A power-law break in the near-infrared power spectrum of the Galactic center black hole
L. Meyer, T. Do, A. Ghez, M. R. Morris, S. Yelda, R. Schoedel, A. Eckart
AA power-law break in the near-infrared power spectrumof the Galactic center black hole
L. Meyer, T. Do, A. Ghez, M. R. Morris, S. Yelda
Department of Physics and Astronomy, University of California, Los Angeles, CA90095-1547
R. Sch¨odel
Instituto de Astrof´ısica de Andaluc´ıa – CSIC, Camino Bajo de Hu´etor 50, 18008 Granada,Spain
A. Eckart
Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 K¨oln, Germany
ABSTRACT
Proposed scaling relations of a characteristic timescale in the X-ray powerspectral density of galactic and supermassive black holes have been used to ar-gue that the accretion process is the same for small and large black holes. Here,we report on the discovery of this timescale in the near-infrared radiation ofSgr A*, the 4 · M (cid:12) black hole at the center of our Galaxy, which is the mostextreme sub-Eddington source accessible to observations. Previous simultaneousmonitoring campaigns established a correspondence between the X-ray and near-infrared regime and thus the variability timescales are likely identical for the twowavelengths. We combined Keck and VLT data sets to achieve the necessarydense temporal coverage, and a time baseline of four years allows for a broadtemporal frequency range. Comparison with Monte Carlo simulations is used toaccount for the irregular sampling. We find a timescale at 154 +124 − min (errorsmark the 90% confidence limits) which is inconsistent with a recently proposedscaling relation that uses bolometric luminosity and black hole mass as parame-ters. However, our result fits the expected value if only linear scaling with blackhole mass is assumed. We suggest that the luminosity-mass-timescale relationapplies only to black hole systems in the soft state. In the hard state, which ischaracterized by lower luminosities and accretion rates, there is just linear massscaling, linking Sgr A* to hard state stellar mass black holes. Subject headings: black hole physics, Galaxy: center a r X i v : . [ a s t r o - ph . GA ] F e b
1. Introduction
Cosmic black holes (BH) show a very wide range of masses: from stellar masses tohundreds of millions of solar masses. An open question has long been whether the accretionand variability processes occurring in the immediate vicinity of the event horizon are thesame over this broad mass range. To test this intriguing possibility, investigators identifiedand studied a characteristic timescale associated with the aperiodic X-ray variability of blackhole X-ray binaries (BHXRBs) and active galactic nuclei (AGN; see, e.g., Uttley et al. 2002;Markowitz et al. 2003; Uttley & McHardy 2005). This timescale corresponds to a break inthe power spectral density (PSD) at a certain temporal frequency where a power-law of slope γ (with P ( f ) ∝ f − γ ) breaks to a steeper slope β > γ .McHardy et al. (2006) proposed a scaling relationship between the break frequency, themass of the black hole and the bolometric luminosity of its accretion flow. This extendedearlier work which hypothesized that the break timescales of AGN scale linearly with BHmass from the timescales observed in BHXRBs, albeit with some scatter. McHardy et al.(2006) found that this scatter can be explained by introducing the bolometric luminosityas a correction factor. This lead them to conclude that AGN are scaled-up galactic BHs.However, their sample consists of only 10 AGN and it is therefore desirable to test theirscaling relation with newly determined AGN break frequencies. The BH in our own Galacticcenter (identified with the radio source Sgr A*) is an an especially interesting object, asit is the most underluminous BH accretion system observed thus far (with a bolometricluminosity nine orders of magnitude lower than its Eddington luminosity). It can thereforetest the McHardy et al. (2006) relation in so far inaccessible regions of the parameter space.The existence of a supermassive BH in the center of the Milky Way has been demon-strated beyond reasonable doubt by the proper motion of stars detected in the near-infrared(NIR) waveband (Eckart & Genzel 1996; Ghez et al. 1998, 2000, 2005, 2008; Genzel et al.2000; Sch¨odel et al. 2002; Gillessen et al. 2008). X-ray and NIR emission associated withSgr A* has been observable since 2001 and 2003, respectively (Baganoff et al. 2001; Genzelet al. 2003; Ghez et al. 2004), which showed that it is highly variable at both wavelengths(e.g. Eckart et al. 2006a,b; Meyer et al. 2006a,b; Trippe et al. 2007). Simultaneous NIR andX-ray monitoring campaigns revealed that each X-ray flare is accompanied by a NIR flarewith zero time lag, which leads to the conclusion that the X-ray photons are being producedby Compton scattering off of the relativistic electrons which radiate in the NIR (Eckart etal. 2004, 2006a, 2008; Marrone et al. 2008; Yusef-Zadeh et al. 2006; Hornstein et al. 2007).The fact that the same bunch of electrons is the source for both the NIR and X-ray fluxmakes it possible to interpret the NIR timing studies reported here in the context of theAGN and BHXRB X-ray results. In fact, since the X-ray background is very high due to 3 –the lack of resolving power of X-ray telescopes (and thus only during the brightest flares isSgr A* actually observed; see, e.g., Baganoff et al. 2001; B´elanger et al. 2005), a detailedstudy of the PSD is more complicated in the X-rays than in the NIR .Recent studies of the NIR properties of Sgr A* showed that the variability on timescalesof minutes to hours is completely described by an (unbroken) power-law PSD with a slopeof -1.6 to -2.5 (Meyer et al. 2008; Do et al. 2008, see also Fig. 18 in Eckart et al. 2006a.).In this Letter , we extend the time baseline up to years by combining Keck and VLT data.We report for the first time the existence of a power-law break frequency in the NIR PSD ofSgr A*, and we show that its deduced range is inconsistent with the McHardy et al. (2006)scaling relation, but fits the expected value when linear scaling with mass is assumed.
2. The data
The NIR adaptive optics instruments at Keck II (NIRC2) and at the VLT UT4 (NACO)have been used for Sgr A* observations since 2003. A time baseline of years and a samplingtimescale of minutes are needed to analyze the broadband PSD of Sgr A* from minutes toyears. Obviously, large gaps in the sampling pattern are unavoidable as the NIR observationshave to be carried out during the night and available telescope time limits realistic data setsto a couple of nights each year.To cover the high-frequency to mid-frequency part of the PSD (meaning timescales ofminutes to days in this context), we looked for Keck and VLT data from consecutive nightswith individual night observations that are as long as possible. We identified the nightsfrom 2004 July 06 – 08 (all times UT) as the most suitable. Beginning on July 06, Sgr A*was observed from 07:50 to 10:35 UT with Keck II, from 23:19 to 04:16 (July 07) with theVLT, then again from 06:35 to 10:30 with Keck II, and finally from 00:53 (July 08) to 06:53with the VLT (the Keck and VLT data have been published in Hornstein 2007; Eckart et al.2006a, respectively). Clearly, this dense coverage is only possible by combining the Keck andVLT data sets. Please note that these alternating observations were coincidental and notarranged. As the Keck data for these two nights were taken at L’ (centered at 3 . µm ) and allother data were taken at K-band (centered on 2 . µm ) , we adopt the finding of Hornstein We want to note that also in the NIR there might be a background due to unresolved stellar sources atthe position of Sgr A*, which has been estimated to contribute up to 30% to the observed flux when Sgr A*is at its lowest flux levels (Do et al. 2008). This low background level is negligible for the purpose of ourstudy. More precisely, Keck uses a K’ filter ( λ = 2 . µm ) and VLT a K S filter ( λ = 2 . µm ). We calibrated α = − . F ν ∝ ν α ). Thus, we scaled the flux at L’ down to the K-band level with this relation. Theresulting lightcurve for the 3 nights is shown in Fig. 1. All fluxes are de-reddened using A K = 3 . α .For the coverage of the mid-frequency to low-frequency part of the temporal spectrum(timescales of days to years) we averaged the data of individual nights. This leads to alightcurve with irregular sampling of down to one day that covers a time baseline of 4 years.The details of the whole data set are given in Table 1.The data reduction was standard, i.e. flat fielding, sky subtraction, and correction forbad/hot pixels. The VLT data have been deconvolved with point-spread functions (PSF)extracted from the individual images (Diolaiti et al. 2000). Aperture photometry was doneon each image and the flux