Determining the Location of the Tip of the Red Giant Branch in Old Stellar Populations: M33, Andromeda I & II

3 MB·English

by  A. W. McConnachie

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
Determining the Location of the Tip of the Red Giant
Branch in Old Stellar Populations: M33, Andromeda I &
II.
A. W. McConnachie1, M. J. Irwin1, A. M. N. Ferguson2, R. A. Ibata3,
4
0 G. F. Lewis4, N. Tanvir5
0
2 1 Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK
2 Max-Planck-Institut fu¨rAstrophysik, Karl-Schwarzschild-Str. 1, Postfach 1317, D-85741 Garching, Germany
n 3 Observatoire de Strasbourg, 11, rue de l’Universite, F-67000, Strasbourg, France
a 4 Institute of Astronomy, School of Physics, A29, Universityof Sydney, NSW2006, Australia
J
5 Physics, Astronomy and Mathematics Dept., Univ. of Hertfordshire, Hatfield, AL10 9AB, UK
2
2

1
2February2008
v
3
5
4 ABSTRACT
1
0 The absolute bolometric luminosity of the point of core Helium ignition in old,
4 metalpoor,red-giantstarsisofroughlyconstantmagnitude,varyingonlyveryslightly
0 with mass or metallicity. It can thus be used as a standard candle. Here, we review
/ the main difficulties in measuring this location in any real dataset and develop an
h
p empiricalapproachtooptimiseitfortipoftheredgiantbranch(TRGB)analysis.We
- go on to present a new algorithm for the identification of the TRGB in nearby metal
o poor stellar systems. Our method uses a least-squares fit of a data-adaptive slope to
r m
theluminosityfunctionin1 windows.Thisfindstheregionoftheluminosityfunction
t
s thatshowsthemostsignificantdeclineinstarcountsaswegotobrightermagnitudes;
a
the base of this decline is attributed as the location of the tip. This technique then
:
v allows for the determination of realistic uncertainties which reflect the quality of the
Xi luminosityfunctionused,butwhicharetypically∼0m.02rms+∼0m.03systematic,
asignificantimprovementuponpreviousmethodsthathaveusedthetipasastandard
r
a candle.Finally,weapplyourtechniqueto theLocalGroupspiralgalaxyM33andthe
dwarf galaxies And I & II, and derive distance modulii of 24m.50±0m.06 (794±23
kpc), 24m.33±0m.07 (735±23 kpc) and 24m.05±0m.06 (645±19 kpc) respectively.
TheresultforM33isinexcellentagreementwiththeCepheiddistancestothisgalaxy,
and makes the possibility of a significant amount of reddening in this object unlikely.
Key words: Local Group - galaxies: general - galaxies: stellar content
1 INTRODUCTION standard candle known in astrophysics is the Cepheid vari-
able,whosewellstudiedPeriod-Luminosityrelationsetsthe
foundation for the entire cosmological distance ladder (see,
The accurate determination of distances in the Universe is
for example, Tanvir 1999).
a fundamental and difficult problem in astronomy. Paral-
lax is the ideal method of distance determination as it is Oneofthelimitations oftheCepheidmethodisthatit
completely independent of the physical nature of the body isconfinedtorelativelymassivePopulationIsystems.Inor-
of interest. However, this geometrical method can only be dertocalculatedistancestoolder,moremetalpoorgalaxies
currently applied successfully in the nearby Galaxy due to someotherstandardcandleisrequired.TheTipoftheRed
limitations in the accuracy of the astrometry required. For Giant Branch (TRGB) offers an ideal alternative for these
systems external to this we are forced to search for some systems.Physically,thisstageinstellarevolutionrepresents
standard candle - an object whose intrinsic brightness we the point of the core helium flash. Here, the temperature
thinkweknow.Measurementoftheapparentbrightnesscan of the degenerate, quasi-isothermal core is only dependent
then, after correction for extinction effects, lead to an esti- uponthepropertiesofthethinH-burningshellarounditand
mateofitsactualdistance.Themostsuccessfulandreliable this varies only very slightly with chemical abundance and
2 McConnachie et al.
surrounding mass. The TRGB is thus of roughly constant the previously used one, as it allows for some smoothing
intrinsicbrightness. Iben& Renzini(1983) realised that for of the luminosity function over a small range. This allowed
low mass, metal poor stars ([Fe/H]< 0.7 dex) the TRGB them to find the position of the TRGB, as they had de-
varies by only 0m.1 for any star old−er than 2 Gyrs. The fined it, to within an uncertainty of 0m.1. More recently,
∼ ∼
absolute I band magnitude is also found to be constant for Mendez et al. (2002) have refined this method further, and
these stars as absorption effects in the stellar atmosphere theyhavealso used it in conjunction with a maximumlike-
are minimal, butfor stars more metal rich than this molec- lihood method.Thisinvolvedmodelling theRGBinthelu-
ularabsorption causestheTRGBtoappear dimmerinthis minosity functionas atruncatedpowerlaw andattributing
band.Amorerecentstudyofthetheoreticalpredictionscon- the TRGB as the location at which the power law is trun-
cerningthisstageinstellarevolutionhasbeenconductedby cated. We believe that such a technique is a significant im-
Salaris & Cassisi (1997) and a comparison with other dis- provement over the Sobel edge-detection algorithms and it
tance indicators, such as Cepheid and RR Lyrae variables, offers many advantages - for example it is much more sta-
wasmade.Theyfoundagreementbetweenthesemethodsat ble against noise effects. However, a step model (an abrupt
thelevel of 0m.1. cut-off in number counts) is still implicitly assumed as the
∼
DaCosta&Armandroff(1990)comparedthebrightest intrinsic shape of theTRGB in their analysis.
