e. old) stars will be prefer-entially found at higher altitudes. Indeed, this picture is consistentwith Figure 10, which shows an L-shaped distribution of red-giants,with low-mass stars extending from low to large heights and moremassive stars being preferentially close to the Galactic plane. Be-cause of the aforementioned power-law relationship between ageand mass, one might wonder whether a linear fit is appropriate forquantifying the mass gradient shown by red giants. In fact, a changeof say 0 . (cid:12) translates to a few 100 Myr in a 2M (cid:12) star, but corre-sponds to several Gyr at solar mass. Here, our goal is not to providea value for the mass gradient –which given the above discussionwould be of limited utility– but simply to use the masses of our redgiants as a model independent signature of the vertical age gradi-ent. The above fits of the mass gradient su ffi ce for this purpose, andin the following discussion we will focus only on the vertical agegradient. c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 10.
Raw age and mass gradients ( top panels ) and age-metallicity relation ( lower panel ) before correcting for completness and target selection. Opencircles are all giants in SAGA, while filled circles (colour coded according to their seismic classification) are stars satisfying the seismic target selectiondiscussed in Section 4.1.
In Section 3 we have studied the
Kepler selection function to deter-mine the colour and magnitude ranges in which the SAGA aster-oseismic sample reflects the properties of an underlying unbiasedphotometric sample of red giants. However, to derive the Galac-tic age gradient we must assess how target selection systematicallya ff ects our gradient estimates (i.e. once a clear selection functionis defined, we must assess its e ff ect). To avoid our results beingtoo depend on particular model assumptions, we use various ap-proaches to understand how our selection criteria and survey ge- ometry will bias our sample, and to what extent the ages of a pop-ulation of red giants are representative of the ages of a full stellarpopulation. We first want to examine the probability that a star with specificstellar parameters will be observed given our target selection crite-ria. We generate a data-cube in age, metallicity and distance whereeach point in the age and metallicity plane is populated according c (cid:13) , 000–000 Casagrande et al.
Figure 11.
Probability of a star passing the “Giants” target selection to be observed given its height, metallicity and age (see description in Section 4.3.1). Allprobabilities are normalized to an arbitrary scale. to a Salpeter IMF over the BaSTI isochrones. For each of thesepopulations we then assign apparent magnitudes by running overthe distance dimension in the cube. Thus, for each combination ofage, metallicity and distance we can define the probability of a starbeing observed by SAGA as the ratio between how many stars pop-ulate that given point in the cube, and how many pass our sampleselection (i.e. our color, magnitude and gravity cuts, see Section4.1). This approach naturally accounts for the e ff ects of age andmetallicity on the location of a star on the HR diagram. Via theIMF it also accounts for the fact that stars of di ff erent masses havedi ff erent evolutionary timescales, and thus di ff erent likelihood ofbeing age tracers of a given population. This approach is the leastmodel dependent, and provides an elegant way to gauge the selec-tion function.Figure 11 shows the probability of each star being observedgiven its height, metallicity and age. Our sample is biased againststars at large distances (and thus altitudes), low metallicities, andold ages.We can then apply methodology i) and ii) described in Sec-tion 4.2, where in the boxcar-smoothing / fitting procedure we assignto each star a weight proportional to the inverse of its probability.Stars with low probability will be given larger weight to compen-sate for the fact that target selection is biased against them. Figure11 indicates that probabilities are non linear functions of the inputparameters; for some targets the combination of age, metallicityand distance results in a probability of zero, which then translatesinto an unphysical weight. Observational errors are mainly respon-sible for scattering stars into regions not allowed in the probabilityspace. To cope with this e ff ect without setting an arbitrary thresholdon the probability level, for each target we compute the probabil-ity obtained by sampling the range of values allowed by its age,metallicity and distance uncertainties with a Monte-Carlo. Whilethis procedure barely changes the probabilities of targets very likelyto be observed, it removes all null values. Depending on the methodand sample (Section 4.2), factoring these probabilities in the linearfit typically increases the raw age gradient (Table 1). Our second method to explore target selection e ff ects also relies onpopulation synthesis. However, rather than generating a probabil-ity data-cube, we produce a synthetic population with a certain starformation history, metallicity distribution function, IMF, and stellardensity profile. This gives us the flexibility of varying each of theinput parameters at the time, to explore their impact on a popula-tion. We assume a vertical stellar density profile described by twoexponential functions with scale-heights of 0 . . /τ max , where τ max is the maximum agecovered by the isochrones. We also assume a flat metallicity distri-bution function over the entire range of the BaSTI isochrones anda Salpeter IMF. Shallower and a steeper slopes for the IMF are alsoexplored ( α ± ff erence being a burst of star formation centred at 12 Gyr(50 per cent of the stars), followed by a flat age distribution untilthe present day. Note that in both model A and B ages are assignedindependently of their thin or thick disc membership, and no verti-cal age gradient is present.In our last model (C) we describe the ages of thin disc starswith a standard gamma distribution (with γ =
2) having a disper-sion of 2 . . − . For thick disc stars we adopt aGaussian distribution centred at 10 Gyr with a dispersion of 2 Gyr.The metallicity distribution function of thin disc stars is modelledby a Gaussian centred at solar metallicity on the plane, with a dis-persion of 0 . − . − . For thethick disc we assume a Gaussian metallicity distribution centred at[Fe / H] = − . .
