High-frequency X-ray variability as a mass estimator of stellar and supermassive black holes
aa r X i v : . [ a s t r o - ph ] O c t Mon. Not. R. Astron. Soc. , 1–9 (2005) Printed 16 November 2018 (MN L A TEX style file v2.2)
High-frequency X-ray variability as a mass estimator of stellar andsupermassive black holes
Marek Gierli ´nski , ⋆ , Marek Nikołajuk and Bo˙zena Czerny Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krak´ow, Poland Department of Physics, University of Białystok, Lipowa 41, 15-424 Białystok, Poland Copernicus Astronomical Centre, Bartycka 18, 00-716 Warszawa, Poland
Submitted to MNRAS
ABSTRACT
There is increasing evidence that supermassive black holes in active galactic nuclei(AGN) are scaled-up versions of Galactic black holes. We show that the amplitude of high-frequency X-ray variability in the hard spectral state is inversely proportional to the blackhole mass over eight orders of magnitude. We have analyzed all available hard-state data from
RXTE of seven Galactic black holes. Their power density spectra change dramatically fromobservation to observation, except for the high-frequency ( &
10 Hz) tail, which seems to havea universal shape, roughly represented by a power law of index -2. The amplitude of the tail, C M (extrapolated to 1 Hz), remains approximately constant for a given source, regardless ofthe luminosity, unlike the break or QPO frequencies, which are usually strongly correlatedwith luminosity. Comparison with a moderate-luminosity sample of AGN shows that the am-plitude of the tail is a simple function of black hole mass, C M = C/M , where C ≈ . M ⊙ Hz − . This makes C M a robust estimator of the black hole mass which is easy to apply to low-to moderate-luminosity supermassive black holes. The high-frequency tail with its universalshape is an invariant feature of a black hole and, possibly, an imprint of the last stable orbit. Key words:
X-rays: binaries – galaxies: active – accretion, accretion discs
Astrophysical black holes are very simple objects, completely char-acterized by their mass and spin. Hence, the gravitational poten-tial around a black hole simply scales with its mass. An importantquestion in high-energy astrophysics is whether the accretion flowproperties scale with the black hole mass in a simple manner, or,more specifically, whether active galactic nuclei (AGN) are scaled-up versions of Galactic black hole binaries (BHB).One of the ways of tackling this problem is to study X-rayvariability, which is observed in accreting black holes of all masses.Recent advances in mass estimates of AGN central black holes leadto discovery of dependence of the observed variability propertieson mass. Long X-ray monitoring campaigns allowed to constructpower density spectra (PDS) of accreting supermassive black holeswhich turned out to have a roughly of (broken) power-law shape.The variability amplitude (the excess variance; e.g. Lu & Yu 2001;Markowitz & Edelson 2001) and the frequency of the break (e.g.M c Hardy et al. 2004, 2006) can depend on the black hole mass. ⋆ E-mail:[email protected]
In order to use the X-ray variability for mass measurementwe need a property which scales only with the black hole mass,and does not change with accretion rate. The break frequencydoes not satisfy this condition, as it changes significantly with theaccretion rate, in X-ray binaries (e.g. Done & Gierli´nski 2005).M c Hardy et al. (2006) showed that a more general relation holdsbetween the break frequency, ν b , and the black hole mass, M : ν b = AL B bol /M C , where A , B and C are constants. This relationincludes a significant dependence on the source bolometric lumi-nosity, L bol .It was already suggested by Hayashida et al. (1998) that mea-suring the normalization of the high-frequency tail of the powerspectrum, well above the high-frequency break, is an interestingpossibility for black hole mass measurement. Equivalently, onecan use the excess variance, σ , measured for short data sets.This general line was followed by Czerny et al. (2001), Papadakis(2004), Nikołajuk, Papadakis & Czerny (2004, hereafter N04) andNikołajuk et al. (2006, hereafter N06). However, the method wasnot reliably checked against the dependence on the source accre-tion rate.In this paper we put the idea of σ NXS ∝ M − correlation to c (cid:13) M. Gierli´nski, M. Nikołajuk and B. Czerny the test. We use an extensive set of BHB observations to see if andwhen σ is constant for a given mass and whether it anticorre-lates with the black hole mass. Power density spectra of many AGN can be approximated by a bro-ken power law, with power P ν ∝ ν − below and P ν ∝ ν − abovethe break frequency, ν b (e.g. Markowitz et al. 2003b), where P ν is the power spectral density normalized to the mean and squared.A second break at lower frequencies, below which the power isroughly P ν ∝ ν , has been also observed (e.g. Pounds