OPTICAL MONITORING OF THE BROAD-LINE RADIO GALAXY 3C 390.3^*

To study broad emission-line flux variations, it is necessary to calibrate the spectra to a uniform flux level. This can be achieved using the flux of forbidden narrow emission lines from the narrow-line region (NLR). Due to the large spatial extension of the NLR (of the order of 100 pc in Seyfert galaxies) and the long recombination timescales which are due to the low gas density of the NLR (τ_rec ≈ 100 years for n_e ≈ 10³ cm⁻³), light travel-time effects and long recombination timescales will damp out short timescale variability. Thus, it can be assumed that the flux of narrow emission lines can be treated as constant, at least on timescales of years to decades (e.g., Peterson 1993, 2000).

However, 3C 390.3 needs careful additional attention. The NLR of 3C 390.3 seems to be very compact (Baum et al. 1988; Sergeev et al. 2002) and the small emission-line flux ratio of F([O iii]λλ4959, 5007) to F([O iii]λ4363) indicates the presence of high density gas (n_e ≃ 10⁵ cm⁻³). Furthermore, narrow-line variability has been reported for this object. Clavel & Wamsteker (1987) analyzed the strength of the narrow Lyα and C ivλ1549 emission lines. They found that the strength of these two narrow lines decreased by a factor of about 2–3 from 1978 to 1986. On the basis of optical spectra obtained between 1974 and 1990, Zheng et al. (1995) presented evidence that the fluxes of [O iii]λλ4959, 5007 follow continuum variations, although on a longer timescale than the broad emission lines. During the period of decreasing or increasing [O iii] flux, there might also be periods of several months or years of nearly constant [O iii] flux. Hence, we examined the data closely to test for variability of the [O iii] emission lines before we used them for flux calibration.

To test the reliability of the narrow [O iii] emission lines as an internal calibrator, the g-band imaging data are of great value. As noted earlier, these broadband flux measurements were taken nearly simultaneously with the spectra, i.e., within minutes of the spectral observations (Table 1) and the g-band brightness of 3C 390.3 was measured with high precision (see Section 4.2; ∼0.1%). To compare the g-band variations to those which can be derived from the spectral data, the spectra were convolved with the transmission curve of the SDSS g-band filter. The fluxes of the convolved spectra were measured for the transmission wavelength range of the g-band filter curve (λ = 4000–6000 Å). To adjust the flux level of the spectra to the g-band measurements, the spectra were rescaled. These scaling factors are on average (1.10 ± 0.07).

We used these adjusted spectra, based on the observed g-band flux variations, to test whether the [O iii]λ5007 emission-line flux was constant during this monitoring campaign. Although the broadband flux measurements include the broad and narrow Hβ emission-line flux, as well as the [O iii]λλ4959, 5007 line emission, the contribution of emission-line variability to photometric variations can be neglected (Walsh et al. 2009) if it amounts to less than ∼10% of the broadband flux. For 3C 390.3, we find that the Hβ flux amounts to less than 5% of the g-band flux. To test the assumption that the [O iii]λ5007 emission-line flux was constant for the duration of this monitoring campaign, these adjusted spectra were internally flux calibrated by scaling each individual spectrum to a uniform flux of the [O iii]λ5007 emission line. First, we calculated a mean spectrum which was used as a reference spectrum. Next, we applied a modified version of the scaling algorithm of van Groningen & Wanders (1992) to adjust the flux of the [O iii]λ5007 in the individual spectra to the reference spectrum. This routine minimizes the residuals of the [O iii]λ5007 in a difference spectrum which is calculated for each spectrum with the reference spectrum. The intercalibration to a uniform flux level was achieved by allowing different flux scales, small wavelength shifts, and different spectral resolutions. The average scaling factor amounts to (0.99 ± 0.02). This result indicates that the [O iii]λ5007 flux was constant in the adjusted spectra with a flux of F([O iii]λ5007) = (189 ± 4) × 10⁻¹⁵ erg s⁻¹ cm⁻² (Table 2). It is interesting to note that during the monitoring campaign of 3C 390.3 in 1995 (Dietrich et al. 1998), the [O iii]λ5007 emission-line flux was measured to be F([O iii]λ5007) = (161  ±  4) × 10⁻¹⁵ erg s⁻¹ cm⁻² (rest frame flux) which is about ∼15% lower. This indicates that the [O iii]λ5007 emission-line flux was lower in 1995 compared to 2005 when 3C 390.3 was in a less luminous state. Based on the average scaling factor and the small scatter, it is justified to assume that the F([O iii]λλ4959, 5007) emission-line flux was constant during this monitoring campaign.

