Is the anti-correlation between the X-ray variability amplitude and black hole mass of AGNs intrinsic?

0.06 MB·English

by  Yuan Liu

Astronomy&Astrophysicsmanuscriptno.ms c ESO2008
(cid:13)
February2,2008
Is the anti-correlation between the X-ray variability amplitude and
black hole mass of AGNs intrinsic?
(ResearchNote)
YuanLiu1 andShuangNanZhang1
8
0 PhysicsDepartmentandCenterforAstrophysics,TsinghuaUniversity,Beijing,100084,China
0 e-mail:[email protected], [email protected]
2

n ABSTRACT
a
J
Aims.BoththeblackholemassandtheX-rayluminosityofAGNshavebeenfoundtobeanti-correlatedwiththenormalizedexcess

7 variance(σ2 )oftheX-raylightcurves.Weinvestigatewhichcorrelationwithσ2 istheintrinsicone.
rms rms
1 Methods.Wedivideafullsampleof33AGNs(O’Neilletal.2005)intotwosub-samples.Theblackholemassesof17objectsin
sub-sample1weredeterminedbythereverberationmappingorthestellarvelocitydispersion.Theblackholemassesoftheremaining

] 16objectswereestimatedfromtherelationshipbetweenbroadlineregionradiusandopticalluminosity(sub-sample2).Thenpartial
h
correlationanalysis,ordinaryleastsquaresregressionandK-Stestsareperformedonthefullsampleandthesub-samples,respectively.
p
Results.Wefindthatσ2 seemstobeintrinsicallycorrelatedwiththeblackholemassinthefullsample.However,thisseemstobe
- rms
o causedbyincludingthesub-sample2intotheanalysis,whichintroducesanextracorrelationbetweentheblackholemassandthe
r luminosityandstrengthensanycorrelationwiththeblackholemassartificially.Therefore,theresultsfromthefullsamplemaybe
t misleading.Theresultsfromthesub-sample1showthatthecorrelationbetweenσ2 andtheX-rayluminositymaybetheintrinsic
s rms
a oneandthereforetheanti-correlationbetweenσ2rmsandtheblackholemassisdoubtful.
[
Keywords.X-rays:galaxies–galaxies:active–methods:statistical

