Mon.Not.R.Astron.Soc. 000,1–13(2002) Printed13January2015 (MNLATEXstylefilev2.2) Towards a fully consistent Milky Way disc model - III. Constraining the initial mass function J. Rybizki1,2(cid:63) and A. Just1 5 1Astronomisches Rechen-Institut, Zentrum fu¨r Astronomie der Universita¨t Heidelberg, Mo¨nchhofstrasse 12-14, 69120 Heidelberg, Germany 1 2Fellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD) 0 2 n Accepted2014December22.Received2014December18;inoriginalform2014July30 a J 2 ABSTRACT 1 We use our vertical Milky Way disc model together with Galaxia to create mock ob- servations of stellar samples in the solar neighbourhood. We compare these to the ] A corresponding volume complete observational samples of dereddened and binary ac- counted data from Hipparcos and the Catalogue of Nearby Stars. G Sampling the likelihood in the parameter space we determine a new fiducial initial . h mass function (IMF) considering constraints from dwarf and giant stars. The result- p ing IMF observationally backed in the range from 0.5 to 10M is a two slope broken (cid:12) - power law with −1.49±0.08 for the low mass slope, a break at 1.39±0.05M and a o (cid:12) high mass slope of −3.02±0.06. r t TheBesan¸congroupalsoconvergingtoasimilarIMFeventhoughtheirobservational s sample being quite different to ours shows that the forward modelling technique is a a [ powerful diagnostic to test theoretical concepts like the local field star IMF. 1 Key words: Galaxy: disc, evolution, formation, solar neighbourhood; Stars: lumi- v nosity function, mass function; Methods: statistical 0 3 6 2 0 1 INTRODUCTION modelwasconstructedusingdynamicalconstraintsofmain . sequence(MS)stars.Themainadvantageofthismethodis 1 The initial mass function (IMF) has seen numerous re- its weak dependence on the assumed IMF as only the in- 0 determinations since the seminal work of Salpeter (1955). 5 tegrated mass loss goes into the dynamical model. Other The most popular being broken power laws (Scalo 1986; 1 determinationsasforexampleHaywoodetal.(1997)havea Kroupa et al. 1993) or lognormal with a Salpeter-like high : degeneracyoftheirderivedSFHandtheirIMF’shighmass v massslope(Chabrier2003).OriginallytheIMFwasderived slope.AsaresulttheoldBesanc¸onmodel(Robinetal.2003) i by luminosity functions (LFs) of the solar neighbourhood X using the constant SFH from Haywood et al. (1997) had via the luminosity-mass relation and stellar lifetimes. More to be revised recently (Czekaj et al. 2014) to a decreasing r sophisticatedderivationsaccountfortheobservationalsam- a oneinlinewithdifferentdeterminationsfromchemicalevo- ple’sstarformationhistory(SFH)(Schmidt1959)andcon- lution models (Chiappini et al. 1997), extragalactic trends nectedly the scale height dilution (Miller & Scalo 1979). (Ly et al. 2007) and dynamical constraints (Aumer & Bin- Anothermoredirectapproachistolookatyoungstellar ney 2009; Just & Jahreiß 2010). clustersandderiveanIMFfromthesespatiallyandtempo- InpaperIIofthisseries(Justetal.2011)thediscmodel rally confined starburst populations. This yields a range of wasfurtherconstrainedbycomparingSloanDigitalSkySur- differentIMFs(vandenBergh&Sher1960;Dib2014)which vey (SDSS) data with predicted star counts of the north isnotnecessarilyinconflict(Weidner&Kroupa2005)with galacticpole(NGP).ThisyieldedthefiducialmodelA(from a global IMF being able to describe the field stars in the nowoncalledJJ-model)withfixedSFH,awelldefinedlocal MilkyWaydisc.Inthispaperwetrytogetholdofthethe- thick disc model (around 6% of local stars) which showed oreticalconceptofatimeindependentlocalMilkyWayIMF verygoodagreementbetweenourmodelandtheNGPdata byexploringitseffectwithinourdiscmodelintherealmof being confirmed in Czekaj et al. (2014). observables. TheseriesofpapersstartedwithJust&Jahreiß(2010, After fixing the SFH, age-velocity dispersion relation hereafter paperI) where a local vertical Milky Way disc (AVR) and age-metallicity relation (AMR) we now want to further build up a consistent disc model by constrain- ingtheIMFusinglocalstarcountsandtakingintoaccount (cid:63) E-mail: [email protected], [email protected] turn-off stars and giants as well. In paper I the present day heidelberg.de stellar mass function (PDMF) was already converted into (cid:13)c 2002RAS 2 J. Rybizki and A. Just an IMF considering scale height correction and finite stel- Abbreviations larlifetimes.Binarityandreddeningwerenotaccountedfor producingaverysteephighmassslopewithα=4.16which IMF InitialMassFunction isalsorelatedtoarelativelyhighbreakinthepowerlawat PDMF PresentDaystellarMassFunction 1.72M . LF LuminosityFunction (cid:12) In this paper we redo the selection of volume complete VMag AbsoluteVmagnitude B-V B-Vcolour samplesfromtherevisedHipparcosCatalogue(vanLeeuwen CMD ColourMagnitudeDiagram 2007) and an updated version of the Catalogue of Nearby MS MainSequence Stars4 (Jahreiß & Wielen 1997, which will soon be pub- SFR StarFormationRate lished as CNS5) also giving us a sound statistical sample SFH StarFormationHistory to investigate the local stellar mass density (see discussion AVR AgeVelocity-dispersionRelation in sec5.3). We use Galaxia (Sharma et al. 2011) to create AMR AgeMetallicityRelation mock observations from the JJ-model with arbitrary IMFs. CNS CatalogueofNearbyStars WefindanewfiducialIMFdescribingthedatabysampling paperI Just&Jahreiß(2010) the two slope broken power law parameter space using a paperII Justetal.(2011) MarkovchainMonteCarlotechnique.Themodellikelihood JJ-model paperI&IIfiducialmodelA KTG93 Kroupaetal.(1993) given the data is approximated assuming discrete Poisson Cha03 Chabrier(2003) probabilitiesfordifferentmagnitudebinsdifferentiatingbe- BesB DefaultmodelBofCzekajetal.(2014) tween dwarfs and giants. Galaxia Tooltosynthesiseobservations(Sharmaetal.2011) AsimilardeterminationoftheIMFwasdoneintheCzekaj FH06 Flynnetal.