While the Hβ–[O iii]λλ4959, 5007 region is scaled with respect to the narrow [O iii]λ5007 line, we investigated whether this internal calibration holds for the Hα region at longer wavelengths and the Hγ–[O iii]λ4363 region at shorter wavelengths. We intercalibrated the spectra for the Hα region with respect to the fluxes of the relatively strong [O i]λ6300 emission line and the Hγ–[O iii]λ4363 region using the narrow [O iii]λ4363 emission line. We found that the Hα region required small adjustments to achieve a constant [O i]λ6300 flux. The average scaling factor for the Hα region amounts to (1.014 ± 0.021). For the Hγ–[O iii]λ4363 region, minor rescaling of the spectra was also necessary. These scaling factors are on average (1.014 ± 0.004).

A spectrum of an AGN is the superposition of various sources, including the host galaxy, thermal and non-thermal continuum emission from the AGN, which can be described by a power-law continuum, Balmer continuum emission, Fe ii line emission, and emission from other metal line transitions, e.g., carbon, nitrogen, oxygen, magnesium, neon, sulfur. These contributions need to be accounted for in order to obtain reliable line measurements and the strength of the AGN continuum that most of our study relies on. We adopted a multi-component fit approach (e.g., Wills et al. 1985) that we have successfully applied in prior studies of high-z quasars and narrow line Seyfert 1 (NLS1) galaxies (for more details, see Dietrich et al. 2002, 2003, 2005, 2009). We assume that the spectrum of 3C 390.3 can be described as a superposition of three components: (1) a power-law continuum (F_ν ∼ ν^α), (2) a host galaxy spectrum (Kinney et al. 1996), and (3) a pseudo-continuum due to merging Fe ii emission blends.

Neither a Balmer nor a Paschen continuum emission were taken into account because the Balmer edge at λ = 3645 Å is close to the short wavelength end of our spectra of 3C 390.3 and the strength of the Paschen continuum is only weakly constrained (Korista & Goad 2001).

As demonstrated by Bentz et al. (2006b, 2009a), it is crucial to correct for the host galaxy contribution to measure the strength of the AGN continuum flux density. Based on Advanced Camera for Surveys/Hubble Space Telescope (ACS/HST) images which were obtained for 14 AGNs of the AGN-watch sample (Peterson et al. 2004), Bentz et al. (2009a) estimated the host galaxy contribution to the continuum flux density at λ = 5100 Å for those AGNs, including 3C 390.3. For the large aperture of (3'' × 94) which we used, the galaxy contribution amounts to F_gal(5100 Å) = (0.87 ± 0.09) × 10⁻¹⁵ erg s⁻¹ cm⁻² Å⁻¹, based on ACS/HST imaging data (M. C. Bentz 2008, private communication). The average host galaxy contribution which we determined, based on the multi-component fits, is F^obs_gal(5100 Å) = (0.77 ± 0.10) × 10⁻¹⁵ erg s⁻¹ cm⁻² Å⁻¹, i.e., within the errors consistent with the measurements based on the ACS image. To construct representative host galaxy template spectra, we combined several individual spectra of various Hubble types. These galaxy spectra were retrieved from the publicly available sample published by Kinney et al. (1996). The spectral resolution of these galaxy spectra amounts to R ≃ 8 Å which is comparable to the spectral resolution of our spectra.

To account for Fe ii emission, we used the rest-frame optical Fe ii template that is based on observations of I Zw1 by Boroson & Green (1992) which cover the wavelength range from 4250 Å to 7000 Å. An alternative optical Fe ii emission-line template was presented by Véron-Cetty et al. (2004) which in addition takes into account NLR contributions to the Fe ii emission. In general, these two templates are similar but in detail they differ at around λ ≃ 5000 Å and λ ≳ 6400 Å. The optical Fe ii emission in the spectrum of 3C 390.3 is weak, however, and the width of the Fe ii emission is expected to be quite broad, as is seen for other permitted emission lines. Therefore, the choice of the Fe ii emission template has no impact on the results of this study.

