GALEX MEASUREMENTS OF THE BIG BLUE BUMP IN SOFT X-RAY–SELECTED ACTIVE GALACTIC NUCLEUS

We measured GALEX FUV and NUV count rates within 18'' photometric apertures for each AGN in our sample. We converted the measured count rates to magnitudes using the GALEX photometric zero points of Morrissey et al. (2005),

$$\begin{equation} m_{\rm FUV}=-2.5\log (C_{\rm FUV})+18.82 \end{equation} \tag{ 1 }$$

$$\begin{equation} m_{\rm NUV}=-2.5\log (C_{\rm NUV})+20.08, \end{equation} \tag{ 2 }$$

where C_x is the count rate in bandpass x. To compute the Galactic extinction corrections, we averaged the reddening law of Cardelli et al. (1989) across the FUV and NUV effective area curves,

$$\begin{equation} R_{\rm X} = \frac{\int ^{\lambda _{2}}_{\lambda _{1}} R(\lambda)T(\lambda)d\lambda }{\int ^{\lambda _{2}}_{\lambda _{1}} T(\lambda)d\lambda }, \end{equation} \tag{ 3 }$$

where R(λ) is the Cardelli et al. (1989) R-value at wavelength λ, and T(λ) is the filter bandpass. We found R_FUV = 8.24 and R_NUV = 8.10. The R-values and E(B − V) color excesses for each line of sight (Schlegel et al. 1998) were used to compute total extinction and correct the measured fluxes for each object. We used the AB-magnitude relation to convert the de-reddened magnitudes to UV flux densities, and we used PIMMS to solve for F_ν(2 keV) for our AGN using the X-ray count rates and spectral indices recorded in the Grupe catalogs. Grupe et al. fixed column densities at the Galactic value unless N_H,fit > N_H,gal + 2 × 10²⁰ cm⁻², in which case the fitted value was used. Determining F_ν in this way assumes that α_x provides a good description of the X-ray spectrum across the entire ROSAT energy range. This assumption is not perfect and is likely to do worse in objects with steeper spectra, so some results may be biased. However, the number of objects with extremely steep spectra is limited (4 objects with α_x > 3), and the uncertainties on these indices are relatively large, so any biases in the measurements are covered by the error budget.

We use black hole masses and Eddington ratios (L/L_edd) from Grupe & Mathur (2004), which include 30 of our 54 AGNs. These black hole masses were determined using the Kaspi et al. (2000) relations, which was calibrated empirically using reverberation-mapped AGNs. The calibration sample included several AGNs with luminosities similar to the AGNs in our sample, so the Kaspi relation should yield reasonably robust black hole masses. Bentz et al. (2006) reported a different scaling relation with a power-law index of 0.52 ± 0.04, which is in agreement with the value expected from theory. If we use the Bentz relation instead of the Kaspi relation, we find significantly larger black hole masses than computed by Grupe & Mathur (2004), but this is not unexpected since the stellar continuum has not been subtracted from L_λ(5100 Å) for our AGN sample, resulting in an overestimate of the radius of the Broad Line Region (BLR). The alternative masses have a significant impact on some of the measured correlations, which can be seen by comparing the correlation coefficients in Tables 3 and 4. Unsurprisingly, using the alternative L_λ(5100 Å)–R_BLR relation has the largest impact on correlations with L_λ(5100 Å); the other changes are rarely significant. Because we have no information on the host galaxies of our AGN, we have elected to use the empirically calibrated Kaspi relation. This will add scatter to our inferred black hole masses, but based on the few qualitative differences between Tables 3 and 4 we infer that the potential to introduce or hide correlations is limited.

