The local stellar luminosity function and mass-to-light ratio in the NIR
A. Just, B. Fuchs, H. Jahreiss, C. Flynn, C. Dettbarn, J. Rybizki
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 20 August 2018 (MN L A TEX style file v2.2)
The local stellar luminosity function and mass-to-lightratio in the NIR
A. Just (cid:63) , B. Fuchs , H. Jahreiß , C. Flynn , C. Dettbarn , J. Rybizki Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg (ZAH),M¨onchhofstraße 12-14, 69120, Heidelberg, Germany Centre for Astrophysics and Supercomputing, Swinburne University,Hawthorn 3122, Melbourne, Australia
20 August 2018
ABSTRACT
A new sample of stars, representative of the solar neighbourhood luminosity func-tion, is constructed from the
Hipparcos catalogue and the Fifth Catalogue of NearbyStars. We have cross-matched to sources in the 2MASS catalogue so that for allstars individually determined Near Infrared photometry (NIR) is available on a ho-mogeneous system (typically K s ). The spatial completeness of the sample has beencarefully determined by statistical methods, and the NIR luminosity function ofthe stars has been derived by direct star counts. We find a local volume luminos-ity of 0 . ± . L K (cid:12) pc − , corresponding to a volumetric mass-to-light ratio of M/L K = 0 . ± . M(cid:12) /L K (cid:12) , where giants contribute 80 per cent to the light butless than 2 per cent to the stellar mass. We derive the surface brightness of the so-lar cylinder with the help of a vertical disc model. We find a surface brightness of99 L K (cid:12) pc − with an uncertainty of approximately 10 per cent. This corresponds toa mass-to-light ratio for the solar cylinder of M/L K = 0 . M(cid:12) /L K (cid:12) . The mass-to-light ratio for the solar cylinder is only 10 per cent larger than the local value despitethe fact that the local population has a much larger contribution of young stars. Itturns out that the effective scale heights of the lower main sequence carrying mostof the mass is similar to that of the giants, which are dominating the NIR light. Thecorresponding colour for the solar cylinder is V − K = 2 .
89 mag compared to the localvalue of V − K = 2 .
46 mag. An extrapolation of the local surface brightness to thewhole Milky Way yields a total luminosity of M K = − . Key words: solar neighbourhood, Galaxy: stellar content
The stellar luminosity function (hereafter LF), i.e. the in-ventory of stars as a function of their absolute magnitudes,is a fundamental property of a stellar population, and haswide implications for understanding star formation, and theformation and evolution of galaxies. At present, we can de-termine a complete LF on a star-by-star basis, down to thehydrogen burning limit, for stars in the Milky Way only.Such LFs have been determined for stars in the solar neigh-bourhood, and also for open clusters, globular clusters andin the Galactic bulge. Conventionally, the LF refers to lumi-nosities of the stars in the optical bands, such as the V band.One of the most influential determinations of the nearby LF (cid:63) E-mail:[email protected] is that of Wielen (1974), who based the study on the ’Cat-alogue of Nearby Stars’ (Gliese 1969), an extensive compi-lation of data on all stars within 22 pc of the Sun. Morerecently, Flynn et al. (2006) have re-determined the LF us-ing modern data, including in particular the
Hipparcos data(Perryman et al. 1997), but finding only small correctionsto the classic work. Those authors determined not only lo-cal star number densities (as in Wielen 1974), but also de-rived surface densities of stars (i.e. in a column integratedthrough the Galactic disc) as a function of absolute mag-nitude. In surface density terms, light from the Milky Waydisc was found to be dominated by emission from main se-quence stars with M V ≈ M V (cid:38) c (cid:13) a r X i v : . [ a s t r o - ph . GA ] A p r A. Just, B. Fuchs, H. Jahreiß, C. Flynn, C. Dettbarn, J. Rybizki components. From population synthesis it is obvious thatthe mass-to-light ratio (hereafter
M/L , in solar units) inthe optical is very sensitive to the contribution of very youngstellar populations and thus the recent star formation dueto the dominating light of O and B stars (e.g. Into & Porti-nari 2013, for the dependence of
M/L on the star formationtimescale in different bands). Additionally, dust attenuationreduces the observed light in the optical bands significantlyat least in late type galaxies and in the Milky Way. As iswell known, the analysis of galactic rotation curves suffersgreatly from the uncertainty in
M/L V of the stellar disc. Forextragalactic systems this is usually done by adopting or fit-ting a reasonable M/L value of the components. In a newapproach Martinsson et al. (2013) used dynamic disc massesderived by a combination of integral field spectroscopy anda statistical scale height determination to derive
M/L K asa function of galactocentric radius for a set of galaxies.With the increasing number of spatially resolved ob-servations in the Near Infrared bands (hereafter NIR), NIRluminosities are increasingly used for disc mass determina-tions, primarily in the K band. The variation of M/L K fordifferent stellar populations is much smaller than in the op-tical bands and the extinction is smaller by a factor of 10compared to the V band. Recently near- and far- infrared ob-servations are used to determine disc masses of extragalacticsystems (see Martinsson et al. 2013; McGaugh & Schomberg2014, and references therein). But there is still no directmethod to determine for the same galaxy the surface massdensity and the surface brightness independently. Anotherparticular purpose of using M/L K is to construct modelsof the Milky Way’s structure as constrained by star countsin the NIR in various Galactic fields and NIR luminosityfunctions have been measured in many studies, e.g. Gar-wood & Jones (1987); Ruelas-Mayorga (1991); Wainscoat etal. (1992); L´opez-Corredoira et al. (2002); Picaud, Cabrera-Lavers & Garz´on (2003) to probe the Galaxy’s structure.Star count methods are more tractable in the NIR becauseof the greatly reduced extinction compared to optical. A lo-cal K band LF for main sequence stars can also be obtainedby converting the optically determined LF to the NIR using V − K colours averaged over magnitude bins, as in Mamon& Soneira (1982), but the total brightness in the K band isdominated by giants.It is expected that observations in the NIR are bettersuited than those in the optical bands to track the distribu-tion of stars, especially to probe directly the mass-carryingpopulation of late-type main sequence stars (G, K, and Mdwarfs). Since the solar neighbourhood is the only placewhere we can determine directly both the luminosity andthe stellar mass density, it is worthwhile to investigate theproperties of M/L K based on the best available data for thispart of the Galaxy.In this paper we present a fresh ‘ab-initio’ determina-tion of the Milky Way LF in the NIR (we use the K s filter,2.2 µ m, throughout the paper but skip the index ’s’ for clar-ity). Our study is primarily based on Hipparcos stars and atthe faint end on NIR data to be included in the up-comingFifth Edition of the Catalogue of Nearby Stars (CNS5; Justet al., in preparation). These data have been obtained byidentifying the CNS5 stars in the Two Micron All Sky Sur-vey (2MASS Skrutskie, Cutri & Stiening 2006). The data setdrawn from the CNS5 is augmented by samples of stars se- lected from the