Components (1), (2), and (3) were simultaneously fitted to the spectra of 3C 390.3 that were intercalibrated with respect to the [O iii]λλ4959, 5007 emission-line flux, to determine the minimum χ² of the fit. Various host galaxy templates were employed and the best results were obtained using an appropriately scaled spectrum of the E0 galaxy NGC 1407, which is consistent with the morphological type of the host galaxy of 3C 390.3. The width of the Fe ii emission template was allowed to vary from FWHM = 3000 km s⁻¹ up to 9000 km s⁻¹ in steps of ΔFWHM = 500 km s⁻¹. The best-fit results were achieved for Fe ii emission templates whose width was on average FWHM = (6000 ± 1000) km s⁻¹, ranging from FWHM = 5000 km s⁻¹ to FWHM = 8500 km s⁻¹. In Figure 2, we show the fit for a typical spectrum of 3C 390.3 from this study to illustrate the method to determine the different components.

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The uncertainty of the fit was estimated based on the distribution of χ² around the minimum. The spectral slope α of the continuum fit has an average error of Δα ≃ 0.01, while the uncertainties in the continuum flux density at λ = 5100 and the Fe ii emission flux are of the order of ∼5%. The best fit of components (1) to (3) were subtracted before we continued to analyze the emission-line profiles of interest.

In the spectral region of the broad and double-peaked Hα emission line some of the corrected spectra still exhibit a weak residual continuum flux level. To adjust for the residual continua, we fit a linear local pseudo-continuum. The Hα emission-line flux was measured within the wavelength range of 6200–6800 Å. This spectral range contains contributions of several narrow emission lines, including Hα_nar. To correct for narrow-line flux contamination, the average intensities of the narrow emission lines of [O i]λλ6300, 6364, [Fe x]λ6374, [N ii]λλ6548, 6583, [S ii]λλ6716, 6731, and Hα_nar were subtracted (Table 2) and the resulting broad Hα emission-line flux is given in Table 3.

To estimate the uncertainties of the broad Hα flux measurements, first the uncertainties of the multi-component fit were taken into account. As noted earlier, we assumed that the strength of the power law continuum fit, the host galaxy contribution, and the optical Fe ii emission can be determined within 1%. The next source of error is given by the uncertainties of the narrow emission-line intensities (Table 2). The largest source of error is the level of the pseudo-continuum. The strength of the linear pseudo-continuum fit is based on two windows which are located at 6135–6160 Å and 6875–6895 Å, respectively. Using the scatter of the continuum level of these two windows, the error introduced by the pseudo-continuum fit was determined. The individual errors were combined in quadrature to obtain the uncertainty of the individual broad Hα emission-line flux measurements (Table 3). In Figure 6, we present the light curve of the broad Hα emission-line flux. Although the scatter is larger than for the g-band and F_λ(5100 Å) variations, the broad Hα flux shows a similar variation pattern which is shifted to later epochs.

As in the case of the Hα emission line, it was necessary to correct for a residual continuum level of the Hβ λ4861–He iiλ4686 wavelength range. The linear local pseudo-continuum fits are based on two 20 Å wide continuum regions centered at λ = 4520 Å and λ = 5240 Å, respectively. The corrected spectra were used to measure the broad Hβ λ4861 and He iiλ4686 emission-line fluxes. Because the emission-line profiles of 3C 390.3 are so broad (Figure 2) it is necessary to evaluate and to correct for the mutual blending of the Hβ λ4861 and He iiλ4686 emission, i.e., to identify the contributions of the Hβ λ4861 and He iiλ4686 emission in the overlapping region.