Table 3. Physical Correlations (Bentz Masses)

Parameters		NLS1		BLS1		Merged
		rS	Prob.	rS	Prob.	rS	Prob.
νFν(1528 Å)/νFν(2 keV)	MBH	0.47	0.10	0.20	0.54	0.10	0.64
νFν(1528 Å)/νFν(2 keV)	L/Ledd	0.61	0.027	0.36	0.55	0.36	0.054
νFν(1528 Å)/νFν(2 keV)	νLν(1528 Å)	0.92	2.2 × 10−12	0.39	0.052	0.74	2.3 × 10−10
νFν(1528 Å)/νFν(2 keV)	νLν(5100 Å)	0.82	7.2 × 10−8	0.13	0.59	0.60	1.9 × 10−6
νFν(1528 Å)/νFν(2271 Å)	MBH	0.44	0.14	0.48	0.052	0.59	6.6 × 10−4
νFν(1528 Å)/νFν(2271 Å)	L/Ledd	0.53	0.065	0.34	0.54	−0.13	0.56
νFν(1528 Å)/νFν(2271 Å)	νLν(1528 Å)	0.52	3.5 × 10−3	0.61	1.2 × 10−3	0.48	2.4 × 10−4
νFν(1528 Å)/νFν(2271 Å)	νLν(5100 Å)	0.39	0.037	0.37	0.067	0.35	9.7 × 10−3
αx	MBH	0.48	0.10	0.05	0.84	−0.43	0.019
αx	L/Ledd	0.27	0.50	0.14	0.63	0.53	2.7 × 10−3
αx	νLν(1528 Å)	0.55	1.9 × 10−3	0.12	0.62	0.31	0.022
αx	νLν(5100 Å)	0.49	7.0 × 10−3	0.12	0.61	0.28	0.038
FWHM(Hβ)	MBH	0.83	4.7 × 10−4	0.64	5.4 × 10−3	0.90	1.0 × 10−11
FWHM(Hβ)	L/Ledd	−0.54	0.027	−0.20	0.57	−0.80	8.8 × 10−8
FWHM(Hβ)	νLν(1528 Å)	0.32	0.094	0.03	0.88	0.045	0.75
FWHM(Hβ)	νLν(5100 Å)	0.36	0.055	−0.03	0.88	0.08	0.59
FWHM(Hβ)	$M_{\rm BH}/\dot{m}$	0.62	0.023	0.84	2.2 × 10−5	0.94	1.0 × 10−14
MBH	L/Ledd	−0.04	0.89	0.14	0.63	−0.59	5.8 × 10−4
MBH	νLν(5100 Å)	0.90	2.8 × 10−5	0.54	0.025	0.53	2.6 × 10−3
L/Ledd	νLν(5100 Å)	0.044	0.89	0.14	0.63	−0.59	5.8 × 10−4

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Table 4. Physical Correlation Results

Parameters		NLS1		BLS1		Merged
		rs	Prob.	rs	Prob.	rs	Prob.
νFν(1528 Å)/νFν(2 keV)	MBH	0.53	0.065	0.52	0.03	0.28	0.60
νFν(1528 Å)/νFν(2 keV)	L/Ledd	0.55	0.050	0.16	0.59	0.29	0.62
νFν(1528 Å)/νFν(2 keV)	νLν(1528 Å)	0.92	2.2 × 10−12	0.39	0.052	0.74	2.3 × 10−10
νFν(1528 Å)/νFν(2 keV)	νLν(5100 Å)	0.82	7.2 × 10−8	0.13	0.59	0.60	1.9 × 10−6
νFν(1528 Å)/νFν(2271 Å)	MBH	0.47	0.10	0.26	0.50	0.44	0.016
νFν(1528 Å)/νFν(2271 Å)	L/Ledd	0.44	0.13	0.57	0.017	−0.06	0.74
νFν(1528 Å)/νFν(2271 Å)	νLν(1528 Å)	0.52	3.5 × 10−3	0.61	1.2 × 10−3	0.48	2.4 × 10−4
νFν(1528 Å)/νFν(2271 Å)	νLν(5100 Å)	0.39	0.037	0.37	0.067	0.35	9.7 × 10−3
αx	MBH	0.51	0.079	0.28	0.50	−0.28	0.60
αx	L/Ledd	0.21	0.55	0.14	0.62	0.49	5.8 × 10−3
αx	νLν(1528 Å)	0.55	1.9 × 10−3	0.12	0.62	0.31	0.022
αx	νLν(5100 Å)	0.49	7.0 × 10−3	0.12	0.61	0.28	0.038
FWHM(Hβ)	MBH	0.82	6.5 × 10−4	0.42	0.097	0.77	7.7 × 10−7
FWHM(Hβ)	L/Ledd	−0.39	0.53	−0.47	0.060	−0.79	2.7 × 10−7
FWHM(Hβ)	νLν(1528 Å)	0.32	0.094	0.03	0.88	0.045	0.75
FWHM(Hβ)	νLν(5100 Å)	0.36	0.055	−0.03	0.88	0.08	0.59
FWHM(Hβ)	$M_{\rm BH}/\dot{m}_{\rm edd}$	0.87	1.1 × 10−4	0.76	3.6 × 10−4	0.89	6.7 × 10−11
MBH	L/Ledd	−0.14	0.66	0.08	0.77	−0.53	2.8 × 10−3
MBH	νLν(5100 Å)	0.92	7.6 × 10−6	0.63	7.1 × 10−3	0.63	2.1 × 10−4
L/Ledd	νLν(5100 Å)	0.05	0.87	0.63	6.3 × 10−3	0.14	0.55

