Time Delay Measurement of Mg ii Line in CTS C30.10 with SALT

The source was observed in a slit spectroscopy mode, using the Robert Stobie Spectrograph (RSS; Burgh et al. 2003; Kobulnicky et al. 2003; Smith et al. 2006) in the service mode. We collected 26 observations, each consisting of two observing blocks, with exposures lasting ∼800 s. A slit width of 2'' was used, as well as RSS PG1300 grating, which corresponds to a spectral resolution of 1047 at 5500 Å. The dates of the observations are reported in Table 1.

The raw data were reduced by SALT staff, with the help of a semiautomatic pipeline coming from the SALT PyRAF package⁸ (see also Crawford et al. 2010). Two exposures were combined, in order to increase the signal-to-noise ratio (S/N) and to remove the effect of the cosmic rays. The wavelength calibration was performed using the corresponding lamp exposure (argon lamp in most of the observations). One-dimensional spectra of the object and of the background were obtained using the IRAF package noao.twodspec. We used always the same width of the image for spectrum subtraction in order to reduce the unnecessary scatter.

The SALT telescope has considerable vignetting effects, so the spectra required recalibration with the help of a standard star, and for each observation we used the method described in detail by Modzelewska et al. (2014) in their Section 2.1.

We model the spectrum in detail in the 2700–2900 Å wavelength frame, in the rest frame of the source. The decomposition of the spectrum has been done assuming the following components: power law (representing the emission from an accretion disk), Fe ii pseudo-continuum, and Mg ii line itself. The power law is parameterized by its normalization and slope. For Fe ii pseudo-continuum, we use one of the theoretical templates from Bruhweiler & Verner (2008), d12-m20-20-5, which was favored for this source in the previous study (Modzelewska et al. 2014). This template assumes a cloud number density of 10¹² cm⁻³, a turbulent velocity of 20 km s⁻¹, and flux of the hydrogen-ionizing photons of 10^−20.5 cm⁻² s⁻¹. The template was convolved with the Gaussian profile with dispersion of 900 km s⁻¹ representing the kinematic broadening of the Fe ii lines. The assumed broadening and the choice of the Fe ii template have been tested in Modzelewska et al. (2014).

The Mg ii line itself has been modeled using two kinematic components, as in Modzelewska et al. (2014), since a single component does not provide an adequate fit to the data. Each of the components is modeled assuming a Lorentzian shape, since this model minimizes the χ² fit for a given number of free parameters. Gaussian components give higher values of χ² in most of the data sets. Each of the kinematic components is fitted as a doublet, with wavelengths at 2796.35 and 2803.53 Å (Morton 1991). We used the doublet ratio 1.6, which provided the best fit in tested data sets. The best-fit source redshift was derived to be 0.90052, and in our fits it corresponded to the blue kinematic component. The shift of the second (red) kinematic component was a free parameter of the model in each data set. The contribution of the narrow-line region is negligible in this source, and in Modzelewska et al. (2014) we only derived an upper limit of 2%, so we did not include any component representing this emission. An exemplary SALT spectrum and its decomposition into the underlying power law, Fe ii pseudo-continuum, and two kinematic components are shown in Figure 1.

Zoom In Zoom Out Reset image size

The total equivalent width (EW) of the line was calculated from the fitted model, by integration in the 2700–2900 Å band. All four line components are included, i.e., two kinematic components, each with two doublet components. The EW of the line showed significant variations. The values of EW are listed in Table 1. We also calculated the EW of the Fe ii line in the spectral range 2700–2900 Å, and we also give these values in Table 1. The errors to EWs of Mg ii and Fe ii were determined by construction of the error contours and determining the appropriate limit for one parameter of interest (other parameters during the fitting were allowed to vary), and they represent 90% confidence level.

