On the determination of age and mass functions of stars in young open star clusters from the analysis of their luminosity functions

1.1 MB·English

by  A. E. Piskunov

Mon.Not.R.Astron.Soc.000,1–17(2003) Printed2February2008 (MNLATEXstylefilev2.2)
On the determination of age, and mass functions of stars in
young open star clusters from the analysis of their
luminosity functions
A.E. Piskunov,1⋆ A.N. Belikov,1 N.V. Kharchenko,2 R. Sagar,3,4 A. Subramaniam4
4
0 1Institute of Astronomy of the Russian Acad. Sci., 48 Pyatnitskaya Str., Moscow 119017, Russia
0 2Main Astronomical Observatory, 27 Akademika Zabolotnogo Str., 03680 Kiev, Ukraine
2 3State Observatory, Manora Peak, Naini Tal – 263129, India
4Indian Institute of Astrophysics, Bangalore 560034, India
n
a
J

8 Accepted 2003......Received2003.....;inoriginalform2003.....
2

1 ABSTRACT
v BasedontheCCDobservationsofremoteyoungopenclustersNGC2383,NGC2384,
7 NGC 4103, NGC 4755, NGC 7510 and Hogg 15, we construct their observed lumi-
8
nosity functions (LFs). The observed LFs are corrected for field star contamination
5
determined with the help of galactic star count model. In the case of Hogg 15 and
1
NGC 2383we also consideredthe additional contaminationfrom neighboringclusters
0
NGC 4609 and NGC 2384 respectively. These corrections provided the realistic pat-
4
tern of cluster LF in the vicinity of the MS turn on point and at fainter magnitudes,
0
/ revealedthe so called H-feature arising due to transition of the Pre-MS phase to MS,
h
whichisdependentontheclusterage.ThetheoreticalLFswereconstructedrepresent-
p
ing a cluster population model with continuous star formation for a short time scale
-
o andapowerlawInitialMassFunction(IMF)andthesewerefittedtotheobservedLF.
r As a result we are able to determine for each cluster a set of parameters, describing
t
s cluster population (the age, duration of star formation, IMF slope, and percentage of
a fieldstar contamination).Itwas foundthat in spite of the non-monotonicbehaviorof
:
v observed LFs, cluster IMFs can be described as the power law functions with slopes
i similartoSalpeter’svalue.ThepresentMainSequenceturnonclusteragesareseveral
X
times lower than those derived from the fitting of theoretical isochrones to turn off
r region of the upper Main Sequences.
a
Key words: keyword1: specification1 – keyword2: specification2 – ... (see 4th issue
of each volume)
1 INTRODUCTION of cluster reddening, distance and age while the LFs were
transformed to stellar mass spectrum, which can be identi-
Youngclustersarenaturallaboratoriesforthestudyofvari-
fied with the IMF for young(age 6 100 Myr) star clusters.
ousissuesrelatedtostarformationprocesses,e.g.,theinitial
In these conventional studies of young star clusters, cluster
mass function (IMF), duration of star formation etc. In
age, LFs and MFs could not be determined accurately due
recentyears,some remoteyoungclustershavebeenstudied
to reasons discussed below.
using BVRI CCD observations (e.g., see Sagar & Griffiths
1991, Phelps & Janes 1993, Sagar & Cannon 1995, Firstly, for reliable separation of cluster members from
Sagar & Cannon 1997, Sagar & Griffiths 1998, Sagar et al. theforegroundandbackgroundfieldstarspresentinthedi-
2001, Subramaniam & Sagar 1999, Sanner et al. 2001, rection of distant (d > 1 kpc) clusters, accurate kinematic
Pandey et al. 2001, Piatti et al. 2002). These observations (propermotionand/orradialvelocity)measurementsarees-
were used for the construction of color-color diagrams, sential.Unfortunately,evenpresentdaymostaccurateHip-
color-magnitude diagrams (CMDs) and luminosity func- parcos proper motion data (available for brightest V < 13
tions(LFs).Thesediagramswereusedforthedetermination mag stars only) are not sufficient for this purpose. In such
circumstancesthefieldstarcontaminationisremovedstatis-
ticallybyobservingcomparisonareas(blankfields)adjacent
⋆ E-mail:[email protected];[email protected] totheclusterfield.Itshouldbenoted,however,thatinmost
[email protected];[email protected];[email protected] casesoftheconventionalgeneral-purposeCCDphotometry,
2 A.E.Piskunov et al.
whicharenotspeciallyintendedtoe.g.theIMFstudy,such pointinformation hiddenintheexistingCCDobservational
regions are not observed, and hence even statistical means data of 6 young open cluster using a new approach, which
of the data refining could not be used. This leads to a loss does not require observations of blank fields. It should be
of information suitable otherwise for the LFconstruction. noted, that these clusters were subject of a conventional
The other problem in the CMDs of young star clus- study, and many of them already have age and IMF deter-
teris theprecise determination of theMain Sequence(MS) mination.Unlikethesestudies,wewillnotconvertobserved
turn off point, which is used for age determination, since LF to the IMF, but in order to avoid the above mentioned
for young clusters due to their small age, the upper MS is problems, we will construct theoretical LF, which should
practically unevolved. Consequently, it becomes difficult to properly reflect the behavior of the observed ones in the
evaluatetheaccurateclusterage,andinsteadanupperesti- vicinity of the MS turn on point. Wherever the field star
mateof theclusterage is provided.The difficultyis further contaminations could not be determined using theobserva-
increasedbyeffectslikestellarrotationandbinarity,andby tionsofanoffsetfieldregion,theyareestimatedusinggalac-
themorphology oftheupperMS,whichispractically verti- ticmodelfor starcountsfor thesurroundingsofthecluster
cal for early type stars in the colours (B−V), (V −R), or under study. Then the theoretical LF will be fitted to the
(V −I). observations by varying the star formation parameters de-
Theseproblemscanbeavoidediftheclusterageisesti- scribingactualstellarpopulation(clusterage,starformation
matedusingthelocation oftheMSturnonpoint,theplace duration, IMF slope, a percentage of observed field stars).
where Pre-MS stars join theMS in thecluster CMD. For a Consideringitssuccess,wehopethatthisapproachcanalso
young coeval cluster this point resides about 7 mag fainter be implemented to the other young open clusters having
the brightest termination point of the MS. As a result of similar or more deep CCD data, which become presently
the deep CCD observations of the clusters, this region can available (see e.g. Kalirai et al. 2001).