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We might also consider the impact of radiation pressure on the derived black hole masses, as described by Marconi et al. (2008). Qualitatively, the impact of radiation pressure should result in higher masses for systems radiating closer to their Eddington rate, and Marconi et al. (2008) found that, after applying this correction, the Eddington ratios of NLS1 systems are less extreme than result from applying the Bentz and Kaspi relations. Computing black hole masses using the Marconi relation rather than the Kaspi relation yields masses that are, on average, 0.2 dex larger. The new masses are correlated with the old masses with r_s = 0.69. However, Marconi et al. failed to account for the lower Eddington ratios and reduced radiation pressure implied by the adjusted black hole masses. As a result, their results overestimate black hole masses in systems where the corrections for radiation pressure are significant, and the difference between the "true" black hole masses and the results calculated using the Kaspi relation will be less than the 0.2 dex implied by applying the Marconi relation.

Netzer (2009) found that black hole masses determined using the Marconi relation are distributed differently from black hole masses in type 2 AGNs, suggesting that radiation pressure is not important in nearby AGNs. However, Marconi et al. (2009) in turn suggested that the differences found by Netzer (2009) can be attributed to scatter in the underlying scaling relations rather than a lack of radiation pressure support in the BLR. Nevertheless, Marconi et al. (2009) and Netzer (2009) agree that the Marconi et al. (2008) relation is unable to successfully reproduce the "true," underlying mass distribution, indicating that more work is needed. For the rest of this paper, we consider masses resulting from the Kaspi relation with the caveat that the systematic uncertainties associated with the alternative methods for calculating the black hole mass must also be considered.

The masses resulting from applying the Kaspi relation are more properly called virial products, which differ from the true black hole mass by a geometric factor f. There is significant debate in the literature on the proper value of this constant. Using the dispersion of the Hβ emission line to measure the virial products, Onken et al. (2004) found that a statistical correction of f = 5.5 was required to bring their virial masses into agreement with black hole masses predicted by the M_BH–σ_* relation. By contrast, Watson et al. (2007) found a correction of f = 2.2 for AGN in the Grupe et al. (2004) sample using the line dispersion or f = 0.55 using the FWHM. The latter value disagrees with the results of Kaspi et al. (2000), who found f = 0.75. Watson et al. (2007) also found that there is a systematic difference between the geometric corrections required to bring the BLS1 and NLS1 samples into agreement with the M_BH–σ_* relation. Given this disagreement, we choose not to apply a geometric factor and simply use the virial products. As a result, the absolute masses and Eddington ratios we use are incorrect, but there will be little effect on the measured correlations as long as f is a constant. If the geometric corrections required by NLS1s and BLS1s do indeed differ, the impact of using virial products instead of actual black hole masses might be significant.