Hipparcos catalogue, for which 2MASS datais also available.Our paper is organized as follows. In section 2 we de-scribe the construction of our data set and discuss its statis-tical completeness. In section 3 we derive the K band LF andinvestigate its implications. We determine the contributionsby the stars in the various absolute magnitude bins to theluminosity and mass budgets of the Milky Way disc in thelocal volume and in the solar cylinder, the principal resultbeing that even in the NIR the light is dominated by earlytype main sequence stars and late giants. The consequencesof this finding and our conclusions are summarized in thelast section (4).
We have constructed our stellar sample by merging two datasubsets, in order to probe the bright and faint ends of the LFrespectively. For the bright end of the LF we have extractedsamples of stars from the revised
Hipparcos catalogue (vanLeeuwen 2007), using criteria of absolute magnitude M V and the parallax limits summarized in columns 1 and 2 ofTable 1. The first three subsamples fall into the survey partof the Hipparcos catalogue and are thus volume completeby construction. For the last subgroup ( d (cid:54)
25 pc), we sam-pled to apparent magnitude V = 10 . Hip-parcos catalogue is complete down to absolute magnitude M V = 8.3 mag. The numbers of stars in each subgroup aretabulated in the fourth column of Table 1. Stars with rel-ative parallax errors larger than 15 per cent were excluded(except Antares, see below). The numbers of removed starsare also given in Table 1. The vast majority of all the sam-ple stars were then matched to sources in the 2MASS cata-logue and their absolute M K magnitudes and V − K colourwere derived. Only 10 stars did not appear in 2MASS: forthese stars, K magnitudes were found in the literature or es-timated from their known spectral types using the relationof Koorneef (1983). Since the nominal errors for the bright-est stars in 2MASS are of the order of 0.2 mag, we havecompared the 2MASS magnitudes with literature values. Wefind that the differences between 2MASS and literature val-ues are so small that they can be ignored for our purposes.The resulting sample of Hipparcos stars is illustrated as acolour-magnitude diagram (CMD) in Fig. 1. It is completeto M K (cid:54) V band volumes and willbe used down to the M K = 3 ± . M K = 2 . M K = 4 . V − K ) − >
15 per cent) means that we lose some starswhich could significantly contribute to the total luminosity,in particular at the sparse, bright end of the LF. The bright-est among the excluded stars are the M1.5 Iab supergiant α Sco A ‘Antares’ (parallax 5 . ± .
00 mas, M K = − . ± .
37 mag), the M3 III giant Hip 12086 (5 . ± .
81 mas, M K = − . ± .
34 mag), and the K0 III giant Hip 61418 c (cid:13) , 000–000 he local stellar luminosity function and mass-to-light ratio in the NIR Table 1.
Volume complete samples of
Hipparcos stars. M V [mag] d lim [pc] N t ot N (0 . N g N d (cid:54) Note.
Column 1 lists the absolute V -magnitude ranges, col. 2 thedistance limits for completeness, col. 3 gives the total numberof stars, col. 4 the number of excluded stars due to the parallaxcriterion (i.e. the relative parallax error for the stars is greaterthan 15 per cent, except Antares), and col. 5 and 6 the numberof selected giant and dwarf stars (see also Fig. 1). (7 . ± .
74 mas, M K = − . ± .
82 mag). We add Antaresto our sample, because its inclusion corresponds to a factorof two in the star number in the M K = −
10 mag bin and itadds 7 per cent to the total local luminosity. The contribu-tion of all other stars is below 2 per cent and is therefore nottaken into account. The case of Antares also shows, that thesampling of the bright end of the LF has large uncertaintiesdespite the large volume with 200 pc radius.Since we derive distances and also base our absolutemagnitudes on
Hipparcos parallaxes the associated error(which we assume to be Gaussian) can affect our sampleselection. The skewed error distribution after inverting par-allaxes to distances results in the so called parallax bias (Francis 2013) leading to larger mean distances. Anothereffect of the parallax error arising from the volume limit ofthe sample, but of opposite sign, is the Trumpler-Weaver orLutz-Kelker bias (Trumpler & Weaver 1953; Lutz & Kelker1973) which increases star counts as the error volume out-side our limiting distance is larger than inside and morestars should scatter in. In the literature these biases areoften corrected statistically but it was shown that the Lutz-Kelker correction depends strongly on the adopted spatialdistribution of the sample (usually a homogeneous densityis assumed) and the true absolute magnitude distribution(see e.g. Smith 2003). The latter may be well defined forspecial stellar types like red clump giants or cepheids. Inour case for the bright Hipparcos stars there is no simpleway to calculate the Lutz-Kelker correction due to the widespread in luminosities and stellar types. In order to assessthe impact of these biases on our star counts we select allstars in the
Hipparcos catalogue. We sample the parallax ofeach star 100 times randomly as a Gaussian with the meanof the original star’s observed parallax and the standard de-viation of its associated error. Then we perform the samecuts as before, divide the counts by 100 and compare themto our sample. The 40 pc sample decreases by 1 per cent,the 100 pc sample increases by 2 per cent, and the 200 pcsample decreases by 2 per cent. Relaxing the parallax errorcuts yield similar corrections. In effect the corrections aresmall and show no clear trend in star counts as well as inthe luminosity function so that we can safely neglect them.A more severe source of error is dust extinction close tothe Galactic plane. Due to missing 3D maps of the highly in-homogeneous dust distribution in the solar neighbourhood
Figure 1.
