The relationship between X-ray variability amplitude and black hole mass in active galactic nuclei

0.39 MB·English

by  Paul M. O'Neill

Mon.Not.R.Astron.Soc.000,1–13(2004) Printed2February2008 (MNLATEXstylefilev2.2)
The relationship between X-ray variability amplitude and
black hole mass in active galactic nuclei
Paul M. O’Neill1⋆, Kirpal Nandra1, Iossif E. Papadakis2,3 and T. Jane Turner4,5
1Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2BW
2Department of Physics, University of Crete, 71 003, Heraklion, Crete, Greece
3IESL, FORTH-Hellas, 71 110, Heraklion, Crete, Greece
4Laboratory for High Energy Astrophysics, Code 660, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
5Universityof Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
5
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Accepted. Received.
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ABSTRACT
1
2
We have investigated the relationship between the X-ray variability amplitude

and black hole mass for a sample of 46 radio-quiet active galactic nuclei observed
1
v by ASCA. Thirty-three of the objects in our sample exhibited significant variability
1 over a time-scale of ∼40 ks. We determined the normalised excess variance in the
7 2–10keVlightcurvesoftheseobjectsandfoundasignificantanti-correlationbetween
4 excessvarianceandblackholemass.Unlike mostpreviousstudies,we havequantified
1 the variability using nearly the same time-scale for all objects. Moreover, we provide
0 a prescription for estimating the uncertainties in variance which accounts both for
5
measurement uncertainties and for the stochastic nature of the variability. We also
0
presentananalyticalmethodto predictthe excessvariancefroma modelpowerspec-
/
h trumaccountingforbinning,samplingandwindowingeffects.Usingthis,wemodelled
p the variance–mass relation assuming all objects have a universal twice-broken power
- spectrum,withthepositionofthe breaksbeing dependentonmass.Thisaccountsfor
o
thegeneralformofthevariance–massrelationshipbutisformallyapoorfitandthere
r
t is considerablescatter. We investigatedthis scatter as a function of the X-ray photon
s
a index, luminosity and Eddington ratio. After accounting for the primary dependence
: of excess variance on mass, we find no significant correlation with either luminosity
v
or X-ray spectral slope. We do find an anti-correlation between excess variance and
i
X the Eddington ratio, although this relation might be an artifact owing to the uncer-
r tainties in the mass measurements. It remains to be established that enhanced X-ray
a variability is a property of objects with steep X-ray slopes or large Eddington ratios.
Narrow-line Seyfert 1 galaxies, in particular, are consistent with being more variable
than their broad line counterparts solely because they tend to have smaller masses.
Key words:
galaxies:active – galaxies:nuclei – X-rays:galaxies– galaxies:Seyfert
1 INTRODUCTION ated from these light curves were described as a power-
law P ∝ ν−α with a steep ‘red-noise’ index of α ∼ 1.5
Variability was discovered in the X-ray emission from ac-
and an amplitude inversely proportional to the luminos-
tive galactic nuclei (AGNs) roughly three decades ago
ity(Lawrence & Papadakis1993;Green et al. 1993).Itwas
(e.g.,Marshall et al.1981,andreferencestherein).EXOSAT
clear that this power-law must break at some lower fre-
subsequently obtained well-sampled light curves on time-
quency, or the power would diverge, and some evidence
scales of minutes to days, and the power spectra gener-
for this was found using longer-term archival observations
(McHardy 1988; Papadakis & McHardy 1995). It was not
until the launch of RXTE, however, that this break was
⋆ E-mail: [email protected] (PMO); measured definitively(Edelson & Nandra1999).
[email protected] (KN); [email protected] (IEP);
[email protected](TJT) A number of high quality power spectra have now
(cid:13)c 2004RAS
2 P. M. O’Neill, K. Nandra, I. E. Papadakis and T. J. Turner
been obtained, primarily using RXTE and XMM- frequency 20 times higher than that deduced for the other
Newton data (e.g., Uttley et al. 2002; Markowitz et al. theSeyfert 1s.
2003; Vaughan et al. 2003b; McHardy et al. 2004; These works show that excess variance can be a use-
Uttley & McHardy 2004). These have shown breaks to ful complement to full-blown power-spectral analysis, and
be common and emphasized the similarity of AGN power have the advantage that they can be applied to a larger
spectra to that of the black hole binary Cyg X-1. In the number, and wider variety of objects. As has been shown
low/hard state, the power spectrum of Cyg X-1 exhibits a by Vaughan et al. (2003a), some caution must be exercised
twice-broken power-law which breaks from a slope of α∼0 when interpreting excess variance measurements, primarily
to 1 at the ‘low-frequency break’ (νLFB) and from α∼1 to due to the red-noise shape of the power spectra and the
2 at the‘high-frequency break’ (νHFB),with νHFB∼1–6 Hz stochastic nature of the variability. Such effects have not
(e.g.,Belloni & Hasinger1990b).Inthehigh/soft state,the been accounted for in the majority of previous works. The
power spectrum exhibits only a high-frequency break, with intentionoftheworkpresentedhereistoinvestigatethere-
νHFB ∼10–15 Hz (e.g., Cui et al. 1997; Revnivtsevet al. lationshipbetweenexcessvarianceandmassinalargesam-
2000). Though still a subject of debate, the emerging pleofAGN,improvingonthesepreviousstudiesbyfullyac-
consensus is that we usually see the high-frequency break counting for measurement uncertainties, sampling and red-
in the AGN power-spectra, although two breaks are appar- noiseeffectsinthecalculationoftheexcessvarianceandits
ently seen in two objects (viz, AKN 564 and NGC 3783; uncertainty.
Papadakis et al. 2002; Markowitz et al. 2003). In another,
the narrow-line Seyfert 1 (NLS1) NGC 4051, there is
no low-frequency turnover to α = 0 down to very low
2 THE TARTARUS DATABASE AND THE
frequencies,whichledMcHardy et al.(2004)tohypothesise
AGN SAMPLE
that this object (and possibly all NLS1s) resembled Cyg
X-1in thehigh/soft state. The Tartarus1 database contains products for ASCA ob-
Determining accurate power-spectra for AGN is diffi- servations with targets designated as AGN (Turner et al.
cult, as it requires high quality data with near-even sam- 2001).Weselectedradio-quietobjectsthathavedatainthe
pling. Such data are available only for a limited number of Tartarus (Version 3) database and also for which we could
objects and are very costly to obtain in terms of observing conveniently obtain a measurement of the black hole mass,
time.ItisnonethelessveryusefultoquantifytheX-rayvari- M•. Seyfert 2 objects were excluded from our sample, with
ability of AGN to compare with other properties, and nor- the exception of NGC 5506 because for this object we are
malised excess variance, denoted as σ2 , is much simpler confidentofseeingtheX-rayemissiondirectly(Blanco et al.