RGBstarsinvariousglobularclusterstothetheoreticalpre- It is not clear that an edge detection algorithm, or any
dictionsforcoreheliumignitionandfoundverygoodagree- method that assumes a step in the luminosity function, is
mentbetweenthetwo,withonlyaveryweakdependencyon necessarilythebestapproachtothisproblem,asthereisno
metallicity. They were dealing with relatively sparse colour apriorireasonwhytheluminosityfunctionatthelocationof
magnitude diagrams however. A more recent study by Bel- thetipshouldbeasimplestepfunction.Itmayequallywell
lazzini, Ferraro & Pancino (2001) of the globular cluster beaslope,oraparabola,orsomethingelse-theexactshape
ω Centauri came to similar conclusions. They obtained a oftheluminosityfunctionattheTRGBedgeiscurrentlyun-
value of the absolute I band magnitude of the TRGB of known.WenotethatZocacali&Piotto(2000)showthatthe
4m.04 0m.12. We shall later use this value in order to bright end of RGB luminosity function is well modelled by
− ±
calculate the distance modulus to several galaxies (see Sec- asimple powerlaw, butthereisnomodel that predictsthe
tion 3.2 for a discussion of this result). shapeoftheTRGBedge,althoughthisdoubtlesslydepends
Several authors have used various techniques to derive upon the numberof stars that contributeto the luminosity
TRGB distances. For example, Harris et al. (1998) used a function. Generally, we might expect the number of stars
modelfittingtechniquetotheluminosityfunction,although to decrease gradually over a short magnitude range as we
the most influential method to determine the location of approachthetipfromalongtheRGB.Theluminosityfunc-
theTRGBisduetoLeeet al.(1993). Priortothispublica- tion would then exhibit more of a slope at the TRGB, an
tion,estimatesofthelocationoftheTRGBhadmostlybeen abrupt change in the bright end of this slope would then
made by eye. Such a qualitative approach is of course not be, by definition, the TRGB. The Sobel edge detection al-
idealasitisnotreproducible.However,Leeetal.(1993)put gorithm doesnotalways produceapeakinsuchasituation
themethodonaquantitativebasisbyemployinganedgede- (see, for example, Figure 1 of Madore & Freedman 1995).
tection algorithm. Essentially, an edge detection algorithm In practice, the resulting signal is not easy to detect over
inthiscontextcomparesneighbouringstarcountsandlooks and above all the other random variations that the output
for the largest absolute change between neighbouring bins ofthisprocedurecreateswhendealingwithallbutthemost
- in essence, it models the TRGB edge as a step. Specifi- idealised noise-free luminosity functions.
cally, Lee et al. used a zero-sum Sobel kernel [-2, 0, 2] and In Section 2 of this paper, we review the main diffi-
convolveditwiththebinnedluminosityfunction.Therange culties with using the TRGB as a distance indicator. We
over which this kernel is applied is defined by the sampling thendescribeandtestanewalgorithmthatwehavecreated
of the luminosity function. In practise, the output of this in order to measure the TRGB location. This algorithm is
displays a maximum in the bin coincident with the largest baseduponaleast-squaresfittingprocedureandmakesless
absolutechangeinthecounts.Themidpointofthisbinwas assumptions about the shape of the luminosity function at
then assumed to be the magnitude of the TRGB. Typical the TRGB, working equally well for situations where the
uncertainties in thisquantitywere quoted as 0m.1 0m.2. TRGBismarkedbyasuddenorgradualriseinstar-counts.
−
Three years later, Sakai et al. (1996) adapted this This data adaptive slope technique gives uncertainties in
methodtomakeitindependentofbinning.Theyconstructed the distance modulus obtained typically of order 0m.05,
∼
a smoothed luminosity function ℘(m) such that but most importantly reflects the quality of the luminos-
ityfunctionused.Wealsodemonstrateanadditionalsimple
℘(m)= N⋆ 1 exp (mi−m)2 , (1) empirical heuristic scheme that allows a first pass estimate
Xi=1 √2πσi (cid:18)− 2σi (cid:19) oofurthmiseltohcoadtiotonctaolcbuelamteadteh.eIndisSteacntcioens t3owtehegospoinratlogaaplapxlyy
where mi and σi are the magnitude and photometric error M33 and thedwarf galaxies And I & II.
of the ith star from a sample of N⋆. In the limit of large
N⋆,theresultcanbethoughtofasaluminosityprobability
distribution. Hereafter, we adopt the abbreviation LPD for 2 LOCATING THE TRGB
this function, to distinguish it from the binned luminosity
2.1 Data Considerations
function. Sakai et al. then convolved this with a smoothed
Sobel Kernel, this time in the form of [-2, -1, 0, 1, 2] - this Finding the apparent magnitude of the TRGB in a satis-
kernel is less sensitive to spikes due to Poisson noise than factory, quantitative manner has proved a challenge with
TRGB Distance Determinations 3
several different techniques having been used in the litera- trated bylooking at Figure 8, which shows thecolour mag-
ture. Madore & Freedman (1995) have already written an nitude diagram for And I & II. The luminosity function is
excellent review and used computer simulations to analyse onlyplottedforthestarsboundedbythedottedlines.Their
the effects of these difficulties on the Sobel edge detection positionsaredeterminedbytakingstarcountsinstripspar-
algorithm. They concluded: allel to the RGB and placing the boundaries on either side
of the maximum. The exact position or separation of the
(i) low signal to noise (S/N) can hide thelocation of the
boundaries makes no difference to the derived value for the
tip in a luminosity function or lead to the false identifica-
tip,asshouldbeexpected.Alltheotherstarsoutsideofthis
tion of this point, with a jump due to noise possibly being
areaareignored,astheycontributenothingtotheRGBand
attributed to theluminosity of core Heignition;
wouldonlyact tohidethetip.SomebrightAGBstarsmay
(ii) bright Asymptotic Giant Branch (AGB) stars can
still be present but their effect has been lessened. Further,
contaminate the luminosity function in the region of the
as we demonstrate in Section 2.2.1, our technique for find-
tip, and may hide its truelocation;
ing the RGB appears robust even with a significant AGB
(iii) foregroundstarsmasqueradeasmembersofthestel-
population.
lar system and will contaminate theluminosity function.