et al. 2001;Markowitz, Edelson & Vauhgan 2003a). At the zeroth order of ap-proximation this is consistent with the PDS observed in stellar-mass BHB in the hard spectral state. Fig. 1 shows a sample ofPDS from Galactic BHB in the hard state (details of the data re-duction are described in Section 3). Plainly, these spectra are muchmore complex that a doubly-broken power law, with multiple broadand narrow noise components, usually well described by a series ofLorentzians (e.g. Pottschmidt et al. 2003). However, despite thiscomplexity, the entire spectral shape roughly resembles a (doubly)broken power law.N04 assumed that the break frequency is inversely propor-tional to the black hole mass, while the P ν ∝ ν − part of thePDS below the break (the ‘flat top’ in νP ν diagrams) has constantnormalization, independent of the black hole mass. Yet inspectionof several BHB power spectra clearly shows that neither of theseis constant for a given source. The break frequency is known tochange with accretion rate (e.g. Done & Gierli´nski 2005). The ‘flattop’ normalization can change as well, as one can see in GX 339–4spectra in Fig. 1.There is, however, one feature of these power spectra thatremains remarkably invariant: the high-frequency spectral shape,above ν b . For a given source it can be roughly described by a singlepower law with constant index of 1.5–2.0, and constant normaliza-tion for various observations differing in luminosity by more thanone order of magnitude. In this paper we test the idea that the high-frequency part of the PDS remains fairly constant for a given sourceand scales with the black hole mass. This is a simple refinement ofthe idea proposed by N04 and later developed by N06.Here we do not make any assumptions about how the char-acteristic frequencies (e.g. break frequency) depend on black holemass. Instead, we assume that the the PDS above the break fre-quency (the high-frequency tail) has a universal spectral shape(roughly ∝ ν − ) with normalization depending on the black holemass. This can be written as P ν = C M ( ν/ν ) − , (1)where ν is an arbitrary frequency which we chose to be ν = 1 Hz.Thus, C M (in units of Hz − ) is the normalization of the (extrapo-lated) high-frequency tail at 1 Hz.The assumption that C M is unique function of the black holemass would directly correspond to the original assumption of N04about constancy of P ( ν b ) ν b if ν b were constant for a given blackhole mass. Due to limited statistics it is often difficult to study de-tails of the high-frequency shape of the PDS. Therefore, we sim-plify the situation by calculating the amplitude, or the excess vari-ance, of variability in a given frequency band significantly abovethe break. This can be done directly from a light curve or by inte-grating the PDS. The excess variance calculated between frequen-cies ν and ν (both greater than ν b ) is: Source Name Start EndXTE J1118+480 2000-03-29 2000-08-082005-01-13 2005-02-264U 1543–47 2002-06-17 2002-10-11XTE J1550–564 1998-09-07 1999-05-202000-04-10 2000-07-162001-01-28 2001-04-292002-01-10 2002-03-052003-03-27 2003-05-16XTE J1650–500 2001-09-06 2002-04-21GRO J1655–40 2005-02-20 2005-11-11GX 339–4 2002-04-02 2003-05-062003-12-28 2005-08-12Cyg X-1 1996-02-12 2006-01-12
Table 1.
Log of
RXTE observations. Each set of data corresponds to onetransient outburst. For Cyg X-1 we used all data publicly available in Febru-ary 2007. σ = Z ν ν P ν dν = C M ν (cid:16) ν ν − ν ν (cid:17) . (2)The key assumption, which we want to test in this paper, is that C M is inversely proportional to the black hole mass, C M = C/M ,where C is a constant. We note that our C is the same constant asconstant C defined by N04 in eq. 4, divided by ν . We used publicly available
Rossi X-ray Timing Explorer ( RXTE )Proportional Counter Array (PCA) data of seven black hole bi-naries, listed in Table 1. First, we extracted background-correctedenergy spectra from Standard-2 data (top layer, detector 2 only)for each pointed observation. These were used to create hardness-intensity diagrams (HID). Intensity is defined as the total 2–60 keVcount rate and the hardness ratio is the ratio of count rates in energybands 6.3–10.5 and 3.8–6.3 keV. We also calculated power densityspectra (PDS) for each observation from full-band (2–60 keV) datain 0.0039–128 Hz frequency band. We subtracted the Poissoniannoise from the PDS, corrected them for dead-time effects (Revnivt-sev, Gilfanov & Churazov 2000) and background (Berger & vander Klis 1994)We used HIDs to identify hard state spectra. In transients thesespectra lie on the vertical branch of the diagram at hardness ra-tio & σ , with their statistical errors, α , for each pointed obser-vation in the hard state. The power (or excess variance) was mea-sured by integrating the PDS over the frequency band 10–128 Hz c (cid:13) , 1–9 igh-frequency power in black holes Figure 1.