be seen in the CMD, and hence is available for the anal- In Section 2 we detail theoretical approach to LF of a
ysis. Many of the above effects (except that of unresolved young cluster and consider its fine structure in the context
binaries)aremuchweakerintheturn-onpointregion,which ofcurrentstudy.IntheSec.3,wesummarizethedataused
makestheturn-ondatingmethodtobemoreattractivethan in the present work. Sec. 4 describes construction of the
the turn off one. There are, however, practical problems in observed and theoretical LFs. We consider here the major
locating the turn-on point e.g., presence of strong field star effects which should be taken into consideration, define the
contaminationhidesitmorestronglythanthebrightestMS. modelofclusterpopulationanditsparameters,anddescribe
The other limitation of the turn-on method is cluster age. thefittingprocedure.InSec.5wediscussthederivedresults,
As shown by evolutionary calculations, only for extremely and summarize them in Sec.6.
youngclusters with ages of orderof a few Myr,thePre-MS
branch is raised sufficiently above the ZAMS and could be
easilyrecognizedintheCMD.ForolderclustersthePre-MS
2 THEORETICAL APPROACH
branchdeviationfromtheZAMSdiminisheswithincreasing
age and could be seen better as a detail of the LF (as it is Since stellar mass m can be directly measured in very rare
discussed in Sec. 2). At cluster ages exceeding that of the and specific cases, the IMF is not observed directly, but is
Pleiades, the Pre-MS branch could not be identified with converted from the observed distribution of stars overtheir
confidencebothintheCMDandintheLFandtheturnon absolutemagnitudesdN/dM,calledLFφ(M).Foracluster
method cannot be applied (Belikov et al. 1998). of age t (in years throughout the paper if not mentioned
The other related issue is thefinestructure of thestel- otherwise), the LF φt(M) is related to the IMF f(m) via
lar LF, located in the vicinity of the MS turn on point, time-dependentmass-luminosity relation (MLR) mt(M) as
which can mask the IMF shape if one does not take it
dN dlogm dlogm
into account. This feature can produce a false flattening or φt(M)= × =f(mt[M])× .(1)
evendepletionintheLF,whicharefrequentlyconsideredas dlogm (cid:12) dM (cid:12)t (cid:12) dM (cid:12)t
(cid:12) (cid:12) (cid:12) (cid:12)
an evidence of the flattening/turning over of the IMF (see Sinceclustersta(cid:12)rsevolve(cid:12)offorapproacht(cid:12)otheMS(cid:12)with
Piskunov & Belikov1996forreferences).Thisinturnmight a rate which is dependent on their mass, apart from their
haveanimportantconsequencewithrespecttotheIMFuni- stay on the MS, the cluster MLR should also evolve with
versality and othersimilar issues. Sinceluminosity function time, being different at every moment from the standard
of theturn on point dependson cluster age, theposition of one(usuallyadopted tobetheMLRof theMS stars).This
the LF detail also varies with time, and may be used as a deviationisespeciallystrongforredgiantandPre-MSstars.
kindofastandardcandleforagedeterminationfromtheLF Besides,the”instant”MLRhasadefinitefinestructuredue
analysis (Belikov & Piskunov 1997). to presence of quasi-horizontal Post-MS and Pre-MS por-
Thus, theMS turnon region of youngcluster CMDs is tions of corresponding isochrone. The structureitself is not
very important for cluster dating, or should be taken into too prominent to influence the LF, but it produces bumps
accountwhenoneisanalyzingtheLF.Itcanbeeasilyiden- and dips in the MLR derivative, and hence in the LF. The
tified, however, only in the case of a few selected clusters. strongest fluctuation of the derivative occurs near the MS
The well known examples of such clusters are NGC 2264, turnoffpoint,butduetothesmallnumberofstarsobserved
NGC 6530, ONC, and some others. Normally young and theretheycould not beeasily distinguished from statistical
especially remote clusters, which are buried in rich fore- fluctuationsoftheLF.Incontrast,inthevicinityofMSturn
ground/backgroundshow,neitherdistinctturnonsnorPMS on point, a sufficient number of stars coupled with strong
branches. MLR derivative bumping produce a feature, which can be
The aim of present study is to reveal the MS turn on observed in LFs of star clusters, which are young enough
Determination of age, luminosity and mass functions of young stars 3
Figure1.Theoreticalcolor-magnitudediagram(Panela)andluminosityfunction(Panelb)ofayoungopencluster.Solidcurvesarefor
anisochroneoflogt=7.0,includingbothPre-MSandPost-MSevolutionarystages,andcorrespondingluminosityfunction,constructed
from the isochrone. The dotted curves are the ZAMS and ”Main Sequence” luminosity function. Thin curves in panel (a) are Pre-MS
evolutionary tracks of Palla&Stahler (1993). The crosses and adjacent figures separate different evolutionary stages as it is explained
inthetext.TheenlargedMSturnonpointareaisshowninthepanelencapsulated intheCMD.
to display a branch of pre-MS stars. This detail was called Post-MS stages of stellar evolution. We call hereafter for
byPiskunov & Belikov(1996)asH-feature,asitappearsat convenience purposes segment (1,2) as the LF H-dip, and
thelocation,wherethehydrogenburningstartsinthecores segment(2,3)asLFH-maximum.Forcomparison inFig.1,
of the pre-MS stars. weshowpositionofZAMSmodelsofTout et al.(1996),and
ZAMS-LFconstructed from these models. Good agreement
InordertogetanideaongeneralstructureoftheLFsof
betweenZAMS-andLFsisobservedbetweenbothturn-over
young star clusters and their relation to main evolutionary
points, but beyondMS both LFs differ considerably.
stagesofstars,weshowinFig.1theCMDtogetherwithan
isochrone of logt = 7.0, and corresponding LF constructed Since the absolute magnitude of a cluster MS turn on
according to eq. (1) with help of Salpeter IMF (f(m) ∝ point evolves with time, the H-feature should move faint
m−1.35).Theisochronewasconstructedfromasetofstellar ward with increasing of cluster age. The feature not only
modelsdescribedinSec.4.4.Ourcalculationsshow,thatin moves to fainter magnitudes, but also degrades with time.
general this behavior of the LF does not differ for various As calculations of Belikov & Piskunov (1997) show it dis-
setsofmodels,butdetails(thesizeandamplitudeoftheLF appears completely for logt>8.2. This puts a limit on the
fragments) may besomewhat different. H-calibrationapplicability.AgainthetimescaleofH-feature
existence depends on specific models, and can be used for
Asonecan see from Fig. 1for youngclusters theLFis
Pre-MS model verification. In principle this makes the LF
non monotonic even for monotonic IMF, and consists of a
of youngclusters a good tool for cluster age determination.
