A CHANDRA SURVEY OF SUPERMASSIVE BLACK HOLES WITH DYNAMICAL MASS MEASUREMENTS

In Gültekin et al. (2009a), we analyzed archival Chandra data of all SMBHs with dynamical masses. The parent sample was the list of "secure" black hole mass detections in Gültekin et al. (2009c). Of the approximately 50 black holes in that list, 21 had only limits on their X-ray luminosity or else were not observed with Chandra at all. Of these 21, we identified 13 whose nuclear flux could potentially be determined with a 30 ks Chandra observation. The others had large amounts of contaminating hot gas or existing archival data indicated that they were too faint to be observed with 30 ks exposures. We were granted 12 of these observations with joint EVLA time. The rest should be observed with X-ray Multi-mirror Mission–Newton, which has a larger effective area, or are extremely faint objects that have not been detected with deep Chandra exposures. Table 1 summarizes target galaxies and masses of the central black holes. All but one galaxy are within 30 Mpc, and the black hole masses span a wide range: 4.1 × 10⁶–$8.0 \times 10^8\, {{M}}_{\scriptscriptstyle \odot }$. The only galaxy with a nuclear activity classification is NGC 5576, which is classified as a Low Ionization Narrow Emission Region (LINER) with broad Balmer lines (Ho et al. 1997; Véron-Cetty & Véron 2006).

One potential selection effect that arises from requiring a dynamical mass measurement for inclusion in our sample is that it is biased to low Eddington rates. In general, the methods used to measure black holes in this sample are hampered by contamination from a strong AGN contribution in the same bands used for the mass measurement. For this reason, the results and conclusions we draw from this study only apply to the very low Eddington rates (∼10⁻⁵ to 10⁻⁹) that our data adequately cover.

To detect points sources in the field of each galaxy, we used the wavdetect tool on the whole image in the whole Chandra band at full resolution, run with the large_detect.pl wrapper script provided by the Advanced CCD Imaging Spectrometer (ACIS) team.⁵ The wrapper script splits the image into multiple, overlapping sub-images and runs wavdetect⁶ on each region, identifying multiply detected sources in the overlapping portions to minimize edge effects in the wavlet algorithm. Wavdetect was used with default settings for detection and background rejection thresholds and with searches in wavelet radii (i.e., the "scales" parameter) of 1, 2, 4, 8, and 16 pixels. The whole image was used for best possible astrometry and to assess qualitatively whether off-nuclear sources on the galaxy are more concentrated than background print sources.

For each Chandra image we considered only the point sources that were located on the galaxy as determined by comparison to Two Micron All Sky Survey and DSS images of the galaxy. The size of the galaxy was determined manually for each galaxy by looking at the images and is roughly comparable to using isophotes of 25 mag arcsec⁻² in the B band or 20 mag arcsec⁻² in the K band. The range of values for diameters at a particular surface brightness in a particular band for each galaxy could vary by as much as 10%, motivating our choice for a simple, manual estimate. In total, we found 68 point sources over the 12 images. We list all point sources detected in Table 2 with a running identification number, IAU approved source name, J2000 coordinates, and net count rates in the 0.3–1, 1–2, and 2–10 keV bands. We show positions of all sources in the Chandra images as well as locations on the DSS images of the galaxy in Figure 1.

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Because we are principally interested in the nuclear X-ray source of each galaxy, we have taken extra care in identifying which point source is the nuclear source. For most galaxies it was unambiguous: there was only one point source consistent with the optical or infrared center of the galaxy. In three galaxies (NGC 4486A, NGC 4596, and NGC 4742) there were two X-ray point sources that were consistent with the position of the galaxy's center. In a fourth galaxy (NGC 3384), there was faint emission that was not detected by wavdetect as a point source, but which we considered for potential confusion. Typical separation of the confusing sources was 1''–3'', which requires only modest absolute astrometric corrections. For the four galaxies, we registered the Chandra images to Sloan Digital Sky Survey or Deep Near Infrared Survey coordinates using background AGNs that appeared in both the X-ray and optical/infrared images. Each image had between three and six sources for registration. The optical/infrared center was obvious in each image so that we could unambiguously determine which source was closer to the galaxy center. In the end, we were able to determine that the Chandra coordinates were sufficient for determining the nuclear sources. The nuclear source in each galaxy is identified in Table 2 in the classification column as "Nuc."

One possible source of confusion is if an X-ray binary was located at the position of the nuclear center. To estimate this amount of contamination, we adopt the Grimm et al. (2003) universal cumulative luminosity function of high-mass X-ray binaries (HMXBs) in a given galaxy as a function of star formation rate, SFR:

$$\begin{equation} N({>}L) = 5.4 (\mathrm{SFR} / {{M}}_{\scriptscriptstyle \odot }\ \mathrm{\ {yr^{-1}}}) [ (L / 10^{38}\ \mathrm{erg\ s^{-1}})^{-0.61} - 0.038 ], \nonumber \\ \end{equation} \tag{ 1 }$$