We used the broad Hα λ6563 emission-line profile as a template to estimate the shape and strength of the broad He iiλ4686 emission. To extract the broad Hα λ6563 emission-line profile, the mean spectrum of the campaign was used (Section 4.4) and the mean Hα profile was corrected for the narrow emission-line contributions ([O i]λλ6300, 6364, [Fe x]λ6374, [N ii]λλ6548, 6583, Hα_nar, and [S ii]λλ6716, 6731). The width of the double-peaked broad emission-line profile of Hα λ6563 was scaled to have the same width in velocity space at the location of the Hβ λ4861 emission line and the strength was rescaled to fit the central part and red wing of the Hβ λ4861 profile. In Figure 7 we show the Hβ λ4861–He iiλ4686 range corrected for narrow emission of He iiλ4686, Hβ λ4861, and [O iii]λλ4959, 5007. The scaled double-peaked Hα emission-line profile describes the broad Hβ λ4861 emission profile well, except the blue hump which is less prominent in the Hβ profile compared to the Hα λ6563 emission-line profile. This results in an absorption line-like feature in the residual spectrum (Figure 7, bottom panel) which indicates a steeper Balmer decrement for the blue peak in the double-peaked profiles. To measure the He iiλ4686 emission-line flux, the residual emission at the location of the He iiλ4686 line was fit with a rescaled broad Hα λ6563 double-peaked profile and also with a single broad Gaussian profile. The profile in the residual spectrum of the mean spectrum is better represented by a Gaussian profile than using an appropriately scaled double-peaked Hα profile (Figure 7). To determine an appropriate Gaussian profile for the broad He iiλ4686 emission, the blue wing of the He ii line in the residuum was extracted and assuming that the broad He ii profile is symmetric the blue wing was mirrored to represent the red wing of the He iiλ4686 line profile (Figure 8). The resulting profile can be described with a single Gaussian profile at λ_c = 4686.14 Å ± 0.04 Å and a profile width of FWHM = 242.3 Å ± 1.4 Å (15, 500 km s⁻¹ ± 100 km s⁻¹). To measure the He iiλ4686 emission-line light curve (Figure 6) a Gaussian profile of fixed width (FWHM = 15,500 km s⁻¹) was fit to the blue wing of the He iiλ4686 profile in the spectra of 3C 390.3 (Figure 8). The flux of the He iiλ4686 emission was measured from this Gaussian fit for a wavelength range of λ = 4472 Å to λ = 4900 Å. This Gaussian profile fit was subtracted from the spectrum to measure the broad Hβ λ4861 emission flux in the range from λ = 4680 Å to λ = 5000 Å (Table 3). The resulting light curves for the broad Hβ λ4861 emission-line flux, corrected for narrow Hβ emission, and the He iiλ4686 emission line, are displayed in Figure 6.

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To estimate the total error of these flux measurements, we assumed that the uncertainties of the multi-component fit are of the order of 1%. The errors introduced by the correction of the narrow emission lines are taken from Table 2. The uncertainty due to the broad He iiλ4686 emission is of the same order as the total error that is caused by the narrow emission lines. As in the case of Hα λ6563, the dominant source of error is the level of the pseudo-continuum fit. The level of the pseudo-continuum fit was varied within the scatter given by the two ranges centered at λ_c = 4520 Å and λ_c = 5240 Å that are 20 Å wide. The individual contributions to the uncertainty were combined in quadrature to obtain the total uncertainties of the individual broad Hβ and He iiλ4686 emission-line flux measurements (Table 3).

In the case of the Hγ emission-line profile region, we correct for a residual continuum level as well. The linear local pseudo-continuum fits are based on two 20 Å wide continuum regions centered at λ_c = 4160 Å and λ_c = 4530 Å, respectively. The corrected spectra were used to measure the Hγ λ4340 emission-line flux (4220 Å to 4500 Å). These flux measurements were corrected for the emission-line contributions of the narrow Hγ λ4340 emission and of the [O iii]λ4363 emission, using the average narrow line fluxes as given in Table 2. The resulting light curve of the variable broad Hγ λ4340 emission line is shown in Figure 6. The variability pattern is very similar to the broad Hβ variations.

To estimate the total error of the broad Hγ flux measurements (Table 3), we assumed that the uncertainties of the multi-component fit are of the order of 1%. The errors introduced by the correction of the narrow emission lines are taken from Table 2. The dominant source of error is the level of the pseudo-continuum fit. The level of the pseudo-continuum fit was varied within the scatter given by the two 20 Å wide ranges (λ_c = 4160 Å and λ_c = 4530 Å). The contributions of the individual error sources were combined in quadrature to estimate the total uncertainty of the broad Hγ λ4340 emission-line flux.

Based on the calibrated spectra of 3C 390.3, we calculated mean and residual (hereafter simply rms) spectra of the broad emission profiles of Balmer lines Hα, Hβ, and Hγ which are shown in Figure 9. These mean and rms spectra are based on the entire set of MDM spectra, except the spectrum that was recorded at the epoch JD = 244 3696. The signal-to-noise ratio (S/N) for this spectrum is significantly lower than for all the other spectra of 3C 390.3 on account of bad weather conditions (S/N_2443696 ≃ 10, compared to an average S/N for the sample of S/N = 35 ± 7).