1
v
8 1. Introduction black hole mass and σ2 , and some models have been subse-
9 rms
quentlyconstructedtoexplainthiscorrelation.
5 X-rayemissionfromactivegalacticnuclei(AGNs)exhibitsvari-
2 abilityon time scalesfromminutesto days.Thisindicatesthat In this paper we revisit this problem,by studying the sam-
. X-rays are likely to be emitted from the inner most regions of pleofO’Neilletal.(2005),whichincludes33AGNsanduses
1
0 AGNsandthevariabilitymayberelatedtotheimportantproper- nearlythesame timescale foralltheseobjects.Theblackhole
8 tiesofthecentralengine.Lawrence&Papadakis(1993)utilized massesof17objectsinthissampleweredeterminedbythere-
0 long term EXOSAT observations to investigate the power den- verberationmappingorthestellarvelocitydispersion(wedenote
: sityspectraof12AGNs.Theyfoundthepowerdensityspectra theseobjectsassub-sample1inthefollowing).Theblackhole
v
could be described as a power law P ν α with a mean in- massesoftheremaining16objectswereestimatedfromtherela-
i −
X dex α = 1.55, and the amplitude was ∝anti-correlated with the tionshipbetweenbroadlineregionradiusandopticalluminosity
r X-rayluminosity.Detailedstudiesofthepowerdensityspectra (wedenotetheseobjectsassub-sample2inthefollowing).We
a have been performedusing RXTE and XMM-Newton data. The findthattheopticalluminosity(whichisusedindeterminingthe
universal relation between the black hole mass and the ”break blackholemassforsub-sample2)hasobviouscorrelationwith
time”inthepowerdensityspectrawasfoundbothinstellarmass theX-rayluminosityin2-10keVband,asshowninFigure1(a).
andsupermassiveblackholes(e.g.Uttely&McHardy2005).A Thevaluesofthecorrelationcoefficientsbetweenopticallumi-
tighterrelationwasdiscoveredwhenthebolometricluminosity nosity and X-ray luminosity for sub-sample 1 and sub-sample
wasinvolved,i.e.T M2/L(McHardyetal.2006).However, 2 are 0.907and0.910,respectively.Evenif the datumofNGC
B
∝
due to the limited observationdata, the accurate powerdensity 4395isexcludedfromsub-sample2,thevalueofthecorrelation
spectra are only available for a small number of AGNs. As an coefficient is still 0.819. It is well known that there is a strong
alternative,thenormalizedexcessvariance(σ2 )canbeeasily correlation between the optical luminosity and the black hole
rms
calculated,anditwasfoundthatσ2 isanti-correlatedwithX- mass(Kaspietal.2000).Thusifσ2 isintrinsicallycorrelated
rms rms
rayluminosity(e.g.Almainietal.2000). withoneofthem,anartificialcorrelationwiththeotherwillap-
As a result of the progress in determining the black hole pear. However, there will be an additional artificial correlation
massesinAGNs,therelationbetweenthevariabilityandblack forsub-sample2duetotheutilizationoftheopticalluminosity
holemasshasalsobeeninvestigated.Lu&Yu(2001)foundthe indeterminingtheblackholemass,whereastheblackholemass
anti-correlationbetweentheblackholemassandσ2 ,andsug- ofsub-sample1is independentlyobtainedbythereverberation
rms
gested this correlationwas an intrinsic one, rather than the ap- mappingorthestellar velocitydispersion.Therefore,theresult
parentanti-correlationbetween the X-ray luminosity and σ2 . fromthefullsample(especiallyforthatfromthesub-sample2)
rms
SeveralauthorsalsoaddressedthisproblemfollowingLu&Yu’s may be misleading. Furthermore,due to the less reliable black
work.(e.g. Bian & Zhao 2003;Papadakis2004;O’ Neill et al. holemassofsub-sample2,someunclearsystematicbiasesmay
2005).Theseauthorsconfirmedtheanti-correlationbetweenthe be introduced into the analysis. To find out which correlation
2 YuanLiuandShuangNanZhang:Anti-correlationbetweenAGNX-rayvariabilityamplitudeandluminosity(RN)
46 of them indicate σ2 is intrinsically correlated with the black
rms
−1λ L (erg s)λ44442345 °+ SSuubbssaammppllee 21 htthooeltehmemolraiemssri,eterlaidathbseliezresthuoabfn-tshtahemepplrlueems1einnstohssoiawtmys.pHtlheoe,wwceoevnectrro,anrthycelrueadsneualtlthysa.stiDstuhoeef
log −44401 NGC 4395 a breestuwltesenoflusmubin-soasmityplean2diσs2rmdosuibsttfhuel ainntdrinmsaicyboeneth.eMcoorrererloabtiuosnt
−1 conclusionwillbededucedwhenalargersampleisavailable.
2σ)rms−−12. 5
og(−2.5 2.2.Ordinaryleastsquaresregression
l
−3
−3.5 b c To verifythe resultsobtainedin 2.1,we performthe ordinary
−4 40logL241−10 ke4V2 (erg43 s−1)44 5 6 log7( M)8 9 10 leastsquareregressiontothefull§sample,sub-sample1andsub-
sample 2, respectively.The regression equationis, log(σ2 ) =
Fig.1.(a):ThecorrelationbetweenX-rayluminosity(2-10keV) AlogM+BlogL+C. rms
andopticalluminosity(λ=5100 A◦ ).Thepointintheleft-down Theresultsoftheregressionforthethreesamplesaresum-