where the second term in the brackets comes from assuming an upper limit of L_X  =  2.1 × 10⁴⁰ erg s⁻¹ for an HMXB and can be neglected for our calculations. We take the luminosity of each nuclear source; e.g., for the dimmest source, L_X  =  3 × 10³⁸ erg s⁻¹ so that $N(L \,{>}\, 3 \times 10^{38}\ \mathrm{erg\ s^{-1}}) \,{=}\, 2.8 (\mathrm{SFR} / {{M}}_{\scriptscriptstyle \odot }\ \mathrm{\ {yr^{-1}}})$. We calculated the SFR, assuming the (Rosa-González et al. 2002) estimator,

$$\begin{equation} \mathrm{SFR} = 4.5 \times 10^{-44}\ {{M}}_{\scriptscriptstyle \odot }\ \mathrm{\ {yr^{-1}}} (L_\mathrm{FIR} / \mathrm{erg\ s^{-1}}), \end{equation} \tag{ 2 }$$

where the far-infrared luminosity is calculated as L_FIR  =  1.7λF_λ at λ  =  60 μm. This estimator is based on the Kennicutt (1998) prescription and agrees with other, similar prescriptions (e.g., Bell 2003). Using far-infrared may overestimate the SFR in early-type galaxies but is conservative for our argument. We based our calculations on the IRAS 60 μm flux density (Knapp et al. 1989) for all galaxies except NGC 5077 for which we substituted the MIPS 70 μm flux density (Temi et al. 2009). The galaxy with the highest SFR in our sample is NGC 2748 with SFR  =  2.0 M_☉ yr⁻¹, and all but four have $\mathrm{SFR} \,{<}\, 0.1\, {{M}}_{\scriptscriptstyle \odot }\ \mathrm{\ {yr^{-1}}}$. If we assume that the X-ray binaries follow the light in the galaxy, then we can estimate the number of potential contaminating sources. This is an overestimate of central contamination probability for galaxies like NGC 1300 for which X-ray binaries follow the high rate of star formation in the spiral arms. The starlight in the central 1'', roughly our astrometric uncertainty, ranges from approximately 10% for NGC 1300 to less than 1%. Assuming 10% for all galaxies, we can calculate the expected number of contaminating sources at the center each galaxy. The values range from 0.04 for NGC 2748 to less than 0.01 for the majority. These numbers are conservative because the actual amount of light at the center of each galaxy is less than 10%, and the SFR may be overestimated in the early-type galaxies.

For early-type galaxies, a greater concern is the chance positioning of a low-mass X-ray binary (LMXB), which generally traces an older population (Kim & Fabbiano 2004). In this case, we use the results of Kim & Fabbiano (2004), who found the cumulative number of LMXBs greater than a luminosity L_X to scale as N(> L_X) ∼ L⁻¹_X, with a total luminosity of the LMXB population scaling with the galaxy luminosity L_K as L_pop  =  (0.2 ± 0.08) × 10³⁰ erg s⁻¹L_K/L_K☉. This is of most concern for NGC 5077, which has the largest total K-band luminosity of $L_K \,{=}\, 8.5 \times 10^{10}\ L_{K{\scriptscriptstyle \odot }}$ (Skrutskie et al. 2006), and for NGC 4486A, which has the lowest nuclear luminosity (L_X  =  2.5 × 10³⁸ erg s⁻¹) and a K-band luminosity of $L_K \,{=}\, 1.5 \times 10^{10}\ L_{K{\scriptscriptstyle \odot }}$ (Skrutskie et al. 2006). Again assuming that LMXBs follow the light, the expected number of LMXBs that would be as bright or brighter than the identified nuclear sources is less than 0.005 and 0.04 for NGC 5077 and NGC 4486A, respectively. It is lower for all other galaxies. Because each galaxy type is predominantly affected by contamination from only one of either HMXBs or LMXBs and not both, the combined expected contamination by both are then all less than 0.05. Thus, we expect not to have incorrectly identified an X-ray binary as an SMBH source.

We followed the standard pipeline in reduction of all data sets, using the most recent Chandra data reduction software package (CIAO version 4.3) and calibration databases (CALDB version 4.4.3). There was no significant background flaring, so there was no need for filtering. We used the CIAO tool psextract to extract the point-source spectra. Since our observations all used the ACIS, we ran psextract with the mkacisrmf tool to create the response matrix file and with mkarf set for ACIS ancillary response file creation. Source regions were circles with centers at the coordinates given by wavdetect. The radii of the regions were the semimajor axes of the wavdetect error ellipses. This accounts for two effects: (1) the uncertainty in the location of the source and (2) the degradation in the point-spread function off-axis. For background regions, we used annuli with inner radii just larger than the source region radius and outer radii that was typically 14 pixels larger.

We modeled the reduced spectra using XSPEC12 (Arnaud 1996). For some sources we used χ² statistics and for others C-stat statistics (Cash 1979). The decision on which statistics to use was based on the number of photons available for binning in energy bands. If binning the spectra in energy so that each bin contained a minimum of 20 counts resulted in five or more bins, we used χ² statistics; otherwise, we used C-stat statistics with unbinned spectra. There is necessarily a loss of information when binning the spectra, but we found that when using both types of statistics on sources that had enough counts to support it, the resulting parameter estimates were always consistent within 1σ. We report the results from χ² statistics on account of the intuitive nature of the goodness of fit with χ² statistics.