The spectroscopic observations were supplemented with the photometric data from other instruments, whenever possible. High-quality photometry covering the first part of the observational campaign came from the OGLE-IV survey done with the 1.3 m Warsaw telescope at the Las Campanas Observatory, Chile. The exposures were done in the V band, with the exposure time of 240 s. These data have a typical error of 0.005 mag. The relative stability of this photometry for our source has been shown in Modzelewska et al. (2014).

At later epochs OGLE data were not available, but the spectroscopic observations by SALT were usually performed by supplementing photometry with SALTICAM, in g band. The exposure time was 45 s, and two exposures were usually done. The SALT/SALTICAM instrument is not suitable for high-quality photometry. Due to distortions of images taken by SALTICAM, a special, simple method for photometry has been developed. This method allows us to partially reduce the occurrence of significant changes in the number of counts for the background and objects depending on the position in the image. The measurement error is larger in this case than from OGLE, typically of order of 0.012. Since the two photometric sets were done using slightly different filters, we allowed for an arbitrary shift in the SALT photometry, and the amount of shift was optimized in the time span where the two measurements overlapped.

CTS C30.10 has also been observed between 2017 December 2 and 2018 April 1 employing the 40 cm Bohum Monitoring Telescope (BMT) located at the Universitätssternwarte Bochum, near Cerro Armazones in Chile (https://astro.ruhr-uni-bochum.de/Astrophysik/all_infos.html). The observations were carried out with the Johnson broadband filter V_j (λ_eff = 550 nm and zero mag flux f₀ = 3836.3 Jy). Detailed information about the telescope, filters, and data reduction has been published in Ramolla et al. (2013). The light curve for this period of time is built by choosing 20 stars, bright enough and in proximity of the source as calibration stars, and the aperture used for the photometry is 375. The absolute photometric calibration is performed usually by comparison with the PANSTARSS Catalog. Since CTS C30.10 is not available in the PANSTARSS catalog, we used the conversion factor found for other objects that were observed with the same telescope, during the same nights. The photometry is given in Table 2.

Having the photometric coverage of the monitored period, we obtained the photometric light curve. The Mg ii light curve was also obtained with the use of photometry since SALT spectra are not spectrophotometric measurements owing to the construction of the telescope. Having the photometric light curve in V band, we used either SALTICAM measurement or linear interpolation between the nearest available photometric points to obtain the continuum flux at the date of the spectroscopic observation. We corrected the flux for absorption in our Galaxy taken from NED. This allowed us to calibrate the power-law component in our model and to derive the continuum flux value at 2800 Å rest frame. We then calculated the line flux and errors using the numerically determined value of the EW and its error. We integrated the model, not the data, but all four Mg ii components were included (see Section 2.2). Such an Mg ii light curve was ready for further analysis, including time delay measurements. The normalized dispersion of the continuum in the whole period is 6.0%, and the line flux variability, if we neglect observation 6, is at the level of 5.2%, lower (but not much) than the variability amplitude of the continuum.

In the period of 2018 February 3–April 14 we also performed dense photometric observations of CTS C30.10 with the use of the 1.3 m SMART telescope at the Cerro Tololo Inter-American Observatory. Observations were performed three times per night, roughly every week, with an exposure time of 60 s. Photometry from these data was prepared using IRAF daophot software and was typically of accuracy of 0.02, which is lower than what was achieved with other instruments, so we finally used these data only to constrain the variability properties of the quasar in the Appendix.

Since no obvious systematic drift is seen in the Mg ii line, we calculate the mean and the rms variation of the Mg ii line. For that purpose, we normalize each of the SALT spectra using the photometry provided in Table 2. We had to remove observation 6 from the composite plot since these data are particularly noisy, the flux varied up to a factor of 2 between the consecutive wavelength bins, and that considerably affected the mean and rms. This observation was performed in the presence of some thin clouds. The remaining spectra are of much higher quality.