25 dex. While model C pro-vides a phenomenological description of some of the features we c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 12.
Panel a): age distribution input in di ff erent models, as traced by low-mass, unevolved stars. Panel b): the same distributions when restricting tostars with 2 . < log g < .
5. Dotted red line is when changing the IMF in model A to have α = − . Panel c): age distributions when applying the SAGA“Giants” target selection: 2 . < log g < .
5, 10 (cid:54) y (cid:54) .
5, 0 . (cid:54) b − y (cid:54) . / H] > − Panel d): ratio between the outputs in panel c) and the inputsin panel a). All curves are normalized to equal area. Small wiggles are due to realization noise. observe in the Milky Way disc, it is far from being a complete rep-resentation of it, which is not our goal anyway. A more completeMilky Way model is explored in the next Section using Galaxia(Sharma et al. 2011).Here, we simply want to explore selection e ff ects, in par-ticular on stellar ages. Figure 12 shows how the age distributioninput in di ff erent models (traced by unevolved low mass stars,panel a ) is altered when selecting evolved stars (defined as having2 . < log g < .
5, panel b ) or applying the SAGA “Giants” tar-get selection discussed in Section 4.1 (panel c ). It is clear that evenin the simplest case (model A), the age distribution of “Giants” isstrongly biased towards young stars (panel d ), in agreement to whatwe already deduced from Figure 11. This is driven by the combinede ff ect of evolutionary timescales and the slope of the IMF (comparecontinuous and dotted line for model A).We first compute the gradient input in each model using its un-evolved stars , defined here as all stars with masses below 0 . M (cid:12) . Any Salpeter-like IMF breaks at sub-solar mass (e.g. Bastian et al. 2010,and references therein). However, this will only change the density of lowmass stars, but not the underlying age structure they trace.
Because of the large dispersion of ages at each height (also presentin Galaxia, see next Section), we find that fitting heights as func-tion of ages –i.e. to derive a slope in kpc Gyr − – provides a betterdescription of the data. From the population synthesis volume, weextract a pencil beam with Galactic latitudes 8 ◦ < b < ◦ , ap-ply the target selection of “Giants” and “RGB” stars and computethe kpc Gyr − slopes of these sub-samples. The change in slopebetween the unevolved-stars and the target-selected ones definesthe correction that must be applied to the raw data. Thus, we usethe correction in slope determined above to modify the observedSAGA values, by adjusting the height of each star depending onits age. This adjustment increasingly a ff ects older stars, which arelifted in altitude Z after correcting for target selection. In reality,the position of each of our targets is well determined (within its ob-servational uncertainties): the change we introduce here is simplymeant to counteract the bias introduced into the distribution by tar-get selection. This is to say that if our sample were not a ff ected bytarget selection, we would preferentially observe additional stars athigher Z. Once we adjust the height of each of our objects as de-scribed, we then perform a least-squares fit on the shifted points interms of Gyr kpc − . We apply a similar technique to our boxcar- c (cid:13) , 000–000 Casagrande et al. smoothing analysis except here, rather than shifting every star, weshift each median point and re-fit them with a least-squares in Gyrkpc − . Thus, although we apply the same target selection correc-tion, its e ff ect will be di ff erent. Because there is a much widerrange of values star-by-star than in the median points, the gradientfrom the least-squares analysis changes more than for the boxcar-smoothing. Also in this case, correcting for target selection typi-cally increase the raw SAGA gradients by a few Gyr kpc − (Table1). By applying our target selection criteria to a model of the Galaxy,we can determine how well the resulting sample reflects the discbehaviour assumed by the model. For this purpose we simulate theSAGA stripe using Galaxia (Sharma et al. 2011).Galaxia is based on the Besancon analytical model of theMilky Way (Robin et al. 2003); the disc is composed of six di ff erentpopulations with a range of ages from 0 to 10 Gyr. The thick discand