The Fundamental Plane of Accretion Onto Black Holes with Dynamical Masses
Kayhan Gultekin, Edward M. Cackett, Jon M. Miller, Tiziana Di Matteo, Sera Markoff, Douglas O. Richstone
aa r X i v : . [ a s t r o - ph . H E ] O c t D RAFT VERSION N OVEMBER
21, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
THE FUNDAMENTAL PLANE OF ACCRETION ONTO BLACK HOLES WITH DYNAMICAL MASSES K AYHAN
G ¨
ULTEKIN , E DWARD
M. C
ACKETT , J ON M. M
ILLER , T IZIANA D I M ATTEO , S ERA M ARKOFF , AND D OUGLAS
O.R
ICHSTONE Department of Astronomy, University of Michigan, Ann Arbor, MI, 48109. Send correspondence to [email protected]. McWilliams Center for Cosmology, Physics Department, Carnegie Mellon University, Pittsburgh, PA, 15213. and A. Pannekoek, University of Amsterdam, 1090GE Amsterdam, NL.
Draft version November 21, 2018
ABSTRACTBlack hole accretion and jet production are areas of intensive study in astrophysics. Recent work has founda relation between radio luminosity, X-ray luminosity, and black hole mass. With the assumption that radioand X-ray luminosity are suitable proxies for jet power and accretion power, respectively, a broad fundamen-tal connection between accretion and jet production is implied. In an effort to refine these links and enhancetheir power, we have explored the above relations exclusively among black holes with direct, dynamical mass-measurements. This approach not only eliminates systematic errors incurred through the use of secondary massmeasurements, but also effectively restricts the range of distances considered to a volume-limited sample. Fur-ther, we have exclusively used archival data from the
Chandra X-ray Observatory to best isolate nuclear sources.We find log L R = ( . ± . ) + ( . ± . ) log M BH + ( . ± . ) log L X , in broad agreement with priorefforts. Owing to the nature of our sample, the plane can be turned into an effective mass predictor. Whenthe full sample is considered, masses are predicted less accurately than with the well-known M – s relation. Ifobscured AGN are excluded, the plane is potentially a better predictor than other scaling measures. Subject headings: black hole physics — galaxies: general — galaxies: nuclei — galaxies: statistics INTRODUCTION
Accretion onto black holes has many observable conse-quences, including the production of relativistic jets. The phe-nomenon of jet production appears to be universal, as suchjets are observed both in Active Galactic Nuclei (AGN) andstellar-mass black hole systems as well as in neutron stars,white dwarfs, and even young stellar objects. For black holesources, the length scales and relevant timescales of jets ap-pear to approximately scale with mass over 8 orders of mag-nitude, giving rise to the possibility that jet production mech-anisms scale with mass, similar to the way that accretion diskproperties scale. The mechanism by which jets are drivenfrom black holes, however, remains observationally elusive. Itremains one of the most compelling and important problemsin astrophysics, particularly in high energy astrophysics. Theimpact of relativistic jets on the interstellar medium (Galloet al. 2005b), and large-scale structure in clusters of galaxies(Allen et al. 2006; Fabian et al. 2003; McNamara et al. 2006),has become dramatically clear in the era of imaging and spec-troscopy with
Chandra .Virtually all theories of jet production tie the jet to the ac-cretion disk directly or indirectly (see, e.g., Lynden-Bell 1978;Blandford & Payne 1982, see also van Putten 2009). Thus,there is a broad expectation that jet properties might dependon the mass accretion rate ( ˙ M ) through the disk. The blackhole spin parameter ( a ≡ cJ / GM ; 0 < a <
1) may also bean important factor if the black hole and accretion disk arelinked through magnetic fields (Blandford & Znajek 1977).The spin is also important for accretion disk jet-launching be-cause the inner radius of the accretion will decrease, thus in-creasing the launch velocity. This idea may find some supportin the dichotomy between radio-loud and radio-quiet AGN(Sikora et al. 2007). The high flux of stellar-mass black holesfacilitates spin constraints with current X-ray observatories; inthose systems, the most relativistic jets appear to be launchedby black holes with high spin parameters (Miller et al. 2009). Chandra Fellow
One means by which jet production can be examined isto explore correlations between proxies for mass inflow andjet outflow. In stellar-mass black holes, it was found thatradio emission and X-ray emission are related by L R (cid:181) L . X (Gallo et al. 2003). This correlation was quickly extended toalso include super-massive black holes in AGN, resulting inthe discovery of a “fundamental plane” of black hole activ-ity (Merloni et al. 2003; Falcke et al. 2004, also see Merloniet al. 2006). The plane can be described by log L R = . + .
60 log L X + .
78 log M BH , with a scatter of s R = .