The mean spectrum traces the general shape of the Mg ii line very nicely, with very low noise (see Figure 4). The two-component character of the line shape is well seen. The mean spectrum is well fitted by the same model; there are some departures between the data and the model in the Fe ii region, but other Fe ii templates do not provide a better fit. A comparable (but slightly worse) fit was achieved by the Fe ii template d11-m20-21-735 with lower number density (10¹¹ cm⁻³) but higher ionizing flux. Using the mean spectrum decomposition, we measured the total FWHM of the Mg ii line, treated as a single-component asymmetric line. The obtained value, 5009 ± 325 km s⁻¹, locates CTS C30.10 among type B quasars in classification by Sulentic et al. (2000). Since the total line FWHM is not a parameter directly fitted to the data (for those parameters we have errors from the contour error plots), we determined the total FWHM line width from the four-component line model (see Section 2.2), and uncertainties were estimated interpolating a total profile at ±5% with respect to the half-maximum of the profile, which takes into account the possible asymmetries in the total profile. That percentage is appropriate for the high S/N (∼320) of the mean spectrum.

Zoom In Zoom Out Reset image size

We removed the observation 6 spectrum only in the mean and rms computations. Apart from that, this observation, despite a relatively high noise level, gave a meaningful value for the line flux (but with large error; see Figure 2) and was used in further considerations.

Constructing ICCF is the traditionally used method of calculating the time delay in reverberation mapping of AGNs (Peterson et al. 2004). We mostly followed this general approach. However, we do not supplement the data curves with extrapolation when photometry or spectroscopy cannot be obtained through interpolation. Peterson et al. (2004) argue that the extrapolation decreases the error, but our light curves, particularly the spectroscopic one, are so short (only 26 measurements) that introducing not fully justified points may strongly bias the results. Otherwise, we perform the computations in the standard way, subtracting the mean and normalizing the light curves by the variance, but at that stage we use the measurement errors, so our mean and variance are calculated as a weighted quantities. We think that this is important since the errors in our data are not always similar to the typical error.

The best delay from ICCF is determined by the fit of the centroid to the values above 0.8 of the peak values. As a basic output, we treat the solution obtained as a sum of two options: either photometric points are interpolated to match shifted spectroscopic points, or the spectroscopic points are interpolated to match the shifted photometric points. This method is recommended by Peterson et al. (2004). However, we also use each of the two methods mentioned above separately, and they lead to strongly different results, shown in Figure 5 and in Table 3. The asymmetric method, with the interpolation of spectroscopic points, leads to high ICCF cross-correlation values, ∼0.92, close to the peak, implying a time delay of 542 days. On the other hand, interpolation of the photometric points shows a much lower level of correlation, at most ∼0.51, and the highest peak appears at 1073 days and strongly dominates the other peaks. Thus, high values of the ICCF in the previous case simply result from very few spectroscopic points being interpolated to dense photometric coverage that carries redundant information. Symmetrization points toward the lower value of the lag. There is also a clear second peak at 1070 days in the observed frame, but the ICCF peak is slightly lower. Thus, the result from ICCF is not unique, and this problem appears occasionally in reverberation measurements (see, e.g., Du et al. 2016). However, taking into consideration that we use 133 photometric points and 26 spectroscopic points, it is not clear that averaging is more reliable than interpolating only the denser (photometric) light curve. This would favor the longer value of the time delay, just above 1000 days. There is also a third peak, at 1300 days, and its height and position are similar in all three approaches.

Zoom In Zoom Out Reset image size

To obtain the error on the measured time delay, we use the model-independent method based on the bootstrap approach, as described in detail in Peterson et al. (1998). The uncertainties are large. For quoting the bootstrap best value delay and its error, we use the whole histogram, including multiple peaks. Large errors and the sensitivity of the results to the method are due to the fact that the distribution is not a single peak distribution, as clearly seen in Figure 5 from the ICCF distribution, and the same shows up in the histograms obtained through simulations when using the bootstrap method.