In Figure 1, we compare an indicator for the strength of the BBB with respect to the hard X-ray continuum (νF_ν(1528 Å)/νF_ν(2 keV)) with α_x to verify that the correlation between the strength of the BBB and the soft excess, as reported by WF93, also appears in our sample. We find a significant correlation in both the NLS1 and BLS1 samples as well as in the merged sample, as indicated in Table 5. It is apparent that the majority of the BLS1s lie on or near the WF93 relation, but the NLS1s are located systematically above the WF93 best-fit power law. Computing the best-fit relation to our data points in the figure yields

$$\begin{eqnarray} \alpha _{\rm x}&=&(0.84\pm 0.13)\log [\nu F_{\nu }(1528\;\AA)/\nu F_{\nu }(2\;{\rm keV}) ]\nonumber\\ &&+(0.85\pm 0.15), \end{eqnarray} \tag{ 4 }$$

which is steeper than the WF93 best fit, which has slope 0.68 ± 0.1. The two fits overlap in the regime occupied by the BLS1s, and the steeper slope of our fit is driven by the NLS1s in our sample. A two-dimensional Kolmogorov–Smirnov (KS) test confirms that the NLS1 and BLS1 samples occupy a different region in the parameter space at the 99.5% confidence. Figure 2 shows that the NLS1 and BLS1 samples occupy similar ranges in bump strength, but the NLS1 sample shows extended tails at both ends.

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We also computed the ratio (νF_ν(2271 Å)/νF_ν(2 keV)) between the NUV and X-ray fluxes, which is analogous to α_ox. The distributions of the flux ratio in both the NLS1 and BLS1 samples are shown in Figure 3. It is apparent that the NLS1 sample has more objects with high flux ratios, consistent with Figure 2, and a KS test indicates that the two distributions are different at about 97% confidence, which is suggestive but not especially significant. If we assume that the UV continuum is well described by a power law, we can determine α_ox for our AGNs by comparing the FUV and NUV fluxes. We find that the mean and median of the α_ox distribution of the sample are both 1.4, consistent with the results of Elvis et al. (1994). However, we caution that this calculation requires extrapolating the UV power law longward of the NUV effective wavelength, rendering α_ox inherently less robust than the flux ratios shown in Figure 3. In either case, our AGNs appear to be quite typical in this respect, but there is marginal evidence that the NLS1s have slightly stronger BBB than usual, consistent with the results in Figure 1. If BBB photons are reprocessed to form the soft excess, a stronger BBB should be associated with a stronger soft excess, which indeed is observed. (In order to explain the flux ratios in Figure 1, only one in ∼10⁶ BBB photons needs to be reprocessed. The associated UV flux decrement would not be observable.)

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Many of our AGNs occupy the gap between the main locus of WF93 galaxies and their outliers, as shown in Figure 1, indicating a systematic difference between our AGNs and those of WF93. This difference could be caused either by weaker UV at a fixed X-ray flux or by steeper α_x at a fixed BBB strength. It is apparent from Figure 4 that the X-ray spectra of our galaxies are, on average, steeper than the galaxies of WF93 (〈α_x〉 = 2.1, compared to 〈α_x〉 = 1.5 for WF93). Comparing the far-UV (Figure 5) and X-ray (Figure 6) fluxes of our sample with the fluxes of the WF93 AGNs, we find that our AGNs are fainter in both the UV and X-ray, but the difference is larger in the UV. (The median is shifted by a factor of 5.7 in the UV, compared to 5.0 in the X-ray.) This, in combination with the higher average α_x in our sample accounts for the observed differences between our sample and WF93's. The difference between the median νF_ν(1528 Å)/νF_ν(2 keV) in our sample and WF93's might be attributable to their use of IUE observations to measure UV fluxes. The WF93 fluxes show a fractional error near unity for fluxes below ∼3 × 10⁻¹¹, whereas the typical GALEX uncertainty is only a few percent at these flux levels. We are therefore able to obtain significant GALEX detections of all of our sources, and our sample is unbiased with respect to the UV flux.

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While the majority of the BLS1s agree well with the WF93 best fit, the NLS1s are shifted systematically to higher α_x. In combination with the good correlation between νF_ν(1528 Å)/νF_ν(2 keV) and α_x for the full sample, this indicates that the relation between the strength of the BBB and the shape of the soft X-ray continuum is steeper among AGNs with the strongest soft excesses. This in turn suggests the need for a second parameter to account for the variation in α_x at a fixed BBB strength.