NXS
to calculate (Nandraet al. 1997a). An anti-correlation was 1990). This initial sample comprised 68 AGN. We utilised
found between excess variance and luminosity for a sam- theTartarusanalysispipelinetoextractlightcurvesforthe
ple of AGNs observed by the Advanced Satellite for Cos- objectsinthissample.Aswedescribeindetailinthefollow-
mology and Astrophysics (ASCA),confirmingtheEXOSAT ing Section, not all light curves were suitable for our anal-
results but with a larger sample of objects (Nandraet al. ysis. Having screened the available data, there remained 46
1997a).LaterworkalsousingASCAdatarevealedthat,for objects for which we could suitably characterise the X-ray
a given luminosity, the X-ray light curves of NLS1s exhibit variability. These objects are listed in Table 1. Note that,
alarger excessvariancethantheclassical Seyfert1galaxies while a flux limit was not formally applied to our sample,
(Turneret al. 1999; Leighly 1999a). the effect of the screening process was to exclude objects
havinga low countingrate.
Lu & Yu (2001) and Bian & Zhao (2003), again using
Recent progess in measuring black hole masses has
ASCA data, studied the relationship between the excess
made possible the work we present here. We preferen-
variance(on atime-scale ofroughly1d)andtheblackhole
tially used the reverberation-mapping mass estimate from
mass. Those studies revealed an anti-correlation between
Peterson et al.(2004).Ifthiswasnotavailablethenweused
σ2 and mass, which is suggestive that this is the pri-
NXS the mass estimate as determined from either the stellar ve-
mary relationship rather than with luminosity. The NLS1s
locitydispersion (Gebhardt et al.2000)ortheempiricalre-
appearedtofollow thesamerelationship astheotherAGN.
lationship between the broad-lineregion radiusand 5100 ˚A
Papadakis(2004)investigatedtherelationshipbetween luminosity (Wandelet al. 1999). The masses are given in
excess variance and black hole mass on much longer time- Table 1 where we also list the method used to determine
scales(∼300d)usingRXTE dataonasampleof10AGNs. the mass and the corresponding reference. The masses for
The classical Seyfert 1 galaxies followed a variance–mass most objects were available in the literature. For 8 objects
relation that is consistent with a universal power-spectral in Table 1 we obtained optical spectral information from
shape as described above for the low/hard state of Cyg X- Grupeet al. (2004) and utilised eqn. 6 from Kaspi et al.
1. In the universal model used by Papadakis (2004), νHFB (2000) and eqns. 1 and 2 from Woo & Urry (2002) to de-
is inversely proportional to black hole mass, and the am- termine M•.
plitude, when represented in power × frequency space, is The2–10keVluminosityL2−10 keV andhard-X-ray(ei-
assumed tobeconstant.Inagreement withthepowerspec- ther 2–10 keV or 3–10 keV) photon index Γ are also listed
trum analysis of McHardy (2004), Papadakis (2004) found in Table 1 for those objects in which we detected variabil-
thattheNLS1NGC4051didnotfollowthesamevariance– ity. The majority of L2−10 keV and Γ values were taken
mass relationship described by the classical Seyfert 1s. The
excess variance of NGC 4051 was consistent with a singly-
broken power-law, breaking from α=1 to 2, with a break- 1 http://astro.ic.ac.uk/Research/Tartarus
(cid:13)c 2004RAS,MNRAS000,1–13
X-ray variability amplitude and black hole mass in active galactic nuclei 3
Table 1.X-rayspectralandvariabilityinformationforobjects havingatleast1validlightcurvesegment. The2–10keV luminosity
andhard-X-rayphoton indexaregivenforobjectsinwhichvariabilitywasdetected.
Name M• LX Γ Num. Num. σN2XS logσN2XS ∆logσN2XS Refs.
Seq. Seg. ±Boot.Unc. ±TotalUnc. ±Total Unc.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MRK335 7.15 43.07 1.87 1 1 (3.12±1.87)×10−3 −2.51±0.41 −0.44±0.41 R,1,2
PG0026+129 8.59 44.53 1.96 1 3 (1.31±1.92)×10−3 −2.88±0.66 0.61±0.66 R,1
TONS180 7.09 43.58 2.43 2 26 (1.59±0.10)×10−2 −1.80±0.07 0.21±0.07 L,3,4
IZw1 7.20 43.35 2.40 1 1 (1.88±0.92)×10−2 −1.73±0.39 0.39±0.39 L,3,4
F9 8.41 43.91 1.91 8 6 (3.49±5.52)×10−4 −3.46±0.70 −0.16±0.70 R,1,2
RXJ0152.4−2319 7.87 ... ... 1 2 <6.5×10−3 <−1.94 ... L,5
MRK0586 7.86 44.07 2.22 1 3 (2.57±0.75)×10−2 −1.59±0.22 1.17±0.22 L,6,4
MRK1040 7.64 42.40 1.69 1 1 (1.20±0.65)×10−2 −1.92±0.40 0.62±0.40 S,6,7
NGC985 8.05 43.50 1.73 1 2 (3.47±1.76)×10−3 −2.46±0.32 0.48±0.32 L,5
1H0419−577 8.58 ... ... 2 3 <4.31×10−3 <−2.12 ... L,3
F303 6.37 43.03 1.92 1 1 (6.72±6.03)×10−3 −2.17±0.44 −0.74±0.44 L,5
AKN120 8.18 43.88 1.93 1 2 (3.78±7.67)×10−4 −3.42±0.91 −0.35±0.91 R,1,8
PG0804+761 8.84 ... ... 1 2 <3.37×10−3 <−2.23 ... R,1
PG0844+349 7.97 ... ... 1 2 <1.17×10−2 <−1.69 ... R,1
MRK110 7.40 ... ... 1 1 <1.63×10−3 <−2.55 ... R,1
PG0953+415 8.44 ... ... 1 2 <8.18×10−3 <−1.85 ... R,1
NGC3227 7.63 41.66 1.52 2 4 (2.41±0.20)×10−2 −1.62±0.16 0.91±0.16 R,1,2
MRK142 6.76 43.17 2.12 2 1 (4.54±1.33)×10−2 −1.34±0.34 0.37±0.34 L,5,4
HE1029−1401 9.08 44.44 1.83 1 2 (1.02±1.21)×10−3 −2.99±0.56 0.98±0.56 L,6,9