88 dex(where L R is n = − , L X is E = − , and M BH is the black hole’s mass in unitsof M ⊙ ; Merloni et al. 2003; Falcke et al. 2004). Several re-cent works have revisited the original findings with slightlydifferent focuses. K¨ording et al. (2006) found that sourcesemitting far under their Eddington limits followed the relationmore tightly. Wang et al. (2006) found differences in the rela-tionship for radio-loud and radio-quiet AGNs. Li et al. (2008)used a large sample of SDSS-identified broad-line AGNs tostudy a similar relation at lower-frequency (1.4 GHz) radioluminosity and softer-band (0.1–2.4 keV) X-ray luminosities.Yuan et al. (2009) limited the sample to those sources with L X / L Edd < − based on predictions that the correlation be-tween radio and X-ray luminosity steepens to L R (cid:181) L . X atlow accretion rates (Yuan & Cui 2005).It is difficult to overstate the potential importance of thefundamental plane; it suggests that black holes regulate theirradiative and mechanical luminosity in the same way at anygiven accretion rate scaled to Eddington, ˙ m = ˙ M / ˙ M Edd . In thecontext of models that assume jet properties do scale simplywith mass (e.g., Falcke & Biermann 1995; Heinz & Sunyaev2003), the fundamental plane can even be used to constrainthe nature of the accretion inflow. At present, radiatively-inefficient inflow models for X-ray emission, and models as-sociating X-ray flux with synchrotron emission near the baseof a jet, are both consistent with the fundamental plane.To use the fundamental plane as a tool and a diagnostic G¨ultekin et al.instead of as an empirical correlation, however, it must besharpened. Black hole masses represent a significant sourceof uncertainty and scatter in the fundamental plane (Merloniet al. 2003; K¨ording et al. 2006). In this work, we haveconstructed a fundamental plane using only black holes withmasses that have been dynamically determined, the so-called M – s black holes (see G¨ultekin et al. 2009b). Unlike priortreatments, our X-ray data is taken from a single observa-tory and predominately from a single observing mode, and wehave conducted our own consistent analysis of the data. Weanalyzed every archival Chandra
X-ray observation of blackholes with a dynamically-determined mass. Radio data weretaken from archival observations reported in the literature. Byusing a sample of black holes with dynamical masses, we mayprobe the fundamental plane without subjecting the analysis tothe systematic errors inherent in substituting scaling-relation-derived quantities for black hole masses.In Section 2 we describe the sample of black holes used inthis work. We detail our X-ray data reduction and spectral fitsin Section 3. Our fitting methods and results are presented inSection 4. We discuss our results in Section 5 and summarizein Section 6. SAMPLE OF BLACK HOLES
We get M BH from the list of black hole masses compiledin G¨ultekin et al. (2009b), adopting the same distances aswell. This sample of black hole masses includes measure-ments based on high spatial resolution line-of-sight stellarvelocity measurements (e.g., G¨ultekin et al. 2009a), stellarproper motions in our Galaxy (Ghez et al. 2008; Gillessenet al. 2008), gas dynamical measurements (e.g., Barth et al.2001), and maser measurements (e.g., Miyoshi et al. 1995). Itdoes not include reverberation mapping measurements, whichare direct measurements of mass but are secondary in that theyare normalized to the other measurements via the M – s rela-tion (e.g., Peterson et al. 2004; Onken et al. 2004). From thoseavailable black hole masses we use only the measured blackhole masses used in their M – s fits—not upper limits and notthe “omitted sample,” which contains a list of masses with po-tential problems because (1) masses were listed as tentativeby the original study, (2) there was no quantitative analysisof how well the original study’s model fit the data, or (3) thequantitative analysis of the goodness of fit was poor. We re-duce available Chandra data and present X-ray luminositiesfor this collection of potentially problematic masses, but wedo not use them in our fits. Thus, we use only the black holemasses with the most reliable measurements.One of the benefits of using this sample is that most of thedistances to the galaxies are less than 30 Mpc. This distance isclose enough that interestingly low X-ray and radio luminosi-ties will still be measurable. So while this is not a true volume-limited sample, it is insensitive to the potential biases arisingfrom, e.g., a sample limited by X-ray flux. Unlike other sam-ples, however, our sample may be biased to very low nuclearluminosities. The contamination from a bright AGN typicallycauses problems in determining stellar mass-to-light ratio atthe center so that most galaxies selected for dynamical mea-surement do not contain bright AGN. Spiral galaxies, whichare less massive on average than early-type galaxies, may alsobe underrepresented in this sample, and thus low-mass blackholes may also be underrepresented.The radio data we use are 5 GHz peak power measurementsfrom the Ho (2002) compilation of nuclear radio sources. Thedata were compiled to probe whether there was a correlationbetween M BH and L R and thus are ideal for our purposes. X-RAY ANALYSIS
X-ray Data Reduction
The high spatial resolution of
Chandra enables nuclearemission to be isolated best compared to other X-ray obser-vatories. We used
Chandra archival data to obtain accuratemeasurements or tight upper limits of the flux between 2 and10 keV for all galaxies in our sample. This energy range waschosen to probe accretion power rather than total power in-cluding any contaminating diffuse emission and for ease incomparison to previous fundamental plane work.For each source we used a circular extraction region posi-tioned at the brightest point source that was consistent withthe center of galaxy determined by 2MASS images. There areno nuclear point sources in NGC 1399 and NGC 4261, whichwe handled slightly differently as described below. For ex-traction of background spectra, we typically used an annularregion with inner radius slightly larger than the source regionradius. The outer radius was made large enough to encompassa significant number of counts. Point sources were excludedfrom the background region. When there were a large numberof point sources in the annular region surrounding the source,a different region was used, usually an off-nuclear circle. Inthese cases we selected a region where the background lookedto be similar to that surrounding the source.Two cases require special attention to contaminationfrom non-nuclear X-ray emission: NGC 0224 (M31) andNGC 4486 (M87). NGC 0224 has two bright point sourcesnear the center of the galaxy, but neither is the galaxy’s centralblack hole, from which the emission is too dim to be detectedabove the background to high significance ( L X ∼ < erg s − at at assumed distance of D = . Chandra images. In the high spatial resolution
Hubble Space Telescope ( HST ) images, one knot is very close (0 . ′′
85 Harris et al. 2006)to the central engine. As the knot has grown brighter in theoptical by a factor of ∼
100 over the last ∼
10 years, mea-surements of the core X-ray flux become increasingly con-taminated by the knot. We chose the archival
Chandra dataset where the knot was most readily distinguishable from thecore.For the galaxies NGC 1399 and NGC 4261, there is no dis-cernible point source at their nuclei, which are dominated inX-rays by hot gas. For these two sources, we attempt to mea-sure a hypothetical point source at the center. We use a circu-lar region at the center of the diffuse X-ray emission for sourceextraction with an annular background extraction region im-mediately adjacent. For both these sources, X-ray point sourceflux could not be inferred above the background, and they arelisted as upper limits in Table 3.Data reduction followed the standard pipeline, using themost recent
Chandra data reduction software package (CIAOversion 4.1.1) and calibration databases (CALDB version4.1.2). Point-source spectra were extracted using the CIAOtool psextract. Because all observations of interest were donewith the Advanced CCD Imaging Spectrometer (ACIS), weran psextract with the mkacisrmf tool to create the responsematrix file (RMF) and with mkarf set for ACIS ancillary re-sponse file (ARF) creation.