All spectra were modeled with a photoabsorbed power-law model with the intrinsic flux modeled directly as one of the parameters, i.e., the XSPEC model used was phabs(cflux * powerlaw), with the cflux component normalized to the 2–10 keV band. The spectra were fitted from 0.3 to 10 keV. While most of the photon counts above ∼7 keV are probably dominated by background, the fitting methodology takes this into account. We tested our results for sensitivity to the adopted upper energy cutoff and found that as long as it was greater than 5 keV, our results were robust in the sense that they changed by far less than the 1σ uncertainties. When fitting the spectrum, we set a hard minimum for the absorption to be the Galactic value for N_H toward each source (Bajaja et al. 2005), but in calculating the uncertainty in the column, we allowed it to drop below this value. The results of the spectral fits are presented in Table 3 with best-fit parameters and 1σ (68%) uncertainties for N_H, log F_2–10, and Γ (the power-law index). Some sources did not have sufficient counts to produce reliable fits and are not included in the table. Fits to three sources (5, 14, and 62) were unconstrained on at least the spectral index parameter, Γ, and we consider these fits approximate.

Table 3. Results of Spectral Fits

Galaxy	ID	Class.	NH	log F2–10	Γ	χ2/dof
(1)	(2)	(3)	(4)	(5)	(6)	(7)
NGC 1300	3		0.0+0.3 − 0.0	−15.0+0.3 − 0.5	2.6+1.6 − 0.6	⋅⋅⋅
⋅⋅⋅	5		4.4+0.4 − 0.7	−14.5+0.2 − 0.2	$10.0_{\phantom{-0.0}}^{\phantom{+0.0}}$	⋅⋅⋅
⋅⋅⋅	6	ULX	0.6+0.3 − 0.2	−14.6+0.2 − 0.2	4.3+1.1 − 0.9	⋅⋅⋅
⋅⋅⋅	9	ULX	0.9+0.9 − 0.8	−15.5+0.5 − 0.9	5.9+4.0 − 3.4	⋅⋅⋅
⋅⋅⋅	10		0.1+0.3 − 0.1	−15.0+0.4 − 0.6	2.8+1.7 − 0.7	⋅⋅⋅
⋅⋅⋅	12	ULX	0.3+0.3 − 0.2	−15.1+0.4 − 0.5	3.8+1.7 − 1.2	⋅⋅⋅
⋅⋅⋅	13	Nuc.	0.0+0.1 − 0.0	−13.0+0.1 − 0.1	0.4+0.3 − 0.2	4.51/3
⋅⋅⋅	14		0.6+0.4 − 0.6	$-14.8_{\phantom{-0.0}}^{\phantom{+0.0}}$	$4.1_{\phantom{-0.0}}^{\phantom{+0.0}}$	⋅⋅⋅
NGC 2748	18	ULX	1.6+0.6 − 0.5	−13.2+0.1 − 0.1	2.3+0.7 − 0.7	2.09/3
⋅⋅⋅	20		0.1+0.2 − 0.1	−14.4+0.2 − 0.3	1.8+0.9 − 0.5	⋅⋅⋅
⋅⋅⋅	21	Nuc.	1.0+1.2 − 0.5	−15.0+0.5 − 0.6	5.0+4.1 − 2.0	⋅⋅⋅
⋅⋅⋅	22	ULX	1.0+0.6 − 0.4	−15.0+0.3 − 0.4	5.1+2.3 − 1.6	⋅⋅⋅
⋅⋅⋅	23	ULX	2.3+1.9 − 1.0	−14.4+0.2 − 0.4	5.1+6.4 − 1.7	⋅⋅⋅
⋅⋅⋅	24	ULX	1.6+0.4 − 0.3	−13.5+0.1 − 0.1	3.0+0.5 − 0.5	⋅⋅⋅
⋅⋅⋅	25	ULX	0.2+0.1 − 0.1	−13.7+0.1 − 0.1	2.9+0.4 − 0.4	6.23/9
NGC 2778	26	ULX	0.2+0.2 − 0.2	−14.5+0.3 − 0.4	2.4+1.1 − 0.8	⋅⋅⋅
⋅⋅⋅	27	Nuc.	0.2+0.3 − 0.2	−15.7+0.6 − 0.8	4.6+2.6 − 1.4	⋅⋅⋅
NGC 3384	29		0.0+0.2 − 0.0	−13.3+0.2 − 0.3	1.0+0.7 − 0.3	⋅⋅⋅
⋅⋅⋅	30	ULX	0.6+0.4 − 0.3	−15.2+0.4 − 0.4	4.1+1.9 − 1.4	⋅⋅⋅