cornerisNGC4395,whichisadwarfSeyfertgalaxy.Thevalue marized in Table 2. In Figure 2, we show the comparison be-
of the correlation coefficient is not significantly influenced by tween the valuesof σ2 predictedby the resultsof the regres-
rms
whetherthispointisincludedornot(seethetextfordetails).(b): sionandtheobservedones.
ThedataofthenormalizedexcessvariancesandtheX-raylumi- TheresultsofFstatisticdemonstratethehighsignificanceof
nosities(2-10keV).(c):Thedataofthenormalizedexcessvari- the linear correlation.However,the values of χ2 are still large,
ancesandblackholesmasses.Thevaluesofblackholemasses especiallyforthefullsample.Weshouldnoticethatthedepen-
are in units of M . The data are obtained from O’ Neill et al. dence on the black hole mass and the luminosity seems to be
(2005)andtheref⊙erencestherein. different for the two sub-samples. For sub-sample 1, σ2 ap-
rms
pearsto dependweakly on the black holemass, whereasit de-
pendsmorestronglyonthe X-rayluminosity.Due to thesmall
is intrinsic, we perform the partial correlation analysis to sub- sample, the difference between values of A and B are not very
sample1andsub-sample2separatelyin 2.1.Theordinaryleast significant(theyarecoincidencewithinthe95%confidencein-
§
squaresregressionresultsareshownin 2.2asanotherapproach terval).However,ifthesub-sample2isincluded,thedependence
§
tothisproblem.K-Stestsareperformedin 2.3totestwhether ontheblackholemassisstrengthenedandthegoodnessofthe
§
sub-sample1andsub-sample2aredrawnfromthesameparent regression decreases dramatically. The value of the total χ2 of
population.In 3wediscussourresultsandmakeconclusions. thesub-samplesis139(27);therefore,theprobabilityoftheim-
§
provement by chance is only about 10 9 (obtained by F-test).
−
Thustheaboveresultsindicatethatthesub-sample1and2are
2. Dataanalysis likelytoobeydifferentcorrelationrelationshipsanditisnotap-
2.1.Partialcorrelationanalysis propriatetocombinethemintoonesample.
Thepartialcorrelationanalysisisanappropriatemethodtodis-
entanglethecorrelationbetweenvariables.Thedefinitionofthe 2.3.K-Stests
firstorderpartialcorrelationcoefficientbetweenvariablesxand
cyoinst(rKolelneddavlalr&iabSlteuaarntd1r977i)s,trhxey.zc=orr√el(ar1xt−yi−ro2xrznx)z(r1cz−yorez2yffi),cwiehnetrebeztiwsethene sTtwaomoinesvupebas-rtseiganmattdepilswetsrh.iebTtuhhteieorncth,uwemteuwlfiaortsisvtuepbe-drsifasomtrrmipbluethtsieoanr1eDfduKrnac-wtSinotnefssrtoomtof ttthhheee
xy
variablesxandy. twosamplesarecalculatedfirst.Thenthemaximumvalueofthe
We adopt the data of the black hole mass, the X-ray lumi- absolute difference between two cumulative distribution func-
nosityandσ2 fromO’Neilletal.(2005)andpresentthemin tionsisusedasthestatistictoobtainthesignificanceofthedif-
rms
Figure 1 (b) and (c) for clarity. The correlationanalysis is per- ference (see details of the K-S test in Press et al. [1992]).The
formedonthefullsampleandthesub-samples,respectively.The significancesof the differencesare 89%, 25% and 96% for the
resultsareshowninTable1. distributions of the black hole mass, the luminosity and σ2 ,
rms
For the full sample, both the black hole mass and the X- respectively(thecumulativedistributionfunctionsareshownin
rayluminosityshowstrongapparentanti-correlationswithσ2 Figure 3). Clearly exceptfor the X-rayluminositydistribution,
rms
(Figure 1 [b] and [c]). However, it appears that after the black boththe blackhole mass and σ2 for the two sub-samplesare
rms
hole mass is controlled, the correlation between the luminos- not likely drawn from the same parent population. There is no
ity and σ2 is not significant. On the contrary, the correlation obviousreasonaccountingforthedifferences,thereforethisre-
between trhmes black hole mass and σ2 is still significant after sultislikelytobeduetosomeunknownselectioneffects,which
rms
theluminosityiscontrolled.TheseresultsseemtosupportLu& shouldbeinvestigatedfurtherinthefuture.
Yu’ssuggestionthatthecorrelationbetweentheblackholemass Since it seems visually that the difference between the dis-
andσ2 istheintrinsicone.However,asdiscussedin 1,since tributions of sub-sample 1 and sub-sample 2 in Figure 1 (b) is
rms §
any correlation between the black hole mass may be strength- moresignificantthanthatinFigure1(c),weperformthe2DK-S
ened artificially by the sub-sample 2, we should exclude them test to investigate this problem. The results of the 2D K-S test
when investigating the intrinsic correlation. For sub-sample 1, show that the significance of the difference in Figure 1 (b) is
thecorrelationbetweentheblackholemassandσ2 disappears 89.4%,whereassignificanceofthedifferenceinFigure1(c)is
rms