Some foreground stars will have survived the cut that
Poisson noise is usually the dominant uncertainty in we have applied to our data and will be contaminating our
any luminosity function, and becomes a serious issue if we RGB luminosity function. We cannot hope to identify fore-
intendtotryandmeasureaspecificlocationdefinedbysome ground stars individually but with wide field data we are
discontinuityinstar-counts.Althoughthealgorithm wede- able to apply a statistical correction for them by using a
scribe later is less sensitive to Poisson noise than an edge luminosityfunctionofaneighbouringregionconsistingonly
detectionalgorithm,orsuchlike,itisstillnotimmune.The of foreground stars. Then, assuming all the brighter stars
only way to overcome Poisson errors successfully is to get in our fields of interest are foreground objects, we can use
photometry for more stars. As well as decreasing the re- these to scale the reference luminosity function to our data
sultingnoiseinaluminosityfunction,largestar-countsalso if necessary.
ensure that the luminosity function we construct from our Finally, we propose to use, in the same way as Sakai
sample is as close to the underlying luminosity function of et al. (1996), the LPD given by Equation 1 instead of the
the population as possible. Therefore, the most luminous normal binned histogram. This makes the process indepen-
RGBstarsare morelikely tobeat thepointof corehelium dent of any binning considerations and yields a continuous
ignition-althoughwenotethatCrocker&Rood(1984)con- function for the analysis, with the drawback that the error
ductedMonte-Carlosimulationswhichsuggestthatthereis analysisbecomesmorecomplicated thanforanormallumi-
a large probability of observing a star close to the tip lo- nosity function. Taken as a whole, the above methodology
cation even for a sample of relatively few RGB stars. Ad- creates clean LPDs optimised to measure the TRGB. Fur-
ditionally, the effect of Malmquist bias on a well sampled ther specific discussion of our data is deferred until Section
luminosity function is negligible. 3 after we have describing the algorithm we are using to
Many previousTRGBdistance studieshaveused Hub- analyse LPDs for TRGB locations.
ble Space Telescope (HST) WFPC2 data (eg. Kim et
al. 2002, Maiz-Appellaniz et al. 2002). However, despite
2.2 Least-Squares Fitting of an Adaptive Slope
WFPC2’ssuperbresolution,itsrelativelysmallfieldofview
(an L-shaped 2.5’ x 2.5’) may sometimes not allow for as The technique that we use to locate the TRGB involves
large anumberof brightRGBstars tobeobserved asis re- finding the region on a LPD that shows the most signifi-
quiredforarelativelynoise-freeluminosityfunctionsuitable cant decline in star counts. The base of this decline (slope)
forTRGBanalysis.Instead,whereverpossible,tomaximise will be attributable to the TRGB. The main complication
the numberof stars observed and to increase the reliability arises dueto ourlack of knowledge of theintrinsic shapeof
of the TRGB measurement, we advocate the use of ground theLPDaroundtheTRGBlocation,andwhatgradientbest
basedwidefieldcameras, forexampletheWideField Cam- resemblestheslopeinthisregion.Fittinganoptimummodel
eraon theIsaac Newton Telescope (0.29◦ FOV)orMEGA- totheLPDtherefore impliesageneralised least-squaresfit-
◦
CAM on the Canada-France-Hawaii Telescope (1 FOV). ting procedure. Note that the model we use is not defined
The accuracy of the technique that we introduce later is over the full range of the LPD, as it is not our intention to
significantly improved if used on a well-defined luminosity model the LPD but to locate a specific point of the LPD.
function, although realistic error estimations are obtained By making this our goal we minimise theassumptions that
regardless of the number of stars contributing to the mea- we are forced to make concerning the intrinsic shape of the
surement.Inthesesituationshowever,wemayneedtocon- LPD:weinsteadrepeatedlylookattheLPDthroughawin-
cernourselveswithwhethertheluminosityfunctionwecre- dow, and findthe window in which theLPD is best fit as a
ate is representative of the underlyingpopulation. simple slope function, that is, the region that when coming
Having maximised the number of stars with reliable fromthefaintenddemonstratesthemostsignificantdecline
photometry,wethenwanttoincreasetheS/Nfortheresul- in star counts.
tantluminosityfunction.BysignalwerefertoRGBstarsin In least-squares fitting, we seek to find the values of
the system of interest, and by noise we refer to everything model parameters which for independent errors minimise
else - AGB stars and foreground stars, in particular. The thefunction χ2, given by
most obvious way to increase the fraction of RGB stars on
the luminosity function is to take the luminosity function χ2 = N (di−mi)2 (2)
along the red giant branch only. This is most easily illus- Xi=1 σi2
4 McConnachie et al.
where di is the ith data point with an uncertainty σi, and
mi is the corresponding point in the model. Assuming the
modelchosenisagoodone,thenthebestfitshouldproduce
a χ2 which is equalto thenumberof degrees of freedom.