Finally, we also tested the sensitivity of the results to the few spectroscopic points of likely lower quality. As a test, we removed observations 6, 9, and 17. All of them are clear outliers in Figure 2, the third panel (the first two cases) and the bottom panel (the last one, showing an apparent problem with Fe ii fitting in these data). The results are also included in Table 3. The results are qualitatively the same as before.

Table 3.  Measured Delays for the Mg ii Line in CTS C30.10 from Various Methods in Observed Frame

Method	τ
	(days)
ICCF symmetric	${537}_{-274}^{+1036}$
ICCF interp.phot.	${947}_{-401}^{+629}$
ICCF interp.spectr.	${288}_{-34}^{+1274}$
ICCF symmetric (−3 points)	${412}_{-152}^{+224}$
ICCF interp.phot. (−3 points)	${926}_{-228}^{+27}$
ICCF interp.spectr. (−3 points)	${318}_{+225}^{-35}$
DCF (−3 points)	1066 ± 64
zDCF (without Catalina)	${1050}_{-27}^{+54}$
JAVELIN all points	${1067.9}_{-3.6}^{+3.5}$
JAVELIN (−2 points)	${1067.6}_{-3.4}^{+3.7}$
JAVELIN all points (bootstrap)	${898.6}_{-410.8}^{+373.2}$
χ2 symmetric	${1191}_{-53}^{+101}$
χ2 interp.phot.	${1148}_{-209}^{+133}$
χ2 interp.spectr.	${1182}_{-18}^{+42}$
χ2 symmetric (−3 points)	${1185}_{-25}^{+75}$
χ2 interp.phot. (−3 points)	${1155}_{-202}^{+80}$
χ2 interp.spectr. (−3 points)	${1203}_{-36}^{+102}$
VN	948+90−90

Download table as:  ASCIITypeset image

Since linear interpolation between points does not fully represent the variability of the source between the measured points, the DCF, which does not require such interpolation, has been proposed in the time delay measurement context by Edelson & Krolik (1988). We thus used this approach as well, removing the same three outliers as before, i.e., observations 6, 9, and 17. We also excluded the more noisy CATALINA survey data from the photometry light curve. The resulting plot for the DCF value as a function of the time lag is shown in Figure 6, where we considered the time lag binning of 100 days. In fact, the DCF method depends quite strongly on a time lag binning. Since the mean time between observations for the continuum and line-emission light curve is Δt_cont = T_cont/(n_obs − 1) = 28.4 days and Δt_line = 99.8 days, respectively, the time lag bin should be at least equal to the larger of the two timescales; hence, we set Δτ = 100 days. Again, we see four peaks, at ∼250 days (DCF ∼ 0.47), ∼450 days (DCF ∼ 0.51), ∼850 (DCF ∼ 0.50), and ∼1050 days (DCF ∼ 0.59), the last being the dominant one. This is consistent with the previous analysis using the ICCF.

Zoom In Zoom Out Reset image size

Next, we focused on the surroundings of the dominant peak at ∼1050 days within the time range of (900, 1200) days and a time step of 40 days. The result is plotted in Figure 7, where the peak is at τ ∼ 1069 days (DCF ∼ 0.71). We estimate the uncertainty of the dominant peak by fitting the Gaussian function to the data in Figure 7, by which we obtain the mean estimate and the symmetric 1σ error of τ_DCF = 1066 ± 64 days.

Zoom In Zoom Out Reset image size

Alexander (1997) introduced the z-transformation of the discrete correlation function (zDCF) to correct several biases of the DCF (see Edelson & Krolik 1988) by making use of the equal population binning and Fisher's z-transform.⁹

In the following analysis, we removed 54 points from the continuum light curve, which corresponded to the CATALINA survey with larger observational errors. In the end, we had 82 continuum points and 26 line-emission points. The calculated zDCF values versus the time lag including the uncertainties are shown in Figure 8. Next, we focused on the largest zDCF peak between 800 and 1400 days in Figure 8. We applied the maximum likelihood function of Alexander (1997) to determine the zDCF peak and its uncertainties, and we obtained ${\tau }_{\mathrm{zDCF}}={1050}_{-27}^{+54}$ days.