was calibrated relative to sources in the field with known flux. SeeMeyer et al. (2008) for more details. For the Keck data the individual PSFs have been usedto fit sources in the field; see Do et al. (2008) for details. The two photometric techniqueslead to indistinguishable results as shown in Meyer et al. (2008).The irregular sampling with the large gaps makes standard Fourier transform techniquesunsuitable for our data set. We therefore employ the first order structure function (SF),defined as V ( τ ) = < [ s ( t + τ ) − s ( t )] > with s ( t ) being a measurement at time t , to look fora break in the power-law PSD (a power-law in the PSD translates into a power-law in theSF albeit with a different slope, see, e.g., Simonetti et al. 1985; Hughes et al. 1992; Do et al.2008). The SF for our complete data set is shown in Fig. 2. The diamonds represent the high-to mid-frequency data shown in Fig. 1, and the crosses show the mid- to low-frequency part.The overlapping region was used to normalize the low-frequency part such that a continuousSF emerges . It shows the superposition of a constant at short lags ( (cid:46) (cid:38) both data sets in the same way and thus the difference is negligible. Equivalently, we could renormalize the high-frequency part. The reason why the SF is not automaticallycontinuous goes back to the fact that we average the data of a whole night for the low-frequency part (whichreduces the variance), whereas the high-frequency data are not averaged. Contrary to the periodogram(Markowitz et al. 2003), the SF does not have any normalization factor which accounts for that, and thuswe use the overlapping region to determine the renormalization factor.
3. The simulations
We used the following approach for our MC simulations, which is very similar to thePSRESP method by Uttley et al. (2002) and Markowitz et al. (2003): 1. An intrinsic (brokenor unbroken) power-law PSD model is assumed and corresponding lightcurves are generatedusing the algorithm by Timmer & K¨onig (1995). This is done independently for the high-frequency and low-frequency part. These lightcurves, which are significantly longer and moredensely sampled to account for aliasing and leakage, are subsequently resampled accordingto the sampling function of the real data and uncorrelated measurement noise is added (seeDo et al. 2008, for more details). 2. The SF from the simulated lightcurves are calculated inthe same way as is done for the data (the renormalization is done for each pair of long andshort-term simulations individually). 3. An observed χ value is determined by using theobserved SF, the mean simulated SF, and error bars equal to the rms spread of the individualsimulated SFs at each time lag (Uttley et al. 2002). 4. The goodness of fit is computed bymodeling the χ distribution for each assumed PSD model, i.e. several thousand χ valuesare calculated using the individual realizations of the simulations instead of the observeddata. The probability that the model PSD can be rejected is then given by the percentileof the simulated χ values exceeded by the value of observed χ . 5. The steps above arerepeated to scan a range of break frequencies and power-law slopes.For each model PSD we used 100 simulations for each the high- and low-frequency part.By combining both sets we arrived at 10,000 simulations to determine the χ -distribution.The class of models we employed as the intrinsic PSD is the singly broken power-law: P ( f ) = (cid:26) A ( f /f br ) − γ , f ≤ f br ,A ( f /f br ) − β , f > f br . (1)Here, f br is the break frequency (the characteristic timescale), A the PSD amplitude at f br , β the high-frequency power-law slope, and γ is the low-frequency power-law slope with theconstraint γ < β . We tested a β range of 0 . − γ range of 0 − − − · − min − in multiplicativesteps of 1.2. We also tested unbroken power-law models ( P ( f ) ∝ f − β ) which are, however,rejected with a likelihood of acceptance of 0% (i.e. none of the 10,000 simulations equalledor exceeded the observed χ ). 6 –
4. Results & Discussion
The best fit result with a likelihood of acceptance of 93% can be found at f br = 6 . +5 − . · − min − ( ∼ +124 − min), γ = 0 . +0 . − . , and β = 2 . +0 . − . (errors here are 90% confidencelimits). This solution is plotted over the observed SF in Fig. 2. In Fig. 3 we show theconfidence contours in the β − f br plane at the best fit value for γ = 0 .