numberofmonotonicportionsrelatedtodifferentevolution-
arystages.Comparisonwithevolutionarytracksshowsthat An immediate conclusion, which could be drawn from
segment(0,1)oftheLFcorrespondstoconvectiveportionof the above analysis is that the existence of any bumps/gaps
aPre-MStrack,segment(1,2)toradiativeone,(2,3)canbe intheLFatorneartheMSturnonpointofyoungclusters
identifiedwith MS turnon point intheCMD, andsegment should not be regarded as an evidence of non-monotonous
(3,4) belongs to the Main Sequence. Dueto non-monotonic behavior of the IMF. The existence of the Pre-MS detail
behavior of the isochrone near MS turn on point, the seg- in the LF may also result in the mis-interpretation of mass
ment (2,3) is in fact a superposition of MS and Pre-MS spectra of young open clusters. When the limiting magni-
stellarevolutionary stagesandpureMSstartsatsomewhat tudeofasurveyfalls within theradiativediponecould ob-
brightermagnitudes.The last segment (4,5) corresponds to servetheLFturn-over,whichcanbewronglyinterpretedas
4 A.E.Piskunov et al.
Table 1.Specificationofstudiedclustersandusedobservationdata
Cluster IAU d′ Area Number Vlim Vcmp V −MV E(B−V) logt Refe-
number (⊔⊓′) ofstars (mag) (mag) (mag) (mag) rence
NGC2383 C0722–208 5 31.8 588 21.8 17.6 13.3 0.22 8.45–8.6 [1]
NGC2384 C0722–209 5 27.8 256 20.9 18.9 13.2 0.22–0.28 7.1–7.3 [1]
NGC4103 C1204–609 5 19.4 176 19.8 17.3 12.5 0.31 7.5 [2]
Hogg15 C1240–628 2 6.9 337 21.7 19.6 16.5 1.15∗ 6.78 [3]
NGC4755 C1250–600 10 38.8 576 20.2 18.1 12.9 0.41∗ 7.0 [4]
NGC7510 C2309+603 6 38.4 423 21.1 16.8 16.0 1.12∗ <7.0 [5]
1Subramaniam&Sagar(1999),2Sagar&Cannon(1997),3Sagar,Munari&deBoer(2001),
4Sagar&Cannon(1995),5Sagar&Griffiths(1991)
∗ nonuniformextinction
a consequence of data incompleteness or theIMF turnover. Unfortunatelysuchagoodagreementisnotobservedin
Similarly the steep slope of convective portion of the LF is thecaseofclusterdistances.ComparisonwithLyng˚a(1987)
not an evidence of theIMF steepening. referred distance estimations shows good agreement for all
clusters, except NGC 2383 and NGC 2384, where data of
Table1areoverestimatedbyabout1mag.Thenewscaleof
LGM2.2 indicates even stronger (up to 1.5 mag) disagree-
3 OBSERVATIONAL DATA
ment for these and two more (Hogg 15, and NGC 7510)
We have used CCD photometric data of young (age 6 sev- clusters. Though these disagreements are unpleasant, it is
eral tensof Myr) remote and compact clusters presentedin hardlyavoidableasphotometricdistancesofremoteclusters
papers listed in Table 1 (hereafter referred as original pa- arederivedfromaverysteepupperMS.Inpresentstudywe
pers). The small areas occupied by these clusters and their willuseoriginaldistances,keepingaboveproblemsinmind.
sufficiently deep photometry provide high degree of data The possible uncertainty in cluster age and the MF
completeness, which is very important for LF construction. slope, which can be introduced by the data inaccuracy can
However,theirlargedistancecoupledwithlackofkinematic be estimated from the following considerations. The most
data prevent the separation of cluster members from the importantforourLF-basedanalysisisaccuracy inabsolute
fieldstarsandrequiresspecialapproachtotacklethisissue. magnitude MV. It dependson error in a distance modulus,
In Table 1 the parameters of the clusters under study reddeningcorrection, and on acorrection for metallicity ef-
are listed. Clusterdesignations areshown in columns1 and fect. Since no metallicity measurements are published for
2, in column 3 we list values of cluster angular diameters clusters under study we can estimate this correction only
taken from Lyng˚a (1987). The most of the other parame- indirectly.As the cluster galactocentric distances span over
ters are taken from the original papers mentioned in the arangeof7–11 kpc(withthesolar valueassumedtobe8.5
last column. Columns 4 through 9 contain area covered by kpc), the metallicity radial gradient in the galactic disk is
CCD frames, number of stars within the frames, limiting absent according to data on Cepheids of Andrievskyet al.
and completeness V-magnitudes, apparent distance moduli (2002). Thuswe can accept solar heavyelement abundance
andreddeningvalues.Colorexcesses,markedwithasterisks and avoid metallicity corrections. The spread of cluster
representaveragevaluesinvariableextinctionfields.Incol- metallicity can only be be due to galactic disk inhomo-
umn 10 we show log(age) as it was determined in referred geneity, which according to Vereschagin & Piskunov (1992)
studies.Wedefinelimitingmagnitudesofasurveyasfaintest is about ∆[Fe/H] ≈ 0.1. The corresponding absolute mag-
magnitudeofastarinclustersample.Datacompletenessin nitude uncertainty due to metallicity effect is of order of
magnitude are estimated following the procedure described ∆MV ≈0.1mag.Thestandardaccuracyofclusterapparent
in Sec.4.3. distance modulus is about 0.2 mag (Subramaniam & Sagar
WecomparedoriginaldataofTable1withLyng˚a(1987) 1999). The reddening uncertainty according to original pa-
catalogueandrecentlistofreddeningvalues,distancemod- pers is between 0.03 and 0.06 mag and only for NGC 7510
uli and ages, provided by Loktin1 (private communication, it exceeds (σ = 0.09 mag) the upper limit. These
E(B−V)
referred hereafter as LGM2.2). We find that the extinction cause the MV uncertainty to be typically about 0.1 – 0.2
valuesfromTable1areinexcellentagreementwiththedata mag and less than 0.3 mag for the complete sample. Thus
ofbothcataloguesandtheaveragecolorexcessesfromdiffer- we believe that aggregated uncertainty in absolute magni-
entlistsagreewithinafewhundredthofmagnitude,except tude due to metallicity, interstellar extinction and distance
inthecaseofNGC7510.Inthiscase,theextinctiondataas modulus errors does not exceed 0.5 mag. This corresponds
inLGM2.2differfromthatofTable1by0.26mag.Original toa random error in logt of theorder of 0.15 both for turn
cluster ages also show reasonable agreement (with average off and turn on dating methods. According to simulations
spread of the order of a few tenths in logt) for all clusters of Belikov (1999) one can expect an error in the MF slope
except NGC 2383, which according to both Lyng˚a (1987) caused byσMV =0.5 mag is of order of 0.2.
and LGM2.2 has an age of a few tens of Myr.