NGC3516 7.63 43.08 1.83 5 18 (3.70±0.45)×10−3 −2.43±0.10 0.10±0.10 R,1,2
PG1116+215 8.21 ... ... 1 1 <1.06×10−2 <−1.73 ... L,6
EXO1128.1+6908 7.02 ... ... 1 1 <1.78×10−2 <−1.51 ... L,5
NGC3783 7.47 42.90 1.70 9 8 (3.91±0.51)×10−3 −2.41±0.13 −0.03±0.13 R,1,2
NGC4051 6.28 41.21 1.92 2 6 (8.62±0.66)×10−2 −1.06±0.09 0.31±0.09 R,1,2
NGC4151 7.12 42.62 1.53 13 29 (2.79±0.22)×10−3 −2.55±0.07 −0.51±0.07 R,1,2
PG1211+143 8.16 ... ... 1 1 <2.39×10−2 <−1.38 ... R,1
MRK766 6.54 42.73 2.16 1 2 (4.02±0.48)×10−2 −1.40±0.16 0.15±0.16 L,6,2
NGC4395 4.11 39.99 1.7 5 6 (1.13±0.14)×10−1 −0.95±0.10 0.17±0.10 L,10,11
NGC4593 6.73 42.98 1.81 2 1 (1.42±0.21)×10−2 −1.85±0.33 −0.16±0.33 R,1,8
WAS61 6.66 ... ... 1 1 <6.95×10−3 <−1.92 ... L,5
PG1244+026 6.07 43.03 2.46 1 2 (2.60±0.62)×10−2 −1.59±0.18 −0.31±0.18 L,5,12
MCG−6-30-15 6.19 42.72 2.00 6 48 (4.16±0.13)×10−2 −1.38±0.03 −0.05±0.03 L,3,2
IC4329A 7.00 43.59 1.71 5 6 (2.36±2.44)×10−4 −3.63±0.47 −1.70±0.47 R,1,2
MRK279 7.54 43.66 1.99 1 1 (2.32±0.84)×10−3 −2.63±0.36 −0.19±0.36 R,1
NGC5506 7.94 42.73 2.08 1 2 (1.06±0.14)×10−2 −1.97±0.23 0.87±0.23 S,13,8
NGC5548 7.83 43.41 1.79 11 16 (9.42±2.67)×10−4 −3.03±0.14 −0.30±0.14 R,1,2
MRK1383 9.11 ... ... 1 1 <6.33×10−3 <−1.96 ... R,1
MRK478 7.34 43.50 2.06 1 2 (6.14±3.75)×10−3 −2.21±0.35 0.04±0.35 L,3,4
MRK841 8.10 43.54 2.00 3 5 (1.14±0.93)×10−3 −2.94±0.38 0.05±0.38 L,6,2
MRK290 7.05 43.22 1.77 1 2 (4.11±2.15)×10−3 −2.39±0.32 −0.41±0.32 L,3,7
IRAS17020+4544 6.77 43.73 2.37 1 2 (5.47±2.00)×10−3 −2.26±0.28 −0.54±0.28 L,14,4
MRK509 8.16 44.03 1.82 11 2 (5.75±7.17)×10−4 −3.24±0.59 −0.19±0.59 R,1,2
AKN564 6.06 43.38 2.58 13 70 (5.34±0.14)×10−2 −1.27±0.03 0.00±0.03 L,3,4
RXJ2248.6−5109 7.67 ... ... 1 1 <1.08×10−2 <−1.73 ... L,5
NGC7469 7.09 43.25 1.84 3 2 (4.68±1.60)×10−3 −2.33±0.27 −0.32±0.27 R,1,2
MCG−2-58-22 8.54 ... ... 2 4 <1.53×10−3 <−2.58 ... L,3
TheobjectsarelistedinorderofR.A.(1)Objectname.(2)LogofblackholemassinunitsofM⊙.(3)Logof2–10keVluminosityin
unitsofergs−1.(4)Hard-X-rayphotonindex.(5)NumberofavailableASCAobservingsequences.(6)Numberofusablelightcurve
segments. (7) Mean normalised excess variance with the uncertainty or upper limit as determined from the bootstrap simulations.
(8) Log of the mean normalised excess variance with the uncertainty as determined by combining the bootstrap uncertainty and
the derived red-noise scatter. (9) Residuals from the best-fitting universal model with the uncertainty as determined by combining
the bootstrap uncertainty and the derived red-noise scatter. (10) Method used to determine the black hole mass and references for
the mass and X-ray spectral properties. The methods, in order of preference, are as follows: R, reverberation mapping; S, stellar
velocity dispersion; L, relationship between broad-line region radius and optical luminosity. References: 1, Petersonetal. (2004); 2,
Nandraetal.(1997b);3,Bian&Zhao(2003);4,Leighly(1999b);5,Grupeetal.(2004);6,Woo&Urry(2002);7,Reynolds(1997);8,
Nandra&Pounds(1994);9,Reeves&Turner(2000);10,Filippenko&Ho(2003);11,Iwasawaetal.(2000);12,Georgeetal.(2000);
13,Papadakis (2004);14,Wang&Lu(2001).Thereferencefortheblackholemassofeachobjectislistedfirst.
(cid:13)c 2004RAS,MNRAS000,1–13
4 P. M. O’Neill, K. Nandra, I. E. Papadakis and T. J. Turner
from Nandra et al. (1997b), Leighly (1999b), Reynolds rejectentireobservingsequencessimplybecausesomeofthe
(1997), Nandra& Pounds (1994), Reeves& Turner (2000), bins have a low exposure, but those in which fully-exposed
Iwasawa et al.(2000),andGeorge et al.(2000).Fortheob- bins have < 20 counts should be excluded. If we were to
jects PG 0026+129, NGC 985, F 303, and MRK 279, we remove bins simply because they had a low counting rate,
fitted the available ASCA data to obtain L2−10 keV and we would be biased against observing objects when their
Γ. The SIS0, SIS1, GIS2, and GIS3 spectra were fitted intensity is low. Selecting according to fractional exposure,
simultaneously in the 2–10 keV rest-frame energy range. on the other hand, can remove bins having too few counts
We used an absorbed power-law, with NH constrained to without introducing this bias, as fractional exposure is not
be greater the galactic value which we obtained using the related to the intensity of the source. If, for example, there
NASA HEASARC ‘nH’ tool.2 For our luminosity calcula- arefullyexposedbinswith<20counts,theentiresequence
tions we obtained redshifts from the NASA/IPAC Extra- was discarded. This is the case when we are dealing with a
galactic Database3 and used H0 =75 km s−1 and q0 =0.5. weak source. For brighter sources where only underexposed
AllL2−10 keVvaluescollectedfromtheliteratureweretrans- bins have < 20 counts, we excise all bins below some min-
formed to this cosmology as required. imal fractional exposure. This gets rid of the non-Gaussian
bins,allowingustokeeptheremainderofthelightcurvefor
furtheranalysis, butintroduces no bias against those times
when the source is weak dueto trueflux variability.