X-ray Spectral Fitting
We modeled the reduced spectra using XSPEC12 (Arnaud1996). If binning the spectra in energy so that each bin con-tained a minimum of 20 counts resulted in five or more bins,we did so and used c statistics; otherwise we did not binthe data and used C -stat statistics (Cash 1979). Each spec-trum was modeled with a photoabsorbed power-law model.he Fundamental Plane with Dynamical Masses 3If such a model did not adequately fit the spectrum for datasets that were strong enough to support a more complicatedmodel, we added additional model components. Galaxiesthat were identified as Seyfert 2 or transitional Seyferts inV´eron-Cetty & V´eron (2006) were modeled with a partiallyphotoabsorbed power-law, representing intrinsic absorptionplus another photoabsorbed component, representing Galac-tic absorption. Galaxies with obvious diffuse hot gas towardstheir nucleus were modeled with photoabsorbed Astrophys-ical Plasma Emission Code (APEC Smith et al. 2001) andpower-law components. Regardless of the continuum model,for spectra that showed an obvious Fe K a line, we added aGaussian for each line. All spectra were fit from E = . c of c / n ≤ E = E = F X , tot , was determined from the model and the1 s errors derived from covariance of the model parameters.We then calculated the unabsorbed flux arising from just thepower-law component between E = F X . Thatis, we de-absorbed the flux and removed contributions fromlines and other model components. We assume the fractionalerror in F X , tot is the same as in F X .For sources that did not constrain the flux from the cen-tral point source, we used the total count rate between E = . s (99.7% confidence) up-per limit to F X with PIMMS assuming a power-law with in-dex G = H ”.Because we are ultimately interested in an accurate mea-surement of F X , it is more important that our models charac-terize the spectrum well over the 2 to 10 keV band than it isto reproduce the underlying physics. We tested this approachby fitting several different models to the same spectrum andrecovered consistent values for F X . The results of fits are dis-played in Table 2, and we show four example spectra withmodels in Figure 1.For many galaxies, multiple Chandra observations wereavailable in the archive. We reduced and analyzed the avail-able data and censored the resulting data by (1) choosing thosethat yielded flux detections as opposed to upper limits, (2)choosing those with smaller values of c / n , (3) preferringhigher precision measurements over lower precision, and (4)observed more closely in time with the available radio datasince variable sources will have L R and L X change in concerton the fundamental plane (see Merloni et al. 2006).We compare our results with results from the literature forthe same data sets in Figure 2. The literature values werescaled to our assumed distances and, in some cases, convertedto the 2-10 keV band with PIMMS and the published spectralfits. The comparison reveals good agreement with no partic-ular bias with exception of a single outlier, NGC 1068. Weexpand on NGC 1068 and Compton-thick sources in generalbelow.For the Milky Way (Sgr A*) we used the literature resultfrom Baganoff et al. (2001) during quiescence. The data weuse are displayed in Table 3 along with other galaxies withdynamically measured black holes without measurements of L X , L R , or either. A summary of the X-ray analysis may begleaned from Figure 3, which shows a histogram of valuesof Eddington fractions f Edd = L X / L Edd for all objects thatresulted in an X-ray measurement. The distribution showsthat while most are accreting at a small fraction of Edding- ton, there are still a wide range of values encompassed in thesample. ANALYSIS
Fitting Method
For our measurement of the relation between M BH , L R and L X , we considered the formlog L R , = R + x m log M BH , + x x log L X , , (1)where we have normalized to L R = erg s − L R , , M BH = M ⊙ M BH , , and L X = L X , in order to minimize in-tercept errors. To find the multi-parameter relation, we mini-mized the following statistic˜ c = (cid:229) i ( R i − R − x m µ i − x x X i ) s r , i + x m s m , i + x x s x , i , (2)where R = log L R , , µ = log M BH , , X = log L X , , and thesum is over each galaxy. The s terms are scatter terms that re-flect deviation from the plane due to intrinsic scatter and mea-surement errors. This statistic is the same statistic used byMerloni et al. (2003). We considered two cases. For the first,we assume that the intrinsic scatter is dominant and isotropicand thus use a total scatter projected in to the R direction: s = s r , i + x m s m , i + x x s x , i . To determine s , we use a trialvalue of s and increase the value until the reduced c isunity after fitting with the new value. For the second, we usethe measurement errors in M BH and L X , assumed to be nor-mally distributed in logarithmic space, for s m and s x respec-tively. The measurement errors in L R are likely the smallest,and thus intrinsic scatter is likely to dominate. Here we as-sume s r = s . In this final case, our fit method is no longersymmetric, but it includes measurement errors and does notassume that the intrinsic scatter is isotropic. Both methodsgive nearly identical results, and we report only results fromthe latter method, which includes measurement errors. The er-rors on fit parameters come from the formal covariance matrixof the fit. Fundamental Plane Slopes
Our best-fit relation for the fundamental plane is R = − . ± . x m = . ± . x x = . ± . . (3)The scatter we find in the L R direction is s = .