Power density spectra of black hole X-ray binaries. Each panel shows three spectra in the low/hard spectral state, selected from the data analyzedin Sec. 3. The selection demonstrates that despite dramatic changes in the PDS shape at lower frequencies (including variable QPO), the high-frequency part( &
10 Hz) of the PDS remains relatively constant. (eq. 2). Then, we computed the mean amplitude, h C M i for a givensource (or a particular outburst of the source) weighted by errorsof C M . The error of h C M i was estimated from χ statistics for ∆ χ = 2 . , i.e. corresponding to 90 per cent confidence limits.Despite the initial count rate selection, some data sets (in par-ticular from short observations) gave high-frequency power withlarge errors. We discarded all these, setting an arbitrary upper limiton error, α < . σ . Figs. 2 and 3 show selection of hard-state data (hardness-intensitydiagrams on the left) and measured amplitude of the high-frequency tail, C M in the right-hand panels. It is obvious from thesediagrams that C M is not constant for a given source and varies fromone pointed observation to another. These variations are not verysignificant, though. With very few exceptions (2 observations ofGRO J1655–40 and 3 of GX 339–4) individual C M measurementsare within 3 σ of the mean. The dispersion is higher in Cyg X-1where statistics is better and errors on C M smaller. Generally, wefind that C M does not significantly depend on the source brightness(count rate).More significant differences can be found for different out-bursts of a given source. We have analyzed five outbursts of XTEJ1550–564. The three hard-state outbursts between 2001 and 2003(see Table 2) were similar in all properties, so we analyzed themtogether. They yield mean h C M i = 0 . ± . Hz − . The hardstate in 2000 outburst gave higher h C M i = 0 . ± . Hz − .However, the 1998 outburst was very different, with much lowerand quickly changing high-frequency power, C M ∼ . Hz − .We have excluded the onset of 1998 outburst from further analysis,as it might have represented a different accretion state (we discuss Source Name Outburst h C M i (Hz − ) T hard (days)XTE J1118+480 2000 . ± . . ± . . ± . –XTE J1550–564 1998 ∼ . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . –Cyg X-1 (soft) – . ± . – Table 2.
Mean amplitude of the high-frequency power, h C M i (eq. 1). Weshow data for each outburst separately. T hard is approximate duration ofthe initial hard state during the rise of the outburst. Asterisks denote hard-state only outbursts. 4U 1543–47 was not observed in the initial hard stateand Cyg X-1 is a persistent source. this in detail in Sec. 7). Similarly, GX 339–4 and XTE J1118+480showed different C M during different outbursts, though at least inthe case of the latter one this could have been due to systematiceffects at low count rates, which we discuss later in this section.We looked into the dependence of C M on the hardness ratio,which can be regarded as a crude indicator of the spectral state. Wedid not find any clear general trend, as illustrated in Fig. 4.Additionally, we found variability amplitude for three otherblack hole candidates with no black hole mass estimates. The rel-atively high value of h C M i = 0 . +0 . − . Hz − obtained for XTEJ1720–318 seems to be consistent with a rather low black hole mass c (cid:13) , 1–9 M. Gierli´nski, M. Nikołajuk and B. Czerny
Figure 2.
Panels on the left show hardness-intensity diagrams. Solid lineshows the entire outburst of a given source, while the data points correspondto the hard-state selection of pointed observations, used for further analy-sis. Diagrams on the right show the high-frequency variability amplitude, C M (eq. 1), as a function of count rate, for selected hard-state observa-tions. The horizontal line shows the weighted average of C M for the entireoutburst. XTE J1118+480 shows data from two outbursts: 2000 (red trian-gles) and 2005 (black filled squares). XTE J1550–564 diagram shows datafrom five outbursts: 1998/1999 (red open triangles), 2000 (blue squares)and 2001, 2002 and 2003 together (green crosses). The circled point in XTEJ1118+480 panel was used to estimate background effects (see Section 4). of 5 M ⊙ , as also suggested by Cadolle Bel (2004) from disc spec-tral fitting. For H1743–322 (=XTE J1746–322) and XTE J1748-288 we found h C M i = 0 . ± . and . ± . Hz − ,respectively.The hard X-ray spectral state, which we study in this paper,is dominated by Comptonized emission. It would be very inter-esting to see if similar behaviour can be seen in the other spec- Figure 3.
