The nearest neighbor statistics for X-ray source counts I.The method

0.2 MB·English

by  Andrzej M. Sołtan

Astronomy&Astrophysicsmanuscriptno.nnst˙aeg1 (cid:13)c ESO2011
January18,2011
The nearest neighbor statistics for X-ray source counts
I.The method
A.M.Sołtan
NicolausCopernicusAstronomicalCenter,Bartycka18,00-716Warsaw,Poland
e-mail:[email protected]
Received /Accepted
1 ABSTRACT
1
0 Context.MostoftheX-raybackground(XRB)isgeneratedbydiscreteX-raysources.Itislikelythatstillunresolvedfractionofthe
2 XRBiscomposedfromapopulationoftheweaksourcesbelowthepresentdetectionthresholdsandatrulydiffusecomponent.Itisa
matterofdiscussionanatureoftheseweaksources.
n
Aims.Thegoalistoexploretheeffectivenessofthenearestneighborstatistics(NNST)ofthephotondistributionfortheinvestigation
a
J ofthenumbercountsoftheveryweaksources.
Methods.Allthesourcesgeneratingatleasttwocountseachinduceanonrandomdistributionofcounts.Thisdistributionisanalyzed
7
by means of the NNST. Using the basic probability equations, the relationships between the source number counts N(S) and the
1
NNSTarederived.

Results.ItisshownthatthemethodyieldsconstraintsontheN(S)relationshipbelowtheregulardiscretesourcedetectionthreshold.
]
O TheNNSTwasappliedtothemediumdeepChandrapointingtoassessthesourcecountsN(S)atfluxlevelsattainableonlywiththe
verydeepexposures.TheresultsareingoodagreementwiththedirectsourcecountsbasedintheChandraDeepFields(CDF).
C
Conclusions.InthenextpaperofthisseriestheNNSTwillbeappliedtothetheCDFtoassessthesourcecountsbelowthepresent
. fluxlimits.
h
p Keywords.X-rays:diffusebackground–X-rays:general
-
o
r
st 1. Introduction togram.Acontributionofpointsourcestothecountdistribution
a onecanestimatealsousingtheauto-correlationfunction(ACF).
[ The X-ray background (XRB) is mostly generated by discrete Since the integral of the ACF is directly related to the second

extragalacticsources(e.g.Lehmannetal.2001;Kimetal.2007, momentoftheP(D)distribution,(e.g.Sołtan,1991)bothmeth-
2
andreferencestherein).Thus,thesourcecountsprovidethees-
v odsarecloselyrelated.
sentialinformationontheconstituentsoftheXRBandtheX-ray
6 Aninnovativemethodtoassessthenumberofweaksources
N(S)relationshiphasbeenasubjectofnumerousstudiesforthe
5 wasproposedbyGeorgakakisetal.(2008).Intheirapproachthe
2 last40years. countdistributioninthedetectioncellisexplicitlyexpressedas
0 The individual point-like source is detected if a number of a sum of the source and background counts. As a result of a
1. counts within a specified area exceeds the assumed threshold. rigorousapplication of the Poisson statistics, a flux probability
0 SizeofthedetectionboxisdefinedbythePointSpreadFunction distributionisderivedasafunctionofthetotalandbackground
1 (PSF), whilethe detectionthresholdisusuallyselected to min- countsobservedin the detectioncell. This probabilitydistribu-
1 imizenumberoffalsedetectionsandatthesametimetomaxi- tioncombinedwiththe adequatelydefinedsensitivitymapof a
: mizenumberoftherealsources.Thedetectionthresholdistyp- givenobservationisthenusedtoestimatethethesourcenumber
v
i ically set at the level of 4−5σ above the local average count counts.
X density.Apresenceofweakersources,belowtheformaldetec-
The countfluctuationsare proportionalto the source inten-
tionthreshold,ismanifestedbytheincreasedfluctuationsofthe
r sities. Thus, the observed fluctuation amplitude is dominated
a count distribution as compared to the fluctuations expected for
by the sources just below the detection threshold set for the
therandomcounts.
individual objects, whereas it is only weakly sensitive to the
Acommonapproachtoassesscountsofsourcesweakerthan fainter sources which produce smaller number of counts. One
thedetectionlimitisbasedonthecountdensityfluctuationanal- should note, however, that every source which produces more
ysis. To quantify signal generated by the discrete sources one than onecountgeneratesdeviationfromthe randomcountdis-
should determine the intensity distribution P(D), i.e. the his- tribution. In the present paper we investigate the efficiency of
togram of the number of pixelsas a function of the numberof thenearestneighborstatistics(NNST)fortheweaksourceanal-
counts. The observedfunction P(D) is then comparedwith the ysis and we show that the NNST is a powerful tool to esti-
functions obtained from the simulated count distributions (e.g. mate the source counts, N(S), down to very low flux levels.