halo are modelled as single-burst, metal-poor populations of11 and 14 Gyr, respectively. For our analysis, we limit ourselves tothe six thin-disc populations in Galaxia; this age range is represen-tative of the bulk of the SAGA sample with a more continuous dis-tribution in age and chemistry than if we used also the single-burstpopulations. Although the origin of the thick disc is still unclear, itis unlikely to consist of stars having a single age and it might alsospan a large metallicity range (see discussion in the Introduction).Galaxia itself is a sophisticated –yet simplified– representation ofthe Galaxy, which assumes a certain age and metallicity distribu-tion for each Galactic component. Among other things the metal-licity scale, the stellar radii, gravities, synthetic colours, model T e ff along the red giant branch and mass-loss prescription will also de-pend on the isochrones implemented in the model, which are fromPadova in the case of Galaxia (Bertelli et al. 1994; Marigo et al.2008). We do not attempt to vary any of the Galaxia ingredients,and we have already explored the e ff ect of changing some of thoseassumptions using the population synthesis approach described inthe previous Section.Here, we want to further assess how a known input populationfrom a realistic Galactic model will appear once filtered throughour target selection algorithm. We adopt the same technique de-scribed in Section 4.3.2. We calculate the input gradient using un-evolved stars, implement the “Giants” and “RGB” target selectionon the Galaxia simulated stars to derive corrections in kpc Gyr − ,and apply those to the data before re-fitting the gradient in Gyrkpc − . The age distribution input in Galaxia is rather di ff erent fromthat traced by our simplistic population synthesis models, and itdoes not extend beyond 10 Gyr because of the thick disc exclusion(Figure 12).The Galaxia model shows a wide range of ages at each heightabove the Galactic plane; however, the proportion of young starsdiminishes as the height increases, resulting in typically older agesfar away from the Galactic plane. The SAGA cuts in colour andmagnitude remove many of the older stars at large heights: thisboosts the fraction of young stars and skews the sample to lowerheights in accordance to what we already derived in Section 4.3.1and 4.3.2. Target selection corrections are similar to what we de-rived previously, and of the order of few Gyr kpc − . In a pencil-beam sample such as SAGA, the average altitude Z = D sin( b ) will, by the geometry, rise almost linearly with distance D ; hence the two quantities are strongly correlated. Thus, any cor-relation for example of age with distance, will bias the gradientderived as function of Z. This e ff ect can be accounted for by in-troducing the dependence on distance in the least-squares fit whenderiving the gradients (e.g. Sch¨onrich et al. 2014). This techniqueprovides a model-independent check (modulo the degree at whichgiants trace the ages of an underlying stellar population). Assumingthat a (multi) linear dependence provides a reasonable descriptionof the underlying structure of the data (which over the range of dis-tances studied here is appropriate for ages), one can expand the fitinto τ i = d τ d Z Z i + d τ dD D i + (cid:15) (2)where i is the index running over the stellar sample, d τ/ d Z and d τ/ dD are the free fit parameters measuring the correlation be-tween age τ , altitude Z and distance D , and (cid:15) is the intercept ofthe fit. When we apply this technique to SAGA, the significanceof the derived slopes is usually above three sigma for the “Giants”sample, whereas it is below 1 sigma for “RGB” stars due to thesmaller sample size and range of distances. Thus, we apply thismethod only to “Giants”.Accounting for the distance dependence returns a least-squares gradient of 6 . ± . − . The increase with respectto the value of 4 . ± . − obtained with a simple linearfit (Section 4.2) tells us that the survey geometry is indeed biasedagainst old stars, and thus any fit of the raw data underestimate thetrue age gradient. In Section 4.2 we have used two di ff erent methods and samples tomeasure the raw vertical age gradient with SAGA. We have then as-sessed target selection e ff ects using di ff erent approaches. Althoughthey return a range of values for the correction, they all consistentlyshow that