00 dex, equiv-alent to 0.70 dex normal to the plane. These results are con-sistent with the findings of Merloni et al. (2003) and of Falckeet al. (2004). We plot several views of the fundamental planein Figure 4 and the edge-on view in Figure 5. It is also inter-esting to note that for a fixed value of M BH our relation finds L R (cid:181) L . X , consistent with the findings of Gallo et al. (2003). M BH as the Dependent Variable We are using black hole masses that have been measured di-rectly. This approach allows us to use L R and L X as predictorvariables for M BH . We perform a multivariate linear regres-sion on L R and L X by assuming a formlog M BH , = µ + c r log L R , + c x log L X , (4)and minimizing c = (cid:229) i ( µ i − µ − c r R − c x X ) s m , i + s , (5) G¨ultekin et al. − − − N o r m a li ze d c oun t s s − k e V − IC4296 0.5 0.8 1 2 5 8Energy (keV)10 − − − N o r m a li ze d c oun t s s − k e V − NGC30310.5 0.8 1 2 5 8Energy (keV)10 − − − N o r m a li ze d c oun t s s − k e V − NGC4151 0.5 0.8 1 2 5 8Energy (keV) 10 − − − N o r m a li ze d c oun t s s − k e V − NGC4594 F IG . 1.— Example Chandra spectra with best-fit models. The models have been folded through the instrument response. The horizontal error bars show thebinning used for the fits. These four galaxies were chosen to show a variety of different models used to fit the data. All spectra included Galactic absorption and apower-law component. NGC 3031, NGC 4151, and NGC 4594 included intrinsic absorption; IC 4296 and NGC 4151 included an APEC model; and NGC 4151included a Gaussian component to model the Fe line.
36 38 40 42 44 46log( L X / erg s − ) [Literature]363840424446 l og ( L X / e r g s − ) [ T h i s w o r k ] F IG . 2.— Comparison of results of X-ray analysis in this work to resultsfrom the literature. All values have been scaled to our adopted distances.Squares indicate straight-forward comparisons. Diamonds indicate that wehave converted the literature result to an unabsorbed 2–10 keV luminosityusing the published spectral fit and absorption. − − − − − −
2 0log( f Edd )0 2 4 N F IG . 3.— Histogram of Eddington fractions defined as f Edd = L X / L Edd . Thecontribution to the histogram from Seyfert galaxies is colored red, from otherSMBH sources is colored blue, and from stellar-mass sources considered insection 5.3 is colored gray. The galaxy with the smallest f Edd is Sgr A*. Awide range of values are present in the sample even if most are found between f Edd = − and 10 − . As expected, galaxies classified as Seyferts are, onaverage, emitting at a higher fraction of Eddington than other sources, andthe stellar-mass sources are emitting at a higher fraction still. where s m , i is the measurement error in M BH and s is an in-trinsic scatter term in the log ( M BH ) direction. As before, theintrinsic scatter term is increased until the resulting best fithe Fundamental Plane with Dynamical Masses 5 F IG . 4.— Four views of the fundamental plane. Data are as described in Sections 2 and 3. Red points are galaxies classified as Seyferts. Blue points areLLAGNs and LINER galaxies. The varying views clearly show that as a whole the points lie on a plane in the dimensions shown. It is especially clear in thetop-right panel that the LLAGN/LINER subsample appear to lie on a one-dimensional manifold. gives c =
1. We find a best-fit relation of µ = . ± . c r = . ± . c x = − . ± . , (6)with an intrinsic scatter of s = .
77 dex in the mass direction.The intrinsic scatter is larger than other scaling relations (e.g., s = . ± .
06 for the M – s relation and s = . ± . M – L relation; G¨ultekin et al. 2009b). We plot projec-tions of fit in the left panel of Figure 6. DISCUSSION
Using A Black Hole’s Luminosity to Estimate Its Mass
By using a sample of galaxies that have directly measuredblack hole masses, we are able to investigate the correlationbetween X-ray and radio luminosity and black hole mass. Themeasure of any correlation’s worth as a predictor is the scatter,and we consider the scatter here. The scatter in the full relationis considerable (0 .
77 dex = . M – s and M – L relations that relate M BH and host galaxy velocity dispersion and bulge luminosityhave intrinsic scatters of 0 .
44 dex = .
75 and 0 .
38 dex = .
28 30 32 34 36 x X L X + x M M BH L R F IG . 5.— The fundamental plane relation. This figure shows the edge-onview of our best-fit relation: x m = .
78 and x x = .
67. Error bars on the x -axisare calculated as s i = x m s m , i + x x s x , i . This view is primarily for comparisonwith Merloni et al. (2003) and with Falcke et al. (2004). Red circles areSeyferts. Blue circles are LINERs and unclassivied LLAGN. holes with mass M BH > × or M BH > M ⊙ , the intrin-sic scatter drops to s = .
45 or 0 .