3 EXCESS VARIANCE ANALYSIS
Finally,wefurtherrequiredthetruncatedandscreened
The number of ASCA observing sequences available for light curve segments to have at least 20 bins, so that the
each object is shown in Table 1. We extracted a 2–10 keV variance could be determined accurately.
combinedSIS0+SIS1+GIS2+GIS3lightcurvefromeachse- This procedure resulted in 46 objects having at least 1
quence. These initial light curves had a resolution of 16 s valid light curvesegment,and a totalof 305 valid segments
and each bin was required to be fully exposed. The light in all. The number of segments for each object is given in
curveswere then rebinned to a resolution of 256 s. Table 1. The mean durations of the light curve segments
for each object were in the range 35–40 ks in the observers
frame. The 48 objects in our sample have redshifts in the
3.1 Excess variance calculation range 0.001–0.234. Taking into account the redshift of each
object,therest-framemeandurationswereintherange30–
Forared-noiseprocess,thevarianceinalightcurvedepends
40 ks. We expect the effect of these slightly different dura-
bothonthepowerspectrumofthevariationsandalsoonthe
tions to be small. For a power spectrum with a power-law
timeresolution and duration of thelight curve.This means
slope of α = 2, the worst case we expect, a ∼25 per cent
that different σ2 measurements are only strictly compa-
NXS reduction in the light curve duration (i.e., from 40 ks to
rable if the durations of the light curves are equal. There-
30 ks) results in a reduction in the σ2 of only ∼0.1 dex.
fore,wesub-dividedthelightcurvefromeachsequenceinto NXS
AspresentedlaterinthisSection,theuncertaintiesinmost
many segments of similar duration. The advantage of using
ofourobservedσ2 valuesareafewtoseveraltimeslarger
longdurationsisthattheamplitudeofvariabilityincreases, NXS
than0.1dex.Therefore,the∼25percentdifferencebetween
and the number of points used to calculate σ2 is also
NXS the shortest and longest mean light curve duration can be
larger,reducingthemeasurementuncertainty.Ontheother
neglected and allows us to use more data than would have
hand, using short durations has the advantage that more
been available if we had imposed a strict limit on duration.
light curve segments can be included. We chose a nominal
Wetestedtoseewhichobjectsexhibitedsignificantvari-
segment length of 40 ks for our analysis as a tradeoff be-
abilitybyperformingachi-squaredtest.Theχ2correspond-
tweentheseconsiderations.Inrealitywechoseadurationof
ingtothehypothesisofaconstantcountingratewasdeter-
39936 kswhich is an integer multipleof our 256 s timebin.
mined for each ∼40 ks light curve segment. Then, for each
The sub-dividing of the light curves proceeded as fol-
object, we summed all of the χ2s and degrees-of-freedom
lows. First, the earliest 40 ks segment of the light curve for
(DOFs)totestwhetherthatobjectisvariable.Wedetected
a certain observing sequence was selected. Then, beginning
variability in 33 objects at the 95 per cent confidence level.
with the next exposed bin following the end of this first
We then calculated the excess variance in each light curve
segment, another 40 ks segment was selected. This contin-
segment with thefollowing expression:
ued until the light curve had been completely sub-divided.
Note that the actual duration of these light curves, which N
1
wedefineasthetimebetweenthefirstandlastexposedbin, σ2 = [(X −µ)2−σ2] (1)
can be less than 40 ks because the dividing point between NXS Nµ2 i i
i=1
X
segments can occur when there is a gap in the data train.
where N is the number of bins in the segment, X and σ
We accepted all resulting light curve segments that had a i i
arethecountingratesanduncertainties,respectively,ineach
duration >30 ks.
bin,and µ is theunweighted arithmetic mean of thecount-
To ensure Gaussian statistics, we required each 256 s
ingrates.Forobjectswithmorethanonevalidsegment,the
bintocontainatleast20counts.Thenumberofcountsina
unweighted average excess variance was determined. A ma-
certain 256 s bin dependsboth on the source counting rate
jor advantage of our work is that, given the large number
and the fractional exposure of that bin. We do not wish to
of light curves available, there is often more than one valid
segment per object (see Table 1). Taking the mean σ2
NXS
2 http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl ofthesemultiplesegmentsreducesthepotentially largeun-
3 http://nedwww.ipac.caltech.edu/ certainty owing to the stochastic nature of the variability
(cid:13)c 2004RAS,MNRAS000,1–13
X-ray variability amplitude and black hole mass in active galactic nuclei 5
(see below). When calculating the mean excess variance we MCG−6-30-15, TON S180, NGC 4151, NGC 3516, and
used all valid light curve segments for a particular object, NGC 5548) with a sufficient number of light curve seg-
includingthosesegmentsthatdidnot,inthemselves,exhibit ments (>15) to make a meaningful estimate of the frac-
variability based on the χ2 test. tional uncertainty in σ2 owing to noise nature of our
NXS
light curves. First, we determined the standard deviation
σ of the observed σ2 values for each object. We then
3.2 Estimating the uncertainties in σ2 obs NXS
NXS determined, from the bootstrap uncertainty, the standard
Estimating theuncertainty for excess variance is somewhat deviation σmeas that we would expectto observein thedis-
complicated. Analytical prescriptions have been given in tributionoftheσN2XS valuesifthescatterwasowingonlyto
theliteraturebyNandraet al.(1997a,theircorrectformula measurementuncertainties.Wesubtractedσmeas from σobs,
is given by Turner et al. 1999), Edelson et al. (2002) and inquadrature,toobtainthestandarddeviationσnoise inthe
Vaughan et al. (2003a). The latter authors also discussed observedσN2XSvaluesthatisowingonlytothestochasticna-
theuncertaintiesinσ2 onthebasisofsimulatedred-noise tureofthevariability.Wethendetermined,foreachofthe6
NXS
light curves. These uncertainties depend both on measure- distributions, the ratio between σnoise and mean σN2XS. We
ment uncertainties (e.g., Poisson noise) in the light curve shallrefertothisratioasthe‘fractionalstandarddeviation’
data,andthestochasticnatureofthevariability: anygiven anddenoteitasσfrac.Thefractionalstandarddeviationsfor
light curvesegment representsjust one realisation of a ran- the 6 objects were: 0.49 (AKN 564), 0.47 (MCG−6-30-15),
domprocess,andthuscanexhibitadifferentmeanandvari- 0.69 (TON S180), 0.79 (NGC 4151), 0.82 (NGC 3516), and
ancefrom thetruevalue, oranother random segment. This 0.61 (NGC 5548). These values of σfrac show that, even in
‘noise’ uncertainty can be very large, especially for a single the absence of measurement uncertainties, one can expect
realisation.Onemust,therefore,accountforthisuncertainty noiseuncertaintiesintherange∼50–80percentforindivid-
beforeapparentdifferencesinσN2XS,eitherinagivensource ual measurments of σN2XS (see also Vaughan et al. 2003a).
(Nandra& Papadakis 2001) or in comparing sources (e.g., Thishighlightstheneedofobtainingmanyrealisations(i.e.,
Turneret al. 1999) canbeconsidered robust.Themeasure- manymeasurementsofσN2XS),regardlessofthelevelofPois-
ment and noise uncertainties on σ2 are unrelated,so can son noise in thedata.