⋅⋅⋅	32	Nuc.	0.1+0.2 − 0.1	−13.8+0.2 − 0.3	2.0+0.7 − 0.4	⋅⋅⋅
⋅⋅⋅	34	ULX	0.8+0.8 − 0.5	−14.9+0.5 − 0.7	3.8+3.3 − 1.8	⋅⋅⋅
⋅⋅⋅	35	ULX	0.3+0.3 − 0.2	−15.2+0.4 − 0.4	4.2+1.8 − 1.4	⋅⋅⋅
NGC 4291	36		0.1+0.3 − 0.1	−14.5+0.3 − 0.5	1.8+1.2 − 0.6	⋅⋅⋅
⋅⋅⋅	37	ULX	1.6+1.1 − 0.6	−14.8+0.2 − 0.3	6.1+2.6 − 1.7	⋅⋅⋅
⋅⋅⋅	38		0.6+2.2 − 0.5	−14.0+0.3 − 0.4	1.4+2.5 − 0.9	⋅⋅⋅
⋅⋅⋅	39	Nuc.	0.1+0.1 − 0.1	−13.9+0.1 − 0.1	1.7+0.4 − 0.4	⋅⋅⋅
⋅⋅⋅	40	ULX	0.0+0.1 − 0.0	−13.6+0.1 − 0.1	1.4+0.3 − 0.3	1.07/3
⋅⋅⋅	41		1.1+2.2 − 0.7	−15.6+0.5 − 0.7	6.3+3.7 − 2.7	⋅⋅⋅
NGC 4459	42	Nuc.	0.0+0.1 − 0.0	−13.6+0.1 − 0.1	1.8+0.3 − 0.2	4.28/4
⋅⋅⋅	43	ULX	1.7+0.9 − 0.8	−14.6+0.2 − 0.2	4.7+2.1 − 1.7	⋅⋅⋅
⋅⋅⋅	44		0.3+0.2 − 0.1	−13.6+0.1 − 0.1	1.8+0.4 − 0.3	⋅⋅⋅
NGC 4486A	46		0.4+0.3 − 0.2	−14.2+0.2 − 0.2	2.5+0.8 − 0.6	⋅⋅⋅
⋅⋅⋅	47	Nuc.	0.3+0.6 − 0.3	−14.4+0.4 − 0.3	2.0+1.4 − 1.0	⋅⋅⋅
NGC 4596	48	Nuc.	0.0+0.2 − 0.0	−14.4+0.2 − 0.3	2.0+0.9 − 0.4	⋅⋅⋅
⋅⋅⋅	49		0.0+0.2 − 0.0	−13.9+0.1 − 0.2	1.5+0.6 − 0.3	⋅⋅⋅
⋅⋅⋅	50		0.1+0.1 − 0.1	−14.0+0.2 − 0.2	2.0+0.6 − 0.5	⋅⋅⋅
⋅⋅⋅	53		0.0+0.0 − 0.0	−13.9+0.1 − 0.2	1.1+0.3 − 0.3	⋅⋅⋅
NGC 4742	54		0.3+0.1 − 0.1	−13.8+0.1 − 0.1	2.2+0.4 − 0.4	⋅⋅⋅
⋅⋅⋅	55	ULX	1.1+0.7 − 0.5	−14.5+0.3 − 0.5	3.5+2.4 − 1.2	⋅⋅⋅
⋅⋅⋅	56	Nuc.	0.1+0.1 − 0.1	−13.6+0.1 − 0.1	1.7+0.4 − 0.2	7.01/4
⋅⋅⋅	57	ULX	0.4+0.5 − 0.3	−15.4+0.5 − 0.6	4.3+2.6 − 1.7	⋅⋅⋅
⋅⋅⋅	58		0.0+0.1 − 0.0	−13.5+0.1 − 0.2	1.3+0.3 − 0.3	⋅⋅⋅
NGC 5077	59	Nuc.	0.0+0.1 − 0.0	−13.5+0.1 − 0.1	1.5+0.2 − 0.2	2.18/4
⋅⋅⋅	60		0.2+0.4 − 0.2	−15.1+0.5 − 0.6	3.1+2.2 − 1.2	⋅⋅⋅
NGC 5576	61	ULX	0.5+0.6 − 0.4	−14.7+0.5 − 0.7	3.0+2.9 − 1.4	⋅⋅⋅
⋅⋅⋅	62		$4.8_{\phantom{-0.0}}^{\phantom{+0.0}}$	$-14.5_{\phantom{-0.0}}^{\phantom{+0.0}}$	$9.5_{\phantom{-0.0}}^{\phantom{+0.0}}$	⋅⋅⋅
⋅⋅⋅	63	Nuc.	0.5+0.2 − 0.2	−14.5+0.2 − 0.3	3.5+1.1 − 0.8	⋅⋅⋅
⋅⋅⋅	64	ULX	0.4+0.6 − 0.4	−15.7+0.9 − 0.8	5.1+2.8 − 2.2	⋅⋅⋅
NGC 7457	67	Nuc.	0.3+0.2 − 0.2	−14.4+0.3 − 0.3	2.5+1.0 − 0.8	⋅⋅⋅
⋅⋅⋅	68		0.3+0.3 − 0.2	−14.5+0.3 − 0.4	2.8+1.2 − 0.9	⋅⋅⋅

Notes. This table lists the best-fit parameters with 1σ uncertainties of our spectral fits to sources with sufficient counts to warrant fitting. If we could not reliably obtain uncertainties, we omit the listing of errors and consider the results approximate. Columns list: (1) the name of the galaxy in which the source appears to lie; (2) our running identification number; (3) classification of source; (4) total absorption column toward the source in units of 10²² cm⁻²; (5) logarithmic normalization of the power law in unabsorbed (intrinsic) flux in the 2–10 keV band in units of erg s⁻¹ cm⁻²; (6) power-law slope; and (7) χ² per degrees of freedom if there were sufficient counts to use χ² statistics.