(2006) et al. (2014) Besanc¸on update with matching results. In- NGP NorthGalacticPole steadoflocalluminosityfunctionstheyusedthecolourpro- SSP SimpleStellarPopulation jectionsofTycho2(Høgetal.2000)dataindifferentdirec- CEM ChemicalEnrichmentModel tionstodeterminetheirbestfit.Theirmodellingmachinery ∆mag Magnitudedifferenceofstarsinabinarysystem is very modular being able to incorporate different stellar ISM InterStellarMedium evolutionary- and atmosphere models as well as account- BD BrownDwarf WD WhiteDwarf ing for binarity or reddening from the model’s side. On the MCMC MarkovchainMonteCarlo other hand they have so many degenerate free parameters PDF ProbabilityDistributionFunction that they do not explore the full IMF parameter space but use pre-determined ones from literature. Overall the holistic approach with a concordance Galaxy model in the background able to produce mock observations to test theoretical concepts like for example the IMF is a very promising technique especially with the 2.1 The disc model locally underlying physical models getting more and more refined with increasing observational evidence. One unsolved prob- As we want to compare our JJ-model to volume complete lem which is also pointed out by Czekaj et al. (2014) are stellar samples in the solar neighbourhood we construct missing detailed 3d extinction data of the local ISM. 7spheresthataredisjunctinabsoluteVmagnitudeandhave In section2 we explain how to create mock observa- heliocentric distances from 20 up to 200pc (see table1&2 tions from our vertical disc model using Galaxia. Section3 fordetailedlimits).Inthefollowingweexplainhowwepre- describesthereductionofHipparcosandCNS5datatoob- pare n-body particle representations of our model for each tain our observational sample, followed by section4 on the spheresuchthatGalaxiacanturnthemintomockobserva- likelihood determination in the IMF parameter space. The tions. results are then presented in section5 together with a com- Asourmodelisevolvedover12Gyrin25Myrstepswe parisontowidelyusedIMFs.Insection6theresultsareput construct 480n-body particles (plus one for the thick disc) into context followed by the conclusions in section7. with specific ages, masses and metallicities for each sphere separately.ThesearepassedtoGalaxiawhereeachparticle isturnedintoasimplestellarpopulation(SSP)accordingto a prescribed IMF. The two youngest SSPs are split up age- 2 SYNTHESISING A LOCAL DISC MODEL wise so that the youngest stars coming from Galaxia are In paperI a self-consistent dynamical vertical Milky Way 6.25Myr old. We assign arbitrary phase-space information disc model was developed. Combined with constraints from to the particles since the final observable is a local colour starcountsoftheNGPinpaperIIafiducialdiscmodel(JJ- magnitudediagram(CMD)basedonvolumecompletesub- model) was chosen. The JJ-model fixes the SFH, the AVR, samples. a simple metal enrichment law and from that predicts the Wewillnowillustratetheconstructionofalocaln-body stellar vertical disc structure in terms of kinematics, star representation of our model for the 25pc sphere following counts,agesandmetallicitiesasfunctionsofdistancetothe figure1: Galactic plane. The upper panel depicts the global SFH in units of surface In order to obtain mock observations of the solar neigh- densitypertimeandtheeffectivescaleheight,h ,overage eff bourhoodweturnourverticaldiscmodelintoalocalrepre- for the JJ-model. From that a local mass density, ρ , for 0 sentation used as an input for Galaxia to synthesise stellar each SSP can be calculated which then is multiplied with samples. the volume of the 25pc sphere, V , to obtain 25 (cid:13)c 2002RAS,MNRAS000,1–13 Milky Way disc model - III 3 introduceamassfactor(mf)scalingourmodel’stotalmass 7 600pc] (M ) to fit the observed stellar mass density 6 500ht [ IMF g 2M/pcSFR []flGyr4352 234000000ve scale hei MWalshoIMennFeg=eodimntogf·ctMoorlrIaeMrcgFte,dfroisrhcsed.iegchrtesasaibnogvevetrhtiecaGladleanctsiictypplarnoefil(we4se). 1 100ecti For the 25pc sphere the deviation from homogeneous den- M]fl102 0 eff slaitrygerdisspthriebruetsioanndisespsteicllianlleygwligitihblyeoubnugtsttheilslarchpaonpguelastiwonitsh. ss [10 For example when rescaling the star count in the highest a e m 8 magnitude bin in table1 with a mean age of 0.1Gyr the particl 46 nscuamlebhereigmhutsotfbteheisnecrsetaasresd(cbfy. fi5g4u%re2toinacpcaopuenrtIIf)o.r the low y od 2 0.50 0.25 0.00 n-b 0 metallicity [Fe/H] 12 10 8 6 4 2 0 2.2 Mock observations with Galaxia age [Gyr] Galaxia(Sharmaetal.2011)isatooltogeneratemockcata- Figure1.TheupperpanelshowsindashedbluetheglobalSFH loguesfromeitheranalyticmodelsorn-bodydata.Italready andinsolidredtheeffectivescaleheightoftheJJ-model(atsolar has a default Besanc¸on-like model (Robin et al. 2003) but galactocentricdistance). the updates (Robin et al. 2012; Czekaj et al. 2014) are not In the lower panel the input n-body data which is passed to implemented yet. Galaxia to create the 25pc sphere is displayed. The age, mass Intheprevioussubsectionweconstructedparticlesrep- andmetallicityofeachSSParevisualised. resenting our model locally which we now pass for each sphere separately to Galaxia together with the disjunct M (t )=V ρ (t )= 4πr3 SFH(ti)· 0.025Gyr (1) VMag limits building up a CMD successively. 25pc i 25pc 0 i 3 25pc 2heff(ti) So far Galaxia uses Padova isochrones (Marigo et al. 2008)whichhaveproblemsreproducingthelowerendofthe which is shown in the lower panel of figure1 as resulting MS and the red clump. We include their revised templates n-body particle masses. Colour-coded the chemical enrich- (PARSEC version 1.2S1) where only minor differences at ment law (AMR) is depicted which is also coming from the low mass stars persist (cf. lower MS in figure2). This re- JJ-model as an analytic function but with an added Gaus- mainingdiscrepancyisalsopointedoutinthereleasepaper sianscatterof0.13dexstandarddeviation(testedwithGCS Chen et al. (2014, fig.A3) but should have negligible effect data, cf. fig. 15 of paperI). on the star counts in our used magnitude range. For the thin disc 3552M (M ) in particle (cid:12) IMF,thin,25pc Binaries,whitedwarfsorotherremnantsarenotimple- mass is passed to Galaxia for the 25pc sphere mentedinGalaxiayetbutanupdateisbeingplanned(pri- 12(cid:90)Gyr 480 vate communication, Sharma 2015). When inspecting the (cid:88) MIMF,thin,25pc = M25pc(t)dt(cid:39) M25pc(ti). (2) CMDinfigure2asecondblue-shiftedMSinthesynthesised 0Gyr i=1 catalogueisvisiblewhichcomesfromourdistinctthickdisc metallicity. This is by construction of the JJ-model the