when the luminosity is controlled, whereas the correlation be- 97.6%.Thisunexpectedresultisduetotheexistenceofthepoint
tweentheluminosityandσ2 isstillsignificant.Theresultsof of NGC 4395. After this pointis removed,the 2D K-S test re-
rms
sub-sample2areconsistentwiththoseofthefullsample.Both sultsofFigure1(b)and1(c)are96.5%and92.1%,respectively.
YuanLiuandShuangNanZhang:Anti-correlationbetweenAGNX-rayvariabilityamplitudeandluminosity(RN) 3
Table1.Resultsofpartialcorrelationanalysis.
r r r r
Lσ Mσ Lσ.M Mσ.L
Fullsample -0.636(>99.99%) -0.697(>99.99%) -0.277(87.6%) -0.452(99.1%)
Sub-sample1 -0.856(>99.99%) -0.560(98%) -0.781(>99.99%) -0.003(0.8%)
Sub-sample2 -0.605(98.7%) -0.784(>99.99%) 0.238(60.8%) -0.653(99.2%)
Table 2. Resultsofordinaryleast squaresregression.Thesignificanceofthe linearcorrelationis obtainedbythe F statistic. The
value of χ2 is calculated from the regression result to estimate the goodnessof the regression. The errorscorrespondingto 95%
confidenceintervalsareshown.
A B C Fstatistic χ2(dof)
Fullsample -0.38 0.28 -0.22 0.30 10.4 11.4 16.6(>99.99%) 708.6(30)
± ± ±
Sub-sample1 0.00 0.45 -0.71 0.33 28.3 12.2 19.2(>99.99%) 46.4(14)
± ± ±
Sub-sample2 -0.56 0.39 0.19 0.45 -6.06 17.3 11.4(99.86%) 92.6(13)
± ± ±
0 thesub-sample2.IftheX-rayluminosityistheprimaryquantity,
−0.5 a b c then this will artificially strengthen any correlation with black
2σ)og(rms−1−.15 hinoglethmeaisnst.riWnseicthceorrerfeolaretioshnowulidtheσxc2rmlus.dAecthceomrdiwnghetno itnhveersetsigualtts-
d l−2 fromthesub-sample1,weconcludethatthecorrelationbetween
e
dict−2.5 σ2 andtheX-rayluminositymaybetheintrinsicone,whereas
Pre−3 thremsapparentcorrelation between σ2 and the black hole mass
rms
−3.5 isdoubtful.OurK-Stestsalsosuggestthatsub-samples1and2
−−44 −3 −2 −1 −−44 −3 −2 −1 −−44 −3 −2 −1 0 arenotlikelydrawnfromthesameparentpopulation.
Observed log(σ2 ) As discussed in Lu & Yu (2001),several mechanisms may
rms
be responsible for the correlation between σ2 and the X-ray
rms
Fig.2.Thecomparisonbetweenthevaluesofσ2 predictedby luminosity,suchasthehot-spotmodel,theobscurativevariabil-
rms
the resultsof the regressionandthe observedones. The results ity and so on. After the apparentcorrelation between σ2 and
rms
fromthe fullsampleare shownin (a). (b):The same as(a) but the black hole mass was discovered, some models accounting
forthesub-sample1.(c):Thesameas(a)butforthesub-sample for this correlation were proposed (e.g. O’ Neill et al. 2005,
2. Pessah2007).However,itneedstobeverifiedwhetherthecorre-
lationisintrinsic.Althoughtheblackholemassesofaboutthree
1 dozen AGNs have been determined by the reverberation map-
0.9 a b c
ping method, the size of our sample is still limited due to the
0.8
lack of long enoughand highquality observationdata of these
0.7
0.6 objects. More conclusive results could be obtained when more
F
CD0.5 andhigherqualitydatabecomeavailable.
0.4
0.3
Acknowledgements. We thank the referee, Andy Lawrence, for valuable
0.2
comments and suggestions. SNZ acknowledges partial funding support by
0.1
DirectionalResearchProjectoftheChineseAcademyofSciencesunderproject
0
4 5 6log7(M)8 9 3l9ogL402−1410 ke42V (e43rg 4s4−−14) −3.5−3 l o−g2.5(σ−22r m −s1).5−1 −0.5 Nunod.eKrJpCroXje2c-tYnWo.-T100532a1n0d01b,y1t0h7e3N30at1i0onaanldN1a0tu7r2a5l3S1c3i.enceFoundationofChina
Fig.3.Thecumulativedistributionfunctions(CDF)oftheblack
hole mass(a), the X-ray luminosity(b)and σ2 (c). The solid References
rms
linesare the dataof the sub-sample1, andthe dashedlinesare
Almaini,O.,etal.2000,MNRAS,315,325
thedataofthesub-sample2.Thevalueofblackholemassisin
Bian,W.,&Zhao,Y.2003,MNRAS,343,164
unitsofM . Kaspi,S.,etal.2000,ApJ,533,631
⊙ Kendall, M.,&Stuart, A.1977,TheAdvanced TheoryofStatistics (London:
Griffin)
Althoughthesignificanceofthedifferenceineachfigureishigh, Lawrence,A.,&Papadakis,I.1993,ApJ,414,L85
thevisualdifferencebetweentwothefiguresisnotsignificant. Lu,Y.,&Yu,Q.2001,MNRAS,324,653
McHardy,I.,etal.2006,Nature,444,730
O’Neill,P.,etal.2005,MNRAS,358,1405
Papadakis,I.2004,MNRAS,348,207
3. Discussionsandconclusions Pessah,M.2007,ApJ,655,66
Press, W., et al. 1992, Numerical Recipes in C, Cambridge University Press,
In 2,wehaveperformedthepartialcorrelationanalysisandthe Cambridge
§
regression on the sample and found that the apparent intrinsic Uttely,P.,&McHardy,I.2005,MNRAS,363,586
correlationbetweenσ2 andtheblackholemassislikelytobe
rms
causedbyincludingthesub-sample2intotheanalysis.Because
theblackholemassesofAGNsinsub-sample2wereestimated
from their optical luminosity which in turn is positively corre-
lated with their X-ray luminosity, an extra correlation between
theblackholemassandX-rayluminositywillbeintroducedby