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The primary scientific goal of our Chandra program is to measure the luminosities of SMBHs with direct, primary mass measurements. For each nuclear source we list the flux and luminosity (assuming isotropic radiation and the distance to each galaxy listed in Table 1) for the 0.3–2, 2–10, and 0.3–10 keV bands in Table 4. The fluxes listed are intrinsic (unabsorbed). Note that the fluxes in each band are derived from fits to the X-ray spectrum using different bands as normalizations, and, therefore, the uncertainties are correlated. In Figure 2, we show plots of the spectra of the nuclear sources in all 12 galaxies, along with the best-fit model spectrum for each source. The parameters of the best-fit model spectra are in Table 3. The spectra have been binned for visualization purposes. The spectra are all well fit by a power law, and none of the spectra requires a more complicated model.

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Figure 3 is a histogram that we call a "distributed histogram" of the values of the power-law exponent (Γ) inferred from our spectral fits. The uncertainties in Γ in some spectral fits are much larger than some of the others and in some cases much larger than the size of the histogram bins in which we are interested. To account for this we assume that the errors in Γ are distributed normally and plot the contribution to each bin according to the value and error. That is, for each value of Γ, the histogram in a bin of width w that is xσ away from the central value is $w (\sigma \sqrt{2\pi })^{-1} \exp (-x^2/2)$, where σ is the error of the measurement. We also include analogous data from Gültekin et al. (2009a). There is an obvious peak at Γ  ≈  2. Related to the distributed histogram we introduce a "distributed median," which we define as the point at which the area under the distributed histogram is half of the total area. The distributed median of Γ is 1.86, very close to the canonical 1.7 power law for AGNs (e.g., Mushotzky 1984).

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Because we have direct, primary measurements of the mass of the central black hole in each of these galaxies, we can report luminosities as true Eddington fractions. In Figure 4, we plot a distributed histogram of log (L_2–10/L_Edd). We include uncertainties in the black hole mass. As can be seen in the figure, the hard X-ray Eddington fractions are small, in the range of 10⁻⁸  <  L_2–10/L_Edd < 10⁻⁶, and the distributed median of the logarithmic Eddington fraction is −7.2. The reason that all sources are at such low accretion rates is that the sources were selected based on having a dynamical mass measurement of the central black hole. Most of the black hole masses in Gültekin et al. (2009c) are based on stellar dynamical models (e.g., Gültekin et al. 2009b) and gas dynamical models (e.g., Barth et al. 2001), and these methods are best when there is no contamination from AGN light. Typical bolometric corrections for low-luminosity sources such as these is ∼10 (Vasudevan & Fabian 2007), so that the bolometric Eddington fractions of these sources are in the range 10⁻⁷–10⁻⁵.

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Note that the power of using dynamical mass measurements instead of secondary mass estimates is evident in our tabulation of L_2–10/L_Edd values. The median error in log (L_2–10) is 0.25 dex, the median error in log M_BH is 0.17 dex, and the median error in secondary mass estimates is 0.51 dex. Thus, the uncertainty in secondary mass estimates dominates the total uncertainty in log f_Edd and is completely subdominant when using primary measurements.

In Figure 5, we plot the spectral power-law slope as a function of log (L_2–10/L_Edd) for the current data and for the data in Gültekin et al. (2009a). We also fit for a linear relation between these two quantities. There is one source, Cen A, with L_2–10/L_Edd  >  10⁻³ and Γ  <  0, that is likely Compton thick so that a power-law index at energies below 10 keV possibly probes different energetics and/or accretion physics than the rest of the sources. Because of this and the potential lever arm it could have on the fit, we fit both with and without this source, but it had no effect on our results. Since there is a strong covariance between Γ and L_2–10, we do not include measurement uncertainties on L_2–10 but instead bootstrap (resample with replacement) the sample to estimate uncertainties, expressing our result as the median with 68% interval. A histogram of the bootstrap results showed roughly Gaussian distributions, indicating reliability of results. Finally, we include a scatter term, assumed normally distributed in Γ because the data clearly deviate from any single line by more than their measurement uncertainties. The fitting method and code are the same as that used and described in Gültekin et al. (2009c). The method uses a generalized maximum likelihood method that is capable of including upper limits, arbitrary error distributions, and arbitrary form of intrinsic scatter. Here, we assume Gaussian errors in Γ and Gaussian scatter in the Γ direction. We plot the best-fit linear relation, which is Γ  =  1.8 ± 0.2 − (0.24 ± 0.12)log (L_2–10/10⁻⁷L_Edd) with an rms intrinsic scatter of 0.65 ± 0.20 (uncertainties are 1σ here and throughout this paper). Our results are inconsistent with a slope of zero or larger at the 2σ level.