any raw measurement of the vertical age gradient usingred giants underestimates the real underlying value.We summarize the raw gradients obtained using di ff erent sam-ples and methods in Table 1, along with the target selection cor-rections discussed in Section 4.3.1 to 4.3.3. For each method andsample listed in the table, we compute the median target selectioncorrection and standard deviation as a measure of its uncertainty.This is added in quadrature to the undertainty derived for each fit,after which the weighted average of all gradients is computed, ob-tainining a value of 4 . ± . − . If we instead replace the“RGB” slopes with those obtained without forcing the intercept,then we obtain a weighted average of 3 . ± . − . Hence,the gradient does not change dramatically, but its uncertainty is in-creased.While all the above values clearly indicate that the age of theGalactic disc increases when moving away from the plane, the con-sistency among di ff erent samples, methods and target selection cor-rections vary. It should also be kept in mind that mass-loss changesour age estimates. If we were to adopt the ages derived for SAGAassuming an e ffi cient mass-loss ( η = . . − . Since the e ff ectof mass-loss for the SAGA “RGB” stars is negligible, their gradientdecreases by only 0 . − .Based on the above discussion, we conclude that in the region c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc Figure 13.
Age and mass gradients ( top panels ) and age-metallicity relation ( lower panel ) after correcting for target selection the “Giants” sample. Contourlevels have been obtained by convolving each star with its age, distance and metallicity uncertainties and assigning a density proportional to the logarithm ofthe inverse probability of being observed. Probabilities have been computed as described in Section 4.3.1. of the Galactic disc probed by our sample, the vertical age gradi-ent is on the order of 4 . ± . Gyr kpc − , which also encompassesthe uncertainty stemming from mass-loss. In particular, it shouldbe stressed that at any given height there is a wide range of ages.Figure 13 shows such overdensities in the vertical age, mass andage-metallicity relation when including observational uncertaintiesand correcting for target selection.To our knowledge, the present study is the first of this kind,quantifying the in situ vertical age gradient of the Milky Way disc.While the origin of this age gradient is beyond the scope of this paper, its existence has long been known by indirect evidence suchas e.g. the age-velocity dispersion relation (e.g. von Hoerner 1960;Mayor 1974; Wielen 1977; Holmberg et al. 2007), the chemistry inred giants (Masseron & Gilmore 2015) and the change in fractionof active M dwarfs of similar spectral type at increasing Galacticlatitudes (e.g., West et al. 2011, and references therein). Howevernone of these studies is able to provide a direct measurement as wedo here. c (cid:13) , 000–000 Casagrande et al.
Table 1.
Target selection e ff ects. Corrections are intended to be summed to the raw vertical gradients. All values are in Gyr kpc − .Corrections from Corrections from Corrections fromRaw gradients target selection population synthesis Galacticmodelling modelling modellingA B Cboxcar + . ± . + . + . + . + . + . + . ± . + . + . + . + . + . + . ± . + . + . + . + . + . + . ± . − . + . + . + . + . − . ± . ± . − . An important constraint for Galactic models is provided by the timeevolution of the metal enrichment, the so-called age-metallicity re-lation. The strength or even the existence of this relation amongdisc stars has been largely debated in the literature because of theintrinsic di ffi culty of deriving reliable ages for field stars, as well asissues with sample selection biases (e.g., McClure & Tinsley 1976;Twarog 1980; Edvardsson et al. 1993; Ng & Bertelli 1998; Rocha-Pinto et al. 2000; Feltzing et al. 2001). We can now take a fresh lookat this issue, with the first age-metallicity relation from seismologyshown in Figure 10 for the entire dataset, as well as when restrictingonly to “Giants”. The SAGA target selection intrinsically favoursmetal-rich, young stars thus flattening the overall age-metallicity.If we adopt the age ( (cid:39) ± − ) and metallicity gradients( (cid:39) − . ± . − , Schlesinger et al. 2015) measured over theSAGA stripe we obtain a shallow slope of − . ± .