Colour-magnitude diagram of the
Hipparcos stars usedfor this study. The colour coding is as follows – green: stars with M V (cid:54) < M V (cid:54) < M V (cid:54) < M V (cid:54) Hipparcos stars with M K (cid:54) we cannot correct for this error and only give rough esti-mates on its magnitude. For that we use the analytic ex-tinction model from Rybizki & Just (2015) based on Vergelyet al. (1998) which gives a good estimate for the mean ex-tinction. It represents a slab of constant dust density up toa distance of 55 pc from the midplane with a hole at thesolar position representing the local bubble. The resultingextinction vanishes for stars closer than 70 pc (inside thelocal bubble) and at Galactic latitudes | b | > o . The ex-tinction reaches its maximum of about A V = 0 .
19 mag atthe limiting distance of 200 pc in the Galactic plane. Ap-plying the same procedure as before but now also correct-ing for extinction nothing changes for the 40 pc sample. Forthe 100 pc and 200 pc samples the star counts increase by2.4 per cent and 6.6 per cent, respectively. Overall, neglect-ing extinction leads to a slightly underestimated luminosityfunction for bright stars with M V < . c (cid:13)000
19 mag atthe limiting distance of 200 pc in the Galactic plane. Ap-plying the same procedure as before but now also correct-ing for extinction nothing changes for the 40 pc sample. Forthe 100 pc and 200 pc samples the star counts increase by2.4 per cent and 6.6 per cent, respectively. Overall, neglect-ing extinction leads to a slightly underestimated luminosityfunction for bright stars with M V < . c (cid:13)000 , 000–000 A. Just, B. Fuchs, H. Jahreiß, C. Flynn, C. Dettbarn, J. Rybizki
Figure 2.
Colour-magnitude diagram representing our CNS5stars. Shown here are only stars with reliable distance estimates:For 3520 stars (black) trigonometric parallaxes are available,while for the remaining 365 stars photometric distances wereadopted (red symbols). In the actual construction of the LF wehave used CNS5 stars fainter than M K = 3.5 mag (dashed line). larger scale height and therefore being less obscured by ex-tinction.The faint end of the LF has been determined using anupdated version of the Fourth Catalogue of Nearby stars(hereafter CNS5, in prep.; CNS4, Jahreiß & Wielen 1997).All CNS5 stars within 25 pc were cross matched with the2MASS catalogue. From the 4622 CNS5 stars within 25 pc135 stars got K-magnitudes from other sources. Only 6 closebinaries were removed at all as well as 9 brown dwarfs andone white dwarf below the magnitude limit of the 2MASSsurvey. For the missing stars, K -magnitudes were obtainedfrom the literature, from spectral types, or applying an M K –( V − K ) relation based on CNS5 stars with accurate par-allaxes and reliable colours. The resulting CNS5 sample isillustrated in Fig. 2 as a colour-magnitude diagram M K vs( J − K ). Our
Hipparcos samples are volume complete by construc-tion. But beyond about ±
40 pc perpendicular to the Galacticplane, stellar densities decline with increasing distance fromthe plane, and lead to significant correction factors convert-ing the mean luminosity density in the observed volume to the local luminosity density at the Galactic midplane. It canbe easily shown that the impact of a (distance dependent) in-completeness, if present, on the cumulative number of stars N z as function of z is mainly an apparently reduced localdensity n , but the shape N z /n is essentially unaffected.We thus use N z to determine the vertical density profilesand correct for the local volume density at the midplanefor the r =100 pc and the 200 pc samples, separately forboth giant and dwarf stars. Within a sphere of radius r ,a constant density is obtained if the cumulative number ofstars N z follows the relation N z = 2 πn z ( r − z /
3) (greydotted lines in Fig. 3). We tested different vertical densityprofiles (linear, exponential with and without a shallow coreor Gaussian) and investigated the impact of an offset of thesolar position z (cid:12) from the midplane. It turns out that theprofiles of the dwarfs in both samples can be better fit with z (cid:12) = 15 pc.For the 200 pc sample, the cumulative star counts devi-ate significantly from the constant density fit of the inner 80and 50 pc for giants and dwarfs, respectively, which yields alocal number density of 8 . × − pc − for the giants and1 per cent less for the dwarfs. Instead the giants are bestfit by an exponential profile with a local number density n = 9 . × − pc − and an exponential scale height of456 pc. The dwarfs are better fit by a cored exponential pro-file with flat density at the midplane and local density of n = 9 . × − pc − and an exponential scale height of z exp = 44 pc at large | z | (cid:29) z exp . The flat profile near themidplane and the small scale height as well as the corre-sponding half-thickness of h = 88 pc are expected for B andearly A stars with velocity dispersions σ ≈ − (seeFig. 9). The result is also consistent with the vertical den-sity profile of the A star population derived in Holmberg &Flynn (2000). We note that the midplane densities are 5 - 10per cent larger than the value from the linear fit at small | z | .The conversion factors f = n / (cid:104) n (cid:105) from the mean numberdensities (cid:104) n (cid:105) = N f in /V ( V is the volume of the 200 pcsphere) to the local number densities n are f ,g = 1 . f ,d = 1 .