Hasingeretal., 1993; Miyaji&Griffiths, 2002). It is assumed We apply this technique to one of the Chandra AEGIS fields
thatthesimulatedsourcecountsrepresenttheactualsourcedis- with the exposure of 465ks. The NNST allows us to obtain
tribution if the model P(D) function mimics the observed his- the N(S) relationshipextendingdown to 2×10−17ergcm−2s−1
and 7×10−17ergcm−2s−1in the 0.5−2keV and 2−8keV en-
Sendoffprintrequeststo:A.M.Sołtan ergybands, respectively,i.e. a factor of 5−10 below the stan-
2 A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod
dardsensitivitythresholdcorrespondingtothisexposure.Since where the left hand side sum extends over all the sources and
the presentcountestimates are containedwithin the flux range k isthenumberphotonsproducedbythebrightestsourcein
max
covered by the direct source counts derived from the deepest thefield.Thenumberofphotonsn isrelatedtothenumberof
k
Chandrafields(such as CDFS), the effectivenessofthe NNST sourcesinanobviousway:kN(k) =n ,whereN(k)denotesthe
k
methodcouldbedirectlyassessed. numberof“k-photonsources”.
Theorganizationofthepaperisasfollows.Inthenextsec- Using the basic relationships of the probability theory, one
tion, the method and all the relevant formulae are presented. cancalculateP(r)–theprobabilitythatthedistancetothenear-
Then,inSect.3,theobservationalmaterialisdescribedandthe estneighborofarandomlychoseneventexceedsr:
computationaldetailsincludingquestionsrelatedtothePSFare
given. Results of the calculations, i.e. estimates of the source p P(r|1)+p P(r|2)+...+p P(r|k )= P(r), (2)
1 2 kmax max
counts below the nominal sensitivity limit are presented in
Sect. 4. The results are summarized and discussed in Sect. 5. where p denotestheprobabilitythattherandomlychosenpho-
k
ProspectsfortheapplicationoftheNNSTtothedeepChandra ton is produced by the “k-photon source” (k = 1,2,...,k ),
max
fieldsarepresented. and P(r|k)istheconditionalprobabilitythattherearenoother
counts within r provided the selected event belongs to the k-
photonsource.
2. Thenearestneighborstatistics
TheprobabilityP(r)canbeestimatedforthegivendistribu-
In this section a generalformulaebased onthe theoryof prob- tion of counts by measuring the distance to the nearest neigh-
abilityarederived,whileallthedetailsrelatedtotheactualob- bor for the each photon in the field. Similarly, P(r|1) is given
servations(e.g.relationshipbetweencountsandphotonenergy, by the distribution of distances to the nearest photon from the
instrumentsensitivity)arediscussedinSect.3. randomly distributed points. Assuming that the distribution of
In the present consideration we assume that counts in the “singlecounts”isnotcorrelatedwiththedistributionofphotons
given Chandra ACIS1 observation are distributed according to from k ≥ 2 sources, and that sourcesare distributed randomly,
thefollowingmodel:someaprioriunknownfractionofcountsis theprobabilityP(r|k)fork ≥ 2arerelatedtoP(r|1)andtothe
randomlydistributed,i.e.thepositionsofthecountsarefullyde- PSFbytheexpression:
scribedbythePoissonstatistics,whilealltheremainingphotons
areclusteredinthepoint-likesources.Suchmodelcorresponds P(r|k)= P(r|1)·P(r|k), (3)
to the observation obtained using the idealized telescope with-
outvignettinganddetectorwithperfectlyuniformresponseover where P(r|k) is the probabilitythat the distance from the ran-
thefieldofview.Therealtelescope-detectorcombinationintro- domly chosen photon produced by the k-photon source to its
duces numerousdeformationsto this ideal picture. We address nearestneighborfromthesamesourceexceedsr.Thisquantity
thisquestionbelow. isfullydefinedbythePSF.
Wefurtherassume,thatthepositionsofsourcesarealsoran- Toestimatetheprobabilitiesp wenowusearation /n and
k k t
domlydistributed.Smoothlydistributedcountsconstitutephys- theEq.2takestheform:
ically heterogeneous collection which contains both the parti-
cle background and various components of the foreground X- n kmax n
ray emission including scattered solar X-rays, the geocoronal 1 P(r|1) + k P(r|1)P(r|k)= P(r). (4)
oxygen lines as well as the thermal emission of hot plasma nt Xk=2 nt
within our Galaxy (e.g. Hasinger, 1992; Galeazzietal., 2007;
Henleyetal.,2007)andtheemissionbytheWHIM(e.g.Sołtan, ItisworthtonotethatEq.4islinearinthephotoncountsnk,
2007).Trulydiffuseextragalacticbackground(Sołtan,2003)and whatmakesitparticularlysuitablefortheestimatesofthecontri-
weakdiscretesources,eachproducinginthefinalimageexactly butionofweaksourcestothetotalcounts.Boththesecondmo-
onephoton,contributealsotothesecounts.Aseparationofthe mentof the countdistributionin pixelsand the autocorrelation
extragalacticcomponentfromallthecountscanbeachievedin function depend on squares of the photon counts. Substituting
statisticaltermsusingspectralinformation,however,individual successivevaluesofrintoEq.4,asetoflinearequationsiscon-
eventcannotbedefinitelyclassifiedaslocalorextragalactic. structedwhichallowsustoestimatetheunknowncountsnk.