41, respectively. There areseveral possible interpretations for the decreased scatter whenrestricting the sample by mass. One possibility is that the re-quirement of detection in both radio and X-rays translates toa requirement of high Eddington fraction for low-mass blackholes at a fixed distance. The mean values of f Edd for thewhole sample, for the sample with M BH > × M ⊙ , and forthe sample with M BH > M ⊙ are approximately 6 × − ,6 × − , and 3 × − , respectively. It is possible that whensources accrete at a higher rate, the fundamental plane relationmay no longer apply.Another possible explanation for the smaller scatter in thehigh-mass sample is that the low-scatter trend is real, and thatthe scatter estimated from the entire sample is skewed by afew data points. The most obvious outliers from the left panelof Figure 6 are Circinus and NGC 1068. If these two are elim-inated, the scatter becomes s = .
50 dex. The derived intrin-sic luminosities of these sources may be difficult to determinebecause of obscuration. In these sources we have a poor viewof the central engine and are seeing reflected, rather than directX-ray emission (Matt et al. 1996; Antonucci & Miller 1985).If the intrinsic X-ray luminosity of these sources is higher,then they would lie closer to the best-fit plane than they donow.AGN classification for each galaxy of the sample is listedin Table 3. The distinction between Seyferts and LINERs isjudged from the line ratios with the usual diagnostic and divi-sion set so that Seyferts have [OIII] l b > . f Edd (Ho 2008). The distinctionbetween Seyfert types is determined by the ratio of broad-lineand narrow-line emission. LLAGN are defined by having anH a luminosity smaller than L ( H a ) ≤ erg s − (Ho et al.1997). The difference between Seyfert types is understood tobe due to differing viewing angles with respect to an obscuringdusty torus that surrounds the broad line region (with type 1unobscured and type 2 completely obscured). For a review of the observational differences among the different classes andthe current physical explanations for the differences see thereview by Ho (2008).We may give special consideration to all non-Seyfert AGNsin our sample. Since all Seyferts in our sample are at leastpartially obscured, obscuration is one potential issue that isaddressed. Obscuration will naturally lead to an underesti-mate in X-ray luminosity. We minimize this by fitting forthe absorption across the 0.5–10 keV band. Since the softerphotons are more readily absorbed, the shape of the spectrumgives an indication of the level of absorption. We also use thehard X-ray flux, which is least affected by absorption, for ourX-ray luminosity. Nevertheless, the most heavily obscuredsources may be intrinsically brighter than our fits indicate.We attempt to isolate this issue below by removing Compton-thick sources. In addition to obscuration, as mentioned above,Seyferts also accrete at higher fractions of Eddington and mayaccrete in a mode different from LINERs. In addition, sinceSeyferts are thought to be dominated by thermal output, theirradio luminosities may be poor probes of the power in out-flows and thus not belong on the relation considered here.Thus, there is a physical motivation to separate them from therest of the sample.When we only use the 8 LINER and unclassified LLAGNsources, our fit becomes µ = . ± . c r = . ± . c x = − . ± . , (7)with a scatter of s = .
25, substantially smaller than otherintrinsic scatter measurements found for this relation and ac-tually smaller than the scatter in the M – s and M – L relations.K¨ording et al. (2006) similarly found a substantially reducedscatter in fundamental plane fits to a sample of only stellar-mass black holes, Sgr A*, and LLAGNs. The fit we find issignificantly different from the other fits, notably that it is con-sistent with no dependence on X-ray luminosity ( c x = n H ≥ s − T = . × cm − ). Compton-thick sources will be heavily ob-scured and the intrinsic luminosities may be much higher thanthe observed flux would imply (Levenson et al. 2002, 2006). Ifwe conservatively omit the sources from Table 2 intrinsic ab-sorption larger than 10 cm − (NGC 3031, NGC 4374, andNGC 6251) as well as the sources determined to be Comptonthick from Fe K a modeling (Circinus and NGC 1068; Leven-son et al. 2002, 2006), we obtain µ = . ± . c r = . ± . c x = − . ± . , (8)with a scatter of s = .
53. This result is consistent with theSeyfertless sample at about the 1 s level, though with a largerscatter.he Fundamental Plane with Dynamical Masses 7 c R L R + c X L X M B H c R L R + c X L X M B H F IG . 6.— Best fit linear regression of M BH on L R and L X for (left) all galaxies and for (right) LLAGN and LINER galaxies only. The relation on the right isconsiderably tighter but may be affected by the small number of sources. Red circles are Seyferts. Blue circles are LINERs and unclassivied LLAGN. Sgr A*
Sgr A*, the central black hole in the Galaxy, is a uniquesource in many ways. Its extremely low accretion rate( L X / L Edd ≈ × − ) is two orders of magnitude below thenext lowest in our sample. An analog to Sgr A* could not beobserved outside of the local group.When using only the two nearby super-massive black holeswith extremely well-determined mass and distance (Sgr A*and NGC 4258) and the X-ray binary in which the correla-tion extends over several orders of magnitude (GX 339 − R = − . ± . x m = . ± . x x = . ± . , (9)with an intrinsic scatter of s = .
02, which is not a signifi-cantly different fit.