Continuation of Fig. 2. GX 339–4 diagrams shows two outbursts:2002/2003 (black squares) and 2004 (red triangles). tral state dominated by Comptonization, the very high (or steeppower law) state. We have selected the very high state data fromhardness-intensity diagrams of XTE J1550–564 and GRO J1655–40, as shown in Fig. 5. These observations have significantly lessvariability at frequencies >
10 Hz then the hard state data, so the re-sulting C M is small (black squares in Fig. 5). But we must remem-ber that the soft X-ray part of the spectrum (where PCA has high-est sensitivity) has strong contribution from the cold accretion disc,which can dilute the variability coming from Comptonization (e.g.Done & Gierli´nski 2005) hence suppressing the observed power.Therefore, we extracted additional power spectra at higher ener-gies (above PCA channel 36, roughly corresponding to energy of14 keV), where the disc influence is negligible. The high-frequencyvariability amplitude from these data is much higher, as shown inblue triangles in Fig. 5. Though there is a large scatter in C M fromindividual observations, the mean h C M i is of order of 0.1 Hz − inboth sources, consistent with the hard state results. Therefore, thehigh-frequency variability from Comptonization appears to be verysimilar in both hard and very high states (but see discussion in Sec.7). The soft-state data from our BHB sample are strongly domi-nated by the disc emission. The only source with reasonable countrates at higher energies in the soft state is Cyg X-1. We appliedthe same approach to Cyg X-1 soft-state data (arbitrarily chosenhardness ratio in the range 0.45–0.55), as to the very high statedata in the previous paragraph. The result can be seen in Fig. c (cid:13) , 1–9 igh-frequency power in black holes Figure 4.
Dependence of amplitude of high-frequency variability, C M , onthe hardness ratio. There is no clear trend of C M as a function of spectralstates.
6. The scatter in individual data points in significant, with mean h C M i = 0 . ± . Hz − . Since the fit of the constant to thedata is rather poor, χ ν = 156/73, the error on h C M i is indicativeonly. We tested our results for possible systematic effects. Many of ourobservations have low count rate, which can affect the resultingpower spectra. At low count rates, PCA channels 0–7 can cre-ate artificial power due to problems with the on-board computer(Revnivtsev, private communication). To test the effect of this wehave removed PCA channels 0–7, where PCA configuration al-lowed for that, and calculated new values of C M . We found thatthis had a negligible effect on our results.The white noise level was subtracted from the power spectraduring data reduction. If the white noise level was not estimatedcorrectly, this could have influenced the amplitude of variabilityand the resulting C M . To test this we have selected observationswhere high timing resolution was available and calculated powerspectra up to 1024 Hz. As we do not expect any significant powerabove a few hundred Hz from black holes (Sunyaev & Revnivtsev2000), we assumed that the 512–1024 Hz power could be used asa good white noise estimator. We reanalyzed these data and calcu- Figure 5.
Same as in Figs. 2 and 3, but for the very high (steep power law)spectral state. Black filled squares represent C M calculated from 2–60 keVPCA data, grey (blue in colour) open triangles show the high-energy data(roughly 14–60 keV). Figure 6.