Theoriginal CCD photometrywascomplemented with
photoelectric magnitudes of brighter stars from the sources
1 LoktinA.V.,GerasimenkoT.P.,MalyshevaL.K.,2001,Homo- referred in original papers and/or from the WEBDA
geneous CatalogueofOpenClusterParameters,Release2.2. See database.Thisextendstherangeofdatacompletenessfrom
alsoWEBDAdatabaseatURLhttp://obswww.unige.ch/webda/ a faint cluster-specific limit, discussed in Sec. 4.3 to the
Determination of age, luminosity and mass functions of young stars 5
Figure 2. Maps of stellar fields in the vicinity of studied clusters are constructed with the help of ASCC-2.5, UCAC1, and FONAC
catalogues. Size of the stellar images corresponds to stellar magnitude (RFONAC for the case of NGC 7510 and Ru for the others),
magnitude scale is shown in the upper right corner. Crosses mark cluster centers. Large circles denote core and corona areas, solid
quadranglesshowusedCCD-frames.
brightest in the observed field stars. Since the data in the (Kislyuk et al. 1999) in NGC 7510 case] for fainter stars.
V-magnitude are the most complete we use in this study In order to give a reader an idea on a relation of observed
LFs constructed in V only. framestoclustergeometryweshowinthemaps,positionsof
As one can see from Table 1, stellar fields cov- cluster centers and borders of cluster cores and coronae, as
ered by the frames are comparable with the cluster determined by Kharchenkoet al. (2003) from data on spa-
sizes provided in Lyng˚a (1987) catalogue. It should be tialdistributionofclusterpropermotionmembersfrom the
noted however, that Lyng˚a (1987) provides only sizes ASCC-2.5.ForNGC2383Kharchenkoet al.(2003)failedto
of central parts of open clusters, which according to detectthecoreradiusandthecoreisnotshownintheclus-
Kharchenko,Pakulyak & Piskunov (2003) are 2 to 3 times ter map. Since brightest stars of Hogg 15 are fainter than
lowerthantheirfullsize.Similarconclusionshavealsobeen limiting magnitude of the ASCC-2.5, the data on the clus-
drawn by Nilakshi et al. (2002) based on the spatial struc- tercenterandradiusweretakenfromDias, L´epine & Alessi
ture study of the open star clusters. Thus we should con- (2002) and Moffat (1974) respectively.
sider our samples only as inner population of the clusters,
enhanced with massive stars (Kharchenkoet al. 2003), and
having flatter luminosity functions and mass spectra than
outer population (deGrijs et al. 2002a,b,c).
InFig.2weshowwideneighborhoodsoftheclustersun-
derstudy.Theyareconstructedwiththehelpofall-skycat-
One can see from Fig. 2 that regions of two clusters
aloguesASCC-2.52(Kharchenko2001)forbrightest(V <13
(NGC 2383 and Hogg 15) are overlapped with their neigh-
mag) stars, and UCAC13 (Zacharias et al. 2000) [FONAC4
boring clusters. This should be regarded as a potential
source of cluster sample contamination and should be kept
2 availableatftp://cdsarc.u-strasbg.fr/pub/cats/I/280A inmindinfurtherdiscussions.NGC4755istheonlycluster,
3 availableatftp://cdsarc.u-strasbg.fr/pub/cats/I/268 where observations are carried out in two separated frames
4 availableatftp://cdsarc.u-strasbg.fr/pub/cats/I/261 located on both sides of thecluster center.
6 A.E.Piskunov et al.
4 OBSERVED AND THEORETICAL
Table 2.Cluster-specificmodelparameters
LUMINOSITY FUNCTIONS
Cluster l◦ b◦ aV
Before comparing the empirical and theoretical LFs, it is (mag/kpc)
necessary to correct observed LF for different biases men- Hogg15 302.0 -0.2 1.00
tioned earlier. Among the most serious effects influencing NGC2383 235.3 -2.4 0.38
observed LFs are contamination of a cluster sample with NGC2384 235.4 -2.4 0.41
non-members and data incompleteness. Generally the non- NGC4103 297.6 +1.2 0.59
NGC4755 303.2 +2.5 0.92
membersarefield stars projected ontotheclusterarea, but
NGC7510 111.0 +0.1 1.13
insomecasestherewouldbestarsfromaneighboringcluster
also. The data incompleteness has a strong dependence on
thecrowding/densityofstarsinthefieldandthestellarmag- isdescribedbyitsownstellarLFandspatialdistribution.It
nitudesuchthatthedataislesscompleteincrowdedregions isacceptedthatpopulationLFdoesnotdependonposition
and towards fainter magnitudes (see e.g. Sagar & Richtler within the Galaxy and is equal to its solar neighborhood
1991).Allthese, as well as details of procedure for thecon- counterpart. Density distributions of different populations
structionoftheoreticalLFandfittingoftheoreticalandem- are represented by Bahcall (1986) expressions for disk-like
pirical LFs will bedescribed below. and spheroidal subsystems.
The LF for MS stars brighter than MV = 13 mag is
taken from Scalo (1986) and Murray et al. (1997), while
4.1 Field star de-contamination with galactic star for fainter stars (MV 619 mag) we used data given
count model by Jahreiß & Wielen (1997) for nearby stars. The frac-
tion of evolved stars, which have left MS phase (red gi-
Depending on cluster distance and on the properties of ob- ants and super giants) as a function of MV was taken
served sample, one could apply different methods for the from Scalo (1986). These values were modified at MV ≈
selection of cluster members. For nearby clusters, the kine- 0.75 mag, (B − V) ≈ 1.0 mag according to the data of
maticmethodisthemost common andreliable. Forremote Holmberg, Flinn & Lindengren (1997) taking into account
clusters the statistical selection with the help of adjacent red giant clump, an equivalent of Population II horizon-
stellarfieldsareused.Sincethereisnoblankfieldsobserved tal branch for disk-population core helium burning stars.
in the neighborhood of the most clusters under study, we The LFs of the thick disk and spheroid and the star num-
usetheGalacticstarcountmodelstoestimatethefieldstar berdensitynormalization{thindisk:thickdisk:spheroid}are
contamination in their directions. Due to various perturba- {1:0.02:0.00125} taken from Gilmore (1984). Additionally,
tions(e.g.duetospiral arms) alonglineof sight,whichcan resultsbyDaCosta(1982)forglobularclusterswereusedto
be hardly taken into account in details by the model, this modeltheinfluenceofthehorizontalbranchonthespheroid
approach appears to be crude at the first glance, especially LF.