85 for giants and dwarfs, respectively.Investigating the 100 pc sample yields an almost con-stant density for the giants. The best fit with an exponentialdensity profile yields a local density n = 1 . × − pc − and an exponential scale height of 500 pc (see lower panel ofFig. 3). The dwarfs of this sample are best fit by an exponen-tial profile with a local number density n = 4 . × − pc − and an exponential scale height of 173 pc. The correspond-ing conversion factors are f ,g = 1 .
07 and f ,d = 1 .
23 forgiants and dwarfs, respectively.Our faint star counts are based on the CNS5 and sufferfrom incompleteness, which we assume to depend on dis-tance only. We have assessed the completeness of the CNS5by carrying out radial cumulative star counts in each magni-tude bin (with a width of 1 mag). These are shown in Fig. 4.In a spatial homogeneous sample the cumulative number ofstars n grows with distance d as n ∝ d . In a double loga-rithmic log( n ) vs ( − log( π )) representation of Fig. 4 – where π is the stellar parallax – a homogeneous sample would ap-pear as a straight line with slope 3 (green lines).The completeness limit of the CNS5 in each magnitudebin can be seen by a deviation of the actual star counts fromthe line of slope 3. From Fig. 4 we read off the completenesslimits summarized in Table 2. The magnitude bins M K = 13 c (cid:13) , 000–000 he local stellar luminosity function and mass-to-light ratio in the NIR Figure 3.
Top panel: Cumulative distributions N z in | z − z (cid:12)| ofthe 2 660 giants (red) and 1 796 dwarfs (blue) in the 200 pc-sphere.Best fits of the vertical density profiles are shown in green andpink lines, respectively. For comparison the grey dotted line showsthe expectation for a constant density. The differences of modeland data ∆ N are also shown in the inset. The slopes at small z of giants and dwarfs determining the local number density differby a few per cent only, which is by chance due to the sampleselection in the 200 pc sphere. The lower panel shows the samefor the 100 pc sample with 541 giants and 1477 dwarfs. and 14 mag are dominated by white dwarfs. For magnitudesfainter than M K = 14 mag the completeness limit cannotbe reliably determined. In order to derive a lower limit forthe star number densities we have considered a volume witha radius of 10 pc. White dwarfs in the solar neighbourhoodare completely sampled out to a distance of d ∼
13 pc fromthe Sun (Holberg et al. 2008). The stellar number densitiesgiven in the next section have been determined within eachcompleteness limit and were then converted to a standard(spherical) volume with a radius of 20 pc.The luminosity function of the CNS5 stars alone is il-lustrated in Fig. 5. In order to demonstrate how the volumecompleteness of the CNS5 influences the predictions of theLF, we have split up the original volume of the CNS5 (witha radius of 25 pc) into spherical shells of 5 pc width. The lu-minosity functions derived from each shell are over-plottedonto each other in Fig. 5. As can be seen here all shells
Figure 4.
Radial cumulative star counts of CNS5 stars in mag-nitude bins (bin width 1 mag) ranging from M K = 4 mag andbrighter to M K = 10 mag (grey points). For clarity log(n) hasbeen shifted arbitrarily along the vertical axis for all but the M K = 7 mag sample which is plotted in black. The green linesindicate spatially homogeneous samples with the adopted localdensity. give within statistical errors consistent results up to an ab-solute magnitude of M K = 5 mag. Beyond that magnitudethe outer shells yield reduced star numbers, because theybecome increasingly incomplete. We constructed the local NIR luminosity function Φ( M K ) interms of star numbers in the 20 pc sphere V = 33 510 pc asdescribed in the previous section by combining the samplesof stars brighter than M K = 3 . Hip-parcos catalogue and fainter stars from the CNS5. We willnow discuss Φ( M K ), the luminosity budget, and the massfunctions that result. We show the results of our determination of the local lu-minosity function in Table 2. Given errors indicate the(usually dominating) Poisson errors. We first note howsmoothly the
Hipparcos and CNS5 samples join together inthe M K = 2 and 3 magnitude bins, see columns 3 and 5. The c (cid:13)000
Hipparcos and CNS5 samples join together inthe M K = 2 and 3 magnitude bins, see columns 3 and 5. The c (cid:13)000 , 000–000 A. Just, B. Fuchs, H. Jahreiß, C. Flynn, C. Dettbarn, J. Rybizki
Figure 5.
Illustration of the distance effect on the NIR LF. Thevolume of the CNS5 has been split into spherical shells of 5 pcwidth. The LFs derived from each shell are over–plotted in colourcoding onto each other.