Same as in Fig. 5, but for the soft spectral state of Cyg X-1. lated new values of C M . Again, the effect on our results turned outto be negligible.Background effects can be potentially important for estimat-ing the amplitude of variability, which is defined as (rms / mean) .Our power spectra were calculated from light curves not correctedfor background, so their power is [rms / ( R s + R b )] , where R s and R b are source and background mean count rates, respectively.These PDS were then multiplied by [( R s + R b ) /R s ] , where sourceand background count rates were estimated from light curves. Inthe dimmest observations R s and R b are comparable. To estimatepossible background inaccuracy (which is modelled in the PCArather then measured) we extracted one PCA and HEXTE spec-trum (using standard HEASARC reduction techniques, adding 1per cent systematic errors in the PCA) of XTE J1118+480, cor-responding to a high- C M and low count rate point circled in Fig.2 (ID 90111-01-02-07, observed on 2005-01-24). We have fittedthe joined PCA/HEXTE spectrum with a simple Comptonizationmodel (see Done & Gierli´nski 2003 for details), using PCA in3–40 and HEXTE in 20–200 keV band. The fit was good ( χ = c (cid:13) , 1–9 M. Gierli´nski, M. Nikołajuk and B. Czerny
Source Name M (M ⊙ ) ReferenceXTE J1118+480 8.5 (7.9–9.1) Gelino et al. (2006)4U 1543–47 9.4 (7.4–11.4) Park et al. (2004)XTE J1550–564 10 (9.7–11.6) Orosz et al. (2002)XTE J1650–500 5 (2.7–7.3) Orosz et al. (2004)GRO J1655–40 6.3 (5.8–6.8) Greene, Bailyn & Orosz (2001)GX 339–4 6 (2.5–10) Cowley et al. (2002)Cyg X-1 20 (13.5–29) Zi´ołkowski (2005) Table 3.
Black hole masses in X-ray binaries, used in this paper. / ) with no strong residuals. Then we changed the level ofPCA background by ±
10 per cent. The fit with 90 per cent of back-ground was only marginally worse ( χ = 142 / ), while the 110per cent background resulted in a rather poor fit ( χ = 171 / ).In both cases there were strong residuals in the PCA and disagree-ment with HEXTE data above ∼
25 keV. This shows that the back-ground in the low count rate data is estimated with accuracy muchbetter then 10 per cent. Hence, the uncertainty on C M due to back-ground estimation is no more than a few per cent. The increase of C M by factor 2–5 at low count rates seen in Fig. 2 cannot be causedby incorrect background. Since we see this effect in most sourcesbelow the same count rate of ∼
20 s − (regardless of the distance,hence at different luminosities) it must be of (unknown) instrumen-tal origin. We would like to stress, however, that the increase is notstatistically significant, typically less then 2 σ .The high-frequency variability is known to depend on energyin some sources (e.g. Nowak et al. 1999). This is important whencomparing BHB with AGN, as we look at different parts of theComptonized spectrum, AGN data showing higher scattering or-ders than BHB. We have tested our data for energy dependence.This was possible only in bright observations from XTE J1550-564, GX 339–4 and Cyg X-1, where statistics at higher energieswas good enough. The high-energy (above ∼
14 keV) data give h C M i = 0 . ± . Hz − for the 2000 outburst of XTE J1550–564 and h C M i = 0 . ± . Hz − for the 2002 outburst of GX339–4. These values are consistent with the broad-band data, whichin the PCA is dominated by soft X-rays (see Table 2), which sug-gests that the high-frequency amplitude is not energy-dependent inthese sources. On the other hand, similar approach to Cyg X-1 gave h C M i = 0 . ± . Hz − , higher by about 25 per cent higherthen the broad-band data. Fig. 7( a ) shows the dependence of C M on black holes mass. Weused best currently available mass estimates of black holes in X-raybinaries, as summarized in Table 3. XTE J1650–500 does not havegood mass estimate, though Orosz et al. (2004) found an upper limitof 7.4 M ⊙ . We assumed the mass function, 2.7 M ⊙ , as the lowerlimit and we adopted the actual mass in the middle of this interval,at 5 M ⊙ . We show data from different outbursts of XTE 1118+480,XTE J1550–564 and GX 339–4 separately.Clearly, there is no apparent correlation between black holemass and C M , though we must bear in mind that mass estimates inX-ray binaries are not very accurate. Moreover, different outburstsgiving slightly different h C M i create additional dispersion. Hence,the overall uncertainties are rather large, so the expected relation C M = C/M cannot be robustly confirmed from X-ray binaries.To do this, we need to extend our studies to supermassive black holes. N06 compared masses of a sample of Seyfert 1 galaxies mea-sured