Zoom In Zoom Out Reset image size

To measure the time lag between the continuum emission and the line's response, we also use the JAVELIN¹⁰ software (Zu et al. 2011, 2013, 2016). It models the continuum light curve as a stochastic process, here the DRW (Kelly et al. 2009; Kozłowski et al. 2010; MacLeod et al. 2010), and assumes that the light curves of the emission line are the lagged, smoothed, and scaled versions of the continuum light curve. The Markov chain Monte Carlo (MCMC) posterior probabilities produced by JAVELIN include such parameters as the two DRW model parameters (the variability timescale and amplitude), the time lag, the smoothing width of the top-hat function, and the amplitude scale factor of the emission line to the continuum (A_line/A_photo).

In Figure 9, we present the posterior probability distribution in the lag–A_line/A_photo plane. Several probability maxima are readily visible, of which the one at 1068 days is the strongest. A by-eye inspection of the line light curve uncovered two plausible outliers at JD = 2,456,886.5 and 2,457,110.5. We considered two cases here, where the line data were modeled in full (top panel of Figure 9) and with the two outliers removed (bottom panel of Figure 9). The best-measured observed-frame time lags obtained from the strongest peak are $\tau ={1067.85}_{-3.63}^{+3.45}$ days for the full-line light curve and $\tau ={1067.64}_{-3.42}^{+3.74}$ days for the cleaned light curve, where the error bars are asymmetric 1σ uncertainties. The rest-frame lags are then, respectively, $\tau ={561.76}_{-1.80}^{+1.97}$ days and $\tau \,={561.87}_{-1.91}^{+1.82}$ days, fully consistent with each other.

Zoom In Zoom Out Reset image size

In order to understand the probability structures in Figure 9, we performed simulations of 100 light-curve pairs for (1) the input time lag of 1000 days and (2) the input time lag of 500 days. We used the DRW model with the typical input AGN variability parameters, the timescale of 1 yr, and the asymptotic variability of 0.25 mag (Kozłowski 2016). The cadence and photometric noise were set to be identical to the real data. Then, we modeled the simulated data with JAVELIN in the same fashion as the real data, including the cadence and the error bars.

We find that for the simulations with the lag of 1000 days, 10 cases (10%) show incorrect measurements, other than the input 1000 days. In 25 (25%) cases we also find secondary probability peaks in the vicinity of 500 days (similar to what is observed in real data), in 44 cases (44%) probability peaks at 1200–1300 days, in 27 cases (27%) peaks at about 800 days, and in 22 cases (22%) peaks at about 200 days. Generally, the probability structures in the simulated data resemble quite well what we observe in the real data, and the additional peaks to the main one at 1000 days are expected based on simulations (even if there is a single lag present in the data).

Analysis of simulations with the 500-day lag returns only 2 cases (2%) with peaks in the vicinity of 1000 days, as well as41 cases (41%) at about 1200 days. General inspection of the output appears to be different than for the real data and simulations with the input lag of 1000 days.

JAVELIN-claimed errors are usually very small since the software focuses on the error determination of the position of the most significant peak. In order to make JAVELIN errors comparable to other measurement errors, we performed the bootstrap approach, calling JAVELIN software for 100 bootstrap-generated curves, and at the end we performed the search for the best solution and its error using the whole distribution of time delays instead of focusing on the strongest peak. Now the error is much larger, giving ${898.6}_{-410.8}^{+373.2}$ days in the observed frame. The error is still smaller than in ICCF but larger than in zDCF. For completeness, we quote this value in Table 3, marked as bootstrap.