gas mass that BesidebeingabletochangetheIMFaccordingtowhich wasusedtocreatethethindiscstars(andinthemeanwhile Galaxiaisdistributingtheparticlemassesintostarsweare also remnants) still residing in the 25pc sphere. Nowadays using it as a black box. Specifying a photometric system onlyafractionofthatisleftinstarsduetostellarevolution will already yield a detailed stellar catalogue2 in terms of a (M , cf. section5.3). In the mass-age distribution PDMF,25pc random realisation of our local model representation as for of the n-body data the peak from the global SFH (dashed exampledepictedinfigure2forournewlydeterminedIMF. blueline)around10Gyrcanstillberecognised.Theincrease foryoungerstellarpopulationsstemsfromthedecreasingef- fectivescaleheightconfiningthesestarsclosertotheGalac- ticplane(i.e.abiggerfractionofthemisfoundinthelocal 3 OBSERVATIONS sphere, cf. fig. 14 of paperI). TheanchoringpointforeveryGalaxymodelintermsofob- Thethickdiscisimplementedbyinsertingasinglestar- servationalconstraintsisthestellardistributioninthesolar burst (i.e. one SSP and equivalently one n-body particle) neighbourhoodsincedetailedandvolumecompletesamples with 6.5% of the thin disc mass can only be obtained here. After using the vertical com- M =M +M =1.065·M (3) ponent of the velocity distribution of MS stars in paper I IMF,discs IMF,thin IMF,thick IMF,thin (dynamicalconstraint)andtheNGPstarcountsinpaperII 12Gyr ago with a metallicity of [Fe/H]=−0.7 resulting in (vertical density distribution constraint) we now use abso- a present day thick disc mass fraction of around 5%. Since lute local stellar densities for dwarf and giant stars. the local density of the stellar halo is negligible compared tothediscwedonotconsideritasaseparatecomponentin this work. 1 http://stev.oapd.inaf.it/cgi-bin/cmd A change in the IMF will affect the mass fraction re- 2 See http://galaxia.sourceforge.net/Galaxia3pub.html for de- maininginthestellarcomponent(cf.sec.2.5paperI)sowe tailedinstructions (cid:13)c 2002RAS,MNRAS000,1–13 4 J. Rybizki and A. Just Table 1.Observationalsampleandmockcatalogues-dwarfstars Catalogue d MV-limits Nfin σππ >15% CNS5 N25 JJ25 log-likelihood Meanmass Meanage (pc) (Mag) # #lost 25pc rescaled† to25pc ln(L/Pmax) M(cid:12) Gyr 200 ],−1.5] 98 0 0 0.28 0.13 -21.9 6.4 0.1 200 [−1.5,−0.5] 233 3 1 0.62 0.63 0.00 4.0 0.2 200 [−0.5,0.5] 901 12 4 2.26 2.60 -9.20 2.9 0.3 Hipparcos 100 [0.5,1.5] 520 2 15 8.61 9.09 -0.76 2.2 0.5 75 [1.5,2.5] 677 1 27 25.6 24.4 -0.81 1.7 1.0 50 [2.5,3.5] 518 1 62 65.1 59.9 -1.79 1.4 2.3 30 [3.5,4.5] 200 0 110 115.7 146.5 -5.90 1.1 5.3 25 [4.5,5.5] 191 4 191 191 190.3 0.00 0.9 6.1 25 [5.5,6.5] 198 11 198 198 207.9 -0.22 0.8 6.4 CNS5 25 [6.5,7.5] 193 16 193 193 196.0 -0.02 0.7 6.5 25 [7.5,8.5] 207 13 207 207 205.3 -0.01 0.6 6.6 20 [8.5,9.5] 139 15 245(cid:63) 271.5 258.0 -0.20 0.5 6.7 Σ ],9.5] 4075 78 1253 1278.8 1300.8 -40.8 0.8 6.0 Catalogue shows from which source the observational sample is drawn, d gives the heliocentric distance of stars included (sphere size), MV-limits gives the magnitude range of each bin, Nfin is the final star count in each bin, σππ >15% is the number of stars thrown out due to high distance errors, CNS5 gives the number of stars within the volume complete 25pc sphere, the next two columns give the star counts of the observations and our JJ-model rescaled to 25pc. The JJ-model with newly determined IMF is averaged over 400 random realisations, log-likelihood shows the probability of each bin after equation6 normed with the maximal possibleprobability(cf.section4.1)innaturallogarithmwhichindicateseachbin’simpactonthelikelihoodfunction(wewillcallit penaltybecauseourlikelihoodcanbeseenasarewardfunctionfortheMCMCsimulation),themean mass andmean age showthe valuesforthecorrespondingJJmagnitudebinswherethesumatthebottomisanaverageofallstarswithin25pc. (cid:63)forthismagnitudebinvolumecompletenessisnotgivensothe139starsfromthe20pcspherehavebeenrescaledto25pcyielding 271.5starswhichis10%morethanthe245starsobservedinthe25pcsphere †rescalingthevolumeandaccountingforthedensityprofileofthemagnitudebin’smeanagepopulation Table 2.Observationalsampleandmockcatalogues-giantstars Catalogue d MV-limits Nfin σππ >15% CNS5 N25 JJ25 log-likelihood Meanmass Meanage (pc) (Mag) # #lost 25pc rescaled† to25pc ln(L/Pmax) M(cid:12) Gyr 200 ],−1.5] 74 1 0 0.16 0.14 -0.87 3.4 1.6 200 [−1.5,−0.5] 375 2 1 0.77 0.84 -1.47 1.8 4.4 200 [−0.5,0.5] 1341 20 3 2.78 2.29 -23.4 1.7 3.9 Hipparcos 100 [0.5,1.5] 526 0 6 8.33 9.27 -3.03 1.5 4.8 75 [1.5,2.5] 126 0 5 4.70 3.76 -2.99 1.2 6.3 50 [2.5,3.5] 62 0 7 7.77 11.6 -5.55 1.2 6.6 30 [3.5,4.5] 9 0 6 5.21 7.11 -0.34 1.0 8.8 Σ ],9.5] 2513 23 28 29.7 35.0 -37.7 1.3 6.3 Thisisachievedbyconstructingdifferentsamplescombining then-bodyparticleswehavetotreatbinariesandderedden- absolutemagnitudecutsandheliocentricdistancessuchthat ing from the observational side. theselectedstarsrepresentavolumecompletesphere.Atthe brightendwegoupto200pcinordertoobtainenoughmas- sivestarsandgiantstohavereliablestatistics.Ourobserva- 3.1 Hipparcos tionalsampleconsistsofstarsfromtheextendedHipparcos WeusetheextendedHipparcoscompilation(117955entries) catalogue (Anderson & Francis 2012) and the Catalogue of whichcross-matchestheoriginalstarsfromtherevisedHip- NearbyStars(CNS).CNS5whichweusehereisanupdated parcos catalogue (van Leeuwen 2007) with a large selection versionofCNS4(Jahreiß&Wielen1997)andwillgetpub- of different catalogues. lished soon. Byusingheliocentricdistanceandabsolutemagnitudecuts Fundamentallywewouldliketoimplementallobservational similar to paperI (Just & Jahreiß 2010, table 1) we obtain biasesonthemodelssideandcomparethesynthesisedmock volume complete observational spheres for different stellar observationstounalteredobservables.Inthisrespecttheup- magnitudes going down to 4.5VMag. The only further se- dated Besanc¸on model (Czekaj et al. 2014) has pushed the lectioncriteriatoeliminatemiss-identificationsisadistance linkbetweenmodelanddataintherightdirectionbyimple- errorbelow15%whichreducesthesampleinsignificantlyas mentingextinctionmodelsandaschemeforbinarysystems visible in table1&2. intotheirmodel.AsGalaxiaisnotabletoaccountforbina- Before the VMag and distance cuts are applied all stars