min
Ifweusedasinglemodelshape,thenwewouldrequireonly
one parameter, τ, which would govern the x-axis offset be-
tween data and model. However, we do not want to specify
the slope to be fitted, and so instead we use a family of
templates, φs, where s is a parameter which governs the
gradientofthetemplate.Inall,thetemplateisdefinedover
a1m range,chosenempiricallyastherangethatisgenerally
sufficienttoshowthefeatureweareattemptingtofit,while
still allowing our model to be a reasonable approximation
tothefeature.Sincethisextendsoverashorterx-axisrange
than the complete LPD, then the y-axis range of the data
thatwefittochangeswithτ.Wethusneedtointroducetwo
scaling parameters to deal with this; a vertical scale factor Figure1.Theconvergenceofthemeasuredvalueofthetipwith
kτ,s and a ’dc offset’ aτ,s. For any values of s and τ, the its known location as the S/N is increased (solid line). Here we
relationship between the actual model fitted, mτ,s, and the definetheS/NtobetheratioofthenumberofstarsontheRGB
to the background contaminants. The method is accurate even
template, φs, is then atlow (∼1)S/N. Asecond setof tests whichalsoconsiders the
mτ,s=kτ,sφs+aτ,s (3) effectofanAGBpopulationonthealgorithmshowscomparable
behaviour (dashed line), where this time the S/N is defined as
and so we are left to minimise theexpression the ratio of stars on the RGB to the sum of the background
contaminants andtheAGBpopulation.
χ2(s,τ,k,a)= N (di+τ −(kτ,sφs,i+aτ,s))2 (4)
σ2
Xi=1 i+τ such cases are rare for a well sampled luminosity function,
whereN nowsignifiesthenumberofpointsdefiningourtem- andaregenerallyobvious.Theyareusuallyeasilyidentified
plate φs,and we havedefinedτ such that it corresponds to byvisualinspectionofthefitandappropriateconstraintsto
anintegernumberofpointsinthedata.Simpleexpressions theparameters can be applied if neccessary.
for aτ,s and kτ,s may be derived by looking for the values
whichproduceaminimuminχ2.Bydefiningourtemplates
suchthat Ni=1φs,i =0andassumingaconstanterrorterm 2.2.1 Testing the Data Adaptive Least Squares Method
we then finPd that
To test the robustness of the data-adaptive slope method,
1 N we generated model CMDs that had a TRGB at 20th mag-
aτ,s= di+τ (5) nitude,markedbyasteadyriseinstarcounts.Photometric
N
Xi=1 noisewas addedbyselecting avaluefrom agaussian distri-
kτ,s= PNi=1Ndi+φτ2φs,i (6) binutoiuorndwaittahaσt=200tmh.m02a,gtnhietutdyep.icIatlsphhooutlodmbeetrnicotuendcetrhtaatinwtye
i=1 s,i experimented with different error weightings but found it
P
A constant error term is areasonable approximation to the had a negligible effect on our results, as in the range of
error distribution oftheLPD,although itsactual error dis- interestaroundtheTRGBthephotometricerroriswellap-
tribution is complicated by the covariance of neighbouring proximated by an error of 0m.02. We used a background of
regions. Every point on the LPD is a sum of contributions 8000starsrandomlydistributedbetween19thand23rdmag-
from every star and so a simple Poisson distribution is not nitude and populated a red giant branch between 20th and
valid.However,giventhattherearealargenumberofstars 23rd magnitude.Ourmethodwasthenappliedtothismodel
contributing to every point, then the central limit theorem CMD.MeasuringtheS/Nastheratioofthenumberofstars
implies that the errors can be well approximated by a con- ontheTRGBtothoseinthebackground(ie.unwantedcon-
stant error in this limit. A check of the validity of this ap- taminants), the results as the S/N increases can be seen as
proximation and its effect on the resulting uncertainties on the solid line in Figure 1. For a S/N greater than approx-
the parameters is conducted with the M33 data in Section imately one, the measured position of the tip differs from
3.3, and it is shown to be robust. the actual position by, at most, 0m.02. This nicely shows
Ourmethodisthenasfollows:wecalculatemτ,sforev- that themethod is able to identify and fit theregion of the
eryvalueofτ andsbyusingtheaboveprescriptiontocalcu- TRGB to a high accuracy.
latekτ,sandaτ,s.Wethenconductasimpleleast-squaresfit Weranasecondsetoftestswhichalsoincludedalarge
ofthesemodelstotheLPD,in1m windows,andwefindthe populationof1000AGBstarsrandomlydistributedbetween
valuesofsandτ thatminimiseχ2(s,τkτ,s,aτ,s).Thisthen 19m.5and20m.5,whichwillacttomaskthetip(dashedline
gives us the TRGB. Examination of the 1σ contours in the in Figure 1). The behaviour of convergence with increasing
τ splanealsogivesustheuncertaintyassociatedwiththis S/N remains virtually unaltered for S/N > 1 even with a
−
value.Duetothegeneralnatureofthisfit,itispossiblethat significant AGB population.
attimesthemethodmaylocateafeaturethatisnothingto A third set of tests were conducted on the M33 data.
do with the TRGB, but is perhaps due to noise. However, In Section 3.2 we calculate that the TRGB for this galaxy
TRGB Distance Determinations 5
al. 2001). Over the past 3 years we have been using this to
conduct a survey of the halo and outer disk of our near-
est neighbourM31. Thissurveynowextendsout toover55
kpcinprojection fromthecentreofthisgalaxy.In2002,we
extended this project to include M33 which has now been
mapped out to a radius of 20 kpc. Even the early re-
∼
sults of these surveys were startling, with the halo of M31
showing significant and unexpected substructure; the most
spectacularofthesebeingamassivetidalstream stretching
out as far as we have surveyed (Ibata et al. 2001, Ferguson
et al. 2002, McConnachie et al. 2003). As part of the M31
and M33 surveys we also took fields covering two of M31’s
dwarf spheroidal satellites, AndI & II.