Stellar-mass Sources
Our initial sample includes only the supermassive blackholes in galactic centers. There are, however, several Galacticstellar-mass black holes with dynamically measured masses.If accretion onto black holes is driven by the same physi-cal processes at all mass scales, then the stellar-mass sourcesshould obey the same relation, which is what Merloni et al.(2003) and Falcke et al. (2004) found. So while our focus hasbeen on super-massive black holes, we may revisit our calcu-lations with the sample of stellar-mass black holes given inTable 1. This sample was selected from stellar-mass blackholes with dynamically determined masses with simultane-ous X-ray and radio data. In addition to the sources listed,there were two stellar mass black holes that had adequate data(4U 1543 −
475 and GRO J1655 −
40) but whose jets may notbe in a steady state and thus skewing the relation.The stellar-mass systems, with the possible exception ofGRS 1915, are in the low/hard state, which is characterizedby a hard X-ray photon index (1 . < G < . f < .
2; Remil-lard & McClintock 2006) and is usually seen at low Eddingtonrates. This state is also typically associated with a steady ra-dio jet whereas jets in the high/soft state are quenched (Fender2001). By requiring radio emission, we essentially requirea low/hard state. If such a state can be extended to SMBHsources, it would naturally compare with the similarly lowEddington rates in LLAGNs in which jet emission is moreprominent compared to Seyferts. The mapping of X-ray bi-nary states to accreting SMBHs is complicated by the fact thatno comparable transitions are seen in SMBHs.These three accreting black holes have masses measuredfrom period measurements of the donor star’s orbit. The massof the donor star is estimated based on spectral type, and theinclination of the orbit for systems such as these is generallyderived from modeling the star’s change in flux, assumed tobe from the change in viewing angle of a tear-drop-shapedobject (ellipsoidal modulation). For two of the three stellar-mass sources we are using, however, the inclination is con-strained by other means. For GRS 1915+105 the inclination isconstrained from the apparent superluminal motion of ejectedjet material that is assumed to be perpendicular to the orbitalplane based on the lack of observed precession (Mirabel &Rodr´ıguez 1994; Greiner et al. 2001b). For Cyg X-1, the in-clination has been estimated in several ways, including UVline modeling and X-ray polarization (Ninkov et al. 1987, andreferences therein).The luminosity data from each source is simultaneous,which is important for these highly variable sources. For twoof the sources, we use two sets of simultaneous observations.Using more than one observation of a particular source in thefit over-weights that source and will skew the fit if it is atypi-cal. Under the assumption that each source belongs in the fitin all of the epochs used, however, they provide valuable extrainformation of possible accretion states in the same relation.The results of our fundamental plane fits become: R = − . ± . x m = . ± . x x = . ± . , (10)with an intrinsic scatter of s = .
88. The uncertainties inslopes have decreased because of the increased range in thevalues present, especially for x m . It is interesting to note thatwhile the best-fit parameters do not significantly change fromour fits to central black holes, the intrinsic scatter does. Thisdecrease can be attributed to the fact that these sources liecloser to the plane. It is also worth noting that the fits do not G¨ultekin et al. TABLE 1S
TELLAR -M ASS B LACK H OLE D ATA
Name D log ( M BH ) log ( L R ) log ( L X ) Refs.GRS 1915+105 11 1.15 ± a ± ± a ± ± ± ± b c ± d ± b c ± d EFERENCES . — (1) Fender et al. 1999; (2) Greiner et al. 2001a; (3)Muno et al. 2001; (4) Shahbaz et al. 1994; (5) Gallo et al. 2005a; (6) Bradleyet al. 2007; (7) Bregman et al. 1973; (8) Herrero et al. 1995; (9) Stirling et al.2001.N
OTE . — Stellar-mass black hole data used in section 5.3. Distancesare given in units of kpc. Black hole masses are in solar units. Radio andX-ray luminosities are in units of erg s − . All values are scaled to the dis-tances given. The sources were in low/hard state for the epochs listed withthe exception of GRS 1915, which may be in a plateau state (Muno et al.2001). The numbers in the reference column give the number of the origi-nal reference for the distance, mass, radio luminosity, and X-ray luminosity,respectively. X-ray luminosities have been converted to the E = a Interpolated from n = .
25 GHz and n = . b Mass uncertainty was estimated from the range of values found in theliterature (McClintock & Remillard 2006). c Extrapolated from n = . n F n . d Data come from Rossi X-ray Timing Explorer (RXTE) All-Sky Monitor(ASM) assuming a standard spectral form. change even though two of the stellar-mass sources are accret-ing at a much higher fraction of Eddington than the supermas-sive sources. GRS 1915+105 is accreting at f Edd ≈ .
06 to 0 . f Edd ≈ .
004 to 0 . f Edd < .
001 (Fig. 3).It should be noted that there are different systematic er-rors in the stellar-mass and central black holes. The massmeasurements are from completely different methods. TheX-ray extragalactic sources may be contaminated from pointsources and may be more heavily obscured than the stellar-mass sources. The extragalactic sources may also be contam-inated by supernova remnants along the line of sight, thoughthis can be mitigated by going to higher frequencies. Stel-lar mass uncertainties are dominated by uncertainties in dis-tance, inclination, and light from accretion (see Reynolds et al.2008).