riesyetandwearenotprovidingpositionalinformationwith which have both a binary flag and a ∆mag entry in the (cid:13)c 2002RAS,MNRAS000,1–13 Milky Way disc model - III 5 Table 3.Effectofbinarityanddereddeningonthestarcounts 4 Radiusofsphere[pc] 30 50 75 100 200 Magnitudebin[VMag] 4th 3rd 2nd 1st <0.5 2 200 Nocorrection 204 591 793 1052 2798 Binarycorrection 209 580 803 1046 2756 0 Extinctioncorrection 204 591 793 1053 3060 100 Bothcorrections(Nfin) 209 580 803 1046 3022 MV 2 75 50 4 Obs dwarfs 30 Obs giants original Hipparcos catalogue (ESA 1997) are split up into 6 JJ dwarfs twocomponents.Thebinarycorrectionchangesthenumber JJ giants 25 ofstarsineachdistancebinslightlyascanbeseenintable3. Cuts 8 With the binary correction we have 5394 stars in the Hip- r in pc sphere 20 parcossampleofwhich552entriescomefromsplitupbinary 0.0 0.5 1.0 1.5 systems. On average the bins lose stars when correcting for B-V binaries because the single components get fainter than the system bringing these stars below the magnitude limit. As Figure 2. CMD of the observational sample (Hipparcos stel- lar systems are not split up) and one random realisation from the fainter mag bins have smaller limiting radii only some our model with the newly determined IMF. The cuts with con- lost stars fall into the fainter magnitude bin. On scales of the investigated volume the ISM is in- nected triangles at (B-V,MV) = (-0.3,-5.5), (0.5,0.5), (0.75,4.5) and (3,4.5) for the division into dwarf and giant sample are in- homogeneously distributed with Ophiuchus and Taurus dicated in magenta. In the right the different sphere radii are molecular cloud being the biggest absorbers in the 200pc writteninunitsofpc. sphere (Schlafly et al. 2014). 3d extinction maps with good enough resolution are getting published (Lallement et al. 2014) but the data are not available yet. In order to dered- Thedistanceofthestarsiscalculatedfromparallaxes(when denourstarsweadoptananalyticmodelfromVergelyetal. advisablethephotometricwerepreferredtothetrigonomet- (1998) describing a homogeneous extinction depending on ric) with a correction for the Lutz-Kelker bias from Ander- the distance and Galactic latitude. Due to the local bubble son & Francis (2012, eq. 1). we set the extinction to 0 below 70pc distance and above The20pcsampleincludesstarsfrom8.5to9.5VMag.After 52◦ Galactic latitude (cf. Vergely et al. (1998, fig. 4, 11)). excluding 2WDs and 15stars with relative distance errors Aumer & Binney (2009) also adopted this model but above 15% we are left with 139 stars. For the 25pc sphere onlywithin40pcoftheGalacticmid-planeresultingin4% the absolute Vmagnitude ranges from 4.5 to 8.5. Here 44 less stars in the 200pc sphere compared to our adaptation stars have a too high error so that 789 stars remain in the of the extinction model. We use the Vergely et al. (1998, final sample. p.548) cosecant law: All together 928 stars come from the CNS5 of which 372  areinresolvedmultiplestellarsystems(24oftheremaining 0, d<d ∨ b>52◦  0 556 single systems are detected spectroscopic binaries). Of E(B−V)= E0((cid:16)d−d0), (cid:17) if d< |sihn0(b)| (5) the372starsinmultiplestellarsystems117mostlyprimary  E0 |sihn0(b)| −d0 , d> |sihn0(b)| and87mostlysecondarycomponentsshareajointB-Vvalue but have individual VMag entries so that they can be cor- with h0 = 55pc, d0 = 70pc, E0 = 4.7×10−4mag/pc rectedbyputtingthemontheMS.TheMSwasempirically and d,b being the heliocentric distance and the Galactic assigned from MS stars with low parallax error. latitude. To transform from E(B−V) to A(V) we adopt In the end three stars reside in the CNS5 sample as R =3.1. 0 wellasintheHipparcossamplebecauseofslightlydifferent In essence the dereddening leaves the closer samples unal- VMag values directly at the VMag borders of 4.5 VMag. teredonlyincreasingthe200pcsamplestarcountbyaround ThosestarsareexcludedfromtheHipparcossamplesothat 10%.Thisisduetostarswithheliocentricdistancebetween in our joint sample every star has a unique entry. 100and200pcthatareslightlytoofaintwithoutderedden- As a comparison to the Hipparcos sample we also list the ingbecomeabitbrighterandtherebysatisfythemagnitude 247 volume complete (within 25pc) CNS5 stars which are limits. brighterthan4.5VMagintable1and2.Theonlypeculiarity As the ISM is highly inhomogeneous adopting an analytic is the 2σ outlier with 15 against 8.65 expected stars in the modelisonlyacrudeapproximationbutthebestwecando 100pcbinofthedwarfsample.StilltheCNS5samplecanbe atthemomentandwearecurioustoredotheanalysiswith seenasavalidrandomrealisationoftheenlargedHipparcos upcoming 3d extinction maps. sample. 3.2 Catalogue of Nearby Stars 5 4 STATISTICAL ANALYSIS The Hipparcos sample is supplemented with stars from CNS5(7251entries)atthefaintend(4.5-9.5VMag)which ForouranalysiswedividethederivedCMDsintodwarfand is a volume complete catalogue for stars brighter than 8.5 giantstarswiththecutsspecifiedinfigure2.SinceMSstars (9.5) VMag up to a distance of 25 (20)pc. haveatightcorrelationbetweenluminosityandstellarmass (cid:13)c 2002RAS,MNRAS000,1–13 6 J. Rybizki and A. Just the dwarf sample contains information on the PDMF. The reducedbyafactorof20(cf.tab5)andshouldbeasecond giant sample adds constraints for the integrated SFH and order effect compared to the Poisson noise in the data (if the IMF of higher mass stars (above 0.9 M ). we assume the observed stars have been randomly realised (cid:12) To compare our observable (the LF) to a theoretical IMF from an underlying probability distribution). we feed local representations of our model together with different IMFs to Galaxia leaving us with synthesised star To sample the PDF of our parameter space we use a counts from which we construct a likelihood assuming dis- Python implementation (Foreman-Mackey et al. 2013) of crete Poisson processes (section4.1). This is implemented anaffineinvariantensemblesamplerforMCMC(Goodman into a Markov chain Monte Carlo (MCMC) scheme to ob- & Weare 2010) where step proposals using the information tainarepresentationoftheprobabilitydistributionfunction of multiple walkers reduce the autocorrelation time signifi- (PDF) in the two slope IMF parameter space (section4.2). cantly. Since the overall mass turned into stars (M ) could IMF,discs beslightlydifferenttotheJJ-modelweaddthemassfactor 4.1 Likelihood calculation (mf) as a fourth free parameter beside the three two slope We approximate the likelihood of our model given the data IMF parameters, low mass index (α1), high mass index bydividingeachCMDinto12magnitudebinsforthedwarf (α2), and the power law break (m1): sample and 7 for the giants (see table1&2) and calculate the discrete Poisson probability distribution. The expected dn (cid:26) α=α , m <m<m =k m−α 1 if low 1 (9) valueiscomingfromourmodel(mi)andthenumberofoc- dm α α=α2, m1 <m<mup currencesisthestarcountobservedineachbin(d )leading i with the lower and upper mass limit of the IMF m = to the likelihood low 0.08M andm =100M beingfixedandk normalising (cid:12) up (cid:12) α 1(cid:89)2+7 mdie−mi the IMF to be a continuous function that represents the L = L , where L = i . (6) total i i d ! mass turned into stars i i=1 (cid:90) mup The log-likelihood then follows as: M =mf·M = k m−α+1dm. (10) IMF IMF,discs α 1(cid:88)2+7(cid:16) (cid:17) mlow logL = d log(m )−log(d !)−m . (7) For every set of parameters we use the product of the like- total i i i i i=1 lihoods from dwarf and giant sample For the calculation of the log factorial a very accurate ap- logL =logL +logL (11) total dwarf giant proximationforn>0fromAiyangar(1988,p.339)isused: to sample the parameter space. In this way the probability (cid:18) (cid:16) (cid:17)(cid:19) log n 1+4n(1+2n) for each bin (12 from the dwarf sample and 7 from the gi- log(π) ant sample) being an independent discrete Poisson process logn!≈nlogn−n+ + .(8) 6 2 is weighted equally into the final likelihood (L ). total Wewillnormalisethelog-likelihoodwithitsmaximalpossi- We also investigated three slope IMFs but the gain in like- blevalue,P =−68.5,occurringwhentheobservedsam- lihood is small. Usually a second break in the IMF is in- max ple is tested with itself. troduced to fit the low mass regime (Kroupa et al. 1993, Itshouldbekeptinmindthatalinearincreaseinstarcounts m<0.5M(cid:12)).Withourobservationalevidencerangingfrom results in exponentially increasing penalties for our likeli- 0.5to10M(cid:12) asecondbreakintheIMFbelowthislimitcan hoodfunctionwhenthedeviationinpercentstaysthesame. notbetested.Butanempiricalextensionofour2slopeIMF For example if the expected value is 10 and the number of tofulfillocalmassdensityconstraintsforlowmassstarswill occurrences is 9 then we have a ln(L/P ) of −0.04. For be introduced in section6.6. max 100 expected stars and 90 occurrences ln(L/P ) equals max −0.46,for1000and900itis−5.1andsoforth.Thisonthe one hand takes into account that bins with a lot of stars 5 RESULTS get a higher statistical weight but could also be dangerous whensmallerrorsinrescaling(whichcouldcomefromabad Here we present the newly determined fiducial IMF for our AVRorconnectedlyisochronesindicatingwrongages)result model.Forcomparisonwealsoshowthelog-likelihoodofthe inlargepenaltiesforthelikelihoodpotentiallypointingour observationaldatawithsynthesiseddatageneratedfromour MCMC simulation to a biased equilibrium IMF parameter model but using common IMFs from the literature. configuration. 5.1 New IMF parameters 4.2 Sampling the likelihood distribution Multiple burn-ins from different starting points all settling The variability of the outcome of Galaxia is twofold. First inthesameequilibriumconfigurationsuggestawell-behaved the IMF parameters can be varied changing the laws parameter space with respect to our likelihood calcula- according to which the mock observations are produced. tion. We use 20walkers each sampling 500steps to gener- Second for fixed parameters the random seed of Galaxia ate the point cloud representing the PDF of the parame- can be changed yielding different random realisations. The ter space. For each step we average over 400realisations. latter can be minimised by averaging. We oversample each Figure3 shows the marginalised likelihood distribution of point in parameter space 400times so that this noise is each parameter as histograms on the diagonal and as point (cid:13)c 2002RAS,MNRAS000,1–13 Milky Way disc model - III 7 Furthermore Pearson’s correlation coefficient for each pa- 1.49 0.08 3000 ± rameter pair is given in the upper right of figure3 which is α1 2000 # -0.02 0.62 0.88 also represented in the 1-3σellipses of the projected point 1000 clouds. The only two parameter which are uncorrelated are 3.2 3.02±0.06 3000 the power law indices. Almost positive linear is the corre- 3.1 α23.0 2000 # 0.55 -0.30 lation of the mass factor with the low mass slope. This is 2.9 1000 due to more mass being put into stars which are not rep- resented in our data (m < 0.5M ) for high α which can 1.39 0.05 3000 (cid:12) 1 1.5 ± be counterbalanced with a high mass factor. Similarly but m11.4 2000 # 0.29 less strong the anti-correlation of the mass factor with the 1000 1.3 highmassslopeisduetomassbeingshiftedoutofourrep- 1.2 1.09±0.04 3000 resented data domain when α2 is getting lower. The reason mf1.1 2000 # for both power law indices being positively correlated with 1000 the power law break is due to the shape of the IMF which 1.0 is sharply decreasing at the position of the break and the 1.2 1.4 1.6 2.9 3.0 3.1 3.2 1.3 1.4 1.5 1.0 1.1 1.2 α1 α2 m1 mf numberdensitiesofthefaintestandbrightestmagnitudebin whichneedtobematchedwhenshiftingm .Thisreasoning Figure 3. Marginalised parameter distribution of the MCMC 1 canbebestvisualisedwhenlookingatthegreendottedline runexploringtheequilibriumdistributionoftheparameterspace with respect to our log-likelihood using 105 evaluations. Each (our IMF) and the blue error bars (our observational sam- ple) in figure4. Additionally when increasing both α and scatter plot shows the projected 2d parameter distribution with 1 pointscolouredbylikelihoodincreasingfrombluetored.Crosses α2 the mass that is gained by the steeper low mass slope indicatethemeanvaluesandellipsesencompassthe1-3σregions. willbecounterbalancedbythemasslossofthesteeperhigh The respective correlation coefficients are given at the position mass slope. mirroredalongthediagonal.Gaussianfitsandhistogramsofthe Figure4 shows the binned luminosity function of our fidu- marginalised parameter distribution