Images were taken in theequivalent of Johnson V (V′)
′
and Gunn i (i) bands. For both surveys, most of the
data was taken in photometric conditions. The few non-
Figure 2.The variationof the measured locationof the TRGB photometric nights were calibrated using overlapping fields
withstarcountsinM33.Despitethestellarsamplesizevaryingby and the whole system was placed on a consistent scale us-
2ordersofmagnitude,thepeak-to-peakvariationinthemeasured
ing all of the field overlaps. The data for And I & II were
TRGBlocationisonly0m.12overtheentirerange.
deliberately acquired under photometric conditions. Expo-
suretimes of 800 - 1000 s perpassband allowed usto reach
liesat20m.54(uncorrectedforreddening),baseduponsome i′ =23m.5andV′ =24m.5(S/N 5),andissufficienttode-
2m0e0a0s0ursetadrvsaclounetorifbtuhteinTgRtGotBhevaLrPieDs.aFsiwgeurceon2tsihnouwalslyhorewdouucer stteacrtsintodivMidVu′alRG1BmsatatrtshteodMistVa≃′n≃ce0omf Man3d1.mSaeinve-sreaqlufieenldces
≃ −
taken in poorer conditions were reobserved and co-added
thisnumber.Itisevidentthatalthoughthenumberofstars
as necessary to give an approximately uniform overall sur-
we use changes by 2 orders of magnitude, the peak-to-peak
variationinthemeasuredvalueofthetipisonly0m.12,de- veydepth.Forthispaper,wehaveconvertedourdatatothe
creasingto0m.05whenmorethan2000starsareconsidered. Johnson-Cousinssystem.Thetransformationsemployedare
′ ′
I = i 0.101 (V I) and V = V +0.005 (V I) and
Thisdemonstratesthestabilityofourtechniqueoveravery − × − × −
have been derived by comparison with observations of sev-
large range in star counts. Thereis also evidenceof atrend
eral Landolt standard fields1.
such that, as we go to larger numbers, the measured loca-
Alltheon-targetdatapluscalibrationframeswerepro-
tion of the tip tends to brighter magnitudes. This is not to
cessed using the standard INT Wide Field Survey (WFS)
be unexpected since the larger sample of stars will ensure
pipeline supplied by the Cambridge Astronomical Survey
that the brightest RGB stars lie closer to the true tip loca-
Unit (Irwin & Lewis 2001), which provides all the usual
tion. This fact, and the decreasing scatter in the measured
facilities for instrumental signature removal and astromet-
valuesaswegotolargernumbers,showsthatlargenumber
ricandphotometriccalibration.Internalcross-calibrationof
statistics are desirable for such a study.
thefourCCDsisachievedatalevelbetterthan1%foreach
pointing.Theoverallderivedphotometriczeropointsforthe
2.3 Relative Increases (Heuristic) entiresurveyaregoodtothelevelof 2%inbothbands.Ob-
±
jectclassification usestheobservedmorphological structure
Finally,wehavedevelopedasecond,muchsimpler, method
ontheCCDsbutisvulnerableatfaintmagnitudestodistant
to estimate the location of the TRGB. This involves mon-
compactgalaxies,whichmaybemisclassifiedasstars.How-
itoring the relative increase in star counts between neigh-
ever,thisdoesnotseriouslyaffectushereaswearegenerally
bouring bins in the foreground-corrected RGB luminosity
dealingonlywithstarsneartheTRGB.Forthepurposesof
function once the counts have exceeded a threshold level.
this analysis, we concern ourselves only with those objects
Since we are only comparing neighbouring bins, then noise ′ ′
classified as stellar in both the V and i passbands.
effects will affect ustoa significant degree. Tocombat this,
As already discussed, the Galactic foreground stellar
we (non-recursively) average each bin with its two imme-
population should be subtractedfrom thedata tominimise
diate neighbours. The output should show a peak near the
contaminants. In order to compensate for this population,
location of the TRGB. It is very useful to have such a sim-
we use a number of outer halo fields from our M31 survey
ple method to provide an independent sanity-check of the
forbothM33(wherethereisnosuitablelocalreferencefield
results from thedata-adaptiveleast-squares technique.
we can use) and And I (which is sufficiently close to M31
fortheforegroundstosimilar).Althoughthereisalowlevel
M31halopopulation in thesereference fields,thedominant
3 DISTANCE DETERMINATIONS TO SOME population is galactic foreground stars, thus making them
LOCAL GROUP MEMBERS suitable as reference fields and allowing a statistical correc-
tiontobeapplied.Thecomputedscalingusingthebrighter
3.1 Data
partoftheCMDscompensatesnotonlyforthedifferencein
The Wide Field Camera on the 2.5m Isaac Newton Tele-
scope consists of a four-chip EEV 4K x 2K CCD mosaic
camera that images 0.29 deg2 per exposure (Walton et 1 http://www.ast.cam.ac.uk/∼wfcsur/colours.php
≃
6 McConnachie et al.
areabutalsotofirstordertheGalacticpopulationgradient.
For And II, we are able to apply a local foreground correc-
tion (see Section 3.5). In this study we typically find that
the foreground contamination in the region of the TRGB
is 15 stars/sq.deg/mag for the selection criteria that we
∼
apply.