Discrete sources in the deep X-ray exposures are predom- InthedeepChandraobservationalargenumberofrelatively
inantly extragalactic. Nevertheless, galactic sources are poten- strongsourcesisdetectedandtherangeofsourcefluxesismuch
tially present in the data and are included in the calculations. toowidetoapplyEq.4intheformgivenabovewherethesource
Photons coming from a discrete sources are distributed in the fluxesarelistedconsecutivelyfromk =2uptosomemaximum
imageinclumpsdefinedbythePSFofthetelescope. valueofkmaxrepresentingthestrongestsourceinthefield.Since
Within the present model, statistical characteristics of the weareinterestedinthecountsatthefaintend(downtok = 2),
countdistributionarefullydefined:therearetwopopulationof thevalueofkmax isselectedattheconventionaldetectionlimit.
events, the first is randomly distributed and the second is con- Allthesourcesabovethisthresholdarepinpointedandremoved
centrated in the PSF shaped clusters. This feature of the count fromthedataandthesubsequentanalysisisconcentratedonthe
distribution is convenientlyformulatedusing the NNST. Let n sourceswhichcannotbeindividuallyrecognized.
t
denotesthetotalnumberofcountsintheinvestigatedfield,n - One can express the number of photons due to k-photon
1
thenumberofeventsdistributedrandomly,or“singlephotons”, sourcesbymeansofthedifferentialsourcecountsN(S):
n - the number of photons due to sources producing each ex-
2
actly 2 photons, n - numberof photonsdue to “three photon” Smax
3 n =k dS N(S)P(k|S), (5)
sourcesandsoon.Thus, k Z
Smin
n +n +...+n +...+n =n , (1)
1 2 k kmax t whereP(k|S)istheprobability,thatthesourcegeneratingflux
1 InalltheanalysisweuseACIS-Ichips0-3. S deliversk-photons,whiletheintegrationlimitsSmin andSmax
A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod 3
definethefullrangeofthesourcefluxes.Itisconvenienttointro- Table1.TheChandraAEGISobservationsusedinthepaper
ducetheinstrumentalcountasafluxunit.Theflux sexpressed
in the ACIS counts in the definite observation is related to the
Obs. Observationandprocessing Processing Exposure
fluxinphysicalunits,S,by:
ID dates version time[s]
s=S/cf, (6) 9450 2007-12-11 2007-12-13 7.6.11.3 29100
9451 2007-12-16 2008-01-02 7.6.11.4 25350
where cf is the conversion factor which has units of 9793 2007-12-19 2007-12-21 7.6.11.4 44750
“ergcm−2s−1/count” and is related to the parameter “exposure 9725 2008-03-31 2008-04-02 7.6.11.6 28050
map” defined in a standard processing of the ACIS data2: ex- 9842 2008-04-02 2008-04-03 7.6.11.6 19450
posure map = cf ·hEi, where hEi denotes the average photon 9844 2008-04-05 2008-04-06 7.6.11.6 34600
9866 2008-06-03 2008-06-05 7.6.11.6 31450
energy.
Forthepowerlawsourcecounts,N(s)= N s−b,wehave: 9726 2008-06-05 2008-06-06 7.6.11.6 39750
o 9863 2008-06-07 2008-06-07 7.6.11.6 24100
Γ(k−b+1,s )−Γ(k−b+1,s ) 9873 2008-06-11 2008-06-12 7.6.11.6 30850
nk = No minΓ(k) max . (7) 9722 2008-06-13 2008-06-15 7.6.11.6 19900
9453 2008-06-15 2008-06-17 7.6.11.6 22150
If the slope of the counts b is constant over a sufficiently 9720 2008-06-17 2008-06-18 7.6.11.6 26000
widerangeoffluxes(i.e.s <<2ands >k ),onemight 9723 2008-06-18 2008-06-20 7.6.11.6 30950
min max max
replacetheintegrationlimits s and s by0and∞,respec- 9876 2008-06-22 2008-06-23 7.6.11.6 25050
min max
9875 2008-06-23 2008-06-25 7.6.11.6 33150
tively,toget:
Totalexposure 464650
Γ(k−b+1)
n = N . (8)
k o Γ(k)
3.1.Theexposuremap
Thus, for the source counts represented by a single power
lawoversufficientlywiderangeoffluxes,Eq.4takestheform: Theobservations,listedinTable1weremergedtocreateasin-
glecountdistributionandexposuremap.Acircularareacovered
n1 P(r|1) + No kmax Γ(k−b+1) P(r|1)P(r|k)= P(r). (9) byallthepointingswitharelativelyuniformexposure,centered
nt nt Xk=2 Γ(k) atRA = 14h20m12s,Dec = 53◦00′ withradiusof6.′0hasbeen
selectedforfurtherprocessing.Theexposuremapoftheindivid-
The extensionof thisexpressionforthe sourcecountswith ualobservationresultingfromvariousinstrumentalcharacteris-
varying slope is straightforward. In particular, for the broken tics4,hasacomplexstructure.Ineffect,theexposuremapofthe
power law, the function Γ(k − b + 1) is replaced by a proper mergedobservationisevenlessregularandisdevoidofanyclear
combinationoftheincompletegammafunctions.Substituting symmetries.Ontheotherhand,aroughtextureoftheindividual
exposureis reducedandsmoothedin a sumof16components.
kmax Toreducefurtherthevariationsoftheexposuremapoverthein-
n =n − n (10)
1 t k vestigatedarea,a thresholdofthe minimumexposurehasbeen
X
k=2 setseparatelyforbothenergyband.Pixelsbelowthisthreshold
andusingEq.8wefinallyget: havenotbeenusedinthecalculations.