3. When the threedimensional distribution is marginalized over two parameters, we arrive at the followinglikeliest values and their 90% confidence limits for the individual parameters: f br = 7 . +11 . − . · − min − ( ∼ +329 − min), γ = 0 . +0 . − . , and β = 2 . +0 . − . .The existence of a break in the power-law PSD of Sgr A* can also be inferred by asimple argument: Table 1 reveals that the mean flux of Sgr A* during one night staysroughly the same over the 4 years of observations. If the power-law PSD extended unbrokenover decades, one would expect to observe very different mean fluxes over a time baselineof years . Our detailed MC simulations serve the purpose of quantifying the location of thepower-law break. The same simple argument also holds true for the mean X-ray flux ofSgr A*.Our result of a break frequency at ∼ . · − min − is strikingly different from theprediction of the scaling relation proposed by McHardy et al. (2006). Using a mass of M = 4 · M (cid:12) (Ghez et al. 2008) and a bolometric luminosity of 2 · erg/s (Narayan etal. 1998), the relation by McHardy et al. (2006) predicts a break frequency at the order of10 − min − for Sgr A*.Interestingly, if the term with the bolometric luminosity is neglected in the McHardyet al. (2006) relation, i.e. the bolometric index (called ’B’ in their paper) is set to zero,their relation predicts a break timescale of ∼
110 min for Sgr A*. Also, our deduced breaktimescale fits the expected value if linear scaling with BH mass is assumed from the typicalbreak timescales observed in the high/soft and low/hard states of Cyg X-1; see Fig. 4. Thissuggests that, while the inclusion of the bolometric luminosity as a free parameter improvesthe fit in a limited luminosity range, it is in fact not the true physical correction factor thatcan explain the scatter around the linear mass scaling.It is, however, uncertain how meaningful a comparison of Sgr A*’s break timescale withthe breaks of the much higher luminosity AGN used to derive the mass-luminosity-timescalerelation of McHardy et al. (2006) is. These Seyferts are almost certainly in a different The high-frequency power-law slope of Sgr A* is close to 2. In this simple case of Brownian motion –assuming there is no break in the PSD – the intrinsic standard deviation of the flux scales with the squareroot of time.
Facilities:
VLT:Yepun (NACO), Keck:II (NIRC2)
REFERENCES
Baganoff, F. K., et al. 2001, Nature, 413, 45B´elanger, G. et al. 2005, ApJ, 635, 1095Diolaiti, E., Bendinelli, O., Bonaccini, D., Close, L., Currie, D., Parmeggiani, G., 2000, A&ASuppl., 147, 335Do, T., Ghez, A., Morris, M., Yelda, S., Meyer, L., Lu, J., Hornstein, S., 2008, ApJ, in pressEckart, A., Genzel, R., 1996, Nature, 383, 415Eckart, A. et al. 2004, A&A, 427, 1 8 –Eckart, A. et al. 2006a, A&A, 450, 535Eckart, A., Sch¨odel, R., Meyer, L., Trippe, S., Ott, T., Genzel, R., 2006b, A&A, 455, 1Eckart, A. et al., 2008, A&A, 479, 625Genzel, R., Pichon, C., Eckart, A., Gerhard, O. E., Ott, T. 2000, MNRAS, 317, 348Genzel, R. et al. 2003, Nature, 425, 934Ghez, A. M., Klein, B. L., Morris, M., Becklin, E. E. 1998, ApJ, 509, 678Ghez, A. M., Morris, M., Becklin, E. E., Tanner, A., Kremenek, T., 2000, Nature, 407, 349Ghez, A. M. et al., 2004, ApJ, 601, L159Ghez, A. M. et al. 2005, ApJ, 620, 744Ghez, A. M., et al. 2008, ApJ, in