3.2 The Adopted Absolute I Magnitude of the
TRGB
In order to calculate distances to these galaxies, we must
adopt a value for the I-band absolute magnitude of the
TRGB.Weusethemostrecentobservationaldetermination
of this position, given byBellazzini et al. (2001). These au-
thorsstudied thegalactic globular cluster ω Centauriusing
averylarge photometicdatabase and an independentlyde-
riveddistanceestimatetothissystemwhichusedadetached
eclipsingbinarystar.Theyderiveavalueof 4m.04 0m.12
− ±
for the I-band magnitude of the TRGB at a metallicity
of [Fe/H] 1.7 dex, which agrees extremely well with
≃ −
theoretically-derivedestimatesofthispoint.Althoughthere
is a small variation of this magnitude with metallicity, it is
significantly less than the quoted uncertainty for the range
of metallicities that we will be considering, given the stel-
lar systems we are analysing. As such, we adopt the value
asconstant. Itis important tonotethat theuncertaintyon
this value as it stands would be the single biggest contrib-
utor to the uncertainties in our final derived distances, and Figure 3. The spatial distribution of RGB stars in M33, from
is approximately 4 times bigger than our systematic errors. our INT WFC survey of this galaxy (Ferguson et al. 2004, in
preparation). Only stars brighter than I = 22m.5 lying in the
This degree of uncertainty is somewhat conservative, and
stripindicatedintheCMDareplotted.Starsaroundtheedgeof
the error quoted is primarily due to the uncertainty in the
thedisklyingintheellipticalannuluswillbeusedinouranalysis
truedistancemodulusofω Cen,whichisoforder0m.1.We
(0.72 sq. degrees). The position angle of the disk to the vertical
therefore adopt MTRGB = −4m.04±0m.05, and note that is∼23o.
although it is possible that the scale may be shifted by up
to0m.1,thiswillaffectallofourmeasurementsinthesame
way. The relative distances, to say M31, will essentially re-
RGBluminosityfunctionwhoseoverallshapeisverysimilar
main unchanged.
to the outermost RGB luminosity function, given that the
outer RGB luminosity function samples fewer stars and is
necessarily more noisy. In fact, the location of the TRGB
3.3 M33
fortheouterannulusisat20m.56 0m.02, whichcompares
TheRGBdistributionfromthecentralpointingsofoursur- very well with our final derived v±alue of 20m.54 0m.01.
±
vey of M33 (Ferguson et al. 2004, in preparation), located This implies that theeffect of crowding on ourM33 data is
at 1h 33m 50.9s, 30◦ 39’ 35.8”, is shown in Figure 3 for all negligible for a>0◦.5.
stars brighter than I= 22m.5 in the strip indicated in Fig- TheCMDofthestarsweanalysetomeasuretheTRGB
ure5.TheorientationsofM33’s axesareindicated,asisan are shown in Figure 5. It is obvious from the width of the
elliptical annulus around the edge of the disk, selected so RGB that M33 contains a larger spread in age/metallicity
as to avoid theeffects of crowding in theinnerregion while than the Andromeda dwarfs (Figure 8), and the region of
still sampling stars across the entirety of M33’s disk. The the tip lacks a definite edge with many bright AGB stars
innerellipsehasasemi-majoraxisa=0◦.5,whiletheouter visible in that region. This is illustrated in the RGB lumi-
ellipse has a = 0◦.8. To provide a check that our resulting nosity function (Figure 6) where we see that the tip of the
RGBluminosityfunctiondoesnotsufferheavilyfromblend- luminosityfunctiondelimitsachangeinsloperatherthanan
ingeffects,wehavecomparedtheRGBluminosityfunctions edge,andtheAGBstarsarealsovisiblehereforI<20m.6.
forellipticalannuliatdifferentgalactocentricradius.There- A weak blue plume representative of a youngerstellar pop-
sultsareshowninFigure4andareallforeground-corrected. ulation is also evident in theCMD at V I 0.
− ≃
TheouterannulussamplesaregionofM33whereweexpect A least-squares fit to the RGB LPD gives a TRGB of
negligible crowding, while the innermost annulus might be I = 20m.54 0m.01. The uncertainty in this measurement
±
expected to sample many blended stars. Indeed, the inner- is given by the 1σ contour in the τ s plane (see Figure
−
mostRGBluminosityfunction isseentoextendtobrighter 7). This contour plot was constructed assuming a constant
magnitudes than the outermost RGB luminosity function error term in the LPD (Section 2.2). To provide a check of
suggestingthatblendingmaybeaffectingtheinnerfieldsto this approximation, we also conduct a similar χ2 fit to the
someextent.Inthisrespect,themiddleannulusresultsina binned luminosity fuction, where we know the errors to be
TRGB Distance Determinations 7
Figure 4. RGB luminosity functions for M33 as a function of
galactocentric radius. Each RGB luminosity function is created
fromstarswithinellipticalannuliiwithsemi-majoraxesofa◦and
have been foreground-corrected. The innermostRGB luminosity Figure 5. The colour magnitude diagram of the spiral galaxy
functionmaysufferfromstellarblending.However,thesimilarity M33, using stars in the elliptical annulus shown in Figure 3. A
oftheoutertwoRGBluminosityfunctionssuggestthatblending bright AGB star population is obvious above the TRGB. Only
ishavinganegligibleeffectonourTRGBmeasurementforM33, stars in the strip indicated were used in the construction of the
whichusesstarslocatedat0◦.5<a<0◦.8. luminosityfunction.SelectingsuchanarrowstripalongtheRGB
still gives us ample stars for our analysis and significantly de-
creasestheAGBcontamination.