A threshold has been defined at ∼ 75% of the maximum
No P(r|1)kmax Γ(k−b+1) [1−P(r|k)]= P(r|1)−P(r). (11) valueoftheexposuremapforthebothenergybands.Ineffect,
nt Xk=2 Γ(k) themaximumdeviationsoftheexposurefromtheaveragevalue
donotexceed15%and18%inthebandsSandH,respectively,
Equation 11 contains two parameters which fully describe and the correspondingrms of the exposuresamountto 5.9 and
thepowerlawsourcecountsintheinvestigatedfluxrange,viz. 6.2%.InTable2theconversionfactorscalculatedusingtherel-
thenormalizationNoandtheslopeb.Sincethenormalizationat evantamplitudesof the exposuremaps are given.In the calcu-
the flux s = kmax is defined by the actual source counts above lations “from counts to flux” a power spectrum with a photon
thisthreshold,onlythesloperemainsunknown. indexΓ =1.4wasassumed(Kimetal.,2007).
ph
Variationsoftheconversionfactor,cf,overtheinvestigated
3. Observationalmaterial area alter the source fluxes via Eq. 6 and, consequently, the
sourcecountsN(S).Forthepowerlawcountsthecfuncertainty
TotestefficiencyofthepresentmethodIhaveselectedasetof16 affectsthecountsnormalizationanddoesnotchangetheslope.
closeChandrapointingswithingtheAEGIS3.Theobservations It is shown in the Appendix that – as long as the cf variations
span a periodof 6 monthsandthe data havebeenprocessedin remainsmall–theymodifytheprobabilitydistributionsP(r|1)
auniformwaywiththerecentpipelineprocessingversions.The andP(r)insuchawaythatthesolutionofEq.11isnotaffected.
detailsof16observationsusedinthepresentpaperaregivenin
Table1.Alltheexposureshavebeenscrutinizedwithrespectto
thebackgroundflaresandonly“goodtimeintervals”wereused 3.2.ThePointSpreadFunction
in the subsequent analysis. The data have been split into two
The Chandra X-ray telescope PSF is a complex function of
energybands:S–soft(0.5−2keV)andH–hard(2−8keV).
sourcepositionandenergy(e.g.Allenetal.,2004).However,to
2 Seehttp://asc.harvard.edu/ciao.Fortherealobservations,boththe computethenearestneighborprobabilitydistributionforcounts
cfandexposure maparefunctions oftheposition. Atthisstagethese generatedbyapoint-likesource,P(r|k),wedonotneedafull
parametersareassumedconstant.
3 All-wavelength Extended Groth strip International Survey, see 4 To name the most obvious: vignetting, gaps between chips, tele-
http://aegis.ucolick.org/index.html. scopewobblingandallkindsofchipimperfections.
4 A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod
Table2.Energybandsandconversionfactors
Energyband Conversionfactorsa
[keV] Average rms minimum maximum
S 0.5−2 1.469 0.087 1.279 1.687
H 2−8 5.546 0.346 4.862 6.511
aTheconversionfactor(cf)hasunitsof10−17(erg·cm−2·s−1)/count.
model of the PSF shape. The aim is to find a convenient ana-
lyticPSFapproximationadequatelyreproducingtheNNSTover
the investigated area for each energy band. The model should
beapplicabletothemergedAEGISobservationsprocessedina
standardway.
ToeffectivelycomputetheP(r|k)wefirstconstructamodel
PSF.Then,therelevantprobabilitydistributionsareobtainedus-
ingtheMonteCarlomethod.We notethattheanalyticformof
the PSF should mimic the radial distribution of counts, while
somedeviationsfromtheazimuthalsymmetryareoflesserim-
Fig.1.Encircledcountfraction(ECF)asa functionofdistance
portance.Severalsimpleanalyticmodelshavebeentestedtofit
from the count centroid. Example distributions are shown for
theobserveddistributionofcountsanditwasfoundthatafunc-
4 sources at 1.′0, 3.′2, 5.′0, and 6.′1 from the field center. Solid
tionoftheform
curves – observed count distributions, dashed curves – fits ob-
rα tainedusingEqs.12and13.