This method is frequently used in quasar lensing studies, and Czerny et al. (2013) found that it works better than ICCF for the red-noise AGN variability. Thus, we also apply this method. The light curves are again prepared as for ICCF, by subtracting the mean and renormalizing by variance. Then, the photometric or spectroscopic light curve is shifted with respect to the other one, and, as for ICCF, we can use the asymmetric approach (interpolating only photometry, or interpolating only spectroscopic light curve), or we can use both and average the results. We use only the linear interpolation, as in ICCF. We estimate the similarity of the shifted curves by simply calculating the χ² value. Thus, the minimum of the χ² indicates the most likely time delay. The results are shown in Figure 10. The minimum is always at the location indicated by the JAVELIN method (see Section 4.4).

Zoom In Zoom Out Reset image size

We assign the error again by performing Monte Carlo simulations as in Section 4.1. The error bars given in Table 3 are much larger than in the standard case of the JAVELIN method, comparable to ICCF. However, the distribution is more concentrated around the best solution, without an extended tail as in the ICCF method. What is more important, the resulting most probable time delay is the same, independently from the choice of the data set for interpolation.

Also in this case we tested the sensitivity or the removal of the outlying spectroscopic points. We removed again SALT observations 6, 9, and 17. The results for the best value of the time delay in this case did not change (see Table 3), so the method seems quite stable.

Finally, we complement the previous ways of time delay determination by the method of randomness measure or the complexity measure of the data. This method has the advantage that it does not require either the interpolation of data, as for the ICCF method, or the binning in the correlation space, and it is largely model independent. The measures of the randomness or the complexity of the data are based on different kinds of estimators, among which the optimized von Neumann's scheme turns out to be most robust (Chelouche et al. 2017). The von Neumann (VN) estimator of the randomness (or regularity) of data is defined as the mean-square successive difference,

$$\begin{eqnarray}&&E(\tau )\equiv \displaystyle \frac{1}{N-1}\displaystyle \sum _{i=1}^{N-1}{[F({t}_{i})-F({t}_{i+1})]}^{2},\end{eqnarray} \tag{ 1 }$$

where $F(t,\tau )={\{({t}_{i},{f}_{i})\}}_{i=1}^{N}={F}_{1}\cup {F}_{2}^{\tau }$ is the combined continuum and time-delayed line light curve. When the searched time delay τ is close to the actual time delay between the light curves, τ ≃ τ₀, the VN estimator reaches the minimum. We applied the optimized scheme of the VN estimator based on the python script of Chelouche et al. (2017).¹¹ We show the dependence of the VN estimator on time delay in Figure 11. In the search interval [0, T_max] = [0, 1300] days, there are three distinct minima, with the global one at τ₀ ≃ 945 days, followed by τ' = 239 days and τ'' = 573 days. In the further analysis, we assume that the time delay τ₀ ≃ 945 stands for the actual time delay, but this needs to be evaluated further statistically.

Zoom In Zoom Out Reset image size

To determine the uncertainty of ${\tau }_{0}$, we perform bootstrapping by generating subsamples of 10,000 continuum and line-emission light curves based on the actual observed light curves. This way we obtain a distribution of the time delay around the global minimum; see Figure 12. The distribution is broad, with a mean and standard deviation of ${\overline{\tau }}_{0}=952\pm 90$ days, which is consistent within the uncertainty with the previously determined time delays via the JAVELIN and χ² methods, further strengthening the time delay of ∼1000 days. The advantage of the optimized VN method is that no stochastic model and assumption of quasar variability are needed.

Zoom In Zoom Out Reset image size

The quality of our data is not good enough to give always the same result, independently of the method. Three candidate time delay values appear: one around 1070 days, one at 550 days, and one at 1250 days in the observed frame. JAVELIN gives the smallest errors, and this method favors the intermediate delay, with or without three outliers removed.