NXS
and must beestimated separately. Vaughan et al. (2003a) showed that the uncertainty in
To estimate the uncertainty in σ2 owing to mea- the estimated variance of a red-noise light curve increases
NXS
surement uncertainties, we used bootstrap simulations (the with the steepness of the power spectrum slope. Power-
reader is directed to Press et al. 2001, for a discussion on spectral analyses of AGN have revealed that the value of
bootstrap simulations). Suppose that the observed light νHFB generally decreases with increasing black hole mass.
curvecontainsN bins.ThislightcurveisadistributionofN This means that the shape of the power spectrum in the
counting rates and corresponding Poisson-noise uncertain- frequency range probed by our light curves (∼2.5×10−5
ties from which we calculate σN2XS. Note that calculating to 4×10−3 Hz) is expected to vary as a function of M•,
σN2XS does not depend on the bins being in time-order. A such that the objects with the highest M• should exhibit
bootstrapsimulationinvolvesrandomlyselecting,fromthat the steepest (α ∼ 2) power spectra. We expect, then,
distribution, a new set of N bins. The duplication of bins that the scatter in σ2 owing to red-noise fluctuations
NXS
ispermittedduringtheselectionprocess.This,then,results shouldalsoincreasewithmass.Thelowest-massobjectsare
inthecreationofaslightlydifferentdistributionofcounting AKN 564 and MCG−6-30-15, and the observed values of
rates,andσN2XScanbedeterminedforthisnewdistribution. νHFB for these fall within the frequency range of our data
If one repeats the entire process many times, the resulting (Papadakis et al.2002;Vaughan et al.2003b).Wetherefore
distribution of simulated σ2 values provides an estimate expect σ for this pair of objects to be less than the oth-
NXS frac
of the uncertaintyin σ2 . ers.Thisdoesindeedappeartobethecase:theσ values
NXS frac
We performed a series of 10000 bootstrap simulations of AKN 564 and MCG−6-30-15 are both less than those of
to determine the uncertainty in the mean observed σ2 TONS180,NGC4151,NGC3516,andNGC5548.However,
NXS
for each object in our sample. Each of thesesimulations in- we possess only a limited number of individual σ2 mea-
NXS
volved: simulating a new ‘light curve’ from each valid light surementstoestimateboththemeanandstandarddeviation
curve segment, determining σ2 for those simulated light of each distribution, so it is possible that this apparent dif-
NXS
curves, and then determing the mean of these simulated ferenceisnotstatisticallysignificant.Wedecided,therefore,
σ2 values. We were thus able to generate 10000 simu- tocompare the6distributionsofσ2 valuesusingaseries
NXS NXS
lated values of the mean σ2 . The standard deviation of of Kolmogorov-Smirnov (K-S) tests. Before we could com-
NXS
these values was taken to be the measurement uncertainty pare the distributions, we first had to correct each of them
in the mean observed σ2 . We refer to this value as the for the effect of measurement uncertainties. To do this, we
NXS
‘bootstrap uncertainty’and denote it as ∆ (σ2 ). scaled the deviations of the observed σ2 values so that
boot NXS NXS
Estimates of the uncertainty owing to the noise pro- the standard deviation of the ‘corrected’ distribution was
cess have been presented by Vaughan et al. (2003a), based equal to σnoise, with the mean σN2XS remaining unchanged.
onlightcurvesimulations,whoshowedthatthenoiseuncer- We then normalised each of the corrected distributions by
taintyisproportionaltothemeanvalueofthevariance.The dividing the σ2 values by the mean. The corrected and
NXS
constant ofproportionality dependson thepower spectrum normalised σ2 distributions for AKN 564 and MCG−6-
NXS
shape,whichwedonotknowapriori.Therefore,aspointed 30-15areconsistentwithbeingdrawnfrom thesamedistri-
out byVaughan et al. (2003a),it is preferable todetermine bution. The same is true when comparing the other 4 dis-
theuncertainties in σ2 directly from thedata. tributions with each other. We then created two, combined
NXS
Our large database contains 6 objects (viz, AKN 564, distributions:oneforAKN564andMCG−6-30-15; andan-
(cid:13)c 2004RAS,MNRAS000,1–13
6 P. M. O’Neill, K. Nandra, I. E. Papadakis and T. J. Turner
otherforTONS180,NGC4151,NGC3516,andNGC5548.
Thefractionalstandarddeviationsfromthesetwocombined
distributionswere0.48(AKN564,MCG−6-30-15) and0.74
(TONS180, NGC4151, NGC3516, NGC5548), andaK-S
test showed them to be different at the 95 per cent con-
fidence level. The cumulative distribution functions of the
combined distributions are presented in Fig. 1. Combining
the normalised, corrected distributions of all 6 objects re-
sulted in a σ of 0.61.
frac
For the objects AKN 564, MCG−6-30-15, TON S180,
NGC 4151, NGC 3516, and NGC 5548, we determined the
total uncertainty [∆tot(σN2XS)] in the mean excess variance
directly from their respective values of σ . For each of
obs
the other objects, we estimated the noise uncertainty and
combined it in quadrature with the bootstrap uncertainty
[∆ (σ2 )] using thefollowing expression:
boot NXS
Figure 1. Cumulative distribution functions of the combined
2
σ σ2 normalised σ2 distributions of AKN 564 and MCG−6-30-15
∆tot(σN2XS)=v frac NXS +[∆boot(σN2XS)]2 (2) (solidline)anNdXTSONS180,NGC4151,NGC3516,andNGC5548
uu Nseg ! (dahed-line).
t
p
where σN2XS is the mean excess variance and Nseg is the
number of available light curve segments. For objects with uncertaintyand3σ upperlimitasdeterminedfromonlythe
log M• > 6.54 we adopted a fractional standard deviation bootstrap simulations. The uncertainties and upper-limits
of σfrac =0.74, while for theobjects with log M• 66.54 we given in the column with log σ2 include also the noise
NXS
adopted a value of σfrac =0.48. These ranges in mass were uncertainty.
selected on the basis that the object MRK 766, which has
log M• = 6.54, is the most massive object that has an ob-
served νHFB in the frequency ranged probed by our ∼40 ks 3.3 The variance–mass relation
lightcurves(e.g., Papadakis et al.2002;Vaughan & Fabian
2003;Marshall et al.2004;Vaughan et al. 2004,andseeIn- Therelationship between log σN2XS and log M• is presented
troduction). For objects more massive than this we expect in Fig. 2. It is clear that there is a strong anti-correlation
νHFB to be less than our observed frequency range. In the between the two quantities. This is confirmed using both a
absenceofameasurementofM• oranyinformationregard- Spearmanrank-ordercorrelation testandKendall’sτ,both
ing the shape of the power spectrum, the mean value of of which show the anti-correlation to be significant with
σ =0.61 can be adopted. >99.99 per cent confidence. The upper limits to the vari-
frac
Theσ2 upperlimitsforthenon-variableobjectswere ance in the case where no variability is detected, which are
NXS
also estimated by combining the two components of uncer- shown in the upperpanel of Fig. 2, are generally above the
tainty. We multiplied the 1σ bootstrap uncertainty by the measuredvalues(foragivenmass).Thismeanstheyareun-
appropriatefractionalstandarddeviationofthenoiseuncer- likelytosignificantly affect anymodel fittingandweignore
tainty.Thisvaluewasthenmultipliedby3toprovideanesti- them in theanalysis below.
mateofthe3σ upper-limit.Wealsoestimatedthe3σ ‘boot- While there is a strong general trend for objects with
strap upper-limit’ by multiplying the bootstap-uncertainty higher mass to be less variable, there is clearly substantial
∆ (σ2 ) by 3. scatterinthevariance–massrelationship.Aswehave,forthe
boot NXS
The distributions of the σ2 values for AKN 564 and first time, presented realistic estimates of the uncertainties
MCG−6-30-15 are quite asymNmXeStric, with each having an on σN2XS we can be confident that this scatter is not owing
extended tail towards high values of σ2 . However, the only to these uncertainties.