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The mean spectra of the Balmer emission lines show a characteristic double-peaked emission-line profile with a strong blue hump while the red hump is very weak. The rms spectra of the Balmer lines show a similar shape. The blue hump is the dominant feature for the Hα and Hβ rms spectra while it is less pronounced for Hγ. Only weak residuals of narrow emission lines can be seen in the Balmer rms spectra of Hα, Hβ, and Hγ. The rms spectra of the Balmer emission lines also display extended red wings.

The line width can be characterized by its FWHM and by the line dispersion σ_line (the second moment of the line profile; e.g., Peterson et al. 2004). In order to characterize the uncertainties in each of these parameters we employed a procedure described by Peterson et al. (2004). For a set of N spectra, N spectra are randomly selected, in particular without regard whether the spectrum has been previously selected or not. These N randomly selected spectra are used to construct mean and rms spectra from which the line width measurements are made. This process yields one Monte Carlo realization, and a large number (N ≃ 10, 000) of these realizations yield a mean and standard deviation for each of the measurements of the line width. The results of the line profile measurements and the corresponding errors are presented in Table 4.

We find that the mean spectra of the Balmer emission lines show a similar profile width of FWHM = 11,900–13,200 km s⁻¹ and the second moment of the line profiles σ_line ranges from 4000 km s⁻¹ to 5400 km s⁻¹, respectively. The comparison of the rms spectrum profile widths shows basically the same result. The Balmer emission-line rms spectra have an average profile width of FWHM = (11430 ± 790) km s⁻¹ and an average second moment of the line profiles of σ_line = (5160 ± 310) km s⁻¹, respectively.

The mean and rms profiles are a Gaussian due to our approach to measure the He iiλ4686 emission-line flux and hence also the rms spectrum (Section 4.3.2). However, the rms spectrum of the He iiλ4686 variations is contained in the rms spectrum of the broad Hβ emission line (Figure 9). To recover the He iiλ4686 rms spectrum we examined two approaches.

First, the fit of the He iiλ4686 emission line was subtracted from each spectrum of the Hβ–[O iii]λλ4959, 5007 emission-line region. This results in an uncontaminated broad Hβ profile. These spectra are used to calculate a mean and rms spectrum which represent the Hβ variation alone. In Figure 10 the mean spectrum of the Hβ–[O iii]λλ4959, 5007 emission-line region with and without correction for the broad He iiλ4686 emission is displayed, as well as the comparison of the rms spectra. The difference of these rms spectra can be associated with the He iiλ4686 rms spectrum (Figure 10).

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As an alternative approach to recover the He iiλ4686 rms spectrum, we assumed that the rms spectrum of the broad Hα emission line can be used as a template for the rms spectrum of the Hβ emission line. The Hα rms spectrum was re-binned that the width is unchanged in velocity space at the location of the Hβ emission line. In Figure 11, the rms spectra of the broad Hβ and Hα emission lines are shown. Next, the re-binned Hα rms spectrum was rescaled by dividing by 3.50 to fit the red wing of the Hβ rms spectrum. The difference between the Hβ rms spectrum and the scaled, re-binned Hα rms spectrum can be associated with the He iiλ4686 rms spectrum as well. This difference is displayed in the bottom panel of Figure 11. It can be seen that this representation of the He iiλ4686 rms spectrum is nearly identical to the rms spectrum which is obtained by comparison of the rms spectra of the Hβ–[O iii]λλ4959, 5007 spectral region, corrected and uncorrected for He iiλ4686 emission, as described above.

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The comparison of the He iiλ4686 rms spectrum with the mean Gaussian profile fits to the He iiλ4686 line profile indicates that the recovered rms spectrum of the broad He iiλ4686 emission line tends to show less variability in the far blue wing than the mean spectrum and thus the FWHM is smaller. This result is consistent with the properties of the mean and rms spectra of the Balmer emission lines which also show narrower profiles in the rms spectra compared with the corresponding mean spectra (Table 4). We used both He iiλ4686 rms spectra to measure their FWHM. While the overall shapes of the rms spectra are very similar, the S/N is different. Hence, we used the average of the FWHM, as well as the second moment σ_line of the profile. The results are given in Table 4.