for a study of the fine structure of the LF. A careful anal- We adopt the following general parameters of the
ysis indicates contrary to this apprehension however. It is model. The galacto-centric distance of the Sun is equal to
well known that the leading role among a number effects 8.5 kpc; theratio of axes of thespatial density distribution
influencing the predicted apparent stellar density, is played forthespheroidisequalto0.85;thelengthscaleofthedisk
by interstellar extinction, as it is uneven over the galactic subsystems and the height scale of the thick disk are equal
disk. For example the patchy behavior of the extinction is to 4 kpc and 1.3 kpc respectively. We assume according to
themajor reason enabling observation of remote clustersin Schmidt(1963)andScalo(1986)thattheheightscaleofthe
transparencywindows.Thisisabasicpointofourapproach: disk for MS stars rises from 90 pcto350 pcat MV interval
since the extinction in a cluster direction could be derived [2.3,5.4]mag,whereasforredgiants,itchangesfrom250pc
with much more confidence when compared to an average to 400 pc at MV =[−0.75,2.6] mag. The height scale of 90
extinctionvalueinthedisk,morerealisticmodelpredictions pcwasusedforextremelyyoungpopulationofsupergiants.
can be made in a cluster field than in an arbitrary stellar Youngopenclustersarelocatedinthegalacticdiskwith
field. Below wedescribe the model components, discuss the irregularinterstellarabsorptionandtheirextinctionparam-
degreeofthemodelstabilityagainsttheextinctionandstel- etersdifferfromaverageGalacticvalues.Thisisamajorrea-
lardensityfluctuations,andcomparethemodelpredictions sonforusinginmodelcalculationsacluster-specificabsorp-
with data on selected test fields. tion values aV, which were derived from Parenago (1940)
For each cluster, we compute a theoretical distribu- formula
tion of field stars with apparent magnitude ψ(V), which
is specific to the cluster parameters, including galactic co- AV = aV ·hZ 1−exp −d·|sinb| ,
ordinates, interstellar extinction towards the cluster, and |sinb| (cid:20) (cid:26) hZ (cid:27)(cid:21)
normalized to the area of observed cluster region. We use
wheredisclusterdistancefromtheSun,bisclustergalactic
themodeldescribedbyKharchenko& Schilbach(1996),and
latitude,AV =3.1E(B−V)withE(B−V)takenfromTa-
Kharchenkoet al. (1997). The model treats the Galaxy as
ble1,andhZ isheightscaleoftheextinctionlayerassumed
a system, which is symmetric both with respect to rota- to be equal to 100 pc5. Values of aV together with cluster
tion axis, and to the equatorial plane, and consists of three
galactic coordinates are shown in Table 2.
populations (thin disk, the disk hereafter, thick disk and a
spheroid). The population of thin disk consists of several
sub-populations(MSstars,diskredgiants,andsupergiants 5 See paper of Kilpio&Malkov (1997) for recent discussion of
–starsofIandIIluminosityclasses).Eachpopulationgroup theextinction formulaparameters.
Determination of age, luminosity and mass functions of young stars 7
calculations show that for all clusters in the working mag-
nitude range, the influence of the spiral arms on the stellar
density is negligible. The corresponding increase of ∆logN
is about 0.05 for γ = 0.3, and less than 0.02 for γ = 0.1.
These values indicate that the effect of density fluctuation
is much weaker than theextinction.
At last we compare our model calculations with ob-
servedstellar countsinselected stellarfields.Unfortunately
among the clusters of our sample only observations of
NGC2383/4aresuppliedwithphotometryintheblankfield
containing about 300 stars. The field is about 5′away from
theclustersandthusispartlyoverlappedwithNGC2383/4
areas (cf. Fig. 2). For model calculations we used data for
this cluster from Table 2. Corresponding distributions are
compared in Fig. 3(a). One can observe reasonable quali-
tative agreement of empirical and model distributions. Ac-
cordingtoχ2-testthebothdistributionsagreeat80-percent
significancelevel.Smallandgenerallyinsignificantaccessof
brighter stars (V < 16 mag) can be attributed to the ad-
mixtureof NGC 2384 stars.
Anexcellentopportunityforverifyingofourmodelpro-
videswide-field CCD-photometryof morethan17000stars
inthefieldofNGC7654,publishedbyPandey et al.(2001).
Thephotometry covers an area of 40×40 arcmin2, centered
at the cluster with a size of about 24′. This leaves a wide
circular blank field area, enabling not only to confront the
model counts and observations, but also to estimate the
Figure 3. A comparison of stellar counts in test fields, neigh- stochastic variations of circumcluster fields themselves. It
boring to NGC 2383/4 (a), and NGC 7654 (b) clusters, with a is worth to note, that NGC 7654 is located only about 1
predictionofourmodel.Thehistogramsshowempiricaldata,the degree North of NGC 7510, a cluster of present sample.
barsarethePoissondeviations,indicatingdatauncertainty.The In Fig. 3(b) we show observed and model distribu-
curvesarethemodelcounts.
tionsforNGC7654blankfieldarea.AlthoughPandey et al.
(2001) do not provide explicit value of the cluster radius,
their density profiles show that there is no sufficient clus-
In order to illustrate the decisive role of the reddening
ter population at r > 13.′3. For construction of the ob-
effect inthemodelcountsconstruction, wehaveconfronted 0
served distribution we use all stars located at r > r , but
ourresults to calculations with a standard valueof galactic 0
specificextinctionaG =1.6mag/kpc,adoptedforthegalac- thestarsfromtheSEsegmentofthecircumclusterareaare
V
not considered. This is where the neighboring cluster Cz-
ticdisk.Asshown bythecomparison, forall clustersin the
ernik43havingradiusofaboutof10′islocated,accordingto
working magnitude range of V = 10−20 mag, the overes-
Kharchenko& Piskunov (2003). The cluster-specific model
timation of specific extinction leads to considerable under-
estimation of the model density. The corresponding model parameters (d = 1380 pc, and aV = 1.80 mag/kpc) were
takenfromdataforNGC7654ofPandey et al.(2001).Since
curves differ in working magnitude range for different clus-
thereare indications of outward increase of thecluster red-
tersby∆logN =0.3–0.8(i.e.by70to180%).Themodel
dening, we used the upper limit provided by Pandey et al.
stability with respect of observederrors inthevalueof spe-
(2001) values. As one can see from Fig. 3(b) the agreement
cific extinction was evaluated by variation of values of aV
between the model and observations is good. According to
fromTable2by0.2mag/kpc.AccordingtoSec.3thisvalue
χ2-test the both distributions agree at 90-percent signifi-
canberegardedasupperlimitoferrorinspecificextinction
cance level in the range V =12–19.5 mag. In order to es-
values in our sample. We have found that in this case the
timate the degree of stochastic variations of circumcluster
model counts variations are of order of |∆logN|<0.15 for
population we arbitrarily dividedthewhole blankfield into
all clusters understudy.