Future Work
In this paper, we have only included the 18 black holes withmeasured masses, radio fluxes, and X-ray fluxes. This sam-ple makes up slightly more than one third of the entire sampleof black holes with measured masses. There are 11 withoutnuclear radio data or only with upper limits on one or moreof these quantities. There are a further 16 sources with no
Chandra
X-ray fluxes measured because either there are no
Chandra data or merely insufficient data. Many of the sourceshave masses M < M ⊙ . By completing the sample of M – s black holes with further X-ray and radio observations, the in-creased number of data points should be especially helpful indetermining whether the large scatter at the low-mass end andthe small scatter at the high-mass end are actual differences orjust artifacts of a few outliers.Another place for future work is in understanding the ap-parent special place that Seyfert galaxies occupy in the fun-damental plane. If one were to na¨ıvely assign accretionstates used for stellar-mass black holes to Seyfert galaxies,they would be considered in the thermally dominant/high–softstate. For stellar-mass black holes in this state, jets are not measured. That the Seyfert galaxies are an apparent sourceof scatter in the relation may be an indication that they arediverging away from the fundamental plane relation. To bet-ter understand the differences between Seyfert galaxies andthe other sources, a future theoretical work will consider justthese types of sources, including physical modeling of the datasets presented here. CONCLUSIONS
In this paper we analyze the relationship among X-ray lumi-nosity, radio luminosity, and the mass of a black hole. Distinctfrom previous studies of this relationship, we use only blackhole masses that have been dynamically measured. Because ofthe relatively small distances to the objects in this sample, weavoid potential biases arising from flux limited samples. Us-ing the most recent compilation of black hole masses, we ana-lyzed archival
Chandra data to get nuclear X-ray luminositiesin the E = n = L R , = R + x m log M BH , + x x log L X , to find R = − . ± . x m = . ± . x x = . ± .
12 (11)with a scatter of s = .
00 in the log L R direction, consistentwith previous work. We also fit a relation to be used as anestimation for black hole mass based on observations of L X and L R of the form:log M BH , = µ + c r log L R , + c x log L X , , (12)finding µ = . ± . c r = . ± . c x = − . ± . , (13)with an intrinsic scatter of s = .
77 in the log M BH direc-tion. This intrinsic scatter is larger than other scaling relationsinvolving M BH , but decreases considerably when only usingthe most massive black holes or when eliminating obscuredcentral engines from the sample. Both of these issues requirefurther investigation and could be answered by completing thesample with more Chandra observations.We thank the anonymous referee for useful comments thathave improved this paper. KG thanks Fill Humphrey andTom Maccarone for helpful comments. EMC gratefully ac-knowledges support provided by the National Aeronautics andSpace Administration (NASA) through the Chandra Fellow-ship Program, grant number PF8-90052. This work made useof the VizieR catalog access tool, CDS, Strasbourg, France;data products from the Two Micron All Sky Survey (2MASS),which is a joint project of the University of Massachusetts andthe Infrared Processing and Analysis Center/California Insti-tute of Technology, funded by NASA and the National Sci-ence Foundation; NASA’s Astrophysics Data System (ADS);and the NASA/IPAC Extragalactic Database (NED), whichis operated by the Jet Propulsion Laboratory, California In-stitute of Technology, under contract with NASA. Three-dimensional visualization was made possible by the S2PLOTprogramming library described in Barnes et al. (2006).
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UMMARY OF
Chandra S PECTRAL F ITS
Galaxy Obs. ID Exp. c / n Galactic absorption Intrinsic absorption Power-law APEC Gaussian[ks] n H [cm − ] n H [cm − ] f cov G A pl kT APEC [keV] A APEC E line [keV] s line [keV] A line Circinus 356 24.7 261 . /
166 1 . + . − . × . . . . . . − . + . − . . + . − . × − . + . − . . + . − . × − . + . − . . + . − . × − . + . − . × − CygnusA 1707 9.2 143 . /
112 1 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . 6 . + . − . . + . − . × − . + . − . × − IC1459 2196 58.8 189 . /
178 2 . + . − . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .IC4296 3394 24.8 85 . /
74 1 . + . − . × . . . . . . 0 . + . − . . + . − . × − . + . − . . + . − . × − . . . . . . . . .N0221 5690 113.0 19 . /
22 8 . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N0821 6313 49.5 . . . 8 . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N1023 8464 47.6 6 . /
17 1 . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N1068 344 47.4 217 . /
125 1 . + . − . × . . . . . . 3 . + . − . . + . − . × − . + . − . . + . − . × − . . . . . . . . .N1399 a
319 57.4 17 . /
12 4 . + . − . × . . . . . . 4 . + . − . < . × − . . . . . . . . . . . . . . .N2787 4689 30.9 18 . /
21 1 . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3031 6897 14.8 119 . / < . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N3115 2040 37.0 5 . / . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3227 860 49.3 316 . / < . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . 6 . + . − . . + . − . × − . + . − . × − N3245 2926 9.6 . . . 1 . + . − . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3377 2934 39.6 1 . / . + . − . × . . . . . . 3 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3379 7076 69.3 3 . / . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3384 a . + . − . × . . . . . . 3 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3585 2078 35.3 19 . / . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N3607 a . + . − . × . . . . . . 7 . + . − . < . × − . . . . . . . . . . . . . . .N3608 a . / . + . − . × . . . . . . 5 . + . − . < . × − . . . . . . . . . . . . . . .N3998 6781 13.6 421 . /
