An I-band calibration of the SBF method at blue colours

S.Mieskeetal.:SBFcalibrationatbluecolours5

Normalisethese uctuationsbydividingtheseblankimagesbythemeangalaxyintensityintheregionwhereSBFaremea-sured.SubtractthisnormalisedbackgroundPSBGfromthePSoftheSBFimage.

9.ObtaintheazimuthalaverageoftheresultingPS.10.FitfunctionoftheformP(k)=E(k)×P0+P1

(3)

totheresult.Here,E(k)=PSF(k) W(k)withW(k)thePSofthemaskusedtoexcisecontaminatingsourcesandrestrictthemeasurementarea.PSF(k)istheundistortedpowerspec-trumofthePSFnormalisedtounityatk=0.PSF(k)isderivedfromonetothreewellexposedstar(s)inthesameimageclosetotherespectivegalaxy.Whenmorethanonestarwasused,theaveragePSF-PSwasadoptedasPSF(k).ThevariationsinPSFFWHMbetweenstarschosenforonegalaxyweresmall,around5%.UsingE(k)insteadofPSF(k)implicitlycorrectsforthepowerspectrumdampingatlowwavenumbersthatiscausedbytheconvolutionoftheundistortedPSF

(

k

)

with

thepowerspectrumofthe

mask(seeMieskeetal.2005).Theamountofpowerspectrumdampingatwavenumberzerowasonaverage0.11magwithagalaxy-to-galaxydispersionof0.04mag.Thesignal-to-noiseratiooftheSBFmeasurementisthende nedasS/N=P0

mIfrom

10 0.4 (MV+15)

=5(Milleretal.1998).Thecalculatedval-uesfor GCarelisted

inTable1,rangingbetween0.01magand

0.42magwithameanof0.06magandmedianof0.03mag.Thecolour(V I)0ofeachgalaxywasadoptedasthemeanoftwovalues: rst,the(V I)0estimatesfromtheIMACSdataintheregionwhereSBFweremeasured;second,thecoloursde-rivedbyusforthesamegalaxiesinHilkeretal.(2003).TheselatterdatawerebasedonWFCCDphotometrywiththe2.5mduPonttelescope,alsoatLasCampanas.Thosecolourshadbeendeterminedincircularaperturesof4′′radius,whichisincloseagreementtothesizeoftheregionwhereSBFweremeasured.ForthosegalaxiesenteringourSBFdataset,thesystematicdif-ferencebetweentheIMACSandWFCCDcoloursis0.017±

0.021mag,withascatterof0.10mag.Thisshowsthatourpho-tometriccolourcalibrationisaccuratetobetterthan0.02mag.Fig.3showstheCMDoftheinvestigatedgalaxiesforthevar-iouscoloursets.Thermsdispersionincolourofalinear tis0.083magfortheIMACSdata,0.068fortheWFCCDdataand0.058magforthemean,independentofmagnitude.Weadoptthisverydispersionof0.058magascolouruncertainty.Thisistobeseenasanupperlimittotherealcolourerrorbecausethermsdispersioncertainlyincludessome“cosmic”scatter.With(V I)0athand,

mI,

thisyieldedthe“nullhypothesis”distancemodulus(m M)A,seeTable1.Fig.4showsthumbnailimagesillustratingtheSBFmeasurementprocedurefortwooutofthe28dEswithclearSBFsignal.25ofthe28investigatedgalaxieshadS/N>3andenteredoursubsequentanalysis.Theirabsolutebrightnessesareintherange 16.5<MV< 11.2mag,theircentralsurfacebrightnessescovertheregime20.3<µV<24.6mag/arcsec2.Theyoccupyacolourrange0.8<(V I)0<1.10mag,ide-allysuitedforextendingtheTonrycolourrangetotheblue.InFig.5weshowthatthenullhypothesisSBFdistancemodulusdoesnotcorrelatewiththeS/NoftheSBFmeasurementnorwiththeamountofbackground uctuationsBG.

NotethelargerangeofBGvaluesinFig.5.Thishasthreerea-sons:The rstandmainreasonisrelatedtothefactthatBGisdeterminedby ttingequation(3)tothenormalisedskyback-groundPS.Forthis ttingwehavetoexcludethesamelowwavenumberregimethatisexcludedfor ttingthegalaxySBFamplitudeP0.SinceBGisonlyafractionofP0,thisexclusionoflowwavenumbershasastrongere ectonthe ttingpreci-sionforBGthanfor ttingP0.Indeed,thenegativevalueforBGthatismeasuredforFCC215(seeTable1andFig.5)isaconsequenceofthislowwavenumberexclusion.Theback-groundPSintheremainingwavenumbersispredominantlynoisedominatedsuchthataformallynegativevalueforBGis tted.Asaconsequenceofthat,theSBFdatawithpoorseeing(FWHM>0.85′′)showascatterinBGof0.16magwhichisal-mosttwiceaslargeasthescatterof0.09magforthedatawithgoodseeing(FWHM<0.85′′),seeFig.5.ThesecondreasonforthevariationofBGisthatthegalaxiesinvestigatedspanaconsiderablerangeinsurfacebrightness(>4mag).ThisleadstovaryingrelativeamountsofBG,giventhattheSBFsignalinlowersurfacebrightnessregionsismorestronglya ectedbyskybackground uctuations.Thethirdreasonisthesomewhatvaryingfringingamplitude,bothbetweendi erentchipsanddi erentnights,seeforexampleFig.2.

We nallynotethattheskybackgroundcorrectionofthegalaxySBFdataisdonebydirectlysubtractingtheskyback-groundPSimagefromthenormalisedgalaxylightPSimage.Onlythequanti cationofthecorrectionisachievedby ttingaPS.Totaketheuncertaintyarisingfromthepower-spectrum tandfromthesky uctuationvariabilityintoaccount,weadoptthescatterofP0resultingfromthesubtractionofthreedi er-entskybackgroundPSimagesaserrorestimateoftheSBFmeasurement,seethefollowingSect.3.2.1.