f(<r)= (12)
z+rα+y·rα/2
adequatelyrepresentstheencircledcountfraction(ECF),where
α,z,andyareparametersdependingonthesourcepositionand
energy. Fits of Eq. 12 to the observed count distribution for
several brightest sources have allowed us to find simple rela-
tionshipsbetween these parametersand the off-axisangle.The
wholeprocedurelooksasfollows.WehavenoticedthatthePSF
parametersα,y,andlogzaresatisfactorilyapproximatedbylin-
earfunctionsoftheoff-axisangleθ:
α=a ·θ+b y=a ·θ+b logz=a ·θ+b , (13)
α α y y z z
wherea andb (s = α,y,z)aresixparameterswhicharesub-
s s
stitutedinEq.12andsimultaneouslyfittedtotheobserveddis-
tributionofcountsinanumber(25and31intheSandHband,
respectively)ofthestrongestpoint-likesources.InFig.1asam-
pleoftheresultantfitstotheobserveddistributionsintheSband
isshown.Sincenotallthefitsareofequalquality,theeffectsof
the P(r|k) approximation are carefully examined. As a good
envelopeoferrorsgeneratedbytheimperfectionsofourfitting
procedureswehaveconstructedtwo modelP(r|k)distributions Fig.2. ECF as in Fig. 1 for sources at 3.′2 and 6.′1 from the
using the ECF functionssystematically wider and narrowerby fieldcenter.Solidcurves–observedcountdistributions,dashed
15%ascomparedtothebestfit. curves–bestfitsobtainedusingEqs.12and13,dottedcurves–
Example results of this procedure are illustrated in Fig.2, theECFdistributionswiththeradiusrscaledby±15%.
wheretheenvelopeECFsfortwosourcesat3.′2and6.′1offaxis
angleareshown.Althoughsomedeviationsare quitelarge,the
withineachsourcewasrandomizedaccordingtothemodelECF.
observed ECFs are predominantly contained within the ±15%
Then,foreachsourceadistributionofthenearestneighborsep-
distributioninawiderangeoftheseparationsr.
arations was determined and used to obtain the corresponding
In the observational material used in the present investiga-
amplitudesof P(r|k). The procedurehas been executedfor the
tion the maximum separations found in the NNST very rarely
exceed2′′.Thus,ourfitsappearadequatefortheNNSTanditis bestfitand±15%ECFdistributions.
assumedthatthe±15%limits(indicatedbythedottedcurvesin
Fig.2)definethemaximumsystematicerrorsassociatedwiththe 3.3.The“afterglow”correction
PSFfittingprocedureandwillbeusedtoassessuncertaintiesin
ourfaintsourcecalculations. A charge deposited by a cosmic-ray in the ACIS CCD detec-
In the Monte Carlo computationsof P(r|k) a populationof tor may be released in two or more time frames generating a
108 “sources” of k = 2,3,...,k counts were distributed ran- sequenceofevents,so-called“afterglow”.Theeventsspantyp-
max
domly over the investigated area5. The distribution of counts ically several seconds and do not need to occur in consecutive
frames6. Asa result,the datacontainclumpsofcounts,which
5 Thevalueofk isrelatedthesensitivitythresholdandisdifferent
max
foreachenergyband;seebelow. 6 Seehttp://cxc.harvard.edu/ciao/why/afterglow.htmlfordetails.
A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod 5
mimic very weak sources. Fortunately, the time sequences of
such afterglowcountsspanshorttime intervalsas comparedto
exposure times of all the observations. This allows us to un-
ambiguouslyidentify practically all the afterglowsand remove
themfromtheobservation.
3.4.Pixelrandomization
PositionsofcountsinthestandardACISprocessingarerandom-
izedwithintheinstrumentpixelapproximatedbyasquare0′.′492
a side. Since the typical nearest neighbor separationsare com-
parable to the pixel size, the NNST at small angular scales is
smoothed by the countrandomization.The effect is significant
andgeneratesmostofthestatisticalnoiseinourcalculations.To
assess uncertaintiesintroducedby the countrandomization,12
sets of randomizedobservationswere producedusing the orig-
inaleventdatawithnon-randomized(integer)countspositions.
Then, the NNST was determined for each observation and the
datawereusedtoobtaintheslopebofthesourcenumbercounts
bymeansof Eq.11. A scatter betweenthe12countslopeesti-
matesrepresentsthestatisticalerrorofthepresentmethod.
3.5.Strongsourceremoval Fig.3. The nearest neighbor probability distributions binned
with ∆r = 0′.′1 for the observed counts (solid histogram), and
TomaximizeeffectoftheweaksourcepopulationontheNNST,
betweenrandomand observedcounts(dottedhistogram);error
thestrongsourcesshouldberemovedfromthedata.Thethresh-
barsinthelowerpanelshow1σuncertainties.
oldsourcefluxisdefinedbyk countsintheEq.11.Thus,the
max
valueofk shouldbesetatalevelsufficientlylargetoensure
max
that all the sourcesproducingmore countsthan k are found
max differentseparationsraredependent.Toobtainasetofindepen-
using standard source search criteria. On the other hand, k
max dent equations, we use the differential probability distributions
shouldbesmallenoughtowarrantthatthenumberofsourcesis
∆P(r)= P(r)−P(r+∆R).Inthecalculationstherelevantprob-
adequatelyrepresentedbythefunctionN(s).
ability distributions have been obtained over the hole range of
A catalog of point sources in the AEGIS field is given by the observedseparationswith ∆r = 0′.′1. Then,using the Least
Lairdetal.(2009).Foreachsourcelistedinthatpaper,aradius
Square method the best fit value of the count slope b has been
r enclosing85%ofcountshasbeencalculatedusingthePSF
85 determinedforeachoftherandomizeddistributions.Finally,the
at the source position. Then, several trial values of k were
max average and rms of b was calculated. The rms obtained in this
appliedtoassesscompletenessthresholdintheinvestigatedarea.
wayresultsonlyfromtherandomizationofcounts.