While the variability characteristics of the integrated broad-line flux can be used to infer the size of the BLR, the analysis of the response of different parts of the broad emission-line profile, e.g., the line core, and the profile wings, yield information on the dominant gas motion, i.e., the kinematics of the broad emission-line gas (e.g., Blandford & McKee 1982; Horne et al. 2004). Previous investigations of 3C 390.3 have found that the profile wings vary in phase (Dietrich et al. 1998; Popovic et al. 2011; Sergeev et al. 2002; Shapovalova et al. 2010) which strongly favors orbital motion, i.e., virial motion dominates the gas kinematics. This is also supported by accretion disk models to describe the double-peaked Balmer emission-line profiles (e.g., Eracleous & Halpern 1994; Flohic & Eracleous 2008). Recently, results for several AGNs have been presented (Denney et al. 2009a; Bentz et al. 2009b) which indicate a more complex picture with signatures for orbital motion as well as for radial motions. In addition, for Arp 151 the analysis of spectroscopically resolved broad emission-line results into velocity-delay maps which indicate the presence of substructures like orbiting hot spots (Bentz et al. 2010a).

As in former studies of 3C 390.3, we extracted light curves for different parts of the broad emission-line profile. For better comparison with prior studies (e.g., Dietrich et al. 1998; Gezari et al. 2007; Shapovalova et al. 2010), the width and placement of those regions are motivated by the location and the widths of the blue and red peaks in the double-peaked line profile.

In Table 7, the location of the blue peak and red peak for the Hα, Hβ, and Hγ emission-line profiles are given. Within ∼100 km s⁻¹ both peaks are at similar velocities. Based on the mean line profiles (Figure 9), the width of the peaks amounts to FWHM ≃ 50 Å and FWHM ≃ 70 Å for Hβ and Hα, respectively. We investigated the profiles of the Hα and Hβ emission lines only because the Hγ emission line is too weak and He iiλ4686 is best represented by a single Gaussian profile. We extracted the light curves of the blue and red peaks for a 3000 km s⁻¹ wide region. In Figure 14 the boundaries of the blue and red wings, the blue and red peaks, and the line center are shown. For comparison the location of those profile sections are given in Table 8 which were used by Gezari et al. (2007). The errors for the extracted light curves were determined following the same approach as for the integrated emission-line fluxes.

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Next, the Hα and Hβ line profile section variations are cross-correlated with the g-band light curve, as well as with the AGN continuum F_λ(5100 Å). In Figure 15, the corresponding ICCF(τ) are shown. While for Hα the light curves of the blue wings, the blue and red peaks, and of the line center result into ICCFs with well-defined structures, the light curve of the red wing of the broad Hα line profile results in an ICCF which provides merely an indication for a possible delay (Figure 15). For the Hβ line profile the ICCFs are broad and in the case of the red wing for Hβ there is only a weak indication for a marginally defined peak (Figure 15).

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The location of the centroid, the peak, and the uncertainties in the ICCFs for the wings, the peaks, and the line center of the Hα and Hβ emission lines are determined using the technique described above (Section 4.3) and the results are given in Table 9. As can be seen already in Figure 15, the blue and red peaks of the Hα line and of the Hβ line vary basically with the same time delay relative to the continuum variations, consistent with the model that the hydrogen Balmer line emission originates in a rotating disk (e.g., Eracleous & Halpern 1994; Flohic & Eracleous 2008). Within the uncertainties, there is also no delay of the profile wings with respect to the peaks or the center of the line profile. To test this result, we also applied the time series analysis to the variations of the blue and red wings and the blue and red peaks to eliminate uncertainties which are introduced by the continuum light curves. The corresponding ICCFs for the Hα and Hβ lines are displayed in Figures 16. Within the uncertainties we detect no time delay between the variations of the blue and red peaks and wings (Table 9).

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To determine the mass of the black hole in the center of 3C 390.3, we assumed that the line-emitting gas is bound, i.e., that the virial theorem can be applied. This approach is well motivated by the results of the fitting of the double-peaked emission-line profiles of 3C 390.3 with an elliptical disk model (Flohic & Eracleous 2008). In addition, for several AGNs it has been possible to measure the time delay of additional broad emission-line flux variations. These studies consistently show that emission lines with higher ionization energies show shorter time delays, in keeping with the observed ionization stratification of the BLR and their closer locations to the central continuum source (Bentz et al. 2009a; Kollatschny 2003; Onken & Peterson 2002; Peterson & Wandel 1999, 2000). Together with broader emission-line profile widths, these results provide strong support that the gas is in gravitationally bound motion. The virial mass of the black hole is given by