pressGierlinski, M., Nikolajuk, M., Czerny, B., 2008, MNRAS, 383, 741Gillessen, S., et al., 2006, ApJ, 640, L163Gillessen, S., Eisenhauer, F., Trippe, S., Alexander, T., Genzel, R., Martins, F., Ott, T.,2008, ApJ, subm.Hornstein, S. D. et al., 2007, ApJ, 667, 900Hornstein, S. D., 2007, Ph.D. dissertation, University of California, Los AngelesHughes, P. A., Aller, H. D., & Aller, M. F., 1992, ApJ, 396, 469Markowitz, A., et al. 2003, ApJ, 593, 96Marrone, D. P. et al., 2007, subm. to ApJ, arXiv:0712.2877McHardy, I. M., Koerding, E., Knigge, C., Uttley, P., Fender, R. P., 2006, Nature, 444, 730Meyer, L., Sch¨odel, R., Eckart, A., Karas, V., Dovciak, M., Duschl, W. J., 2006a, A&A, 458,L25Meyer, L., Eckart, A., Sch¨odel, R., Duschl, W. J., Muzic, K., Dovciak, M., Karas, V., 2006b,A&A, 460, 15 9 –Meyer, L., Do, T., Ghez, A., Morris, M. R., Witzel, G., Eckart, A., Belanger, G., Sch¨odel,R., 2008, ApJ, 688, L17Narayan, R., Mahadevan, R., Grindlay, J. E., Popham, R. G., Gammie, C., 1998, ApJ, 492,554Sch¨odel, R. et al. 2002, Nature, 419, 694Simonetti, J. H., Cordes, J. M., & Heeschen, D. S., 1985, ApJ, 296, 46Timmer, J., K¨onig, M., 1995, A&A, 300, 707Trippe, S. et al. 2007, MNRAS, 375, 764Uttley, P., McHardy, I. M., & Papadakis, I. E., 2002, MNRAS, 332, 231Uttley, P., & McHardy, I. M., 2005, MNRAS, 363, 586Yusef-Zadeh, F., et al. 2006, ApJ, 644, 198
This preprint was prepared with the AAS L A TEX macros v5.2.
10 –Table 1. Summary of the data
Date Telescope a Filter Duration Mean Flux b published in(UT) (min) (mJy, de-reddened)2004 July 06 Keck L’ 165 2.64 Hornstein 20072004 July 07 Keck/VLT L’/K 532 4.05 Hornstein 2007/Eckart+ 2006a2004 July 08 VLT K 360 2.00 Eckart+ 2006a2004 July 26 Keck K 42 4.43 Ghez+ 20082005 July 31 VLT/Keck K 591 4.01 Meyer+ 20082006 May 03 Keck K 140 5.53 Do+ 20082006 June 20 Keck K 125 4.57 Do+ 20082006 June 21 Keck K 164 3.62 Do+ 20082006 July 17 Keck K 189 2.86 Do+ 20082007 May 18 Keck K 84 5.53 Do+ 20082007 August 12 Keck K 57 3.05 Do+ 20082008 May 15 Keck K 153 4.57 unpublished2008 June 02 Keck K 160 4.00 unpublished2008 July 24 Keck K 179 3.51 unpublishedNote. — The first three entries are used for the high- to mid-frequency coverage, see Fig. 1. a Keck means NIRC2 at Keck II; VLT means NACO at VLT (Yepun). b The fluxes are K-band fluxes. When the L’ filter was used, we scaled the flux down to the K-band level usinga spectral index of α = − .
11 –Fig. 1.— The de-reddened flux of Sgr A* on 2004 July 06 - 08. The abscissa shows the hourselapsed since July 06 00:00 UT. Error bars are omitted for clarity; a sample is shown in theupper left corner. It is ± .
75 mJy as determined from the structure function. The two biggaps mark the time of daylight. 12 –Fig. 2.— The structure function of Sgr A*. Diamonds represent the high- to mid-frequencydata shown in Fig. 1, crosses show the mid- to low-frequency data (see Table 1). The solidline is our best fit with f br = 6 . · − min − , γ = 0 .
3, and β = 2 .
1, see Section 4. Pleasenote that these parameters describe the PSD and cannot simply be read of the SF depictedhere. Error bars are determined from our Monte Carlo simulations as described in Section 3. 13 –Fig. 3.— Confidence contours showing the errors on the best fit parameters. A slice throughthe three dimensional parameter space at the best fitting γ = 0 . (cid:12)(cid:12)