twoequalpartsandcompared themtotheunitedfield sep-
In order to estimate the effect of spatial density fluc-
arately.Theχ2-testhasshown thattheirstatistics is some-
tuations, we consider galactic spiral arms as the most
what different: they are the same as the total distribution
pronounced galactic disk sub-structures. We have carried
on 85-percent level. This indicates that the model counts
out model calculations for cluster areas with and without
approachtothememberselectionisatleastnotworsethan
the effects due to spiral arms. We considered three local
the blank-field method based on an arbitrary sampled test
arms (Sagittarius-Carina, Orion and Perseus ones) repre-
area.
sented by logarithmic spirals with parameters taken from
Marochnik & Suchkov(1984).Itwasassumed,thattheden- Theoreticaldifferentialdistributionofstarswithappar-
sityinputfromthespiralstructureisequaltoγD(R),where ent magnitude ψ(V), derived with help of this model were
D(R)isregulardensityprofileinthedisk,andγ isascaling used for construction of reduced LFs as it is described in
factor, taken to be equal to 0.1, 0.2, and 0.3. The model Sec. 4.5
8 A.E.Piskunov et al.
4.3 Incompleteness effects
Table 3. Contributions of field star and neighboring clusters
population are compared with the projected stellar densities of
It is well known, that completeness of a photometric sur-
NGC2383andHogg15clusters
vey is achieved only at about 2 mag brighter than V (so
lim
Population Projecteddensity(1/⊔⊓◦) Designation called completeness limit Vcmp). Below Vcmp the observed
NGC2383 Hogg15 LFdoesnotreflectbehaviorofagenuineLF,andthismag-
nitude range should be excluded from the consideration. It
Observed(total) 2724 6077 Σcluster is also important to be confident, that for clusters, having
Fieldstars 932 1870 Σf observations in two CCD frames both fields have the same
NNeeiigghhbboorr+field 1237431 2450311 Σσnneeiigghhbboouurr completeness limit. Unfortunately, no Vcmp determinations
were made in original papers. This forced us to determine
Cluster 1451 3676 σcluster
the Vcmp within this study. Since no test fields were ob-
servedinoriginalpapersweareonlyabletoestimateroughly
4.2 Contamination by an overlapping cluster
the completeness limit as a position of apparent maximum
Thecontaminationbyanoverlappingclusterisstudiedusing in the distribution of stars with V magnitude within ob-
the projected density derived from stellar counts in recent servedframes.Sincetheincompletenessincreasesgradually,
all sky catalogues. The star counts in the USNO-A26 cata- the above estimate should be considered as upper limit of
logue is strongly affected by crowding effect in the areas of Vcmp.
Hogg15andNGC2383/2384 clusters(whichisnotsurpris- TheestimatedvaluesofVcmp areshowninTable1.For
ing for Schmidt camera based surveys in dense fields). To NGC 4755 we find good agreement between Vcmp in both
avoid crowding effect we used astrograph based catalogue observed frames. Note, however, that the frames do not in-
UCAC1. This catalogue is a part of all-sky survey UCAC, cludethecentralareaofthecluster,containingthebrightest
covers Southern Hemisphere south to δ ≈ −20◦, and clustermembers.Thiscausesanotherkindofincompleteness
2000
contains thefields understudy. i.e. incompleteness for bright stars. Oneshould keep thisin
NGC 2383/2384 lie at the very edge of the survey mind when the final result for this cluster is discussed. In
UCAC1area,whereonlyshortexposuresprovidestarcounts contrast the frames of other clusters are located in central
homogeneity and completeness at Ru <13.7 mag. We con- regionsandnobrightstarincompletenessisexpectedthere.
structeddensityprofilesinafieldcenteredattheNGC2384. Theotherkindofincompleteness,whichshouldbecon-
In order to estimate the proper stellar density in a cluster, sideredisincompletenessduetooverlappingofstellarimages
we counted stars in four quadrants separately. The average in the CCD frame, which we call here as ”crowding effect
observed star densities in the field Σ and in NGC 2384 incompleteness” (see Sagar & Richtler 1991 for details). It
f
corona areas Σ were determined from the counts in the arises due to finite size of stellar images which depend on
2384
firstandforthquadrantsofthecorrespondingareaslocated stellarbrightness.Animageofastarofsomebrightnessoc-
at α > 7h.42 and within 0◦.10÷0◦.15 and 0◦.35÷0◦.60 cupiescertainareawithintheCCDframe,hidessomeneigh-
2000
from the cluster center which are free from contamination boringstarsoflowerbrightness,andexcludesthemfromthe
ofNGC2383members.AsthevaluesofΣ andΣ donot statistics,affectingthebrightnessfunctionψ(V),adistribu-
f 2384
stronglyvaryoverthestudiedarea,thedensityofNGC2383 tionofstarswithapparentmagnitude.Therelationbetween
clustermemberscanbecalculated asσ2383 =Σ2383−Σ2384 correctedforthecrowdingeffectbrightnessfunctionψce(V)
(Σ2383 iscountedwithinNGC2383frame).Thestardensity and observed one ψa(V) is expressed as
ofthecoronaofNGC2384itself,thenisσ =Σ −Σ .
2384 2384 f V
The results are shown in Table 3 and one can see that ψce(V)=ψa(V) 1+ s(V′)ψa(V′)dV′ .
contamination of NGC 2383 member sample with stars of (cid:20) Z (cid:21)
Vmin
NGC 2384 cluster computed as σ /σ is about 24%.
2384 2383
We cannot estimate the contamination of NGC 2384 Here Vmin is the magnitude of the brightest star in the
frame,s(V)isanareaoccupiedintheframebyanimageof
membersamplewithstarsofNGC2383clusterasthereare
nostars with Ru <13.7 mag within NGC 2384 frame piece astarofmagnitudeV.Boths(V)andψce(V)areexpressed
in units of theframe area. Wehaveevaluated thecrowding
overlapping with NGC 2383 cluster. Moreover, this frame
effectforeachclusterofoursample.Thefunctionss(V)were
piece is small and we can assume insignificant number of
found empirically for every frame from the image statistics
NGC 2383 members there.