$$\begin{equation} M_{bh}^{{\rm vir}} = f\,{{c\,\tau \,\,\Delta v^2} \over G} \end{equation} \tag{ 3 }$$

with c τ as the size of the broad emission-line region, Δv which is associated with the line profile width, and G as the gravitational constant. The factor f depends on the geometry, kinematics, and the orientation of the line-emitting region. It has been shown by Onken et al. (2004) that the average of f amounts to 〈f〉 = 5.5 when the virial black hole mass, based on a rms spectrum profile width, is calibrated to the M_bh–σ_* relation for quiescent galaxies (e.g., Tremaine et al. 2002; Gültekin et al. 2009, 2011). In the following, we calculate the pure virial product, i.e., we will assume f = 1.0 because in the case of 3C 390.3 and the successful accretion disk models v_obs = v_intr × sin i can be used to derive the intrinsic velocity, v_intr, of the gas.

$$\begin{equation} M_{{\rm bh}}^{{\rm vir}} = 1.951\times 10^5\,\left({\tau \over {\rm days}}\right) \left({\sigma _{{\rm line}} \over {1000\,{\rm km\ s^{-1}}}}\right)^2 \,\,M_\odot. \end{equation} \tag{ 4 }$$

Using the second moment of the emission-line profiles σ_line (cf. Peterson et al. 2004) for the mean and rms spectra (Table 4) and the measured time delay of the broad emission-line flux response to continuum variations, given by τ_cent (Table 6), derived from a threshold of 80% of the maximum of the ICCF, we calculated the virial black hole mass using Equation (4). The results are presented in Table 10. For comparison we estimated black hole masses for the individual emission lines using σ_line from the the mean and the rms spectra and the time delay which we derived using the AGN continuum variations of F_λ(5100 Å) and of the g band as well (Table 6). We find that within the uncertainties, the virial mass, M^vir_bh, of the black hole of 3C 390.3, based on the mean emission-line profiles, is in the range of 1.0 × 10⁸ M_☉ to 2.6 × 10⁸ M_☉ based on the properties of the mean profiles of the Balmer lines and the He iiλ4686 emission line, independent of the continuum used (Figure 17 and Table 10). However, if only the strong Balmer emission lines Hα and Hβ are considered because the uncertainties for the Hγ and He iiλ4686 emission lines are larger, the virial products are in better agreement, 2.3 × 10⁸ M_☉ to 2.6 × 10⁸ M_☉. The rms spectra yield results which are consistent with the mean spectrum with virial black hole masses of 1.1 × 10⁸ M_☉ to 3.0 × 10⁸ M_☉ (Table 10) and the strong Balmer emission lines Hα and Hβ yield black hole mass estimates of about ∼2.6 × 10⁸ M_☉. Since, the rms spectrum represents the actually variable part of the emission-line profile, we conclude that the virial product of the black hole of 3C 390.3, based on the strong Balmer emission lines Hα and Hβ is M^vir_bh(3C390.3) = (2.60^+0.23_{− 0.31}) × 10⁸ M_☉.

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To estimate the black hole mass M_bh using RM studies, it is still necessary to specify the factor f in Equation (3) which depends on the not well-constrained geometry and kinematics of the gas, as well as on the orientation of the line-emitting region. While it is generally assumed that this factor f is of the order of unity, it has been pointed out that a small inclination of the line-emitting region can result in a significant underestimation of the black hole mass (e.g., Collin et al. 2006) if the gas motions are almost entirely in the disk plane with little motion in the polar direction. In contrast to almost all AGNs monitored so far, the BLRG 3C 390.3 is unique since it is the only radio-loud AGN with extended double-lobed radio emission (Leahy & Perley 1995) whose variability has been studied in detail. This allows us to estimate the impact of the inclination i of the line-emitting region on the black hole mass measurement and hence the factor f, assuming pure disk motion of the gas. Based on very long baseline interferometry observations, Alef et al. (1988) reported indications for super-luminal motion for 3C 390.3. Further radio observations and a detailed analysis provided information on the inclination i of the axis of the radio emission in 3C 390.3 (Alef et al. 1994, 1998), which was found to be i = 28°$^{+5^{\circ }}_{-9^{\circ }}$. The inclination of the radio axis of 3C 390.3 has been also estimated based on X-ray observations (Eracleous et al. 1996). They successfully modeled the resolved Fe Kα line emission with X-ray reprocessing in a cool, dense disk of gas at an inclination of i ≃ 26°. A third way to estimate the inclination of the disk-like line-emitting region is provided by fitting the double-peaked profile. Using a relativistic Keplerian disk, Eracleous & Halpern (1994) derived an inclination angle of i = 26°$^{+4^{\circ }}_{-2^{\circ }}$. Recently, Flohic & Eracleous (2008) analyzed the variations of the double-peaked profile shape of Hα for Arp 102B and 3C 390.3. They found that the best disk models for 3C 390.3 strongly indicate an inclination angle of i = 27° ± 2°.