Hipparcos sample is volume complete down to M K = 4 mag.However, binary stars have been treated more carefullyin the CNS5 than in the Hipparcos sample as evinced byslightly larger star numbers in these magnitude bins (cf. Ta-ble 2). In column 9 of Table 2 the luminosity function of thedwarfs alone (main sequence and turnoff stars) is shown.This was derived by excluding giants according to the di-viding line in the CMD as discussed earlier (cf. Fig. 1) andsimilarly white dwarfs in the bins M K =11–14 mag were re-moved.The total LF and the contributions of dwarfs and giantsare shown in the top panel of Fig. 6. Since the number ofgiants is very small, the giant LF is multiplied by a factor of100 to make it visible. The dwarf LF shows a clear maximumat M K = 7–8 mag beyond which it drops by nearly an orderof magnitude at M K = 13 mag. The faint end of the LF in theL, T dwarf regime is unreliable due to the large error barsand incompleteness. The shape of log (Φ( M K )) is roughlyconsistent with the LF observed in the optical bands as isillustrated in Fig. 6, bottom panel. Both the NIR and V bandmain-sequence LFs are shown together with the analyticalfit by Mamon & Soneira (1982) to the V band LF of Wielen(1974), and its theoretical transformation into the 2.2 µ mfilter band. As can be seen from Fig. 6 the analytic modelsfit very well in the F – K dwarf regime. At the very bright end( M K (cid:54) In the previous section we have given the luminosity func-tion Φ( M K ), i.e. the number of stars in absolute magnitudebins. In order to calculate the contribution of each bin to thetotal local luminosity in V , we now multiply Φ( M K ) withthe mean luminosity L K = 10 − . M K − M K (cid:12) ) L K (cid:12) of the bin,where M K (cid:12) denotes the absolute K -magnitude of the Sun M K (cid:12) = 3 .
27 mag (Casagrande et al. 2012). The result is il-lustrated in Fig. 7 and again the contributions of dwarfs andgiants are shown. The dwarf luminosity distribution peaks at M K = 2 mag corresponding to F type stars and is analogousto the peak at M V = 1 mag in the optical luminosity distri- Figure 6.
Top panel: Histogram of Φ( M K ) for all stars (grey his-togram), dwarf stars (blue) and giants (red). For clarity Φ( M K )of the giants has been enhanced by a factor of 100 in the redhistogram. The contribution of white dwarfs at M K ≈
13 mag isclearly visible. Bottom panel: Comparison of the main sequenceLFs in the V band (triangles) and in the NIR (pink circles). Theblue line is an analytical fit to the optical data and the pink onea theoretical transformation of it to the K band. Φ MS is given asnumber of stars in the 20pc sphere V . bution (Flynn et al. 2006). The contribution of the faint endwith M K > M K = − M K < − M K = − M V < ρ K = 0 . ± . L K (cid:12) pc − , now converted to the local luminosity den-sity. The uncertainty is dominated by the bright end of thegiants. The giants dominate with 0.0971 ± L K (cid:12) pc − (80 per cent) and the dwarfs contribute 20 per cent with0 . ± . L K (cid:12) pc − . A re–calculation of the V band lu-minosity of the same sample yields 0.053 L V (cid:12) pc − (slightlysmaller than the old value ρ V = 0 . L V (cid:12) pc − of Flynn etal. 2006). Combining the local K- and V band luminosityresults in ( V − K ) = 2 .
46 mag in the solar neighbourhood(with ( V − K ) (cid:12) = 1 .
56 mag). This colour is slightly bluer c (cid:13) , 000–000 he local stellar luminosity function and mass-to-light ratio in the NIR Table 2.
Stellar luminosity function Φ( M K ). M K Sp Φ
Hip limits Φ
CNS err Φ comb err Φ MS Mass[mag] [ ] (*) [pc] (*) (*) (*) (*) (*) [ M (cid:12) ](1) (2) (3) (4) (5) (6) (7) (8) (9) (10) −
10 M3,4 I–II 0.0024 0.0024 0.0017 − − − − − − − − − > > > > > > > > > > > > M K magnitude; col 2: spectral type (after Garwood & Jones (1987)); col 3: luminosity function determined fromthe Hipparcos sample; col 4: completeness limits of the CNS5 in abs. magnitude bins 4 to 10 mag, radius for upper limits at faintermagnitudes; col 5: luminosity function determined with the CNS5 stars; col 6: errors to col. 5; col 7: luminosity function combined from
Hipparcos and CNS5; col 8: errors to col. 7; col 9: resulting luminosity function for main sequence stars in the K band; col 10: mainsequence star masses.(*): The luminosity functions in cols. 3, 5, 7, and 9 as well as corresponding errors in cols. 6 and 8 are given in terms of stars per 20pcradius sphere V = 33 510 pc and per magnitude interval. than the value of ( V − K ) = 2 .
55 mag determined in modelA of Just & Jahreiss (2010) based on a Scalo IMF.