The results from various methods are not easy to compare since error determination is very sensitive to the method used. JAVELIN returns the error of the best solution position, while in other methods the errors are based on the whole statistics, including side peaks as well. This basically explains the huge difference in errors between JAVELIN and χ², ICCF and VN. If we force JAVELIN to use the same approach as the other methods do (see Section 4.4), the errors from JAVELIN are still somewhat smaller than from ICCF, comparable to χ² results, but larger than the errors from zDCF and VN. Still, the value of the best time delay in JAVELIN seems very insensitive to the details like removal (or not) of outliers and the way errors are measured.

In the case of the ICCF and χ² methods, the delay can be determined using either a symmetric approach to photometry and spectroscopy or an asymmetric one. Since the photometric coverage is much better, actually the asymmetric method requiring interpolation only for photometric data seems more justified.

Thus, we calculated the weighted average value of the best time delay by including only the full data ICCF and χ² for photometric interpolation and the results from the other three methods. For JAVELIN, we used the standard result. Such an average value is 1068 days in the observed frame, not far from the JAVELIN result itself.

A delay of that order seems to be most consistent with the visual impression of the data. In Figure 13 we show the continuum and the line light curves, average-subtracted and renormalized by variance, with photometric curve shifted by a few trial values of the time delay. The middle one represents best the overall pattern. This additionally supports the adoption of the mean value calculated above.

Zoom In Zoom Out Reset image size

However, such an averaging is not statistically viable (although informative), and determination of the error of this quantity is even more problematic. The bootstrap approach in the case of ICCF gives values that seem too high, and the bootstrap approach to χ² indicates delays in the range from 939 to 1281 days. Such an error is large, but in our opinion this is the most realistic and conservative approach. Simulations of the ICCF and χ² results with the method of Timmer & Koenig (1995) (see the Appendix) give errors of order of 30–60 days, but they do not include the information about the light-curve properties, apart from the power density spectrum (PDS). So finally we adopt, as the best time delay, the value of 1068 days, and for the error we adopt the limits from the χ² bootstrap, i.e., ${1068}_{-129}^{+213}$, in the observed frame. JAVELIN small error results are well within these limits.

Determination of the time delay opens the way to estimation of the black hole mass. Our rms spectrum is not of high quality, but the mean spectrum can be used for that purpose. Black hole mass can be measured as

$$\begin{eqnarray}&&M={f}_{\mathrm{vir},1}\displaystyle \frac{\tau {\mathrm{FWHM}}^{2}}{G},\end{eqnarray} \tag{ 2 }$$

or

$$\begin{eqnarray}&&M={f}_{\mathrm{vir},2}\displaystyle \frac{\tau {\sigma }_{\mathrm{line}}^{2}}{G}.\end{eqnarray} \tag{ 3 }$$

We have measured the FWHM values and the line dispersion σ_line from the fitted mean profile. The corresponding values are 5009 and 3326 km s⁻¹. The ratio of FWHM/σ_line is 1.51. For the virial factor, f_vir, we use the recommendation of Collin et al. (2006): factor of 1.43 in the first case and factor of 3.85 in the second case. The two expressions above give the two values of the black hole mass: 3.9 × 10⁹ M_⊙ and 4.6 × 10⁹ M_⊙. The second method is generally recommended by Collin et al. (2006), and it compares well with the black hole mass of 4.9 × 10⁹ M_⊙ favored by Modzelewska et al. (2014) for this source.