by reverberation method with masses from high-frequencyvariability, using the value of constant C derived from Cyg X-1observations. Here we use the same sample of AGN in order tocompare them with our much larger sample of BHB, constrain the C M = C/M relation better and get a ‘big picture’ overview ofvariability properties for all masses of black holes. In Fig. 7( c ) weplotted the sample of of N06; panel b shows the overview of stellar-mass and supermassive black holes.The red diagonal line in the diagrams represents the best-fitting function C M = C/M , with C = 1.25 ± . M ⊙ Hz − .We would like to point out that this particular value depends on theselection of X-ray binary data, as different outbursts can give dif-ferent h C M i . Also, errors on mass are non-Gaussian in many cases,so the error on C , given here, is indicative only. It is interesting tonotice that the constant C is 1.24 ± ⊙ Hz − for X-ray bina-ries only and 1.51 +0 . − . M ⊙ Hz − for AGN only; both values areconsistent within error limits. We also plotted a line (green dashed)corresponding to the soft state of Cyg X-1, for comparison (with C = 2.77 M ⊙ Hz − ). Some of the AGN are consistent with thesoft-state line.We also tested whether the relation between variability andmass is really linear. We fitted a more general form of a power-lawdependence, C M = C f ( M/ M ⊙ ) − α to all BHB and AGN data andfound C f = 1 . ± . Hz − and index α = 0 . ± . veryclose to linear relation. Hence, we opt for the simpler solution andregard the linear relation as well established.The C M = C/M relation is in excellent agreement with thedata spanning over eight orders of magnitude in mass. It givesus robust confirmation of our hypothesis that the high-frequencypower is inversely correlated with black hole mass. Even morefirm confirmation would come from objects with intermediateblack holes mass, filling the big gap in Fig. 7( b ). One categoryof sources potentially useful for testing this is the ultra-luminousX-ray sources (ULX) with possible masses of hundreds of M ⊙ .Alas, they are most likely in the very high spectral state, with softX-ray emission dominated by the disc. Besides, there are no re-liable high-frequency power spectra available from ULX. An in-teresting source for comparison turns out to be a dwarf galaxyNGC 4395, with black hole mass estimates between 0.13 and3.6 × M ⊙ (Kraemer et al. 1999; Ho 2002; Filippenko & Ho2003; Vaughan et al. 2005; Greene & Ho 2006). The luminosity islow, L/L
Edd . . (Vaughan et al. 2005). We analyzed its ASCA observations of 2000-05-24 and 2000-05-26 and found the excessvariance for this source and C M = 3 . +1 . − . × − Hz − . We showthe range of masses and C M for NGC 4395 in Fig. 7( b ). They arein excellent agreement with our mass-variability relation! We canalso give an independent estimate of the black hole mass in NGC4395 based on the best-fitting value of C : M ≈ . × M ⊙ . We have analyzed all available hard-state data from seven Galac-tic black hole systems and found that the amplitude of the high-frequency tail, C M , is roughly constant for a given source, chang-ing by no more than a factor two. There is no apparent depen-dence of C M on luminosity or hardness ratio. This contrasts QPOor break frequency behaviour, which typically show a strong cor-relation with luminosity. The scale of change of C M is also muchless than the observed span of QPO/break frequency, which can beone order of magnitude within the hard state. This makes C M much c (cid:13) , 1–9 igh-frequency power in black holes Figure 7.
Dependence of C M on black hole mass. Data in the upper left panel show the results of this paper for X-ray binaries. Black hole masses, withreferences, are listed in Table 3. The lower right panel shows the sample of Seyfert 1 galxies from N06. Black hole masses are obtained from reverberationmethod (Peterson et al. 2004). The lower left panel show an overview of stellar and supermassive black holes. The red cross in the middle of the diagramrepresent NGC 4395. The solid red diagonal line shows the best-fitting relation C M = C/M , with C = 1 . M ⊙ Hz − . The green dashed line correspondsto the soft state of Cyg X-1, with C = 2 . M ⊙ Hz − . more invariant feature of a given black hole and a robust estimatorof its mass.There are, certainly, some departures from this rule. Differ-ent outbursts of the same transient can have slightly different high-frequency tail amplitudes. Most of the sources show an increase in C M below count rate of ∼