We identified nine AGNs that lie well away from the "main" relation between νF_ν(1528 Å)/νF_ν(2 keV) and α_x. These outliers are listed, along with a number of important properties, in Table 6. Five of the outliers show UV-optical luminosity ratios less than 1, putting them well below the main locus in Figure 7. This suggests that the primary cause of our outliers is UV absorption. We fit a power law to the objects in Figure 7 with L_FUV ⩾ L_V and found

$$\begin{equation} \log (L_{\rm FUV}/L_{\odot })=1.09\log (L_{V}/L_{\odot })-0.65. \end{equation} \tag{ 5 }$$

Assuming that all of the galaxies with L_FUV < L_V fall exactly on the best-fit line, we require internal E(B − V) between 0.3 and 1.0 to explain the measured luminosity ratios. For the Galactic gas-to-dust ratio, this implies N_H ≈ 5 × 10²¹ cm⁻², which is far larger than the column densities measured from X-ray spectral fits. We note, however, that the implied N_H is degenerate with α_x, so these systems could have larger column densities and steeper spectra than reported, though this seems unlikely given the recorded values of α_x. Also, WF93 found a small number of galaxies with significant internal extinction despite moderate column densities inferred from the ROSAT spectra of those systems. The unusual luminosity ratios shown in Figure 7 might also be an indication that the FUV and V-band luminosities are dominated by young stars rather than by the AGN accretion disk. This hypothesis is supported by the unusually low values of α_ox exhibited by three of these five objects. All three AGNs with such low α_ox have L_FUV < 10⁴³erg s⁻¹, which corresponds to star formation rate (SFR) of 10 M_☉ yr⁻¹ (Salim et al. 2007).

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Of the four outliers in Table 6 that do not appear to be strongly absorbed in Figure 7, three show unusually large α_x, and the fourth (RX J0902 − 07) lies very close to the line dividing the "normal" AGNs from the outliers. This last object does not differ significantly from the "typical" AGNs in our sample for any of the parameters listed in Table 6, suggesting that it should be considered normal. The other outliers can be divided into two classes: objects that show UV absorption and objects that show extraordinarily high α_x.

Most of our AGNs occupy the gap between the WF93 best-fit relation and their outliers. Walter & Fink (1993) explained their outliers as normal objects with strong intrinsic absorption, but few of our AGNs show evidence for UV absorption. Only 3 of the 8 objects with L_FUV/L_V < 1 fall into our main sample, so strong UV absorption cannot be responsible for this difference between our sample and WF93's. We note, however, that weak absorption is difficult to identify from Figure 7 due to the large intrinsic scatter about the mean relation, so weak intrinsic absorption might contribute to the shift in our sample away from the WF93 mean.

We also examine the relation between indicators for the strength of the BBB and its shape (νF_ν(1528 Å)/νF_ν(2271 Å)), shown in Figure 8. Like WF93, we find a plateau accompanied by a sharp drop toward lower values of νF_ν(1528 Å)/νF_ν(2 keV). However, Figure 8 shows a broader scatter in the plateau region, plateaus at a lower ratio, and fills in the red tail of the distribution less completely than seen in the analogous diagram in WF93. Also, the objects occupying the "tail" of the distribution in Figure 8 are all listed as outliers in Table 6, which immediately suggests that the tail in our sample is due to absorption. The correlation between the strength and shape of the BBB disappears if we disregard the outliers, indicating that the shape of the BBB is largely independent of its strength.

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The absence of any correlation between the shape and strength of the BBB is a direct contradiction of the results of WF93. We use very different methods to measure the shape parameter of the BBB, so it is possible that the disparity between our results and theirs are due to systematic biases, particularly since our wavelength baseline is only half theirs. However, the average UV flux ratio in Figure 8(〈νF_ν(1528 Å)/νF_ν(2271 Å)〉 ≈ 1.4) implies a power-law continuum $(F_{\nu }\propto \nu ^{-\alpha _{uv}})$ with 〈α_uv〉 ≈ −0.85, which in turn suggests that 〈νF_ν(1375 Å)/νF_ν(2675 Å)〉 ≈ 1.75. This is consistent with the plateau seen in WF93 Figure 11, which is at approximately 1.8. Given this agreement and the fact that the tail of our distribution is populated by AGNs showing probable absorption, we suggest that WF93 were too quick to dismiss absorption as a potential cause of their correlation.