Itwasfoundthatfork =20allthebrightersourcesareclearly
max To facilitate comparison of the present results with
recognized,whilethisvalueislowenoughtoallowforstatistical
those available in the literature, we have adopted from
treatmentofthepopulationofstillweakersources.
Georgakakisetal.(2008)thenumbercountsmodelforthebright
sources. In effect, we fixed the normalization and the number
counts slope in the flux range not coveredby the presentanal-
4. Thesourcecounts ysis, i.e. at fluxes aboveS = 2.9·10−16cgs or k = 20. The
max
present calculations provide the slope best estimate below that
4.1.Thesoftband
flux down to the level determined by the sources generating 2
UsingtheselectioncriteriagiveninSect.3,theareaofthefield counts.Theaveragefluxofthosesourcesdependsonthenumber
andthetotalnumberofcountsinthesoftbandamounttoto97.5 countsslope and within the powerlaw approximationamounts
sq.arcminand73711,respectively.Aftertheremovalofstrong toS ≈(2−b)·cf,wherecf =1.469·10−17cgsistheaverage
w
sourcesaccordingtotheproceduredescribedabove,theareais conversionfactor(seeTable2).
reducedto 92.7sq.arcminandthenumberofcountsto48737. The probability distributions ∆P(r) and ∆P(r |1) averaged
Theaveragecountdensityamountsto0.146persq.arcsec.and over 12 data sets are presented in the upper panel of Fig. 3. A
theaveragedistancetothenearestneighborfortherandomdis- histogram in the lower panel shows the difference between the
tribution is equal to1′.′22. To eliminate a contamination of the bothdistributions;theerrorbarsindicatethehistogramrmsun-
count distribution by strong source photons in the PSF wings, certaintiesobtainedfromthe12pixelrandomization.Dotsshow
theremovalradiusof∼4·r wasapplied. themodelhistogramobtainedfortheNNSTbestsolutionslope
85
The calculationshavebeenperformedas follows.First, the ofb=1.595.
count distributions were used to calculate the P(r) and P(r|1) In Fig. 4 the results based on the NNST for the S band
probabilities.Thedistributionofadistancestothenearestneigh- are superimposed on the differential number counts presented
borforeachcountdefinestheP(r),whilethedistributionofthe by Georgakakisetal. (2008). The countsare normalizedto the
distancesbetweentherandomlydistributedpointsandthenear- Euclidean slope of −2.5. Full dots and dotted line denote the
estobservedcountisusedtodetermineP(r|1).TheNNSThas counts and model by Georgakakisetal. (2008), open circles –
been formulated in the Sect. 2 using the cumulative probabili- the counts by Kimetal. (2007), and the dot-dash curve – the
ties,andestimatesofthecountslopebobtainedfromEq.11for predicted AGN counts from Uedaetal. (2003). The thick line
6 A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod
A question of systematic errors is less straightforward.
Several potential effects could influence our estimate of the
count slope. The first one, already discussed in Sect. 3.2, is
associated with our PSF approximation. The complex shape
and intricate position dependence of the PSF demands ap-
proximate treatment. Our PSF fits inevitably introduce errors.
Unfortunately,anamplitudeoftheseerrorsisdifficulttoassess.
Becauseofthatweadoptedacautiousapproachtothisproblem.
AlongsidethebestfitPSFmodelwehaveconsideredtwoancil-
larysetsofPSFswhichdelineatetheobservedcountdistribution
oftheobservedstrongsourcesandalsoconfinepossibledevia-
tions introducedby our simplified modelof the PSF variations
overthefieldofview.Visualcomparisonoftheactualcountdis-
tributionsandourmodelPSFshowsthatthe±15%modification
of the PSF width account for any potential deficiencies of our
PSFcalculations.
The ±15% uncertainty of the PSF width introduces a sub-
stantialuncertaintyoftheslopeestimate.ForthePSFwiderthen
thebestfitby15%wegetb=1.744±0.207,whileforthe15%
narrower,b = 1.421±0.267.One can summarizethese results
as follows. Statistical 1σ limits around the best fit solution of
b=1.595aredefinedas1.365<b<1.825,whilethecombined
statistical and systematic uncertaintiesare 1.154 < b < 1.951.
Fig.4.Differentialnumbercountsin the0.5−2keV bandnor- The statistical uncertaintiesare shown in Fig. 4 with thin solid
malized to the Euclidean slope. The data points and the dotted linesandthetotaluncertainties–withdashedlines.Oneshould
line are taken from Georgakakisetal. (2008); dot-dash curve note that these “total” error estimates are highly conservative.
shows the AGN model by Uedaetal. (2003); solid lines - the They have been obtained by simple addition of the systematic
numbercountsestimates basedon the NNST:thick solid line - andstatisticalerrorsassumingtheirhighest“possible”values.
the best power-law fit, thin solid lines - 1σ uncertainty range Another source of the slope estimate error is related to the
due to the counting statistics; dashed lines - maximum total uncertaintyofthenumbercountsnormalization.Anymodifica-
uncertainty including potential errors generated by inaccura- tion of the numberof sources at the strong flux end affects the
cies of the PSF fitting. The count normalization at the bright countslopeas well. However,in the relevantflux rangethere-
end of the NNST model is fixed at the amplitude given by alisticnormalizationuncertaintiesremainsmall(seeFig.4)and
Georgakakisetal.(2008)approximation. donotaffectconsiderablytheslopeuncertaintyrange.