20 s − per PCU. This might be attributedto systematic instrumental effects at low count rates. Due to largeerrors this has a negligible effect on our results. But, if this effectextends to higher count rates, than the difference between 2000 and2005 outbursts of XTE J1118+480 (increase of C M below 40 s − ,see Fig. 2) could be of instrumental origin.XTE J1550–564 gives a much clearer exception to this rule.The first outburst in 1998 had a very short initial hard state withthe high-frequency tail amplitude much lower than in the next out-bursts. In contrast, the three hard-state only outbursts in 2001, 2002and 2003 produced very consistent results. The explanation of thisphenomenon might lie in the stability of the accretion flow. Thespectral state during the onset of the 1998 outburst was changingfaster than in any other outburst analyzed here and took a differ-ent track on the colour-colour diagram (Done & Gierli´nski 2003).It is possible that the accretion flow during such a rapid transitionwas in a different state than in slower transitions, possibly due tohigher ionization state of the reflector (Wilson & Done 2001; Done & Gierli´nski 2003). Clearly, this also affected its variability proper-ties, decreasing the high-frequency power. This particular hard statewas different in many aspects, so we rejected it from our sample.The more stable hard-state only outbursts give probably a betterestimate of C M . Similarly, the persistent source, Cyg X-1, gave avery stable and robust high-frequency power.Thus, fast transitions, characterized by an unstable andquickly changing accretion flow, can apparently break the C M = C/M law. In Table 2 we show the approximate duration of theinitial hard state after the onset of the outburst. XTE J1650–500showed a rather quick transition, so might have suffered from a sim-ilar instability, as XTE J1550–564 in 1998. On the other hand, thetail amplitude was very stable and returned to the same level in thehard state at the end of the outburst (see Fig. 2). The 2004/2005 out-burst of GX 339–4 had a much longer hard state than the 2002/2003outburst, so perhaps it was a better estimate of the high-frequencypower.In this work we assumed the constant power-law high-frequency tail with the spectral index of -2. This is not necessarilytrue and detailed fits to PDS of X-ray binaries, where statistics issufficient, show indices between 1.5 and 2.0. We note that the par-ticular choice of the spectral index does not affect the overall resultof our paper, as different index would only introduced a constant c (cid:13) , 1–9 M. Gierli´nski, M. Nikołajuk and B. Czerny offset in C M . Scatter in spectral indices from one observation toanother might introduce some additional uncertainty, though thiswould be very difficult to estimate for AGN.The luminosities of the Seyfert 1 sample used in this paperare generally low, . L Edd , except for 3C120 andNGC 7469, which are significantly brighter. The 3–10 keV photonpower-law indices are Γ ∼ Γ ≈ ) power-lawComptonized tail, while measuring the intrinsic spectral index inAGN is not straightforward due to presence of complex absorptionand/or reflection (Gierli´nski & Done 2004). We cannot rule out (inparticular for brighter AGN) that some of the sources in the sampleare actually in the soft X-ray spectral state. Despite that the mass-variability amplitude correlation seems to hold well for all of them.Some of the objects are more consistent with the soft-state CygX-1 line in Fig. 7( c ). On the other hand, the brightest 3C120 andNGC 7469 lay below the hard-state line. Clearly, the dependence ofhigh-frequency amplitude on luminosity and spectral state in AGNrequires further studies.Our analysis of the very high (steep power law) state showsthat after a simple bandwidth correction the tail power is consis-tent with the hard-state results. This is very encouraging, show-ing that perhaps the universal shape of the high-frequency tail isthe inherent property of Comptonization, regardless of the spectralstate. Narrow-line Seyfert 1 galaxies (NLS1) are most likely thesupermassive counterparts of Galactic sources in the bright veryhigh state (e.g. Pounds, Done & Osborne 1995). N04, and laterNikołajuk, Gurynowicz & Czerny (2007) showed that the excessvariance from the NLS1 sample is by a factor ∼
20 larger than ex-pected from C M = C/M correlation established for moderate-luminosity AGN. NLS1 have also systematically higher break fre-quencies for a given mass, so the luminosity dependence of thebreak frequency (M c Hardy et al. 2006) can perhaps be related to theincrease in C M . One explanation could be a bandpass effect. Theseed photons in X-ray binaries in the very high state are at ∼ .