To verify that the differences between Figure 8 and WF93's Figure 11 are not caused by systematic differences between νF_ν(1528 Å)/νF_ν(2 keV) and νF_ν(1528 Å)/νF_ν(25 μm), we compare the two strength indicators in Figure 9. Despite the differences between the UV fluxes of the two samples, our galaxies show good agreement with the WF93 best-fit relation. We derive a best-fit relation for our sample, obtaining

$$\begin{equation} \frac{\nu F_{\nu }(1528\;\AA)}{\nu F_{\nu }(25\;{\rm \mu m})}=(0.95\pm 0.08)\frac{\nu F_{\nu }(1528\;\AA)}{\nu F_{\nu }(2\;{\rm keV})}-(1.16\pm 0.09) \end{equation} \tag{ 6 }$$

which is consistent with the WF93 best fit within the uncertainties. Thus, there is no inherent bias in νF_ν(1528 Å)/νF_ν(2 keV) compared to νF_ν(1528 Å)/νF_ν(25 μm), and the absence of a correlation between the shape and strength of the BBB among our AGN sample is not caused by differences between our strength indicator and WF93's. It also indicates that the steeper relation between α_x and νF_ν(1528 Å)/νF_ν(2 keV) among our AGN compared to the WF93 sample is not due to systematic errors. This lends credence to the hypothesis that a factor besides the strength of the BBB must contribute to the shape of the soft X-ray continuum.

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The structure of a standard thin disk, which might be reflected in the UV color of the disk, can depend on both black hole mass and Eddington ratio (see Equation (8)), so We want to know whether νF_ν(1528 Å)/νF_ν(2271 Å) shows systematic differences between the NLS1 and BLS1 samples. A KS test reveals that the distributions differ between the NLS1 and BLS1 samples at about 95% confidence, but this difference disappears when we eliminate the outliers (Table 6). Thus, we measure no intrinsic variation in the structure of accretion disks powering NLS1 and BLS1 AGNs. However, the GALEX bands are sensitive only to variations in disk structure if the Eddington ratio is well below $\dot{m}$. (See Figure 10.) As a result, we expect little intrinsic difference between the BLS1 and NLS1 AGN samples.

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Finally, we look for any relationships of the indicators we have already examined with FWHM(Hβ), which might tend to indicate systematic differences between the two classes of AGNs. We find two trends that might be of interest: a correlation with α_x among the merged AGN sample at >99.9% confidence and another with νF_ν(1528 Å)/νF_ν(2271 Å) among the NLS1 sample at 97% confidence. The second is interesting if true, because it would suggest that the structure of the BLR is related to the UV color of the accretion disk, but the correlation is not strong enough to support such a claim unequivocally. Figure 11 shows the relation between FWHM(Hβ) and α_x and is consistent with the "zone of avoidance," in which BLS1s generally have α_x ≲ 2.0, as reported by Boller et al. (1996). The measured correlation is a result of this effect.

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We also examined the relationships between the observables discussed above and the physical parameters (M_BH, $\dot{m}=L/L_{\rm edd}$ and L_bol) that characterize each AGN, where the Eddington ratios were all determined using L_bol from the Grupe et al. catalogs. We find strong correlations (>99% confidence) of shape, strength and α_x with UV luminosity among the NLS1 sample, as shown in Figure 12. We also find correlations of both strength and shape with luminosity in the merged sample, but the lack of correlation of α_x with luminosity among BLS1s dilutes the correlation in the merged sample to 98% confidence. This is significantly weaker than the same correlation reported by Kelly et al. (2008) for a sample of optically selected, radio-quiet quasars. This suggests that the shape of the X-ray continuum is more tightly coupled to the BBB among optically selected AGNs than among X-ray selected AGNs. The Spearman correlation coefficients and significance values for the various parameters we examined are listed in Table 4.