Variationsoftheconversionfactoroverthefieldofviewalso
do not play significant role in our slope estimates. As pointed
shows the NNST best fit solution with the slope b = 1.595 outinSect.3.1,onecanincorporatethesevariationswithinthe
andthenormalizationfixedatS = 2.9·10−16cgs.Thenumber investigatedareaintouncertaintyofthecountnormalizationN .
o
counts above that flux are described by the Georgakakisetal. Assuming the differential number counts N(S) = NS−b (with
(2008) model, i.e. over a wide range of fluxes the counts are fluxS inergcm−2s−1),arelationshipbetweenthecountnormal-
approximated by the power law with the differential slope of izationN andtheconversionfactorcfis
o
−1.58. It is clear that the NNST solution is in full agreement
N =N ·cf1−b. (14)
withthedirectsourcecountsderivedfromthedeepChandraex- o
posures.OneshouldemphasizethatthepresentNNSTsolution Thus,forb = 1.595andthermsoftheconversionfactoratthe
is based on the relatively shallow exposureas comparedto the level of ∼5.9% (Table 2), the uncertainty of N amounts to ∼
o
Georgakakisetal.(2008)data.Infact,thesourcesatthelowflux 3.5%andissmallincomparisonwiththeuncertaintyofNitself.
end of the presentsolution on the averagegenerate in our data Substantial variations of the PSF width with the distance
essentiallylessthan2countseach.Evidently,theNNSTiscapa- from the telescope axis limit the effectiveness of the NNST
ble to providea sensitive estimationmethodfor the population method. This is because the nearest neighbor distribution P(r)
ofsourceswhichcannotberecognizedasindividualentities. It fortheEq.11isestimatedusingtheactualdata,i.e.actualdis-
isworthtonotethatcountsduetothesourcesgeneratingthesig- tribution of sources, while the probability P(r|k) is determined
nal,i.e.producing2 ≤ k ≤ 20photons,constituteasmallfrac- byaveragingthemodelsourcesoverthefieldofview.Thisfea-
tionofallthecounts.Inthepresentmodeljust∼1080countsor ture introduces additionaluncertainty if the number of sources
2.2% comes fromthese sources while the remaining97.8% is generatingk ≈ k issmall,butshouldbeoflesserimportance
max
distributedrandomly. inthedeepChandraexposures.
4.2.Errorestimates 4.3.Thehardband
Asmallfractionalcontributionofcountsproducedbysourcesto A fraction of counts produced by discrete sources to all the
thetotalnumberistobeblamedfortheratherlargestatisticalun- countsinthe2−8keVbandissubstantiallysmallerthaninthe
certaintiesoftheNNSTmethod.Inourcasethermsoftheslope 0.5−2keVband.Ineffect,theNNSTmethodislesseffectivein
bintheSbandamountsto0.230.Thesituationisevenworsein theHthanintheSband,i.e.theslopeestimatesintheHbandare
theHband,wherethermsuncertaintyoftheslopereaches0.34 subject to higherstatistical uncertainties.Here we briefly sum-
(seebelow). marizetheresultsintheHband.
A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod 7
althoughinthelattercasetheexpectedaccuracyofourestimate
mightbequitelow.
The constraints obtained for the H band are not restrictive.
This is because the contribution of the non X-ray counts in-
creaseswithenergyandthedatabecomestronglycontaminated
bytheparticlebackgroundwhicheffectively“dilutes”thecounts
concentrationsproducedbythesources.Inthenextpapersome
prospectstoimprovetheS/Nratioabove2keVwillbeexplored.
Acknowledgements. Ithank all thepeople generating the Chandra Interactive
Analysis of Observations software for making a really user-friendly envi-
ronment. This work has been partially supported by the Polish KBN grant
1P03D00327.
References
Allen,C.,Jerius,D.H.,&Gaetz,T.J.2004,Proc.SPIE,5165,423
Galeazzi,M.,Gupta,A.,Covey,K.,&Ursino,E.2007,ApJ,658,1081
Georgakakis,A.,Nandra,K.,Laird,E.S.,Aird,J.,&Trichas,M.2008,MNRAS,
388,1205
Hasinger, G. 1992, in X-ray Emission from Active Galactic Nuclei and the
CosmicX-rayBackground,ed.Brinkmann&Tru¨mper,MPEReportno.235,
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Hasinger,G.,Burg.R.,Giacconi,R.,etal.1993,A&A,275,1
Henley,D.B.,Shelton,R.L.,&Kuntz,K.D.2007,ApJ,661,304
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Laird,E.S.,Nandra,A.,Georgakakis,A.,etal.2009,ApJS,180,102
normalized to the Euclidean slope. The data points and the
Lehmann,I.,Hasinger,G.,Schmidt,M.,etal.2001,A&A,371,833
dashed line are constructed using the 2 − 10keV band from Miyaji,T.,&Griffiths,R.E.2002,ApJ,564,L5
Georgakakisetal. (2008); solid lines - the number counts esti- Sołtan,A.M.1991,MNRAS,250,241
matesbasedontheNNST:thicksolidline-thebestpower-law Sołtan,A.M.2007,A&A,475,837
Sołtan,A.M.2003,A&A,408,39
fit, thin solid lines - 1σ uncertainty range due to the counting
Ueda,Y.,Akiyama,M.,Ohta,K.,&Miyaji,T.2003,ApJ,598,886
statistics.