06 dex Gyr − .This is consistent with what is obtained instead if we were fittingthe age-metallicity in Figure 10, and correcting for target selectionafterwards. Seismology thus confirm the rather mild slope and largespread at all ages in the age-metallicity relation of disc stars, asalready derived from turn-o ff and subgiant stars in the solar neigh-bourhood (e.g., Nordstr¨om et al. 2004; Haywood 2008; Casagrandeet al. 2011; Bergemann et al. 2014) and also in agreement with thestudy of Galactic open clusters (e.g., Friel 1995; Carraro et al. 1998,see also Leaman et al. 2013 for the age-metallicity relation of discglobular clusters). It should also be noted that a typical age uncer-tainty of order 20 per cent implies a much larger absolute number atolder ages than at younger ones (i.e. 10 ± ± . / H] > −
1, preventing us from trac-ing the early enrichment expected in the age-metallicity relation(compare e.g. the steep rise in metallicity at about 13 Gyr in figure16 of Casagrande et al. 2011).Figure 14 shows the age distribution for the “Giants” sample.Overall this is similar to what we have already discussed in Section2.3, apart from the fact that we are now applying completness cuts.A significant overdensity seems to appear at the oldest ages, above (cid:39)
10 Gyr, which persist also when adopting ages computed withmass-loss. We know that our target selection is biased against oldstars (Section 4.3), and it would thus be intriguing to interpret thisoverdensity as the signature of a population formed / accreated earlyin the history of the Galaxy. As we have discussed in Section 4.3.2,a constant star formation rate produces an age distribution of red Figure 14.
Panel a): age distribution (with Poisson error bars) for the “Gi-ants” sample.
Panel b) and c): same as above, but for Z (cid:62) . < . giants which peaks at young values, and with a long tail. A strongburst in star formation at a given age manifests instead as a localisedpeak at that epoch (see Figure 12).We only select stars with [Fe / H] > −
1, implying that thisoverdensity is associated with disc stars, rather than the halo, and itcould be the signature e.g., associated to the formation of the thickdisc or enhanced star formation in the early Galaxy (c.f. Haywoodet al. 2013; Robin et al. 2014; Snaith et al. 2014). Because of thevertical age gradient and the survey geometry we must first verifywhether this overdensity could simply stem from stars at the high-est Z. Correcting the histogram for the vertical age gradient is notstraightforward since we have a mixture of young and old stars atall heights, and this would unphysically shift part of the age his-togram at negative values. We therefore split the age distributionbelow and above Z = . c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc ical properties on galactic scales (Torrey et al. 2012; Vogelsbergeret al. 2014), yet the survival of discs seem to critically depend onthe abscence of violent events (Scannapieco et al. 2009); our resultssupport such scenario. The secondary clump is populated by stars which ignite helium in(partly) nondegenerate conditions, and it is a phase relatively well-defined in time (e.g., Girardi 1999). Although the precise mass andhence age, at which this happens depend on the models themselves,secondary clump stars define a nearly pure population of young( (cid:46) i) the intrinsic metallicity spread at young ages, and ii) totrace the aging of the Galactic disc. i) We have already shown that thanks to our precise seismiclog g determinations we can discriminate between primary and sec-ondary clump stars in a field population (see also Paper I, figure 17).Here, we use only stars with secure “RC” classification and adopt afixed log g = . σ [Fe / H] = . ± .
04 dex.The reported uncertainty is twice as large as the variation stemmingfrom a change of 0 .
05 dex in gravity cut, and from the adoption ornot of the completeness cuts (0 . (cid:54) b − y (cid:54) . y (cid:54) . / H] measurements, Paper I), we find that the intrinsic metallic-ity spread of secondary clump stars is 0 . ± .
04 dex (this pro-cedure holds true under the assumption that errors are reasonablyGaussian, and there is no systematic bias). This number is essen-tially unchanged when correcting for the vertical metallicity gradi-ent measured in SAGA (Schlesinger et al. 2015). The intrinsic scat-ter derived here is similar to that obtained using stars in the sameage range from the GCS (Casagrande et al. 2011), which, after ac-counting for the uncertainty in those metallicity measurements, isabout 0 . ± .