are given on the diagonal. cial IMF (averaged over 400realisations) compared to the The mean and standard deviation of each parameter is written observationsinabsolutenumbers.TheCMDrepresentation andalsoindicatedbysolidanddashedgreylines. oftheluminosityfunctionscanbeseeninfigure2wherethe same colours have been used for the different samples. The deviations in each bin look small and systematics are not 20 25 30 50 75 100 200 apparent neither in the dwarf nor the giant sample which showsthatthewholemachinery,consistingofthediscmodel 2000 and Galaxia producing the mock observations, works well r in pc andtheMCMCsimulationhaslikelyconvergedtowardsthe nt1500 Ospbhser edwarfs equilibrium configuration. ou Obs giants Wheninspectingthedetailedlikelihoodcontributionofeach c ar 1000 JJ dwarfs binintable1&2weseethatthelargestpenaltycomesfrom st JJ giants the0thVMagbin.Especiallythegiantswithln(L/P )= max −23.4haveahugeimpact.Thereasonforthatislikelyared 500 clump that is too faint in our mock catalogue. In the dwarf sampletoomanystarsareinthesynthesised0thVMagbin with JJ ·(N /N ) = 1036 compared to 901stars in the 0 25 fin 25 9 8 7 6 5 4 3 2 1 0 -1 -2 Hipparcos catalogue. MV A weakness of our modelling machinery is apparent in the Figure 4. Luminosity function of the observations and the new highln(L/P )valueofthebrightestdwarfbinindicating max IMFoftheJJmodel(cf.table1and2).ErrorbarsindicatePois- that too few bright stars are produced. The reason is most sonnoiseintheobservationalsampleandthestandarddeviation likely that we are not accounting for binaries. Minor effects forthe400timesoversampledsynthesisedcatalogue.Starcounts could be missing high metallicity stars and that our syn- arenotnormalisedforthedifferentdistancelimits.Thelimiting thesised stars are not younger than 6.25Myr. It could also radiiofthecorrespondingmagnitudebinsarewritteninthetop. indicateachangeofthehighmasspowerlawindexforstars more massive than contained in our data. clouds for each parameter pair in the lower left. Just for illustrative reasons each dot is coloured to indicate high 5.2 Tested IMFs (red =−77.2) and low (blue =−86.7) log-likelihood max min BecauseforothershapesoftheIMFpartofthemasscould valueswithgreydotsbeingbelowthis3σ range.Inthehis- also be hidden in the mass range not represented by our togramsthemeanandthestandarddeviationofthenumber observational sample (0.5 to 10M ) we determine the fac- (cid:12) density of each marginalised parameter is given which rep- tor with which the total mass needs to be rescaled in order resents our central result and defines our new fiducial IMF: to maximise the likelihood for a particular IMF when it is α = 1.49±0.08, used with our JJ-model against the observational sample. 1 α = 3.02±0.06, Weagainaverageover400realisationsandgetthestandard 2 (12) m = 1.39±0.05, deviation as the enclosing 68% of the likelihood. 1 mf = 1.09±0.04. Intable4thelog-likelihoodsofthedifferentIMFs(usingour (cid:13)c 2002RAS,MNRAS000,1–13 8 J. Rybizki and A. Just where it produces too few stars within 0.9 - 2M and too Table 4.LikelihoodsofthedifferentIMFs (cid:12) many outside of this range compared to our IMF. IMF JJ JJ3σ BesB KTG93 Cha03 (cid:16) (cid:17) ln L −79.3 −86.7 −96.7 −195.6 −216.1 Pmax Besan¸conB This is one of the new fiducial Besanc¸on model IMFs from Czekaj et al. (2014) tested with Tycho2 all-sky colour dis- Table 5.Variabilityofthelog-likelihood tribution. It is also a three slope broken power law with Sample JJ400 JJ1 JJideal(cid:63) α1 =1.3,α2 =1.8,α3 =3.2,m1 =0.5 and m2 =1.53. (cid:16) (cid:17) ThemassfactorforthisIMFis1.107±0.015whichiscom- ln L −79.3±0.6 −89.5±14 −13.8±2.8 Pmax patible with our own mf. Also the shape of the IMF (cf. (cid:63)ifdatawascomingfromourmodel(seetext) figure5), the likelihood (see table4) and the mass fractions (cf. table8) are similar. This is not surprising since they are also using the local mass density model from Jahreiß & discmodelbutadjustingforeachIMF’sbestfitmassfactor) Wielen(1997).Stillitisreassuringthatfordifferentobserva- are listed with JJ being the log-likelihood value which is 3σ tionaldata(Tycho2colourvs.Hipparcos/CNS5VMag)and lowerthan99.7%ofthepointsrepresentingthePDFinfig- differentmodellingtechniquessimilarresultsareyielded.Al- ure3. thoughthedifferenceof17.4inln(L/P )stillmeansthat Table5 illustrates a few properties of our log-likelihood. max the likelihood for their IMF is 36million times lower than JJ andJJ showthemeanandthestandarddeviationof 400 1 for our IMF. 100log-likelihooddeterminationswithdifferentseedswhich Compared to our IMF the Besan¸conB IMF is producing are averaged over 400 in the former and 1realisation in the slightlylesshighmassstarsandmorelowmassstarswhich latter case. This shows that the averaging is important for could be partly due to their rigorous treatment of binaries the MCMC simulation in order to smoothen the likelihood (see figure5 and cf. section6.3). distribution.Thedeterioratedmeanforsinglerealisationsis duetoaskeweddistributionsinceP isalowerlimitand max the increasing penalty for extreme values. Chabrier03 The last column of table5 (JJ ) gives an ideal log- ideal AnotherwidelyusedIMFcomesfromChabrier(2003).Itis likelihood which is obtained when the data is indeed rep- amixtureofalognormalforminthelowmassandaSalpeter resented by the model. For that we draw 100 single ran- power law in the high mass regime dom samples (JJ ) from JJ with replacement and evalu- 1 400 Eataechthseairmlpolge-lfiuklefililhlionogdthweitohbstehrevaptairoennatldciosntrsitbrauitniotnof(JhJa4v0i0n)g. dn =(cid:40) 0.85m2464 exp(cid:16)−lo2g·20(.609.m2079)(cid:17), if m<M(cid:12) (14) dwarf and giant star counts fixed to N =4075 and dm 0.237912·m−2.3, m>M(cid:12). Obs,dwarf NObs,giant =2513.Thismeansthataln(L/Pmax)ofaround The mass factor for the best likelihood is: 1.317±0.017. It −13.8 would indicate a perfect model. An even lower log- is so high because a huge mass fraction is going into stars likelihood value close to Pmax (ln(L/Pmax) = 0) would be more massive than 8M(cid:12) (cf. table8, highest supernova unrealistic since there is a natural scatter to Poisson pro- rate compared to any other standard IMF). The shape cesses. of the Chabrier03 IMF fits our model worst with respect Related to that we also inspected the distribution of star tothedatascoringalog-likelihoodofln(L/P )=−216.1. max countsinindividualmagnitudebinsforrandomrealisations (withthesameparametersbutdifferentseeds)whichindeed is Poissonian. 