We next examine how the NIR luminosity function relates tothe mass distribution of the stars. For this purpose we havemultiplied the numbers of stars with mean masses of thestars in each M K magnitude bin. For dwarfs, the adoptedmasses are reproduced in the last column of Table 2. Forabsolute magnitudes 3.1 (cid:54) M K (cid:54) M K -mass relation by Henry& McCarthy (1993). For main sequence stars brighter than M K = 3 mag we have used the M V -mass relation compiledby Andersen (1991) which we have transformed to M K byapplying V − K main sequence colours. For red giants weassume a mass of 1.4 ± M (cid:12) as derived by Stello et al.(2008) from asteroseismic observations. Supergiants may be significantly more massive than that (Schmidt-Kaler 1982),but are so few that their mass densities are negligible. Theresulting mass distributions for giants and main sequencestars, which we have been able to estimate without recourseto population synthesis modeling (Bell & de Jong 2001; Zi-betti, Charlot & Rix 2009), are illustrated in Fig. 8 as afunction of absolute M K magnitude. The small contributionof giants is scaled up by a factor of 10 for visibility. In con-trast to the luminosity distribution (see Fig. 7) the localmass density is dominated by the lower main sequence with M K > . . M (cid:12) pc − , respectively. For the to-tal stellar mass density we need to add the contribution ofbrown dwarfs and white dwarfs. For brown dwarfs we add0.002 M (cid:12) pc − assuming a 50 per cent incompleteness inthe observational data of late M, T and L dwarfs. For the lo-cal mass density of white dwarfs we use 0.0032 ± M (cid:12) c (cid:13)000
We next examine how the NIR luminosity function relates tothe mass distribution of the stars. For this purpose we havemultiplied the numbers of stars with mean masses of thestars in each M K magnitude bin. For dwarfs, the adoptedmasses are reproduced in the last column of Table 2. Forabsolute magnitudes 3.1 (cid:54) M K (cid:54) M K -mass relation by Henry& McCarthy (1993). For main sequence stars brighter than M K = 3 mag we have used the M V -mass relation compiledby Andersen (1991) which we have transformed to M K byapplying V − K main sequence colours. For red giants weassume a mass of 1.4 ± M (cid:12) as derived by Stello et al.(2008) from asteroseismic observations. Supergiants may be significantly more massive than that (Schmidt-Kaler 1982),but are so few that their mass densities are negligible. Theresulting mass distributions for giants and main sequencestars, which we have been able to estimate without recourseto population synthesis modeling (Bell & de Jong 2001; Zi-betti, Charlot & Rix 2009), are illustrated in Fig. 8 as afunction of absolute M K magnitude. The small contributionof giants is scaled up by a factor of 10 for visibility. In con-trast to the luminosity distribution (see Fig. 7) the localmass density is dominated by the lower main sequence with M K > . . M (cid:12) pc − , respectively. For the to-tal stellar mass density we need to add the contribution ofbrown dwarfs and white dwarfs. For brown dwarfs we add0.002 M (cid:12) pc − assuming a 50 per cent incompleteness inthe observational data of late M, T and L dwarfs. For the lo-cal mass density of white dwarfs we use 0.0032 ± M (cid:12) c (cid:13)000 , 000–000 A. Just, B. Fuchs, H. Jahreiß, C. Flynn, C. Dettbarn, J. Rybizki
Figure 7.
Local luminosity distribution in the NIR (grey his-togram). The contribution by giants is shown with the red line,the blue line indicates the contribution by main sequence stars.
Figure 8.
Distribution of the mass density ρ for the stellar partof the Milky Way disc (grey histogram). Stellar mass densities aregiven as a function of absolute M K magnitudes (blue lines : mainsequence stars, red lines : giants). For clarity the mass densitiesof the giants have been enhanced by a factor of 10 in the redhistogram. pc − (Holberg et al. 2008), which is similar to our finding of0.0030 ± M (cid:12) pc − . This way, we find a total localmass density of ρ = 0 . M (cid:12) pc − for the stellar compo-nent. The total stellar density is slightly smaller than in ear-lier determinations (0.039, 0.044, 0.0415 M (cid:12) pc − ; Jahreiß& Wielen 1997; Holmberg & Flynn 2000; Flynn et al. 2006,respectively). In all three publications a larger contributionof white dwarfs was adopted. Additionally, in Holmberg &Flynn (2000) and Flynn et al. (2006) the mass of the upperMS ( M V < . M/L K =0 . ± . M (cid:12) /L K (cid:12) . This can be compared with the opticalmass-to-light ratio of 0 . M (cid:12) /L V (cid:12) , which is smaller thanthe value of 0 . M (cid:12) /L V (cid:12) derived in Flynn et al. (2006)mainly due to the smaller local mass density. The results presented so far reflect the NIR LF as well as theluminosity and mass distributions as functions of M K for theMilky Way in the solar neighbourhood. More representativefor the entire Milky Way disc is the local surface brightness,i.e. the local luminosity distribution multiplied by the ver-tical scale heights of the stars. The calculations of surfacebrightness and surface density depend on the adopted discmodel, because for most sub–populations there are no directobservations of the vertical density profile available. It wasdemonstrated in Flynn et al. (2006) that the vertical scaleheights h of the various stellar populations can be roughlyestimated by their vertical velocity dispersions σ , because ina given gravitational potential the vertical scale heights arein lowest order directly proportional to the latter. For an im-proved estimation of the surface brightness we use a higherorder fit of the (half-)thickness h ( σ ) defined by Σ = 2 h n connecting the local volume density n and the surface den-sity Σ of the tracer. For a non–exponential vertical profile ofthe tracer the thickness h differs from the exponential scaleheight. Each stellar subpopulation is approximated by anisothermal component in the total potential characterizedby the local density ρ = 0 . M (cid:12) pc − and effective scaleheight z = 280 pc leading to h ( σ ) = (cid:113) π · z ( σ ) + z ( σ ) z with z ( σ ) = σ √ πGρ . (1)The dotted line in the lower panel of Fig. 9 shows the ap-proximation for the thin disc. The parameters ρ and z are chosen to reproduce the values h ( σ ) of the detailed lo-cal disc model of Just & Jahreiss (2010) as well as the massmodel used in Holmberg & Flynn (2000); Flynn et al. (2006)(triangles in the lower panel of Fig. 9). The parameters areoptimized such that the fit function can also be used for thethick disc with velocity dispersion σ ≈
40 km s − .The upper panel of Fig. 9 shows the measured verticalvelocity dispersions in each M K bin and Hipparcos group forthe giants. At the bright end the velocity dispersions scatteraround a constant value of 17.3 km s − , whereas the faintgiants in the 40 pc group show a large scatter and a largermean value of 22.7 km s − . Since the outliers do not con-tribute much to the total luminosity, we will use the sameconstant σ g = 17 . − for all giants. For the upper mainsequence dwarfs in the Hipparcos samples we show the meanvalues in each M K bin (weighted by the number densities inthe 20 pc volume in order to avoid a bias due to the increas-ing age with increasing V − K ). To all lower main sequencestars we have assigned the mean velocity dispersion of G, Kdwarfs – falling in the M K = 5 mag bin – from Table 4 ofJahreiß & Wielen (1997) because the CNS5 is kinematicallybiased to high proper motion stars at lower magnitudes andwe expect – and assume here – the same kinematics for allthese stars. Their σ values are shown as open symbols inFig. 9. In this figure the blue line shows the analytic fit, us-ing a shifted error function, of σ ( M K ) for the dwarfs, whichwe use to convert the velocity dispersions to the correspond-ing thickness h .Additionally a correction for the thick disc contribu-tion is necessary. Adopting a standard isothermal old thickdisc with a velocity dispersion of 40 km s − corresponds toa thickness h = 1031 pc. In the solar neighbourhood we as- c (cid:13) , 000–000 he local stellar luminosity function and mass-to-light ratio in the NIR Figure 9.