In a simple model of an accretion disk as the dominant source of the observed double-peaked emission-line profile, the observed velocity is actually v_obs = v_intr × sin i. Assuming an inclination angle of i = 27° ± 2° this implies a correction factor of f = 4.85^+0.75_{− 0.60}. Within the uncertainties, this factor is consistent with 〈f〉 = 5.5, determined by Onken et al. (2004) and 〈f〉 = 5.25 by Gültekin et al. (2009) which were determined by comparing the reverberation-based black hole masses with the masses determined using the M–σ relation, with the stellar velocity dispersion σ based on stellar or gas dynamics of the host galaxy's bulge. Applying the correction due to the measured inclination of the accretion disk, the black hole mass of 3C 390.3 based on the Hα and Hβ rms spectrum profile properties and τ_cent yields M_bh(3C 390.3) = (1.26^+0.21_{− 0.16}) × 10⁹ M_☉.

Accretion disk models have been shown to be successful in explaining the observed profiles and even profile variations over longer timescales for 3C 390.3, Thus, the typical velocity of the line-emitting gas can be associated with the location of the blue and red peaks of the double-peaked emission-line profile, i.e., the location in velocity space of the dominant part of the line-emitting gas. Using the separation of the blue and red peaks of the Balmer emission-line profiles as Δv, we estimated the virial product using Equation (4) and τ_cent (80% threshold level). The derived black hole masses are given in Table 10. Based on the Balmer emission lines Hα, Hβ, and Hγ we find virial black hole masses in the range of 1.1 × 10⁸ ≲ M_bh ≲ 2.0 × 10⁸ M_☉ using the mean emission-line profiles and 0.9 × 10⁸ ≲ M_bh ≲ 1.9 × 10⁸ M_☉ based on the rms spectra (Figure 17 and Table 10). However, this wide range is caused by the Hγ-based results which have a larger uncertainty than the black hole mass estimates obtained from the strong Hα and Hβ emission lines which are consistent within the errors. Since the rms spectrum represents the actual variable part of the emission-line gas, we calculated the average virial black hole mass which is based on the rms spectra of the Balmer emission lines Hα, Hβ, and Hγ. We find a virial black hole mass of M^vir_bh = (1.77^+0.29_{− 0.31}) × 10⁸ M_☉. Together with the correction factor f = 4.85^+0.75_{− 0.60} which is given by the inclination of the accretion disk, we find for the black hole mass for 3C 390.3 a value of M_bh = 0.86^+0.19_{− 0.18} × 10⁹ M_☉.

Using the black hole mass estimates based on the broad Balmer and emission-line profiles in the rms spectrum and the optical continuum luminosity of 3C 390.3 at λ = 5100 Å, we computed the Eddington ratio L_bol/L_edd. The value of the conversion factor f_L between the monochromatic luminosity and the bolometric luminosity (L_bol = f_L × λ F_λ(5100)) is still debated in the literature (e.g., Elvis et al. 1994; Laor 2000; Netzer 2003; Richards et al. 2006). Recently, Marconi et al. (2008) even suggested a luminosity-dependent correction factor f_L, due to radiation pressure effects. We assumed that the bolometric luminosity is given by L_bol = 9.74 × λ L_λ(5100 Å) (Vestergaard 2004). We find L_bol = 2.22 × 10⁴⁵ erg s⁻¹. To calculate the Eddington luminosity L_edd, we assumed that the gas is a mixture of hydrogen and helium (μ = 1.15), i.e., L_edd = 1.45 × 10³⁸ M_bh / M_☉ erg s⁻¹. The derived Eddington ratio L_bol/L_edd for 3C 390.3, based on this monitoring campaign, amounts to L_bol/L_edd = 0.018^+0.006_{− 0.005}. This low Eddington ratio is typical for radio-loud AGNs like 3C 390.3 (e.g., Boroson & Green 1992; Boroson 2002).