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The strong positive correlation of νF_ν(1528 Å)/νF_ν(2271 Å) with νL_ν(1528 Å) is in conflict with the results of Scott et al. (2004), who reported that the BBB, as measured in the FUSE band (900–1200 Å) becomes softer in more luminous AGNs. However, the GALEX FUV band does not overlap with the FUSE coverage, so the different variations with luminosity might be strictly a wavelength effect. If this is the case, the peak of the BBB in the average Seyfert AGN must lie somewhere between 1500 Å and 900 Å. The observed correlations are weaker but still significant (>99% confidence) if we consider νL_ν(5100 Å). In Figures 12(a) and 12(b), the relations exhibited by the NLS1 and BLS1 show good agreement, which is consistent with the structure of the accretion disks showing little variation between the two classes. We see very different relations of α_x with luminosity between the NLS1 and BLS1 samples. This could be caused by most NLS1s being within a factor of a few in M_BH, causing variations in $\dot{m}$ to drive a correlation of α_x with luminosity.

From simple virial considerations, we expect that the width of the Hβ emission line should correlate with both M_BH and $\dot{m}$. If we assume the simplest possible relation, R_BLR ∝ L^1/2, which is consistent with the results of Bentz et al. (2006), we find that ${w(H}\beta {\rm)}\propto (M_{\rm BH}/\dot{m})^{1/4}$. Examining the correlations in Table 4, we find that the Hβ line width is indeed correlated with both M_BH and $\dot{m}$, but the correlation is much stronger with $M_{\rm BH}/\dot{m}$. Fitting the FWHM to $M_{\rm BH}/\dot{m}$ yields

$$\begin{equation} \log [{\rm w(H}\beta) ]=(0.24\pm 0.01)\log (M_{\rm BH}/\dot{m})+(1.5\pm 0.1) \end{equation} \tag{ 7 }$$

with χ²_ν = 1.02, which is consistent with the simple prediction above. The result of this comparison is shown in Figure 13. This relationship is also consistent with the results of McHardy et al. (2006), who found that the break timescale of the power density spectrum, which is proportional to $M_{\rm BH}/\dot{m}$, is well correlated with the line width. This good agreement both with theory and with previous observations suggests that the Grupe & Mathur (2004) mass measurements are, on average, robust.

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Our correlation measurements also agree with Piconcelli et al. (2005), who found a strong anti-correlation of Γ_soft with the Hβ line width (r_s = −0.54), where $N_{\nu }\propto \nu ^{-\Gamma _{\rm soft}}$ is measured from 0.3–2.0 keV. In fact, the strength of the correlation they report is very similar to the strength of the correlation we find between FWHM and α_x (our r_s = −0.53), suggesting that much of the scatter between the two variables may be intrinsic. The errors on the line widths of NLS1s are comparable to the errors on BLS1 widths, so the lack of correlations with M_BH or $\dot{m}$ among the BLS1 sample suggests that the geometry factor f varies more among the BLS1 sample than the NLS1 sample. This might happen if the BLR becomes more spherically symmetric at high Eddington ratio.

There is also a correlation of moderate significance (98.4% confidence) between M_BH and νF_ν(1528 Å)/νF_ν(2271 Å), but correlation is positive, which is in opposition to the trend predicted for a standard thin disk (see Figure 10). This correlation is probably spurious, since it is driven by a small number of AGNs with νF_ν(1528 Å)/νF_ν(2271 Å) < 1, three of which are outliers in Figure 1. Two additional AGNs with νF_ν(1528 Å)/νF_ν(2271 Å) < 1 have L_FUV < λL_λ(5100 Å), suggesting that low levels of recent star formation could significantly influence the measured UV flux ratio. After excluding both of these groups, we find no significant correlation of νF_ν(1528 Å)/νF_ν(2271 Å) with M_BH. There is no correlation of νF_ν(1528 Å)/νF_ν(2271 Å) with $\dot{m}$ regardless of whether the AGNs with νF_ν(1528 Å)/νF_ν(2271 Å) < 1 are considered, which is expected given the relatively high Eddington ratios typical of the AGNs in our sample.