Conversion of velocity dispersion to thickness in 2steps. Top: Observed vertical velocity dispersions σ for dwarfsand giants in each magnitude bin together with the analytic ap-proximations. For the meaning of open symbols, see text. Bottom:Vertical thickness h of the analytic fit used for the transformation(blue dotted line), for the best Just-Jahreiss-model (JJ model A,solid orange line), the values for the mass model used in Holm-berg & Flynn (2000) (HF2000, orange triangles) and in Flynnet al. (2006) (red dashed line and F2006 for giants, red circle).For comparison the linear extrapolation based on the total localdensity is also shown (green dashed line). sume a thick disc fraction of 10 per cent for all giants andfor the lower main sequence with M K > . i is given by Σ K,i = 2 h i L K,i .The resulting surface brightness distribution is shownin Fig. 10, where we find similar features as in the lo-cal luminosity distribution (Fig. 7). The distribution isstrongly dominated by the bright end of the giants, the redclump giants are visible in the M K = -2 mag bin, and thedwarfs peak at the F dwarfs. The integrated surface bright-ness is Σ K = 98 . L K (cid:12) pc − = 19 . − in total,composed by 16 . L K (cid:12) pc − (16 per cent) for dwarfs and82 . L K (cid:12) pc − (84 per cent) for giants. Antares alone hasadded 5.3 L K (cid:12) pc − demonstrating the uncertainty due toPoisson noise for the brightest supergiants.A similar determination of the V band surface bright-ness distribution shows a similar shape, but with a less dom-inant red giant contribution (cf. discussion above). We find29 . L V (cid:12) pc − = 22 . − . This value is 19 per centlarger than the value of 24 . L V (cid:12) pc − determined by Flynn Figure 10.
Surface brightness distribution in the NIR. The blueand red lines indicate the contributions by the main sequencestars and giants, respectively. et al. (2006) arising from inconsistencies in the earlier trans-formation to surface brightness as can be seen by comparingtheir Figures 2, 5 and 6. In the I band, which Flynn et al.(2006) have used to derive the location of the Milky Waywith respect to the Tully-Fisher (TF) relation, their Figures9 and 10 seem to be correct. Combining the K and V bandsurface brightnesses yields ( V − K ) = 2 .
89 mag for the solarcylinder.The corresponding stellar mass surface density derivedfrom the local mass distribution (Fig. 8) including whitedwarfs and brown dwarfs is 33.3 M (cid:12) pc − , which is slightlysmaller than the value 35.5 M (cid:12) pc − of Holmberg & Flynn(2004); Flynn et al. (2006). It implies a K band mass-to-light ratio of M/L K = 0 . M (cid:12) /L K (cid:12) . The correspondingoptical mass-to-light ratio is M/L V = 1 . M (cid:12) /L V (cid:12) . The surface brightness of the disc in the solar cylinder can beused to estimate the total K band luminosity of the MilkyWay and compare it with the observed Tully-Fisher (TF)relation of extragalactic systems. We proceed similar as inFlynn et al. (2006) for the I band. The total disc luminosityis approximately independent of the adopted radial scale-length h R in the range of 2.5–5 kpc. For definiteness weadopt h R = 3 . K = 98 . L K (cid:12) pc − at the solar radius of R = 8 kpc a total disc luminos-ity of 8 . × L K (cid:12) . Adding the bulge luminosity of1 . × L K (cid:12) (Drimmel & Spergel 2001; Portail et al. 2015)yields a total luminosity of M K = − . ±
20 km s − and an estimated un-certainty in the total luminosity of 0.2 mag is shown. Forcomparison we plotted two determinations of the TF rela-tion in the K band based on 2MASS data. Karachentsev etal. (2002) used edge-on galaxies to derive the isophotal TFrelation in K band using K = 20 mag arcsec − . We haveadapted their TF relation from Fig. 8 to a Hubble constantof H = 70 km s − Mpc − . For the Milky Way seen edge-on K is at R ≈
20 kpc including more than 98 per centof the total light. The Milky Way is 0.2 mag brighter than c (cid:13)000
20 kpc including more than 98 per centof the total light. The Milky Way is 0.2 mag brighter than c (cid:13)000 , 000–000 A. Just, B. Fuchs, H. Jahreiß, C. Flynn, C. Dettbarn, J. Rybizki
Figure 11.