Similar procedure was applied to Hogg 15 cluster. The as s(V) = πrp2, where rp is average minimum distance in
theplaneofaframefromstarsofmagnitudeV toremaining
counts were performed down to Ru =15 mag. The average
stars. As we found for the considered range of magnitudes
observed density in corona of NGC 4609, Σ was com-
4609
puted from the stellar counts at distance 0◦.15−0◦.20, and this effect only negligibly (less than two percents) changes
average observed density of the field stars at 0◦.35−0◦.60 distributionψa(V).Thusweassumeψce(V)=ψa(V)inour
furtherdiscussions.
from thecenterof NGC4609. Asonecan seefrom Table 3,
Hogg 15 is contaminated with NGC 4609 stars by about
14%. This fraction is smaller than in the NGC 2383 case,
but is not negligible.
4.4 Theoretical luminosity functions
6 MonetD.,BirdA.,CanzianB.etal.,USNO-A2.0:ACatalogof For construction of a theoretical LF φ(MV), we have as-
Astrometric Standards (CD-ROM distribution). US Naval Obs., sumedamodelofcontinuousstarsformation.Itiscomputed
Washington, 1998. as
Determination of age, luminosity and mass functions of young stars 9
Table 4.OpenclusterparametersderivedfromthefitoftheLFs.
Cluster Fitparameters Derivedparameters
MV range χ2 MVH p x logt ∆logt
(mag) (mag)
NGC2383 −6.0,3.0 23.6 2.50 0.1±0.2 2.2±0.4 7.1±0.2 0.0±0.2
NGC2383a −2.0,3.0 37.2 − 0.2±0.2 1.2±0.4 8.5 −
NGC2384 −5.0,3.0 39.6 2.00 0.3±0.2 1.0±0.5 7.0±0.2 0.2±0.4
NGC2384b −5.0,3.0 41.5 2.00 0.2±0.2 1.0±0.5 7.0±0.2 0.2±0.4
NGC4103 −3.5,2.0 24.1 1.25 0.3±0.2 1.5±0.6 6.8±0.1 0.2±0.4
Hogg15 −5.0,2.5 79.2 0.75 0.6±0.2 1.3±0.4 6.7±0.3 0.6±0.6
Hogg15c −5.0,2.5 17.0 0.75 0.5±0.2 1.3±0.4 6.6±0.3 0.2±0.6
NGC4755 −3.0,3.0 10.8 2.50 0.2±0.2 1.4±0.3 7.2±0.2 0.2±0.2
NGC7510 −6.0,1.5 13.8 0.75 0.3±0.2 1.2±0.4 6.8±0.3 0.5±0.7
a correctedforcontamination byNGC2384stars,logt=8.5isaccepted fromtheupperMSisochronefitting
b derivedintheframeareafreefromNGC2383stars
c correctedforcontaminationbyNGC4609stars
t1 of 0.25 mag. The histogram form of smoothing kernel was
φ(MV)=Z φt(MV)λ(t)dt, selected to apply correction for field star contamination.
t0
(ii) Construction of acluster LF:Weassumethatappar-
where t0 and t1 are minimum and maximum ages of the ent LF ψa(V) is a composition of a distribution of cluster
clusterstars,functionφt(MV),computedusingequation(1), starswithapparentmagnitudeψc(V)andfieldstarsbright-
is the LF of stars with age t (t0 6 t 6 t1), λ(t) is the star ness function represented by galactic star count model dis-
formation rate(SFR)ataget.Thevalueoft1,representing tribution ψf(V):
duration since formation of first stars, could beregarded as
thecluster age. ψa(V)=ψc(V)+pψf(V),
The mass – absolute magnitude relation m(MV,t) and where free parameter p denotes percentage of field stars,
its derivative were calculated along the isochrone of age t
contaminating the sample. With help of apparent distance
usingacubicsplineinterpolation.FortheIMFweconsidered
a power-law representation f(m) = km−x, with k to be modulusV −MV thefunctionψc istransformedtoabsolute
magnitude scale, and is called hereafter as observed cluster
normalizing factor, and x the IMF slope to be determined
LFφc(MV).
withinthisstudy.WeassumedaconstantSFRinourmodel,
(iii) Luminosity function fitting: Theoretical LF φ(MV)
λ(t)=const.Theresultingparametersforeachclusterwere
smoothed in the same way as observed LF was fitted to
drawnfromthebestfitofthetheoreticalandobservedLFs.
empiricalfunctionφc(MV)withinacluster-specificrangeof
In order to construct theoretical isochrones and LFs
magnitudes. The entropy
which include both Post-MS and Pre-MS stages for ages
typicaltoclusters ofoursample, wecombinedPopulation I ∆= |φc(MVi)log(φc(MVi)/φ(MVi))|
Pre-MSevolutionarytracksofD’Antona& Mazitelli(1994) X
for masses 0.1 to 0.8 m⊙, and of Palla & Stahler (1993) for wasconstructedtofindthebest-fitparameters{p,x,t0,t1}.
masses 0.8 to 6 m⊙, and Maeder’ group Post-MS calcula- Steps(ii) and (iii) were repeated iteratively by varying
tions(Schaller et al. 1992)for m=0.8−120m⊙.Thegrids puntilthebestagreement(intermsofχ2statistics)between
wereproperlytunedtoprovideacontinuoustransitionfrom
theoretical and observed LFs is achieved. The parameter p
Pre- to Post-MS ages and smooth and uniform mass – lu-
was varied between 0 and 1 in a step of 0.1.
minosity and mass – radius relations along theZAMS. The
Internal accuracy of x,logt ≡ logt , and ∆logt ≡
1
isochrones were computed from the models corresponding
logt − logt was estimated on the basis of the kernel
1 0
tothePopulationIchemicalabundance(Y,Z)=(0.30,0.02)
smoothing theory (Silverman 1986, Lapkoet al. 1996) for
using linear interpolation.
selected histogram grid parameters (range and step). An
In order to convert the theoretical isochrones from
accuracy of thefield stars percentage p was estimated from
logTeff,logL/L⊙planetotheobserved(B−V)0,MV plane,
we used bolometric corrections and (B−V)0−logTeff re- σ2= (pi−p)2/χ2i, p= pi/χ2i,
lations from Schmidt-Kaler(1982) tablesfortheluminosity p P 1/χ2 P 1/χ2
i i
classes I, III and V. assumingPthat σ2 is a seconPd order central moment of a
p
distribution function equal to that of 1/χ2.