10 eV. Therefore,we see a different part of the Comptonized spectrum in X-rays.Gierli´nski & Zdziarski (2005) showed that the variability ampli-tude strongly increases with energy in the very high state, while itis almost constant in the hard state. As we see higher orders of scat-tering in NLS1 than in the very-high-state stellar mass black holes,we expect higher variability in the former ones, as observed. An-other possible explanation is contribution from complex absorptionto the observed variability, which can significantly increase the rmsat energies ∼ C M by factor 4.8 for a maximally spinningKerr black hole with respect to a Schwarzschild one. Alas, theblack hole spin is notoriously difficult to measure (compare, e.g.,McClintock et al. 2006 and Middleton, Done & Gierli´nski 2006).The potentially highly spinning GRS 1915+105 has never been ob-served in the low-luminosity hard spectral state. Another culprit isXTE J1650–500, in which a broad iron line, suggesting high blackhole spin, has been reported (Miller et al. 2002, but see also Done& Gierli´nski 2006). However, as one can see in Fig. 7 C M of thissource is well below the C/M line, so its high spin does not seemto be supported by our data. On the other hand the mass of blackhole in this source is not very well established, so we cannot makeany strong statements about it. Fig. 7 shows that the scatter in C M for all sources is only factor two, so we do not expect large scatterin black hole spins in the sample.Yet another factor that might influence the results is the in-clination of the disc with respect to the observer. If rapid X-rayvariability is produced in flares or fluctuations corotating with thedisc, then it is affected by Doppler effects for highly inclined discs.˙Zycki & Nied´zwiecki (2005) calculated these effects and predictedthat high inclinations would give rise to strong increase in the high-frequency power. However, the additional signal appears above ∼
100 Hz and would not be easily detected in PDS. We calculated C M from 10–128 Hz band, so our results should not be affected bythe inclination. An alternative approach of linking variability with black hole massis the mass-luminosity-break frequency scaling, ν b = AL B bol /M C (M c Hardy et al. 2006; K¨ording et al. 2007), where the three observ-ables: mass, luminosity and break frequency form a ‘fundamentalplane’ along which the BHB and AGN data correlate.We would like to point out few advantages of the method pro-posed in this paper over the break-frequency scaling. Generally, itis easier to find variability amplitude than the break frequency, asthe latter requires modelling of the power spectrum (but see Pes-sah 2007). The amplitude method does not involve luminosity oraccretion rate, so it doesn’t require distance to the source, which incase of many Galactic black holes is poorly constrained. Anotheradvantage of this approach is its simplicity, as it relates amplitudeof variability with black hole mass directly, with just one scalingconstant. The break-frequency method requires three independentconstants. This suggests that high-frequency variability amplitudeis more fundamental in nature.One of the drawbacks of the amplitude method is its limitationto the hard X-ray spectral state in BHB. Another state dominatedby Comptonization, the very high (steep power law) state is po-tentially useful, but its application to bright AGN requires better c (cid:13) , 1–9 igh-frequency power in black holes understanding of energy dependence of rms. Soft-state spectra ofBHB are strongly diluted by the (stable) disc and good-statisticshigh-energy data, required to establish C M reliably, is not avail-able (except for Cyg X-1 and, perhaps, GRS 1915+105). But mostof the AGN spectra in the 2–10 keV band are dominated by Comp-tonization, so this method might be valid in the soft state. The sameproblem seems to affect the break-frequency scaling. K¨ording et al.(2007) point out that their method is mostly limited to the hardstate, as measuring and defining the break frequency in soft andvery high states is very difficult.Both break-frequency and amplitude correlations require ashift in the relation when applied to Cyg X-1 in the soft spectralstate (K¨ording et al. 2007), though this is rather difficult to extendto soft states of other BHB.The high-frequency amplitude depends, to some extend, onenergy, while the break-frequency is energy-independent. Anotherpotential disadvantage of the amplitude method is additional vari-ability introduced by ionized smeared absorption or reflectionin some sources, which is most pronounced around 1–2 keV(Markowitz et al. 2003b; Gierli´nski & Done 2006; Crummy et al.2006). Black hole X-ray binaries have a universal shape of the high-frequency tail (above the break frequency) in their PDS, as il-lustrated in Fig. 1. Though the exact shape of the tail is noteasy to establish, it can be approximated by a power law, P ν = C M ( ν/ν ) − . The amplitude, C M of the tail is remarkably con-stant for any given BHB in the hard state, regardless of the lumi-nosity. When extended to supermassive black holes in moderate lu-minosity AGN, the tail amplitude scales very well with black holemass, C M = C/M . The best-fitting value of the scaling constantfrom our sample of BHB and AGN is C = 1.25 ± . M ⊙ Hz − .This method can be applied to estimate black hole masses in manyAGN. If the universal shape of the high-frequency tail is an inher-ent property of Comptonization, it might be applied to other spec-tral states. We speculate that the constancy of the tail is an imprintof the last stable orbit around the black hole. ACKNOWLEDGEMENTS
We thank the anonymous referee for their valuable comments. MGacknowledges support through a PPARC PDRF and Polish Min-istry of Science and Higher Education grant 1P03D08127. MN andBC acknowledges support through Polish Ministry of Science andHigher Education grant 1P03D00829.
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