The Milky Way (dot with uncertainty bars, MW)with respect to the TF relations of Karachentsev et al. (2002);Masters et al. (2014), respectively. Dotted lines show the 1-sigmascatter in the TFs. the TF relation. For an alternative TF relation we show inFig. 11 also the corrected equations of Masters et al. (2008,2014) applied to an Sbc type galaxy. In this case the MilkyWay is 0.2 mag fainter than the TF relation. The systematicdifferences of the two TFs arise mainly on the inclination de-pendence of the K isophotal luminosity (see also Said et al.2015) and the uncertainties in the extrapolation to the totalluminosity. Deeper K band observations may solve this issuein the future. For the Milky Way seen face-on the isophotalluminosity would be 0.4 mag fainter than seen edge-on. We have constructed a new sample of stars representativefor the solar neighbourhood. The data set is based on twosubsets separated by the intrinsic brightness of the stars.The bright part comprises stars drawn from the survey partof the
Hipparcos catalogue. The faint part M K > . K s ) available.For the Hipparcos stars we have selected volume com-plete samples in M V magnitude bins and corrected for thevertical density profile to determine the midplane density.These subsamples were then resampled in M K magnitudebins for M K (cid:54) . M K bin resulting in a reason-able estimate of the local number density as faint as the M K = 14 mag bin, which is well in the brown dwarf regime.For fainter stars we derived lower limits for the correspond-ing star numbers per magnitude bin. All star numbers havebeen converted to a fiducial spherical volume with a radiusof 20 pc (centered on the Sun) and constant density as ameasure of the local volume density. For a detailed analysiswe have separated the giants and white dwarfs from main se-quence stars including turnoff stars and brown dwarfs. Wehave then determined the NIR luminosity function of the stars in the Milky Way disc by direct star counts in our sam-ple. The K band luminosity function shows a strong maxi-mum at M K = 7 − M K = 13 −
14 mag the whitedwarfs dominate the star counts.The luminosity function has then been converted tothe luminosity distribution of the stars by multiplying thestar numbers with the typical luminosities of the stars ineach absolute magnitude bin. The resulting (midplane) lu-minosity distribution is strongly dominated by the verybright end of giants and supergiants. A secondary peakat M K ≈ M K = − ρ K = 0 . ± . L K (cid:12) pc − , where the giants dominatewith a contribution of 80 per cent. Combined with the Vband luminosity density of ρ V = 0 . L V (cid:12) pc − we find avalue of V − K = 2 .
46 mag for the colour in the solar neigh-bourhood.We have determined the mass distribution of the starsas probed by the NIR luminosity function in the same way.Quite contrary to the NIR light, the mass of the Milky Wayis dominated by K and M main sequence stars. We concludefrom this discussion that the mass–carrying population ofstars in galactic discs cannot be observed directly in theNIR on a star-by-star basis. The total mass density of thestellar component is ρ = 0 . M (cid:12) pc − , which is about 10percent smaller than earlier determinations due to reducedcontributions by white dwarfs and brown dwarfs. The localmass-to-light ratio is then M/L K = 0 . M (cid:12) /L K (cid:12) . Thecorresponding corrected value in the optical is M/L V =0 . M (cid:12) /L V (cid:12) .For a comparison to extragalactic systems it is impor-tant to determine the surface brightness of the disc. Wehave used a detailed vertical disc model to derive the ef-fective thickness of the stellar populations in the magni-tude bins and took into account a correction for the thickdisc contribution with a larger thickness. The resulting sur-face brightness function shows similar features as the lo-cal luminosity distribution. The total surface brightness is99 L K (cid:12) pc − = 5 . × − W m − with 84 per cent result-ing from giants. This value can be compared to the K bandsurface brightness of the disc determined from DIRBE dataafter removing all point sources yielding 68 L K (cid:12) pc − (Mel-chior, Combes & Gould 2007). The difference correspondsto the contribution of all supergiants with M K < − . M (cid:12) pc − and a mass-to-light ratio for the so-lar cylinder of M/L K = 0 . M (cid:12) /L K (cid:12) . The correspond-ing optical surface brightness and mass-to-light ratio is29 . L V (cid:12) pc − and M/L V = 1 . M (cid:12) /L V (cid:12) , respectively.With the redetermination of the surface brightness we havecorrected a bug in the earlier determination by Flynn et al.(2006), which happened particularly in the V band. An ex-trapolation of the local surface brightness to the whole MilkyWay yields a total K band luminosity of M K = − . c (cid:13) , 000–000 he local stellar luminosity function and mass-to-light ratio in the NIR strongly dominated by young stars compared to the popu-lation in the solar cylinder due to the much smaller scaleheights of the young populations. Nevertheless, the mass-to-light ratio in the K band is only 10 per cent larger in thesolar cylinder. The reason for the luminosity of the presentday giants – dominating the light in the K band – being arough measure of stellar mass, carried by F, G, and K starsof the lower main sequence, is the similarity of their age dis-tributions. The birth time distribution for the precursors ofthe giants – mainly F and G dwarfs – is spread over the ageof the disc and similar to that of the F, G, and K dwarfsstill on the main sequence. As a consequence, the dynamicalevolution is similar and produces comparable scale heights.Thus we conclude that the mass-to-light ratio does not varystrongly in disc populations with a long star formation his-tory and a calibration of the absolute value is provided bythe solar neighbourhood properties.The colours and mass-to-light ratios are consistent withthe stellar populations derived in the local disc model ofJust & Jahreiss (2010). Into & Portinari (2013) determinedmass-to-light ratios and colours for disc populations withexponentially declining star formation histories. Our val-ues are roughly consistent with these models for relativelyflat star formation histories. An extrapolation from the so-lar radius to the whole disc would shift the colour andmass-to-light ratios slightly dependent on the disc growthmodel. Our K band mass-to-light ratio of the solar cylin-der of 0 . M (cid:12) /L K (cid:12) is very close to the mean value of0 . M (cid:12) /L K (cid:12) of 30 disc galaxies derived by Martinsson etal. (2013). ACKNOWLEDGEMENTS
This work was supported by the Collaborative ResearchCentre SFB 881 ’The Milky Way System’ (subproject A6)of the German Research Foundation (DFG). This publica-tion makes use of data products from the Two Micron AllSky Survey, which is a joint project of the University of Mas-sachusetts and the Infrared Processing and Analysis Center,funded by the National Aeronautics and Space Administra-tion and the National Science Foundation. This research hasmade use of the SIMBAD and VIZIER databases, operatedat CDS, Strasbourg, France.We thank Laura Portinari for fruitful discussions ac-companying this project.
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