297 5 . + . − . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4026 a . + . − . × . . . . . . 3 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4151 335 47.4 366 . / < . × . + . − . × . + . − . − . + . − . . + . − . × − . + . − . . + . − . × − . + . − . . + . − . × − . + . − . × − N4258 2340 6.9 67 . /
69 2 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N4261 a . /
169 9 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . + . − . . + . − . × − . . . . . . . . .N4303 2149 28.0 19 . / . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4342 4687 38.3 6 . / < . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4374 803 28.5 18 . /
28 2 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N4459 a . + . − . × . . . . . . 3 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4473 a . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4486 2707 98.7 344 . /
216 5 . + . − . × . . . . . . 0 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4486A a . + . − . × . . . . . . 6 . + . − . < . × − . . . . . . . . . . . . . . .N4564 a . + . − . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4594 1586 18.5 110 . /
102 2 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N4596 a . + . − . × . . . . . . 4 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4649 a . /
37 1 . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4697 784 41.4 3 . / < . × . . . . . . 1 . + . − . . + . − . × − . . . . . . . . . . . . . . .N4945 864 50.9 19 . /
15 1 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . 6 . + . − . . + . − . × − . + . − . × − N5128 3965 49.5 255 . /
200 5 . + . − . × . . . . . . − . + . − . . + . − . × − . . . . . . . . . . . . . . .N5252 4054 60.1 676 . /
445 0 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N5845 4009 30.0 0 . / . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N6251 4130 45.4 456 . /
358 9 . + . − . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N7052 a . / . + . − . × . . . . . . 3 . + . − . . + . − . × − . . . . . . . . . . . . . . .N7457 a . + . − . × . . . . . . 2 . + . − . . + . − . × − . . . . . . . . . . . . . . .N7582 436 13.4 131 . / < . × . + . − . × . + . − . . + . − . . + . − . × − . . . . . . . . . . . . . . .N OTE . — Results from X-ray spectral analysis. First column gives galaxy name. The second column gives
Chandra observation identification number. The third column lists exposure time in units of ks. Fourth column lists c / n where n is the number of degrees of freedom. If the fit used C -stat statistics instead of c statistics, then the third column is left blank. Best-fit parameters with 1 s errors for each. A blank entry in a given column indicates that the givencomponent was not part of the spectral model used. Galaxies with superscript “a” were only able to constrain an upper limit to the flux. The model for Circinus also included a pileup model. TABLE 3B
LACK H OLE D ATA
Galaxy AGN Class. D / Mpc log ( M BH ) log ( L R ) log ( L X ) Ref.Circinus * S2 a ± ± ± ± ± ± ± ± ± ± a ± < .
00 10,11NGC 0821 25.5 7.63 ± ± ± ± a ± ± ± ± < .
64 17NGC 2748 24.9 7.67 ± ± ± ± a ± ± ± ± ± ± a ± ± ± ± a ± ± ± < .
55 12NGC 3585 21.2 8.53 ± ± ± < .
60 27NGC 3608 S3 a ± < .
79 12NGC 3998 * S3b 14.9 8.37 ± ± ± < .
53 27NGC 4258 * S2 7.2 7.58 ± ± ± < .
92 31,29NGC 4291 25.0 8.51 ± ± ± ± ± a ± < .
97 18,24NGC 4473 17.0 8.11 ± < .
50 12NGC 4486 * S3 17.0 9.56 ± ± ± < .
96 37NGC 4564 17.0 7.84 ± < .
79 12NGC 4594 * S1.9 10.3 8.76 ± ± a ± < .
72 18NGC 4649 16.5 9.33 ± < .
95 12,40NGC 4697 12.4 8.29 ± ± ± ± ± ± ± ± ± ± ± < .
69 45,46NGC 7457 14.0 6.61 ± < .
28 12NGC 7582 * S2 a ± ± ± ± ± ± ± ± ± ± ± ± ± ± he Fundamental Plane with Dynamical Masses 13 TABLE 3B
LACK H OLE D ATA R EFERENCES . — (1) Greenhill et al. 2003, (2) Turner & Ho 1983, (3) Cappellari et al. 2002, (4) Sadler et al. 1989, (5) Dalla Bont`a et al. 2008, (6) Sambruna et al. 1999, (7) Ghezet al. 2008 and Gillessen et al. 2008, (8) Ekers et al. 1983, (9) Verolme et al. 2002, (10) Bender et al. 2005, (11) Crane et al. 1992, (12) Gebhardt et al. 2003, (13) Bower et al. 2001,(14) Lodato & Bertin 2003, (15) Ulvestad & Wilson 1984, (16) Atkinson et al. 2005, (17) Gebhardt et al. 2007, (18) Sarzi et al. 2001, (19) Heckman et al. 1980, (20) Devereux et al.2003, (21) Ho & Ulvestad 2001, (22) Emsellem et al. 1999, (23) Barth et al. 2001, (24) Wrobel & Heeschen 1991, (25) Gebhardt et al. 2000, (26) Fabbiano et al. 1989, (27) G¨ultekinet al. 2009a, (28) de Francesco et al. 2006, (29) Wrobel & Heeschen 1984, (30) Herrnstein et al. 2005, (31) Ferrarese et al. 1996, (32) Cretton & van den Bosch 1999, (33) Bower et al.1998, (34) Jenkins et al. 1977, (35) Macchetto et al. 1997, (36) Biretta et al. 1991, (37) Nowak et al. 2007, (38) Kormendy 1988, (39) Hummel et al. 1984, (40) Spencer & Junor 1986,(41) Silge et al. 2005, (42) Wright et al. 1994, (43) Ferrarese & Ford 1999, (44) Jones et al. 1986, (45) van der Marel & van den Bosch 1998, (46) Morganti et al. 1987, (47) Wold et al.2006, (48) Gregory et al. 1994, (49) Tadhunter et al. 2003, (50) Onken et al. 2007, (51) Pastorini et al. 2007, (52) Gregory & Condon 1991, (53) listed as in preparation in Tremaineet al. 2002 but never published, (54) Greenhill et al. 1997, (55) Elmouttie et al. 1997, (56) Capetti et al. 2005, (57) Polletta et al. 1996.N