NXS
distributions of log σ2 look much more symmetric. This Thereisalsoevidencefromtheplot–albeit basedsolely
NXS
is not surprising as it is well known that the logarithmic on the lowest mass object, NGC 4395–that the variance–
transformationofarandomvariablewithanextendedtailin massrelationshipisnon-linear.Thisisexpectedinthepres-
itsdistributionbringsthatdistributionmuchcloserto‘nor- enceofbreaksinthepowerspectrum(e.g.,Papadakis2004),
mality’(e.g.,Papadakis & Lawrence1993).Ideally,then,we as we now show by modelling the variance–mass relation-
wouldliketoestimatelogσ2 fromeachsegmentandthen shipusingbothsimpleparametrizations andwithaspecific
NXS
determine the mean of log σ2 for each object. Unfortu- power-spectral form.
NXS
nately,wecannotusethismethodbecauseσ2 isnegative
NXS
for some light curve segments. We did, however, determine
thelogarithm of themean σ2 , which brings thedistribu-
NXS 4 MODELLING THE RELATIONSHIP
tion of the mean σ2 closer to normality. We also trans-
NXS BETWEEN EXCESS VARIANCE AND MASS
formed the uncertainties ∆tot(σN2XS) to be the uncertainty
in thelogarithm of the mean σ2 . Having obtained the mean σ2 for each object, we then
NXS NXS
The mean σN2XS values, uncertainties and upper limits wishedtomodeltherelationbetweenσN2XS andM•.Allfits
are listed in Table 1. The column listing σN2XS gives the wereperformedonlogM• andlogσN2XS.Wefittedthedata
(cid:13)c 2004RAS,MNRAS000,1–13
X-ray variability amplitude and black hole mass in active galactic nuclei 7
using both a simple parametrization and also with a model afunctionalformandisthuscontinous.Wedenotethiscon-
that assumes theexistence of a universal power spectrum. tinuous power spectrum as PM(ν). We need to determine
howthediscretepowerspectrumPD(ν)isrelatedtoPM(ν).
Thefollowing description is appropriate for evenly sampled
4.1 Simple parametrizations light curvescontaining nogapsand havingan evennumber
Weinitiallymodelledthedatausingapower-lawoftheform of bins. Note also that the model power spectrum PM(ν)
σ2 = AM−γ. The index and normalisation of the best- must be defined to be two-sided and, since we are dealing
NXS • with a noise process, we refer to the expectation value of
fiting power-law were γ = 0.570 and A=125, respectively,
each power.
and the reduced chi-squared was χ2/DOF = 8.05/31. This
ν Thefirsteffect toconsiderisbinning.Supposewehave
modelisshownasadot-dashedlineinthetoppanelofFig.2.
Wedonotquoteuncertaintiesinthebest-fittingparameter a continuous process, with power spectrum PM(ν), and we
transform it into a discrete process by binning the signal
values because theχ2 is formally unsatisfactory.
ν over a time period of δt. The power spectrum of the ob-
We then used a singly-broken bending power-law de-
fined as: servedbinnedlightcurve,sayPB(ν),isrelatedtothePM(ν)
through thefollowing relation:
σN2XS=AM•−γlow 1+ MM•b•end γhigh−γlow −1 (3) <PB(ν)>=B(ν)PM(ν) (4)
(cid:20) (cid:16) (cid:17) (cid:21) wherethebinningfunctionB(ν)(van derKlis1989)isgiven
where A is thenormalisation factor and thefunction bends
by:
from a power-law slope of γ to γ at the bend mass
low high
M•bend. sin(πνδt) 2
We fixed the lower index to γ = 0. The best-fitting B(ν)= (5)
low πνδt
bend mass, normalisation, and upper index were M•bend = (cid:20) (cid:21)
5.59×105 M⊙, A = 0.144, and γhigh = 0.836, respectively Thenexteffecttoconsiderisaliasing.Thefactthatthe
(χ2/DOF = 5.99/30). The bending power-law clearly im- observed light curve is sampled at discrete intervals means
ν
provesthe fit statistic substantially, but it is difficult to as- thatpowercanleakintothepowerspectrumfromabovethe
sesstheformal improvementwith,e.g., anF-test asthefits Nyquist frequency νNyq = 1/(2δt). The binned and aliased
are so poor. powerspectrum,sayPBA(ν),isrelatedtotheintrinsicpower
spectrum PM(ν) through therelation (Priestley 1989):
∞
4.2 Predicting σ2 from a power spectrum model
NXS <PBA(ν)>= <PB(ν+i/δt)> (6)
Based on recent power-spectral analyses of AGN, it is pos- i=−∞
X
sible that the power spectra of AGN have the same shape
The power in one of our typical model power spectra de-
with the time-scale of the variations being proportional to
creases sharply with frequency and the data are binned.
black hole mass (see Introduction and references therein).
This means that only a relatively small amount of power
We decided to investigate this possibility by modelling the
is aliased into the observed frequency range. Accordingly,
relationship between σN2XS and M• with the assumption of we found that summing from i = −10 to i= 10 was easily
auniversalpowerspectrum.Amodelestimatecanbemade
sufficient to account for aliasing. Power spectra that are ei-
simply by integrating the continuous power spectrum over
ther flat or increase with frequency might require a larger
some frequency range, for exampleas definedby thelength
range in i.
and the time bin size of the observation (e.g., Papadakis
The final effect to account for is red-noise leak. This
2004).This,however,neglectstheeffectsfrom thesampling
occurswhenvariationsexistatfrequencieslowerthanthose
pattern of the light curve, specifically the fact that it is
sampled by the observed light curve, as is the case for a
binned, may have gaps, and is of finite duration. For rea-
red-noise process. This ‘leakage’ of power from low to high
sons discussed below these effects, particularly that of the
frequenciescanbeseenaseitherarisingorfallingtrendover
finiteduration and subsequent‘red-noise leak’, arelikely to
the duration of the light curve. The power spectrum of the
be more important on the time-scales considered here than
finallight curve,i.e. PD(ν),is related totheintrinsic power
the much longer ones discussed by Papadakis (2004). We
spectrum,i.e.PM(ν),bytheconvolutionofthe<PBA(ν)>
havetakenananalyticalapproachtodeterminingthemodel-
with the so-called ‘window function’ W(ν) of the observed
predictedσ2 ,ratherthanusesimulationsasistypicalfor
NXS lightcurve.Forevenly-sampledlightcurves,W(ν)issimply
power spectrum analysis (e.g., Uttley et al. 2002). Our ap-
Fejer’s kernel (e.g., Priestley 1989):
proach is preferable for two reasons. First, simulations are
far more computer-intensive, and second they rely on the 1 sin(πνT) 2
W(ν)= (7)
simulation technique accurately reproducing the character- T πν
istics of the physical process giving rise to the variability. (cid:20) (cid:21)
While the technique described below applies to calculation where T is the duration of the light curve. We performed
ofmodelσ2 valuesitcanbeadaptedstraightforwardlyto theconvolution with thenumerical integral:
NXS
theestimation of discrete model power-spectra.