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To illustrate what the additional data in this program add as well as how sensitive these results are to the use of dynamical masses, we perform two exercises. First, we repeat the fit without the new data, yielding a slope of −0.33 ± 0.25. So while this is consistent with our full results, it is far less conclusive as to whether the X-ray spectral properties at very low Eddington rates is different from those at high Eddington rates. Second, we fit with the full sample but with masses and uncertainties generated from the M–σ relation:

$$\begin{equation} \log (M_{\mathrm{BH}}/ {{M}}_{\scriptscriptstyle \odot }) = \alpha + \beta \log (\sigma / 200\ \mathrm{km\ s^{-1}}), \end{equation} \tag{ 3 }$$

where σ is the velocity dispersion of the host galaxy, and α  =  8.12 ± 0.08 and β  =  4.24 ± 0.41 are the best-fit M–σ parameters (Gültekin et al. 2009c). Propagating uncertainties, _x, for each quantity, x, the variance in logarithmic mass due to random errors is

$$\begin{equation} \epsilon _{\log M_{\mathrm{BH}}}^2 = \epsilon _\alpha ^2 + [\log (\sigma / 200\ \mathrm{km\ s^{-1}})]^2 \epsilon _\beta ^2 + \frac{\beta ^2}{\sigma ^2} \epsilon _\sigma ^2 + \epsilon _0^2, \end{equation} \tag{ 4 }$$

where ₀  =  0.44 is the observed intrinsic scatter in the M–σ relation. When fitting with these new masses and uncertainties, the slope is −0.27 ± 0.13. So, it appears that without direct masses, we would have come to the same conclusion, though by using masses derived from the M–σ relation for the very galaxies from which M–σ was derived, we necessarily underestimate the amount of systematic errors introduced from using a secondary quantity.

As mentioned in Section 2.1, one consequence of our sample selection is that we only cover SMBHs emitting at very low Eddington rates. A direct consequence of this is that our X-ray observations are more susceptible to contamination from a fixed amount of hot gas than would be an X-ray source that was intrinsically brighter. Since hot gas emission typically peaks at an energy below 1 keV, any contaminating hot gas in our spectral data would tend to make the spectrum softer and thus we would infer a larger value for Γ. This is particularly concerning since our results indicate larger values of Γ at lower Eddington rates, as this selection effect and contamination would manifest. We have mitigated this as much as possible with our selection of background regions, which are annuli that surround the source regions. This effectively removes the contribution of hot gas that is seen in the background region. If, however, the diffuse emission is more centrally concentrated than the background annulus, then there will still be some contamination from diffuse gas. To take this into account we refit all nuclear sources with an additional astrophysical plasma emission code (Smith et al. 2001) component (with abundance fixed to solar). For all but three the results for L_2–10 and Γ are consistent at the 1σ level or better, and the remaining three are consistent at about the 2σ level. This lack of significant change strongly suggests that our results are not affected by the contamination of diffuse gas.

Our results are consistent with previous works looking at the anti-correlation between Γ and Eddington fraction. Using a sample of 55 LLAGNs with a mixture of primary and secondary black hole mass measurements, Gu & Cao (2009) fit a linear relation assuming a constant L_Bol/L_2–10 = 30. For L_Bol/L_Edd  <  10⁻¹, they inferred Γ  =  1.55 ± 0.07 − (0.09 ± 0.03)log (L_Bol/L_Edd). Note that the differences in intercepts between the fits are all a result of defining the independent variable differently. While their results are consistent with ours at about the 1.2σ level, the absence of a scatter term in the fitting function may skew the results since the residuals are larger than would be expected just from measurement errors. As part of the Chandra Multiwavelength Project, Constantin et al. (2009) studied 107 LLAGNs with Chandra observations. They found a strong anti-correlation between Γ and L_Bol/L_Edd, where they assumed L_Bol = 16L_2–10 and used the M–σ relation to estimate black hole masses. Their Spearman rank correlation coefficient was −0.75, and their fit to the relation was Γ  =  0.98 ± 0.13 − (0.27 ± 0.04)log (L_Bol/L_Edd). The goodness of fit, however, was χ²  =  274 for 105 degrees of freedom, indicating that it was very unlikely that the data could come from a scatter-free model. In a study of 153 AGNs detected by the Swift Burst Alert Telescope hard X-ray instrument, Winter et al. (2009) found a no correlation between Γ and L_2–10/L_Edd. It is possible that their sample, which is primarily at L_2–10/L_Edd  >  10⁻⁴ and reaches up to L_2–10/L_Edd  ≈  0.04 is actually measuring an energetically different mode of accretion than our sample. In any case, our study agrees with Winter et al. (2009) that the spectra do not harden with increasing accretion rates. In a study of 13 LINERs with XMM-Newton and/or Chandra observations and assuming very different M–σ relation (due to Graham et al. 2011), Younes et al. (2011) found Γ  =  0.11 ± 0.40 − (0.31 ± 0.06)log (L_2–10/L_Edd). A similar anti-correlation is suggested in X-ray binary data (Corbel et al. 2006). Given the difference in assumptions and fitting techniques, we consider all of these results consistent with each other in the following conclusions: (1) there is an anti-correlation between the hard X-ray photon index, Γ, and Eddington fraction for LLAGNs, and (2) the correlation between Γ and log (L_2–10/L_Edd) has a slope of roughly −0.2.