welldescribedbyaPoissondistribution.Theexpressionsfor
aandkgivenbyEquations3and4becomenecessarilymore
complex, but thecontour levels remain nearly unchanged. our results to the fact that we have better statistics than
From the extinction maps of Schlegel et al. (1998), was possible to achieve with their smaller HST fields - we
the interstellar reddening E(B V) is found to be 0m.042, do not see evidence for a significant jump at 20m.9, as
− ∼
and they state that these values are accurate to within these authors do (see their Figure 4). Of the 10 fields that
16%. Using the relation between reddening and extinction theyanalyse,severaloftheluminosityfunctionsdoshowev-
in the Landolt system (Landolt 1992), given in the same idence for a discontinuity in the slope at 20m.5 20m.6,
paper, AI = 1.94E(B V), implies that AI = 0m.081. whichmaywellbeassociatedwiththedisc∼ontinuity−thatwe
−
We take the absolute I magnitude of the TRGB to be attributeastheTRGBinourluminosityfunction.Sinceour
MTRGB = 4m.04 0m.05asdiscussedinSection3.2..The algorithm defines the location of the TRGB somewhat dif-
− ±
other uncertainty that we take into account are the mean ferentlytoanedgedetectionalgorithm,thenthislocationis
systematicphotometricerrorinourmeasurements,whichis highlighted in preference to any absolute change in counts.
of order 0m.02. Blending has already been shown to have Additionally,it isunlikelythatwehaveinstead measured a
a negligible effect on our measurement of theTRGB. Com- feature due to AGB contamination - an AGB population is
bining the observational errors lead to an overall error of clearly observed in theCMD and RGB luminosity function
0m.03 - 0m.04 in the measured value of the TRGB mag- for this galaxy (Figures 5 and 6) and so there would have
nitude. We calculate the distance, D, to the galaxy from tobeasecond AGBpopulation intheCMDexactly coinci-
I−MTRGB =5log10D−5+AI andconcludethatoverall, dentwiththetopoftheRGB,althoughthisseemsunlikely.
DM33 =794 23 kpc. Application of the heuristic method We would also expect that, if this was the case, their effect
identifies a ti±p at 20m.6, in good agreement with the more wouldbelessened inaluminosityfunction located atlarger
detailed treatment. galactocentric distance. Inspection of Figure 4 shows that
This result brings the TRGB distance to M33 into ex- this is not the case for these stars, even though the con-
cellent agreement with theCepheiddistances (seeTable1). tribution from the known AGB component can be seen to
Thesuggestion ofasignificant amountofreddeninginM33 reducein theexpected way.
byKimetal(2002),toexplainthediscrepancybetweentheir Kim et al. hasalso calculate thedistancetoM33 using
TRGBmeasurementsandtheirCepheidmeasurements(Lee the red clump (RC), finding agreement with their TRGB
et al. 2002), seems unlikely. We attribute the difference in measurement. This feature represents stars with a helium
8 McConnachie et al.
Figure 6. Upper panel: The RGB luminosity function for M33 Figure 9. Upper panel: The solid line shows the (RGB) lumi-
(lower curve is binned luminosity function, upper curve is off- nosityfunctionofAndI.Thedashedlineshowsthe(RGB)LPD.
set LPD)with anarrow indicating the measured positionof the The total are of sky surveyed was 0.47 sq. degrees. The arrow
TRGB using the fit of an adaptive slope. This location is par- indicates the position that we measure the TRGB to be located
tiallymaskedbyanAGBpopulation,althoughitsdetermination using our adaptive slope. Lower panel: Measurement of the lo-
is still possible by use of the least squares method. The area of cation of the tip using the heuristic method, showing excellent
sky used in this analysis is the area contained by the elliptical agreementwiththeadaptiveslope.
annulusshowninFigure3.Lowerpanel:TheTRGBasidentified
bytheheuristicmethod.
Udalski 1998; Sarajedini 1999). However, both observation
and theory show that MRC is significantly fainter for stars
I
core and a hydrogen shell passing through a relatively slow with ages > 10 Gyrs than for their intermediate age coun-
phase of their luminosity evolution, leading to a clump ap- terparts (Udalski 1998, Girardi & Salaris 2001). Kim et al
pearingintheCMD.Theyhaverecentlybeensuggested for (2002) assume that the majority of the stellar population
use as a distance indicator by Paczynski & Stanek (1998). in M33 is old (> 10 Gyrs) in order to associate the mean
They compared the I-band magnitude of OGLE RC stars metallicity oftheRGBwiththemetallicity oftheRC.This
seen throughBaade’s Windowwith asample from theHip- metallicityisthenusedtocalculate MIRC usingthecalibra-
parcos catalogue with parallaxes knownto 10%, allowing tions for MIRC vs [Fe/H] by Udalski (1998) and Popowski
theabsoluteImagnitudeoftheRC(MRC),≤andhenceadis- (2000). However, it is not clear that these calibrations are
I
tance,tobecalculatedforthestarsseentowardstheGalac- validforM33giventheassumptionabouttheageofthebulk
tic Centre. The reliability of this method when applied to of the stellar population. If MIRC is significantly fainter in
other stellar systems relies on the similarity of the Hippar- this galaxy due to age then M33 will be correspondingly
cos sample with theRCstars in thesystem of interest, and closer than these authors calculated.
remainsacontroversialpointduetopossibledependeciesof
MRC onthepropertiesofthepopulation.Theoretical mod-
I 3.4 And I
els seem to predict a relatively strong dependency of MRC
I
withage andmetallicity (eg.Cole 1998; Girardietal. 1998; The dwarf spheroidal galaxy And I was discovered by van
◦
Girardi 2000; Girardi & Salaris 2001) while observational den Bergh (1971) and lies at 0h 45m 39.8s, 38 2’ 28”. It
studies are in apparent disagreement with each other (eg. is a satellite galaxy of M31 at a projected distance of 40
∼
TRGB Distance Determinations 9
Figure7.ContourplotofthemeasuredTRGBlocationversustheslopeparameter,s,forM33.Contoursshowthe1,2and3σconfidence
levels.