As a reference data we used those published by AppendixA: Impactoftheexposurefluctuationson
Georgakakisetal.(2008).Thecountsandtheanalyticfitinthe
theNNST
band2−10keVgiveninthatpaperwereconvertedtotheband
2 − 8keV assuming a power spectrum with the photon index Effects of the small variationsof the exposureoverthe field of
Γ=−1.4. viewontheprobabilitydistributionsP(r)andP(r|1)inEq.11are
After the removalof strong sources, the area and the num- investigated.
berofcountsusedintheNNST amountto94.4sq.arcminand Theobserveddistributionofcountsisconsideredhereasthe
101596,respectively.Applyingthe same procedureas in the S realizationofaPoissonprocess.Accordingtothispremise,the
band,the bestfit slopeb = 1.676±0.340was foundusing the probabilityofgettingan eventinthe x−y planeofthefieldof
12randomizeddistributions.ThisisshowninFig.5withathick view(fov)isdescribedbyasmoothfunctionρ(x,y).Theρ(x,y)
lineandtwothinlines.ThesourcesrepresentedbytheNNSTso- amplitudeis proportionalto the exposuremap (Sect. 3.1). One
lution,i.e.sourcesproducing2 ≤ k ≤ 20counts,contributejust can define the “true” probability density ρ (x,y) which would
◦
1214+979 photonsor 1.19+0.96% of all the counts. Due to large describe the distribution of counts expected for the perfect in-
−427 −0.42
statistical uncertainty in this band the NNST does not provide strumentwiththeflatexposuremapand-consequently-acon-
tightlimits onthe numbercountsandwe havenotplottedhere stantconversionfactor.Thus,theρdistributionisrelatedtothe
linesrepresentingtherangeofsystematicuncertainties. ρ andthefluctuationsoftheexposuremap,EM:
◦
EM(x,y)
ρ(x,y)=ρ (x,y)· , (A.1)
5. Conclusions ◦ EM
◦
WehaveestimatedthesourcenumbercountsintheSbanddown
whereEM istheexposuremapoftheperfectinstrument.Anat-
to ∼2·10−17cgsusingthemergeddata withthe integratedex- ◦
uralnormalizationoftheexposuremapisassumed:hEM(x,y)i=
posuretimeof∼465ks.Thisfluxlevelisbelowthestandardde-
EM ,wherethebracketsh...idenotetheaveragingoverthefov.
◦
tectionthresholdforindividualsourcesinthe deepestChandra
Since thedistributionsρ andEM areindependent,hρi = hρ i.
◦ ◦
exposuresof2Ms(Kimetal.,2007).
To quantifythe amplitude of the EM fluctuationswith a single
Our slope estimate below S = 10−16cgs fits perfectly the
parameterε,wedefineafunction f(x,y):
actualdiscretesourcecountsdeterminedusingsuchdeepobser-
vations.ItshowstheNNST potentialasaneffectivetoolin the EM(x,y)
κ(x,y)= =1+ε f(x,y), (A.2)
investigation of the extremely weak source population. In the EM
◦
secondpaperofthisseriesweplanto applythe NNSTmethod
to the Chandra Deep Fields. With a 2Ms exposure the NNST in such a way that hfi = 0 and σ2 = hf2i = 1. We also
f
willallowtoassessnumbercountsdownto∼4·10−18cgsinthe have hκi = 1 and σκ = ε, where σκ is the rms of κ. Since
0.5−2keV band and to ∼2·10−17cgs in the 2−8keV band, σκ = σEM/EM◦, the data in the Table 2 show the ε parameter
8 A.M.Sołtan:ThenearestneighborstatisticsforX-raysourcecountsI.Themethod
characterizingthepresentobservationalmaterialremainssmall;
in particular, it is equal to 0.059 and 0.062 for energy bandsS
andH,respectively.
The expected nearest neighbor probability distributions
P(r|1) and P(r) are related to the count distribution ρ in a fol-
lowingway:
P(r|1)=he−πr2ρ(x,y)i, (A.3)
1
P(r)= hρ(x,y)·e−πr2ρ(x,y)i. (A.4)
hρi
SubstitutingEqs.A.1andA.2intoEqs.A.3andA.4onecanex-
pandthedistributionsP(r|1)andP(r)inpowersoftheparameter
ε.Sincethedistributionsof f(x,y)andρ (x,y)areuncorrelated,
◦
onefindsthatthelinearterminεvanishes.