5.3 From luminosity function to local stellar mass density KTG93 In figure5 the IMFs are normalised such that their inte- The widely used Kroupa et al. (1993) IMF is a three slope gratedmassisrepresentingthemassofgasthatwasturned broken power law of the form into stars (of which a few already turned into remnants)  α=α , m <m<m still residing in the 25pc sphere (i.e. MIMF,25pc including dn  1 low 1 thethickdiscandeachIMF’smassfactorcf.equation4).Be- =k m−α α=α , if m <m<m (13) dm α 2 1 2 ware that in figure5 actually the number of stars per mass  α=α , m <m<m 3 2 up interval is displayed though normalisation happens in mass with α = 1.3,α = 2.2,α = 2.7,m = 0.5 and m = 1. space. 1 2 3 1 2 Againk isanormalisationconstantensuringacontinuous Theyellowshadedareashowstheincompletenessofourob- α IMF between m =0.08M andm =100M . servationsinlowmassstarsduetothemagnitudecut.This low (cid:12) up (cid:12) Thelikelihoodpeakisobtainedwithamassfactorof1.392± comes from high metallicity stars being redder and fainter 0.019 which is quite high. The reason for that is too much than their equal mass low metallicity counterparts so they massbeingputintolowmassstarswhicharenotrepresented areexcludedanddonotcontributetothesmallestmassbin in our observational sample (see table8). leadingtoadecreaseoftheobservedmasscomparedtothe With ln(L/P ) = −195.6 it scores poorly compared to theoretical IMF. The same is true for the JJ mock dwarfs max our or the Besanc¸onB IMF showing that its shape is not indottedred(asweapplythesamemagnitudecuts)which abletoreproducelocalstarcounts.Thisisvisibleinfigure5 were spawned using the green dashed IMF. We could drop (cid:13)c 2002RAS,MNRAS000,1–13 Milky Way disc model - III 9 (figure1 of paperI adapted for 25pc age distribution yields 56.5% of stars increasing to 70.8% when remnants are in- cludedwhichistheirg )confirmingthatthenewIMFstill eff fitswithinourmodel’sframeworkasSFHandAVRareonly dependent on the integrated mass loss. In order to clarify our remnants which Galaxia is not synthesising we derive from our PDMF and IMF that we should have 707 stellar remnants for stars between 1 and 8M (potentially WDs) and 17 heavier ones (potentially (cid:12) blackholesorneutronstars)inthe25pcsphere.Compared to Sion et al. (2014) we have over a factor of 2 more since they expect 344WDs within the same limits. But their as- sumedvolumecompletenessforthe13pcsampleseemsquite optimisticatleastwhenspeakingofthecoolendoftheWD coolingsequence.Holbergetal.(2008)proposeaWDmean massof0.665M whichforusresultinaWDmassdensity (cid:12) of 0.0072M /pc3 quite close to the Besanc¸on value based (cid:12) Figure 5. 25pc LF of dwarf stars translated into mass space. on Wielen (1974) which was already corrected downwards Our fiducial IMF is plotted in thick dashed green and the Be- in Jahreiß & Wielen (1997) because one out of 5WD left san¸conB and KTG93 IMFs are plotted in dotted and dashed the 5pc sphere. We admit that our high number of WDs is grey. The blue error bars represent the 12magnitude bins from partly due to the two slope IMF structure having a bump theobservationaldwarfsamplewiththeirlimitingradiusgivenin where the power law break lies (when expecting a concave parsec. The x-value and the x-error are associated with the me- functiondescribingtheunderlyingdistributioninlog-space) dian and the range of stellar masses in the corresponding mock whichslightlyexaggeratesthemassfractionoftheIMFgo- magnitude bin synthesised with our fiducial IMF. The y-value ing into planetary nebulae (cf. table8). representsthenumberofstarsnormalisedtothemassrangeand they-erroristhePoissonnoise.Asthemassesfromdifferentmag- With these values (summarised in table6) the over- nitudebinsoverlapthevaluesareaddedupinthesolidcyanline all present day mass fraction of stars and stellar remnants representing a kind of observational PDMF. Below 0.7M(cid:12) the (MPDMF,total,25pc) is yellowshadedareaindicatesthemassthatismissingduetothe M +M magnitude cut. In dotted red the same effect is visible as this g = PDMF,25pc BD&WD25pc =71.7% (15) line represents the synthesised mass function of our IMF aver- eff MIMF,25pc aged over 100realisations. From 0.9M(cid:12) upwards the IMF and which is close to 70.8% for the 25pc sample of paperI. thelocalIMFinsolidbluedeviatesincelocally(i.e.closetothe This also implies a combined BD and WD mass fraction Galacticplane)youngandthereforemassivestarsareoverrepre- sentedwhichisindicatedbytheblueshadedarea.Thereforethe of about 20% of the local mass budget (MPDMF,total,25pc) PDMFinthe25pcspherewouldlooklikethelocalIMFifthere consistent with the Jahreiß & Wielen (1997, tab.3) value wasnostellarevolution(showninmagenta)atwork. and the stellar evolution of paperI. Then again we could also have chosen the proposed WD local mass density of Holberg et al. (2008) M /pc3 = 0.0032 which would (cid:12) themagnitudecutforfaintstarsinGalaxiaandthereddot- decreasethelocaldiscmassdensityandchangeg andthe eff ted line would be perfectly aligned with the JJ IMF in the remnant fraction. lowmassregimewhichcorrespondstothelocalpresentday For the following local mass density test we use our disc mass function (PDMF). In the observational PDMF repre- model and the fiducial IMF to produce all stars within sented by the solid cyan line the little bump compared to 25pc down to 0.08M without any magnitude cuts and (cid:12) the JJ mock dwarfs at around 0.8M could be related to analyse the sample’s properties. In this sample we find (cid:12) the isochrones badly fitting our low mass stars biasing the a local stellar mass density of 0.034M /pc3 compared (cid:12) associated mass ranges. to 0.030M /pc3 for the same selection of stars in Flynn (cid:12) In the high mass regime the blue shaded area shows the et al. (2006, tab.2) (including giants, excluding BDs, WDs, over representation of massive stars in the local IMF of the other remnants and the thick disc component) who use a 25pc sphere as their vertical distribution is more confined similar method. In table6 the detailed comparison reveals totheGalacticplane(ifonewouldnotaccountforthescale that especially the two brightest bins of the FH06 sample height dilution the deduced IMF would look like this). The are 3times denser than the corresponding bins in the JJ magentaareaindicatesthedeviationfromthelocalIMFto sample. For VMag>3 the densities are matched quite well the mock sample due to stellar evolution. except for the faintest mag bin. The over abundance of We can quantify the difference of IMF and PDMF by look- bright stars could be coming from their large value of n in ing at the integrated mass of the two functions represented Holmbergetal.