The anti-correlation between Γ and L_2–10/L_Edd is in contrast to AGNs emitting at higher Eddington fractions. For example, Shemmer et al. (2008) found from fits to a sample of 35 radio-quiet, luminous, and high-Eddington-fraction (L_Bol/L_Edd ∼ 0.01–1) AGNs that there was a strong positive correlation. From their fits, they found log (L_Bol/L_Edd)  =   − 2.4  ±  0.6  +  (0.9  ±  0.3)Γ, corresponding to Γ  =  2.6 + 1.1(L_Bol/L_Edd). Thus, our evidence leads us to conclude that we are seeing a change in the physical processes responsible for emission at low Eddington rates.

The softening of the spectrum with decreasing Eddington fractions at low-mass accretion rates as we find in our sample is predicted by advection-dominated accretion flow (ADAF) models in stellar-mass X-ray binaries (Esin et al. 1997), which should reasonably translate to the low accretion rates seen in our SMBH sources. Esin et al. (1997) models predict a steepening of the 1–10 keV band index of Γ  ≈  1.7 to about 2.2 over mass accretion rates of $\log (\dot{m}) \,{=}\, {-}2.5$ to −4, or a slope of approximately 0.33. The comparison is not direct, but our results are fully consistent with this prediction, which is attributed to the fact that at lower accretion rates bremsstrahlung emission becomes relatively more important compared to Comptonizaion.

ULXs are a category of X-ray emitting point sources that are too bright to be explained by isotropic emission resulting from sub-Eddington accretion onto stellar-mass ($M \lesssim 10\, {{M}}_{\scriptscriptstyle \odot }$) black holes and are also non-nuclear so that they are unlikely to be the galaxy's central SMBH. ULXs are an intriguing class of source because if (1) they are emitting roughly isotropically (i.e., not strongly beamed toward our line of sight), so that we may correctly infer their luminosity, and (2) they are not accreting well above the Eddington limit, so that we may robustly infer a lower limit to the mass, then the most natural explanation is a black hole of mass ∼10²–$10^5\, {{M}}_{\scriptscriptstyle \odot }$. These intermediate-mass black holes (IMBHs) would fill the gap in mass between stellar-mass black holes and SMBHs. IMBHs are interesting because their formation in the local universe requires a non-standard path (Miller & Hamilton 2002; Gültekin et al. 2004, 2006; Portegies Zwart et al. 2004). There are alternative interpretations to ULXs other than IMBHs, including beamed, non-isotropic emission (King et al. 2001). This would change the luminosity inferred from the flux so that it is consistent with sub-Eddington accretion onto stellar-mass black holes. There are at least a few sources (Kaaret et al. 2004; Pakull & Mirioni 2003) where emission from the surrounding medium argues in favor of roughly isotropic emission. Another alternative is to have super-Eddington accretion, which has been invoked in a number of different ways (Begelman 2002, 2006). The very bright source ESO 243-49 HLX-1 is similarly difficult to interpret without invoking an IMBH (Farrell et al. 2009).

Although our survey was not targeted at ULXs, we are sensitive to them and present a list of ULX candidates in Table 5. We adopt the definition of Irwin et al. (2003) that ULXs are sources with L_0.3–10  >  L_ULX ≡ 2 × 10³⁹ erg s⁻¹, which avoids contamination from bright, massive stellar-mass X-ray binaries. We assume that all sources are isotropically emitting at the distance of the host galaxy. Because of measurement uncertainties, there is always a finite probability that a source that appears to have L_0.3–10  >  L_ULX is actually intrinsically too dim to be a ULX. Additionally, sources that are just below L_ULX are still viable ULX candidates. For these reasons, we list all sources that are at least 1σ consistent with being a ULX and list P(L_0.3–10  >  L_ULX), the probability of the source having L_0.3–10  >  L_ULX. This is calculated by finding ΔC, the change in fit statistic, when setting F_0.3–10  =  2 × 10³⁹ erg s⁻¹ (4πD²)⁻¹ and refitting. Then we calculate P(L_0.3–10  >  L_ULX)  =  0.5 ± 0.5 erf [(ΔC/2)^0.5], taking the top or bottom sign for when the best-fit luminosity is above or below L_ULX, respectively.

One source of contamination is background AGNs that appear to be in the galaxy. We have checked the catalog of Véron-Cetty & Véron (2010) and found no known AGNs at the locations of our ULX candidates, but it is clearly possible for a previously unknown AGN to be located at this position. To quantify this effect, we calculate 〈N_BG〉, the expected number of background AGNs in the region of the galaxy target with fluxes greater than 2 × 10³⁹ erg s⁻¹ (4πD²)⁻¹. To do this we use the fits to the background AGN density as a function of flux from Giacconi et al. (2001). Unfortunately, they only provide fits for fluxes in the 0.5–2 and 2–10 keV bands. Since most of the flux in a power-law source comes from the soft band, we use the softer band and assume that all of the flux comes from this band. Since the relations are approximately linear, if only half of the emission comes from this portion of the band, then the average number of background sources would roughly double, still a small number. It is important to use the whole area of the galaxy and the smallest possible flux for the calculation of 〈N_BG〉 because we would have listed any source above the threshold flux apparently within the galaxy as a ULX candidate. The expected background for each galaxy is listed in Table 5, and it is small. The probability of a galaxy having at least one confusing background source is 1 − exp (− 〈N_BG〉), and a lower limit for the probability of an individual ULX candidate's truly being a ULX is P(ULX)  =  P(L_0.3–10  >  L_ULX)exp (− 〈N_BG〉).