Figure 8.Thecolour magnitude diagrams forthe dwarfgalaxies AndI (leftpanel) andAndII (rightpanel). The luminosityfunction
is constructed from the region contained within the dotted lines. The RGB is easily visible in both cases with the tip at I ≃ 20m.45
for And I and I ≃ 20m.1 for And II. The additional red component centred around V−I ≃ 2.4 in the left panel is due to the giant
Andromedatidalstream discoveredbyourINT surveyofM31(Ibata etal2001, Ferguson etal.2002, McConnachie etal.2003) which
extends overtheregionaroundAndI.
10 McConnachie et al.
outer regions of M31, which is sufficiently broad that part
of it lies along the line-of-sight to And I.
Application of the least-squares method identifies the
TRGB to be at I = 20m.40+−00mm..0032, as shown in Figure 9.
Schlegel et al. (1998) gives E(B V) = 0m.056 for the in-
−
terstellarreddening,andsowederiveDAndI =735 23kpc,
±
closer than had previously been determined (Table 2) but
still in reasonable agreement. Application of the heuristic
method identifies the tip to lie at 20m.45 (Figure 9), in ex-
cellent agreement with the adaptive slope, not unexpected
for such a clean LPD.
3.5 And II
And II is another dwarf spheroidal satellite of M31 discov-
◦
eredat thesame timeasAndI.Itlies at 1h16m 29.8s, 33
25’ 9”, approximately 140 kpc in projection from M31, to-
wards M33. Despite the fact that it lies significantly closer
toM33inprojection,thelargermassofM31meansthatitis
most likely a satellite of M31. DaCosta et al. (2000) derive
ameanmetallicityof<[Fe/H]>= 1.49 0.11dexwitha
− ±
significantlylargerabundancespreadthaninAndI.Anold
( 10 Gyrs) population is again implied from the presence
≥
of a blue horizontal branch. The presence of young or in-
termediate populations in this galaxy remains a possibility
fromtheirstudy,butwedetectnosignofsuchapopulation
to MV 0m.
≃
The colour magnitude diagram for And II (right panel
of Figure 8) again shows the upper parts of the RGB very
clearly,withatipvisibleatI 20m.1.Weconstructthelu-
≃ ◦
minosityfunctionforAndIIbyusingstarswithing0.1 from
its centre. This then allows us to apply a local foreground
correctiontotheluminosityfunctionbyusingasareference
populationallstarsonthesameINTfieldoutsideofthisra-
Figure 10. Same as Figure 9, but for the dwarf galaxy And
II. This shows all stars located within 0.1◦ of the centre of this dius, scaled in theusual way. The result is shown in Figure
10, where an unambiguos tip can be observed. We find the
galaxy, allowing the rest of the field to be used as a local fore-
TRGBto lie at 20m.13 0m.02 (20m.15 using theheuristic
groundcorrection. technique).FromSchleg±eletal.(1998),E(B V)=0m.063.
−
This corresponds to DAndII = 645 19 kpc, in reasonable
±
kpc. Mould & Kristian (1990) were the first to study its agreement with estimates that havebeen made using other
RGB using CCD images taken at the prime focus of the methods(Table 3).
Haletelescope.UsinganeyeballdeterminationoftheTRGB
they found it to be at a distance of 790 60 kpc. They
±
also concluded the galaxy to be of intermediate metallic-
4 SUMMARY
ity. Da Costa et al. (1996) obtained HST WFPC2 images
of this system and by comparison with the giant branches Inthispaperwehaveintroducedanewquantitativemethod
of several globular clusters confirmed the metallicity to be forthedetermination ofdistancestometalpoorstellar sys-
<[Fe/H]>= 1.45 0.2 dex. By analysing the morphol- temsusingtheTRGBasastandardcandle,accuratetotyp-
ogy of the hori−zontal±branch they also came to the conclu- ically 0m.02rms, 0m.03systematic.Themethodinvolves
± ±
sionthatthemajorityofthepopulationisapproximately10 a least-squares fit of a data-adaptive slope template to the
Gyrsold,However,thepresenceofabluehorizontalbranch foreground-corrected RGBLPD in 1m windows, tofind the
and RR Lyrae stars shows that there is a minority popula- region of the LPD that is best fit as a simple slope func-
tion at least 3 Gyrs older than this. And I,like many other tion ie. the region that shows the most significant change
dwarf spheroidals, appears to have undergone an extended in slope as we go to brighter magnitudes. This determines
period of star formation. theapparentmagnitudeoftheTRGB,asrepresentedinthe
The colour magnitude diagram for And I is shown in luminosity function. Observations and stellar evolutionary
theleft panelof Figure8.Theupperfewmagnitudesofthe codesagreethatthismagnitudeisvirtuallyconstantforold,
RGB are clearly observed and even by visual inspection we metalpoorpopulationsandsothephotometricparallaxcan
candeterminethatthetipisatI 20m.45.Oneinteresting easily be calculated.
≃
pointofnoteistheadditionalredfeature(V I>2)thatis We believe that our method provides significant im-
−
notpresentintheAndIICMD(rightpanel).Thisisdueto provementsoverpreviousattemptstolocatetheTRGBina
thegiantstellarstreamfirstdiscoveredbyoursurveyofthe luminosity function. By only considering stars along astrip