Table 4.  Measured Delays for the Mg ii Line

Object	z	Tau	Err+	Err−	$\mathrm{log}{L}_{3000}$	Err	Reference Tau	Reference L
		Rest Frame
141214.20 + 532546.7	0.45810	36.7	10.4	4.8	44.63882	0.00043	Shen et al. (2016)	Shen et al. (2018)
141018.04 + 532937.5	0.46960	32.3	12.9	5.3	43.72880	0.00506	Shen et al. (2016)	Shen et al. (2018)
141417.13 + 515722.6	0.60370	29.1	3.6	8.8	43.68735	0.00290	Shen et al. (2016)	Shen et al. (2018)
142049.28 + 521053.3	0.75100	34.0	6.7	12.0	44.69091	0.00090	Shen et al. (2016)	Shen et al. (2018)
141650.93 + 535157.0	0.52660	25.1	2.0	2.6	43.77781	0.00198	Shen et al. (2016)	Shen et al. (2018)
141644.17 + 532556.1	0.42530	17.2	2.7	2.7	43.94795	0.00105	Shen et al. (2016)	Shen et al. (2018)
CTS 252	1.89000	190.0	59.0	114.0	46.79373	0.09142	Lira et al. (2018)	NED NUV GALEX
NGC 4151	0.00332	6.8	1.7	2.1	42.83219	0.18206	Metzroth et al. (2006)	Code & Welch (1982)
NGC 4151	0.00332	5.3	1.9	1.8	42.83219	0.18206	Metzroth et al. (2006)	Code & Welch (1982)
CTS C30.10	0.90052	564.0	109	71	46.023a	0.026	this paper	this paper

Note.

^aObtained from the mean V mag 17.1, for the cosmology H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3, Ω_Λ = 0.7 (Kozłowski 2015).

Download table as:  ASCIITypeset image

We now combine the measurements available in the literature (see Section 1) with our measurement of the time delay in CTS C30.10. The list of the measurements is given in Table 4, and the plot using the final value set in Section 4.7 is shown in Figure 14. We use these data to determine the R–L relation for the Mg ii line. The best fit, taking into account errors in both axes, reads as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\tau & = & 1.458\pm 0.024\\ & & +(0.524\pm 0.027)(\mathrm{log}{L}_{3000}-44.0),\end{array}\end{eqnarray} \tag{ 4 }$$

and it is shown in Figure 14 with a solid line. There is some dispersion in the plot. CTS C30.10 is much brighter than AGNs measured by Shen et al. (2016). On the other hand, it is somewhat less luminous and located at lower redshift than CTS 252, but our time delay is longer than for CTS 252. The black hole mass in CTS 252 is 1.26 × 10⁹ M_⊙, lower than the most probable mass in CTS C30.10, while the luminosity is higher by a factor of 5. The FWHM of the Mg ii line is also much narrower in CTS 252 (3800 km s⁻¹) than the total FWHM of the line in CTS C30.10 (5009 ± 325 km s⁻¹). This strongly suggests that the Eddington ratio in these two objects is different, much lower in CTS C30.10 than in CTS 252. Thus, the difference in the detected lag may be related to the difference in the Eddington ratios, in a similar way to how it is for Hβ line lags, with considerable lag shortening (for a given monochromatic luminosity) with the rise of the Eddington ratio (Du et al. 2016; M. L. Martinez Aldama 2019, in preparation).

Zoom In Zoom Out Reset image size

As a reference, we plot in Figure 14 the radius–luminosity relation of Bentz et al. (2013), their model Clean2. Since now we use L₃₀₀₀ on the x-axis, we transformed the relation by recomputing L₃₀₀₀ from L₅₁₀₀ using the ratio of the bolometric corrections derived by Richards et al. (2006): L₃₀₀₀ = (10.33/5.62)L₅₁₀₀. Both relations have the same slope within the error (slope of the Bentz et al. 2013 relation is 0.546 ± 0.027), and the normalization is also similar.

Our result shows that the Mg ii line kinematically is very similar to the Hβ line. In their recent work Popović et al. (2019) came to a different conclusion on the basis of the line shape study in 284 sources, stressing the difference in the line wings, as noticed also before by Trakhtenbrot & Netzer (2012). However, this difference was mostly seen for lines with FWHM above 6000 km s⁻¹. CTS C30.10 and all of the sources in Table 4, apart perhaps from NGC 4151 (FWHM = 6558 ± 1880 km s⁻¹; Metzroth et al. 2006), have narrower Mg ii lines, between 2391 and 4066 km s⁻¹ in the Shen et al. (2016) sample.