Nf/2
According to Parseval’s Theorem, the variance in a PD(ν)=2 <PBA(iδν′)> W(ν−iδν′)δν′ (8)
binned light curve is equal to the sum of the powers in
the observed discrete power spectrum of that light curve. i=−XNf/2
The model power spectrum, however, is initially defined in (ν =1/T,2/T,...,νNyq)
(cid:13)c 2004RAS,MNRAS000,1–13
8 P. M. O’Neill, K. Nandra, I. E. Papadakis and T. J. Turner
Intheabovesum,N isthenumberofbinsinthelightcurve Using this model, the relation between variance and mass
and f is a positive integer. The frequency step δν′ is given can therefore be described with three parameters: CHFB,
by δν′ = 1/(Tf). The value of f must be large enough so CLFB, and PSDAMP.
that the convolution extends to a low-enough frequency to To determine the best-fitting model, we minimised χ2
account for all of the low-frequency power. Determining a
for grid of values of CHFB, CLFB, and PSDAMP values. We
suitable valueof f required aprocess of trial-and-error. We
foundthatwecouldnotconstraintheparameterCLFB.This
performed the convolution with successively higher values
is because the low-frequency break generally does not fall
off untilfurtherincreasesproducedonlyanegligibleeffect.
within our sampled frequency range. Therefore, we fixed
Wefoundthatf =500wassufficentforallourconvolutions.
this at CLFB = 20. This is roughly the value of CLFB ob-
Note that the introduction of the factor 2 in Eqn. 8 means
served in the AGN NGC 3783 (Markowitz et al. 2003) and
that PD(ν) is single-sided and it is defined only for N/2 inCygX-1inthelow/hardstate(Belloni & Hasinger1990a;
frequencies. Also note that, for iδν′ = ±νNyq the term δν′ Nowak et al. 1999).
was replaced by δν′/2, to account for the end-effectsin the
numerical integral. Thebest-fittingvaluesofCHFBandPSDAMP aregiven
in Table 2. This best-fitting model (for the fit including all
The expected excess variance was then determined by
33objects)isshownasthesolidlineinFig.2(bottom).The
summing thepowers in PD(ν): probability of exceeding the χ2 of the best-fitting universal
ν
N/2−1 model is 2×10−25. This indicates that, while the model
1
σN2XS,model = PD(i/T)δν + 2PD(νNyq)δν (9) appearstodescriberatherwell theoveralltrendof decreas-
" Xi=1 # ing σN2XS, there exists significant scatter not accounted for
by the model. The residuals ∆log σ2 from this model
where δν =1/T. Thefactor of 1/2 is required for PD(νNyq) NXS
are listed in Table 1. We also fitted the universal model
because in a double-sided power spectrum the Nyquist fre-
to the data excluding various objects. As seen in Table 2,
quency occurs only once. The factor δν is required because
neither the lowest mass object (viz, NGC 4395), nor the
thepowerisexpressedinunitsoffractional-rms-squaredper
6 objects with the largest number of light curve segments
Hz.
(viz, AKN 564, MCG−6-30-15, TON S180, NGC 4151,
Each of our 305 light curve segments has its own par-
NGC 3516, NGC 5548), dominate thefit.
ticular duration and sampling pattern, and there are many
gaps in the data train. Therefore, the window function will Thescatterpresentintherelationshipbetweenlogσ2
NXS
bedifferentforeachsegmentandwillnot,ingeneral,berep- andlogM• canbeexplainedwithavariationofeitherCHFB
resentedbyFejer’skernel.However,thepresenceofmissing orPSDAMPfromtheirbest-fittingvalues.Thisisillustrated
bins in the light curve will affect only the scatter in the in Fig. 2 (bottom). We find that a range in CHFB values
σ2 measurements,with themean valuebeingunaffected. between 7.2 and 520 (upper and lower dotted-lines, respec-
NXS
Moreover, we have taken care to use light curve segments tively),orarangeinPSDAMP between0.004 and0.29(up-
ofsimilardurations.Therefore,wewereabletosimplifythe per and lower dashed lines, respectively), can account for
modellingprocedurebyassumingthatourlightcurveswere most of the scatter in thelog σN2XS versus log M• relation.
all fully sampled with the same number of bins. We used
The scatter might also be owing to a combination of
N = 148, as this is the even number-of-bins closest to the the uncertainties in log σN2XS and log M•, the latter of
meansegmentdurationof38143.5 s.Havingmadethissim-
which are typically about 0.5 dex (e.g., Woo & Urry 2002;
plification,wewererequiredtodetermineonlyasinglevalue
Peterson et al. 2004). We performed simulations to inves-
of σ2 for each object (for a certain model power
NXS,model tigate this possiblity, adopting the best-fitting relation be-
spectrum), thusspeeding up themodelling process. tween log σN2XS and log M• as our model. We needed first
to obtain a set of 33 model data points to which we could
4.3 A universal power spectrum model then apply scatter in log σN2XS and log M•. To do this,
we projected each of our 33 observed data points onto the
Motivated by power-spectral analyses of AGN (see Intro- best-fitting relation, minimising the distance between the
duction,in particularMarkowitz et al. 2003),and following observedpointandthemodel.(Thedistancebetweenanob-
therecentworkofPapadakis(2004),wehypothesisedauni- served data point and any particular location on the model
versal power spectrum of the form: relationwascalculated fromthedifferencesinlogσ2 and
NXS
PM(ν)=A (νLFB/νHFB)−1(ν 6νLFB) (10) log M• between the observed point and the model, divided
by the corresponding uncertainty in the observed values.)
PM(ν)=A (ν/νHFB)−1(νLFB<ν <νHFB) (11) Havingthusadoptedasetof33modeldatapoints,wethen
PM(ν)=A (ν/νHFB)−2(νHFB6ν) (12) psceartfoterrmteodt1h0e00msoimdeullaptoioinntss. Eanadchtohfenthdeseeteirnmvionlvinegd tahdediχng2
ν
where the normalisation factor A is the power at the high- between the simulated data points and the model relation.
frequency break νHFB. The value of νHFB is assumed to We found that 79 per cent of the simulations produced a
decrease with black hole mass, according to the expres- χ2 exceeding that found for the observed data. Therefore,
ν
sion νHFB = CHFB/M•, where CHFB is a constant and M• the scatter that we have observed in the relation between
is the mass of the black hole in units of M⊙. The low- log σN2XS and log M• might be owing only to measurement
frequency break is related to the high-frequency break by uncertainties. If this is indeed the case, then we would ex-
νLFB=νHFB/CLFB whereCLFB isaconstant.Thenormali- pect this scatter to be unrelated to other properties of the
sationAvariesasafunctionofνHFBasA=PSDAMP/νHFB, objects in our sample, and we investigate thispossibility in
where PSDAMP is assumed to be the same for all objects. thefollowing Section.