02 dex. ii)
Since primary and secondary clump stars occupy very simi-lar position on the H-R diagram, it is reasonable to assume that theyhave nearly equal probability of being observed by
Kepler . There-fore, the ratio of secondary to primary clump stars is independent ofthe selection function. Most importantly, this ratio is sensitive to themixture between a young population (including secondary clumps)and an old one (including primary only), thus meaning that it canbe used to trace the relative age of a population. Also, althoughages in the clump phase are a ff ected by mass-loss (especially at thelow masses typical of primary clump stars) their number ratio isuna ff ected by this uncertainty, until linked to an age scale.This is explored in Figure 15 (left-hand panel), which showsthe ratio of secondary to primary clump stars as function ofheight from the Galactic plane Z. Again, members of the clusterNGC 6819 have been excluded since we are interested at study-ing properties of field disc stars. Adopting our completness cuts( V (cid:54) . . (cid:54) b − y (cid:54) .
8) is irrelevant here, with little ef- fect aside from changing the limit in heights at which we have tar-gets. The ratio of secondary to primary clump stars will vary whenadopting di ff erent cuts in log g (2 .
45, 2 .
50 and 2 .
55 dex). How-ever, once these ratios are normalized to the value at the maximumheight, they all remarkably overlap. It is clear that at lower alti-tudes secondary clump stars outnumber primary ones, indicatingthat the fraction of young stars decreases when moving away fromthe plane, in qualitative agreement with the vertical age gradientmeasured in Section 4.4.
In this paper we have used the powerful combination of asteroseis-mic and classic stellar parameters of the SAGA ensemble to inves-tigate the vertical age structure of the Galactic disc as traced by redgiants in the
Kepler field. This goal is facilitated by the pencil-beamsurvey geometry analyzed here, which covers latitudes from about8 ◦ to 20 ◦ translating into bulk vertical distances up to ≈ l (cid:39) ◦ implying nearly constant Galactocentric distances and thusminimizing radial variations.For the asteroseismic sample we have complemented the stel-lar masses, metallicities and distances already derived in Paper Iwith stellar ages. For a large fraction of our stars we have seismicclassification available to distinguish between red giants burninghydrogen in a shell and clump stars that have already ignited he-lium in their core, thus greatly improving on the accuracy of agedeterminations. For clump stars, as well as for stars on the upperpart of the red giant branch the largest source of uncertainty in agedetermination stems from mass-loss. We have therefore includedthis uncertainty by deriving stellar ages under two very di ff erentassumptions for mass-loss.The Str¨omgren photometry of SAGA is magnitude completeto y ≈
16, i.e. nearly two magnitudes fainter than the giants selectedto measure stellar oscillations with the
Kepler satellite. This, andthe capability of Str¨omgren photometry to disentangle dwarfs fromgiants, has allowed us to build an unbiased population of giants, thatwe have used to benchmark against the asteroseismic sample. Wehave been able to constrain the thus-far unknown selection functionof seismic targets for the
Kepler satellite (see also Sharma et al., inprep.), by identifying a colour and magnitude range where giantswith oscillations measured by
Kepler are representative of the un-derlying population in the field. This holds true for V = y (cid:54) . . (cid:54) b − y (cid:54) .
8, modulo reddening, which is anyway wellconstrained for our sample. This has been verified to correspondto K S (cid:54) − . J − K S ) and 0 . (cid:54) J − K S (cid:54) .
782 for the2MASS system. These cuts, together with the use of stars with bestquality flags in SAGA, as well as the exclusion of members of theopen cluster NGC 6819 and a handful of the most metal-poor stars([Fe / H] (cid:54) −
1, for which seismic scaling relation might be inaccu-rate) reduce the initial SAGA sample by almost one third, to 373stars.Although we have been able to identify the colour and magni-tude range where our sample is representative of giants in the field,when measuring the vertical age structure of the Galactic disc wemust still correct the raw measurements for the colour and mag-nitude cuts reported above, i.e. for target selection. To control forthese biases, we separately estimated the e ff ects of the selectionfunction from Galaxy models, and from a more simple and straight-forward approach with direct population synthesis.We see a clear increase of the average stellar age at increasing c (cid:13) , 000–000 Casagrande et al.
Figure 15.
Left panel: ratio between the (cumulative) number of secondary to primary clump stars and function of height from the plane Z. Di ff erent coloursindicate the adopted surface gravity cut to discriminate between primary and secondary clump, whether with (w) or without (w / o) completness cuts. Dottedareas indicate Poisson’s errors. Right panel: same plot but normalizing each curve to the value at the highest altitude Z.