4.5 Fitting of theoretical and observed LFs
The following iterative steps are used in the fitting proce- 5 DISCUSSION
dure:
The results of the LF fitting are listed in Table 4. For each
(i) Apparent LF construction: Apparent LF (observed clusterweshowfitparameters:therangeofabsolutemagni-
brightness distribution) ψa was constructed from the data tudesselected for theLFsfit,thebest-fit χ2 valueachieved
available for given frame(s) assmoothed densityestimation in the iterations, and derived parameters: position of H-
with a rectangular one-magnitude widewindow, and astep maximum in theoretical LF MH, indicating absolute mag-
V
10 A.E.Piskunov et al.
nitudeof theturn on point,derived percentage of field star curve, while the input from the NGC 4609 population is
contamination p, the IMF slope x, adopted age of a cluster shown with thedot-dashed curve.
logt,and age spread parameter ∆logt.Note, that parame- As one can see from Table 4, taking into account pop-
ter errors shown in Table 4 are in good agreement with the ulation of NGC 4609 does not strongly change cluster pa-
uncertainty estimations given in Sec. 3 on the basis of data rameters(excepttheagespread),butconsiderablyimproves
accuracyanalysis.Theresultsarediscussedbelowseparately agreementbetweentheoreticalandobservedparametersno-
for clusters understudy. tably reducing χ2 parameter.
In Fig. 4, we show the CMDs of clusters under dis- Fig. 4 shows that ages t ,t derived from the LF fit-
0 1
cussion. The reddening and distance parameters are taken tingareingoodagreementwithclusterCMD.Theobserved
from Table 1 and fine tuned by the variation of the tabu- clusterLFφc alsodemonstratesahighdegreeofconformity
larvalueswithintheiraccuracytoreachthebestagreement with composite theoretical LF. A deep and narrow gap of
of the cluster CMD, empirical ZAMS, and corresponding φc at about MV ≈ 2 mag shows that the derived input of
isochrones. Cluster proper motion members and non mem- field stars p is somewhat overestimated. As shown in Ta-
bers are marked, if cluster stars are bright enough to be ble 4, Hogg 15 is one of the two clusters in our sample,
includedintheASCC-2.5.Togiveanideahowtheadopted whereconsiderable(althoughquiteuncertain)agespreadof
distances and reddening values agree with the photometry ∆logt=0.6 is detected. This value, however, is reduced to
we show in the diagram a position of empirical ZAMS of the insignificant value of ∆logt=0.2, which is common to
Schmidt-Kaler (1982). In order to illustrate how ages de- the whole sample, if one takes into account the contamina-
rived from the LF analyses confirm with cluster CMDs, we tion from NGC 4609. We therefore conclude that the age
show in Fig. 4, theisochrones corresponding to theages, t spread in Hogg 15 is introduced mainly due to admixture
0
and t . of field and overlapping neighbor’ stars, and consider the
1
In Fig. 5, we display the LFs. Observed cluster LF φc, cluster parameters as derived for theHogg 15c case.
constructedasdescribedinSec.4.5isshownwithfilledhis- Hogg 15 is the youngest cluster of our sample, with an
togram, and corresponding theoretical LF φ fitted to φc is age of 4 Myr.How does thisvaluecompare with theearlier
shown with a curve (the fitted portion of φ is shown with published values? For the first time, the age of Hogg 15
heavy curve, while the rest is marked with thin curve). For was estimated by Moffat (1974) as 8 Myr. According to
comparison purposes we show also apparent LFs ψa (thin Lyng˚a (1987) the age is 10 Myr. Recent values come gen-
histogram and hatched area), and contribution from field erally from isochrone fitting of the upper MS: LGM2.2 de-
stars ψ (dotted curve), both displaced by the value of fine it as 6 Myr, Sagar et al. (2001) provide age estimate
f
V − MV. Below we discuss each cluster as per their ap- of 6±2 Myr, Piatti & Clari´a (2001), who found initially
pearance in figures 4 and 5. it as 300 Myr, later reduced this unusually high value to
20±10 Myr (Piatti et al. 2002). Techniqueof matching in-
tegratedtheoreticalandobservedspectraoftheclusterlead
Ahumadaet al. (2000) to the age values 5±2 Myr or 30
5.1 Hogg 15
Myr. Since MS turn off ages are biased toward higher val-
IntheclusterCMD, onecan indicatetheMS turnon point ues,ourageestimate,basedonMSturnonpointtechnique,
at M∗ ≈ 0.5 mag. Note, that the H-feature should be also is in fact the lowest one. Note, however, that the majority
V
found in the LF in the vicinity of M∗, where apparent LF of the above mentioned values agree with the present one
V
of Hogg 15 stars also shows a local maximum. The data within their accuracy. The IMF slope was determined for
incompleteness dominates after MV = 3 mag, where ψa Hogg15starsbySagar et al.(2001),whofindittobeequal
gradually decreases in agreement with Table 1, indicating 1.35±0.21, which again coincides with thepresent result.
that Mcmp = 3.1 mag. The field star contamination for
V
Hogg 15 is found to behighest amongst clusters considered
5.2 NGC 7510
here. This effect, as it is seen from the filled histogram in
Fig 5, keeps the LF practically unchanged at the brighter It can be seen from the CMD that the regular cluster se-
portion (MV < 0.5 mag), and hides substantially the H- quencesareembeddedina”cloud”ofpresumablyfieldstars.
feature region. The astrometric members also show definite spread around
In order to take into account contamination from an theMS turn off point. It may bethat some of them are co-
overlapping cluster NGC 4609 theoretical LF φ(MV) was moving field stars. The MS turn on point can be detected
composed of two populations belonging to these clusters. somewhere between M∗ ≈ 0.5 and 1.5 mag. The apparent
V
SincetheclustersresideatdifferentdistancesfromtheSun, LF ψa shows a broad plateau between MV∗ ≈ 0.5 and 3
the stars of NGC 4609 should be shifted with respect to mag,withweakmaximumatMV =2.25 mag.Asindicated
that of Hogg 15 by the difference of cluster apparent dis- by galactic model star counts, it should be formed mostly
tancemoduli ∆(V −MV).SinceV −MV of theNGC 4609 byfieldstars.Thesampledemonstratescertainincomplete-
according to LGM2.2 is equal to 11.45 mag the value of ness below the maximum, where ψa instead of steady in-
∆(V −MV)istakentobeequalto−5.05mag.Duetosmall creaseslowlyfallsdown.AccordingtoTable1thecomplete-
fraction of NGC 4609 stars projected on the Hogg 15 area ness limit of original data is Mcmp = 0.9 mag. We select
V
(about 15% of Hogg 15 population according to Sec. 4.2) MV = 1.5 mag as faint limit of the LF fitting range based
the specific shape of the corresponding mass spectrum is on the value of χ2 parameter. The agreement between ob-
not important and we assume that it follows Salpeter law. served cluster LF ψc and theoretical LF is one of the best
Theage of NGC4609 was takenfrom Mermilliod (1981) as amongst the studied clusters.
logt = 7.56. Composite LF is shown in Fig. 5 with solid AsonecanseefromFig.4,theagest =2Myrandt =
0 1