Note that the probability of ULX luminosity that we calculate is only accurate if our spectral model is a reasonably good model. In Figure 6, we plot spectra of the six sources (6, 18, 23, 24, 25, and 40) with P(L_0.3–10  >  L_ULX)  >  0.9. Sources 18, 23, and 24 all have N_H  >  10²² cm⁻² and thus while their apparent flux is rather modest, the inferred, intrinsic absorption-corrected luminosity is quite high. This inference depends strongly on the correct estimation of N_H, which derives from the assumption of a power-law spectral form. We have tried fitting with other spectral forms for these three sources but were not able to come up with acceptable fits. In particular, we tried more complicated models with combined disk blackbody and power-law spectral forms, but the disk blackbody temperature normalization was either unphysically high or its normalization was so low as to make the component irrelevant.

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To address the issue of a ULX classification's sensitivity to the unknown intrinsic absorption column, we refit all ULX candidate spectra with N_H fixed to the Galactic value in Table 1 for each galaxy. The resulting luminosities (L_F) are listed in Table 5. We also calculated p-values of a likelihood ratio statistic using Monte Carlo simulations using the method of Protassov et al. (2002). Each simulation synthesized 1000 realizations of the ULX spectra based on the best-fit fixed-N_H model. We then fitted each synthetic data set with a fixed-N_H model and one in which N_H was a free parameter, and calculated a likelihood ratio for each spectrum. This procedure generated a distribution of our likelihood ratio statistic according to the null hypothesis that N_H fixed at the Galactic value is the correct model. We adopt p  <  0.01 as our level of significance for requiring an additional free parameter. Using the fixed N_H spectral model, only four sources (18, 24, 25, and 40) are bright enough to be considered ULXs. We note that source 25 has a luminosity of L_0.3–10 = 1.03 × 10⁴⁰ erg s⁻¹, with P(L_0.3–10  >  L_ULX)  >  0.99 and is very well fit by the absorbed power-law model (p  <  0.002).

Finally, we note that NGC 4291 is also represented in the XMM-Newton catalog of ULXs due to Walton et al. (2011). In NGC 4291 we find two ULX candidates (sources 37 and 39). Given the best-fit absorption column, both of these are likely to have a flux bright enough to be true ULXs (P, (L_0.3–10  >  L_ULX)  >  0.96), and source 40 has L_F  =  2.5^+1.2 _{− 0.9} × 10³⁹. Neither source, however, is listed as a ULX in Walton et al. (2011). These two sources are within 9''of the nuclear source and would not be reliably resolved with XMM-Newton, and Walton et al. (2011) did not consider bright sources within 15'' of the center of elliptical galaxies or any sources within 75 of the center of the galaxy. On the other hand, Walton et al. (2011) list 2XMM J122012.5+752204 as a ULX candidate, using a definition of L_ULX  =  1 × 10³⁹ erg s⁻¹. This XMM source is consistent with the position of our source 36 (CXOU J122012.1+752203) for which we measure a luminosity L_0.3–10 = 4.0^+3.0 _{− 1.6} × 10³⁸ erg s⁻¹. The Walton et al. (2011) classification of this source as a ULX is based on the 2XMMS serendipitous source catalog (Watson et al. 2009) 0.2–12 keV flux of (5.5 ± 1.0) × 10⁻¹⁴ erg s⁻¹ cm⁻², which, at our adopted distance, corresponds to an isotropic luminosity of (4.1 ± 0.8) × 10³⁹. The 2XMMS flux measurement assumes a spectral form of an absorbed power law with Γ  =  1.7 and N_H  =  3 × 10²⁰ cm⁻², very close to the best-fit values we find from our spectral fit. If we fix Γ  =  1.7 and N_H  =  3 × 10²⁰ cm⁻² and fit to our Chandra data, we get a 0.2–12 keV flux of (4.9 ± 1.5) × 10⁻¹⁵ erg s⁻¹ cm⁻². Given that 10 years elapsed between the XMM observation (MJD 51671) and the Chandra observation (MJD 55541), we conclude that the source is variable on these scales.

In addition to the nuclear and ULX candidate sources, we detected 36 off-nuclear sources whose position on the sky is consistent with being in the galaxy. As the flux goes down, the probability for a source to be a background AGN increases. Thus, we cannot be certain that the sources are intrinsic to the galaxy, but, generally, the clustering of point sources within the galaxies' optical extent is much higher than over the entire Chandra image. To summarize the off-nuclear sources, in Figure 7 we plot C_0.3–1/C_1–2, the ratio of the count rates in the 0.3–1 to 1–2 keV bands, as a function of C_2–10, the 2–10 keV count rate. With only two data points (one color and one intensity), it is not possible to break the degeneracy in an absorbed power-law model between N_H and Γ, but the range of parameters needed to reproduce most of the data points is reasonable.

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