Galactic heights, thus indicating the aging of the Milky Way discas one moves away from the Galactic plane. This is also traced bya stellar mass gradient, since the mass of a red giant is a proxy forits age. We have used linear fits to describe these trends; althoughthis allows us to quantify their strength, we are aware that theymight not capture the full complexity of the age and mass structurein the Galactic disc. The bulk stellar age increases with increasingaltitude, but there is a large spread of ages at all heights. This trans-lates into a decreasing stellar mass with increasing altitude; stellarmasses are not linearly mapped into ages, and the overall trend ofthe stellar mass with Galactic heights is rather L-shaped.We have quantified these trends using giants independently oftheir seismic classification (373 stars), as well as “RGB” stars only(48 object), for which the impact of mass-loss on age estimates isnegligible. All the above estimators and samples agree in showingincreasing stellar ages (and decreasing stellar masses) at increasingGalactic heights, albeit the degree of consistency among di ff erentmethods and samples varies. We have argued that our current bestestimate for the vertical age gradient is 4 ± − . Part ofthe scatter might stem from uncertainties related to sample size andtarget selection corrections, although it should also be kept in mindthat part of it is real, an the age gradient we measure is just theheighest overdensity of a wide distribution. We have also used thenumber ratio of secondary clump stars to primary clump stars as anindependent proxy of the aging of the stellar disc, confirming thepresence of preferentially old stars at increasing Galactic heights.Stellar ages show a smooth distribution over the last 10 Gyr,whereas a small overdensity appears at older values, which could be a signature associated with the early phases of the Milky Way.Once age uncertainties are taken into account, this does not appearto be statistically significant. Nevertheless, the smooth distributionof ages over the last 10 Gyr is consistent with a rather constantstar formation history and suggests that the Galactic disc has hada rather quiescent evolution since a redshift of about 2. This is inagreement with scenarios where stellar discs in galaxies form atrelatively early times, and their survival critically depends on theabsence of major mergers.Finally, we derive the first seismic age-metallicity relation forthe Galactic disc. We confirm results from other methods (such asage dating of turn-o ff and subgiant stars, as well as Galactic openclusters) that a metallicity spread exists at all ages, and the over-all slope of the age-metallicity relation is small. Because of theiryoung ages, secondary clump stars can also be used to assess theinstrinsic metallicity spread at almost the present time, which weestimate to be ≈ .
14 dex. We remark that studies of local earlytype stars and gas-phase in di ff use interstellar medium reveal in-deed a high degree of homogeneity in the present day cosmic abun-dances (Sofia & Meyer 2001; Nieva & Przybilla 2012). Thus, de-spite a spread of ages at all heights, and a spread of metallicity at allages, there are well defined and smooth vertical age and metallic-ity gradients, indicating that the disc is generally composed of wellmixed populations that have undergone a largely quiescent evolu-tion. This validates scenarios in which the evolution of the disc islargely driven by internal dynamical processes, and it provides afirst constraint on the disc spatial growth over cosmic time. c (cid:13) , 000–000 ge stratigraphy of the Milky Way disc ACKNOWLEDGMENTS
We thank an anonymous referee for his / her insightful commentsand suggestions which has helped to strengthen the paper and im-prove the presentation of the results. We thank P. E. Nissen andA. Dotter for useful discussions. We thank the nature of who kneweverything upfront for giving a good laugh. Funding for the Stel-lar Astrophysics Centre is provided by The Danish National Re-search Foundation (grant agreement No. DNRF106). The researchis supported by the ASTERISK project (ASTERoseismic Investi-gations with SONG and Kepler), funded by the European ResearchCouncil (grant agreement No. 267864). V.S.A. acknowledges sup-port from VILLUM FONDEN (research grant 10118). A.M.S.is partially supported by grants ESP2014-56003-R (MINECO),EPS2013-41268-R (MINECO) and 2014SGR-1458 (Generalitat deCatalunya). We acknowledge the generous hospitality of the KavliInstitute for Theoretical Physics where part of this work was carriedout. This research was supported in part by the National ScienceFoundation under Grant No. NSF PHY11-25915. REFERENCES
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