(cid:13)c 2004RAS,MNRAS000,1–13
X-ray variability amplitude and black hole mass in active galactic nuclei 9
Table 2.Best-fittingvaluesforfitsusingtheuniversalpowerspec-
trummodel.
Excludedobjects CHFB PSDAMP χ2ν/DOF
(HzM⊙)
(1) (2) (3) (4)
None(allobjects 43 0.024 6.24/31
areincluded)
NGC4395 53 0.021 6.30/30
AKN564, 55 0.033 4.30/24
MCG−6-30-15,
TONS180,NGC4151,
NGC3516,NGC5548
(1) Objects excluded from fit. (2) Scaling constant for the high-
frequencybreakνHFB,whereνHFB=CHFB/M•.(3)Power-spectral
amplitude at νHFB in power × frequency space. (4) Reduced chi-
squaredanddegrees-of-freedomforfit.
Figure 3. Log of excess variance (top), log of the product of
excessvarianceandblackholemass(middle),andexcessvariance
residuals(bottom), versuslogofthe2–10keVluminosity.
5 THE ORIGIN OF THE SCATTER IN THE
VARIANCE–MASS RELATIONSHIP
Previous studies have revealed an anti-correlation between
σ2 and X-ray luminosity, and a positive correlation be-
NXS
tween σ2 and photon index Γ (e.g., Nandraet al. 1997a;
NXS
Turneret al. 1999; Markowitz & Edelson 2001; Papadakis
2004). Given the strong dependence between the σ2 and
NXS
M•,itisofinteresttoseewhetherthesecorrelationsstillex-
ist when this primary dependence is removed. This should
allow us to shed light on the origin of the scatter in the
variance–mass relationship.
InFig.3weplotlogσN2XS,logM•σN2XS,andtheresidu-
als ∆log σ2 from the best-fittinguniversal model, versus
NXS
the logarithm of the 2–10 keV luminosity. The quantities
logM•σN2XS and∆log σN2XS areusefulbecausetheyremove
the mass-dependence. Note that the quantity log M•σN2XS
Figure 2. Log of excess variance versus log of black hole mass. ismodel-independent.InFigs. 4 and5weplot thevariabil-
In the top panel, the dot-dashed and solid lines show the best-
ity parameters versus, respectively, the photon index and
fittingpowerlawandbendingpower-lawmodels,respectively.In
thelogarithm ofthe2–10 keVluminosity normalised tothe
thebottompanel,thesolidlineshowsbest-fittinguniversalpower
spectrummodel.Thedottedanddashedlinesillustratetheeffect blackholemass,log(L2−10 keV/M•).Totheextentthatthe
X-rayluminosityisproportionaltothebolometricluminos-
of varying either CHFB or PSDAMP, respectively (see text for
details).Theσ2 upperlimitsare,forclarity,shownonlyinthe ity, as is commonly assumed, the value log (L2−10 keV/M•)
NXS
upperpanel. isproportionaltotheratiobetweenthemass-accretion rate
and that required to reach the Eddington luminosity (i.e.,
the ‘Eddington ratio’). Note that the correction factor be-
tweenthe2–10keVandbolometricluminositiesisuncertain,
with considerable scatter. The Spearman rank-order corre-
(cid:13)c 2004RAS,MNRAS000,1–13
10 P. M. O’Neill, K. Nandra, I. E. Papadakis and T. J. Turner
Figure 4. Log of excess variance (top), log of the product of
Figure 5. Log of excess variance (top), log of the product of
excessvarianceandblackholemass(middle),andexcessvariance
excess variance and black hole mass (middle), and excess vari-
residuals(bottom), versusthe2–10keVphotonindex.
ance residuals (bottom), versus log of the 2–10 keV luminosity
normalisedbytheblackholemass.
lation coefficient and Kendall’s τ of all 9 relationships are
presented in Table 3.
Aswithpreviousstudies,wefindaverystrongcorrela-
tion between log σN2XS and log L2−10 keV (see Fig. 3). This
correlation disappears when we remove the dependence of nosity are less variable for a given mass. While significant,
σN2XS on M•.It seems most likely that theprimary correla- this relationship should be treated with some caution. The
tion is in fact with mass, and that theapparent correlation presenceofrandomscatterintheblackholemassestimates
with log L2−10 keV is secondary. could possibly induce such an anti-correlation. If M• is un-
A similar situation is present when considering the derestimated then ∆log σ2 will also be underestimated
NXS
photon index (see Fig. 4). Indeed, the correlation between and log (L2−10 keV/M•) will be overestimated. An artifi-
logσ2 andΓisnotverystronginanyevent,beingsignif- calanti-correlation wouldcertainlybeinducedifallobjects
NXS
icant at only the 96 per cent confidence level, though there had the same value of log (L2−10 keV/M•). However, it is
does seem to be an absence of objects having both a steep less clear that this effect could produce an anti-correlation
photon index and low σN2XS. When the mass dependence is between ∆log σN2XS and log (L2−10 keV/M•) in our data
accounted for, however, no residual correlation remains. In because the normalised luminosities in our sample span 3
theplot of ∆log σ2 versusΓ, the steep spectrum objects orders-of-magnitude. We used the simulations described in
NXS
do not have a systematically higher ∆log σ2 than the Section 4.3 to test whether the observed anti-correlation
NXS
others. couldbeowingtotheuncertaintiesintheblackholemasses.
Finally, we consider the relationship between the Foreachofthe1000simulations,wecalculatedlogσ2 and
NXS
variability properties and the normalised luminosity log(L2−10 keV/M•)fromthesimulateddatapointsandmea-
log (L2−10 keV/M•) (see Fig. 5). There is considerable suredKendall’sτ.Wefoundthat,evenwithnointrinsicanti-
scatter, and no strong correlation, between log σN2XS and correlation between σN2XS and L2−10 keV/M•, 57 per cent
log (L2−10 keV/M•). Here, however, we do find a signifi- of the simulations gave a Kendall’s τ that was more nega-
cant relationship between log (L2−10 keV/M•) andboth the tive thantheobservedvalueof−0.31.Therefore,wecannot
mass-normalised excessvarianceandtheresiduals from our rule-outthepossibilitythattheobservedanti-correlationbe-
best-fitting model. The latter correlation is significant with tween ∆log σN2XS and log (L2−10 keV/M•) is an artifact in-
∼99 per cent confidence and, perhaps surprisingly, it is in ducedbythepresenceofuncertaintiesinthemeasurements
thesensethatobjectswithlargervaluesofnormalisedlumi- of black hole mass.
(cid:13)c 2004RAS,MNRAS000,1–13