(1997,eq.3)resultinginnearlyexponential by JJ IMF and JJ mock dwarfs from figure5 (the latter vertical density distribution with high local mass densities being made equal to JJ IMF for the low mass part not though this then should apply to the other mag bins as affected by stellar evolution). The outcome is that 55.8% well. Another indication for an over estimation of their of the mass originally turned into stars is still present in model’s local star counts becomes evident when comparing dwarfstarstoday(M /M ).Includinggi- thenumberdensitiesfromHolmberg&Flynn(2000)(upon PDMF,25pc IMF,25pc antsthisvalueonlychangesslightlyto56.9%whichiscon- which FH06 is based) with the CNS5 ones. For M < 2.5 V sistent with the integrated stellar mass of our disc model they have 0.0013starpc−3 whereas our volume complete (cid:13)c 2002RAS,MNRAS000,1–13 10 J. Rybizki and A. Just Table 6.Localstellarmassdensityofdifferentthindisccompo- Table 7.Localmassdensityoverstellaragein10−4M(cid:12)/pc3 nentsin10−4M(cid:12)/pc3 Age[Gyr] Besanc¸onA Besan¸conB JJ-model Masscomponent Flynnetal.(2006) JJ-model 0-0.15 20 19 9 0.15-1 55 50 36 giants 6 7 1-2 46 41 30 MV<2.5 31 11 2-3 33 28 27 2.5<MV<3 15 5 3-5 58 49 52 3<MV<4 20 19 5-7 61 50 54 4<MV<5 22 26 7-10 117 93 84 5<MV<8 70 66 10-12 - - 46 8<MV 135 205 Thindisc 390 330 338 ρ (z=0pc,t=12Gyr) 299 338 thindisc Thickdisc 29 29 17 ρthickdisc(z=0pc,t=12Gyr) 35 17 WD 71 71 92† browndwarfs 20 20(cid:63) (cid:80) 490 430 447 whitedwarfs 60 72† † includingbrowndwarfs ρ 414 447 PDMF,total (cid:63)takingthesamevalueasFH06 6 DISCUSSION †derivedimplicitlyfromourPDMF(seetext) We discuss our findings with respect to their model depen- dencies as well as our analysis method. Then we provide a comparisonofthemassdistributionfromvariousIMFsand endthissectionwithanempiricallydrivenadaptationofour sample has half of it with 0.0007starpc−3 and this despite IMF to account for missing low mass star representation in the fact that the 25pc from the CNS5 seem to have an our data. overrepresentationoftheupperMScomparedtothelarger sample (cf. table1). The next mag bin 2.5 < M < 3 is V three times denser according to Holmberg & Flynn (2000) 6.1 Mass factor with 0.0010starpc−3 compared to 0.0003starpc−3 we measure for our 25pc sample which strongly indicates a TheoverallmasswithournewlydeterminedIMFcompared necessary revision. Fainter mag bins are much better fit in to the JJ-model SFH increased by the thick disc fraction star counts as well as in stellar mass density. (6.5%)andthemassfactor(mf =1.09).Thenormalisation Comparing our thin disc stellar mass density (excluding inpaperIwasdoneusingρPDMF,total =0.039M(cid:12)pc−3 from thick disc stars and WDs) of 0.0338M /pc3 to the default Jahreiß & Wielen (1997). Since we utilise new isochrones (cid:12) Besanc¸onB model that has 0.0330M /pc3 reveals a sim- together with number densities derived from volume com- (cid:12) ilar discrepancy. Our disc model has 12Gyr of evolution plete star counts a change of about 10% is not unexpected compared to 10Gyr and we make a detailed comparison butourvalueof0.045M(cid:12)pc−3isprobablyexaggerated.The in table7 keeping in mind the different SFH and vertical present day mass fraction (geff) stays similar and also the profiles of each disc model (Czekaj et al. 2014, tab.7 and remnantfractioniscompatiblewithpaperIvaluesasshown fig. 4). The Besanc¸onB model use the same local mass insection5.3thoughWDandfaintstellarlocalmassdensity density as we do from Jahreiß & Wielen (1997) based on are arguably too high. The problem is that we do not have 25pc though they add the thick disc on top of this value observationalconstraintsforthewholemassrangeresulting which is an inconsistency as the Jahreiß & Wielen (1997) in a degeneracy of the mass factor and the low mass slope table3 accounts for all local stars not only the thin disc. (seefigure3,alsovalidforthehighmassslope).Wepropose Overall for the other disc mass models more mass seems a solution to this in section6.6. to be sitting in stellar mass bins of bright stars which mightbepartlyduetothelocaldwarfsamplebeingalmost 6.2 Isochrones 2times denser than the 200 and 100pc sample at the upper MS (cf. table1 Observations & CNS5). Since we use A crucial ingredient for our investigation is the used set of volumecompletesamplestofittheluminosityfunctionand isochrones since it translates our analytical disc model into take into account the scale height dilution according to the the realm of observables. When we were using the option stellarageswetrustourmassdistributioninthemassrange providedoriginallybyGalaxia(Marigoetal.2008)thehigh- that our observational evidence samples (0.5 − 10M ). est likelihood we could score was ln(L/P )=−109 with (cid:12) max Ideally we should have included a constraint for the low slightlydifferentIMFparameters.WiththelatestPARSEC mass stellar mass density from other observations because isochronestheobservationsaremuchbetterfitbythemodel now the mass factor and the low mass power law index increasing the normed log-likelihood by 30. Still a few dis- are strongly correlated and certainly a bit too high. Better crepancies are visible when inspecting figure2. valuesforamorerealistictwoslopeIMFwouldprobablybe The cold dwarf V-Band problem was already mentioned in in the lower left of the 1σ ellipse in figure3 (e.g. mf (cid:39)1.05 section2.2andisdiscussedinChenetal.(2014).Wewould and α (cid:39)1.4). not recommend to go much fainter in V band for such an 1 analysis. (cid:13)c 2002RAS,MNRAS000,1–13