Optical Variability of Eight FRII-type Quasars with 13 yr Photometric Light Curves

The SF provides quantitative measures of how strongly the flux changes as a function of the time interval, i.e., it measures the mean value of the magnitude difference for measurements separated by a given time interval. Several approaches to SF calculations are employed in the literature (e.g., Emmanoulopoulos et al. 2010; MacLeod et al. 2011; Kozłowski 2016); however, in this work we use the formalism described by Simonetti et al. (1985) with SF defined as:

$$\begin{eqnarray}&&\mathrm{SF}{({\rm{\Delta }}t)}^{2}=\displaystyle \frac{1}{N}\sum _{i,j\gt i}^{{\rm{N}}}{\left({m}_{{\rm{i}}}-{m}_{j}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

where the sum is over all N magnitude measurements that are separated by some time lag Δt. To correct for the observed frame variability to the rest frame, we use Δt = ∣t_i − t_j ∣(1 + z)⁻¹ in Equation (1). SF is commonly characterized in terms of its slope α, where SF²(Δt) ∝ Δt^α . Interpreting the specific variability characteristics based on the shape of the SF and its properties sometimes becomes complicated due to the complex shape of the SF. However, some properties can provide information about the typical timescales of observed variability and can be used to search for periodicities. A characteristic timescale of variability, defined as the time interval between the maximum and the neighboring minimum in a light curve, manifests itself as the maximum of the SF, while the periodicity in the light curve is visible as a minimum of the SF (e.g., Heidt & Wagner 1996). Furthermore, the slope of the SF characterizes the process underlying the variability. Based on theoretical models of AGN variability, Hawkins (2002) determined the predicted slopes of the SF, which are equal to 0.83, 0.44, and 0.25 for the starburst, accretion disk instability, and microlensing models, respectively.

Equation (1) does not include the contribution to the SF values due to measurement noise, which should be subtracted to obtain the "true SF" (for a comprehensive review, see Kozłowski 2016). Therefore, to obtain the correct results that characterize the variability, we fit the four-parameter function to the SF calculated by Equation (2) (Kozłowski 2016):

$$\begin{eqnarray}&&{\mathrm{SF}}^{2}({\rm{\Delta }}t)={\mathrm{SF}}_{\infty }^{2}[1-\exp {\left(-| {\rm{\Delta }}t| /\tau \right)}^{2\alpha })]+2{\sigma }^{2},\end{eqnarray} \tag{ 2 }$$

where SF_∞ is the variance on long timescales, τ is the decorrelation timescale at which the SF changes slope—the characteristic timescale of variability, α is the slope of the SF, and σ is the noise term. The obtained SFs with the fitted functions calculated for our quasar light curves are presented in Figure 2, and the basic information on the light curves and the SF fitted parameters is listed in Table 2.

The SF slopes obtained for the studied quasars are relatively steep. According to the model predictions (Hawkins 2002) almost all have values between 0.44 ± 0.03 and 0.83 ± 0.08, corresponding to the theoretical values of the SF slopes for the accretion disk instability model and the starburst model, respectively. Therefore, it is difficult to unambiguously designate the mechanism of variability in the case of our quasar sample. However, the starburst model is sufficient to explain the magnitude changes in low-luminosity AGNs. It becomes problematic to explain magnitude changes in moderate and most luminous quasars (e.g., Hawkins 2002). It should also be stressed that, as all of the observed quasars have radio lobes lying close to the sky plane, the optical variability mechanisms are expected to originate at the accretion disk. Nevertheless, it cannot be excluded that observed variability is to some extent modified by the Doppler-boosted emission from radio jets. The large-scale radio emission manifested by the radio lobes (Figure 1) was generated in a different epoch from the optical variability observed during the last 13 yr. The orientation of the quasar could change during that time, and now the radio jets can propagate closer to the line of sight. Therefore, the explanation of the observed SF slopes for our quasar sample is difficult and requires a complex elaboration of theoretical models to comprehensively account for the variability phenomenon.

As shown by Kozłowski (2016), the distribution of SF slopes for AGNs peaks for α = 0.55 ± 0.08 with a typical decorrelation timescale τ = 354 ± 168 days. Five of the eight quasar light curves from our sample have SF slopes higher than α = 0.55 and the timescales of variability are also larger in most cases. However, based on 20 yr light curves Smith et al. (1993) found that typical timescales for quasars peak between 2 and 6 yr in a rest frame. It should be noted that to obtain a reliable estimate of characteristic timescales, the length of the data set should be at least a few times longer than the decorrelation timescale (Kozłowski 2017). In the case of our quasar sample, we obtained a sufficiently low τ compared to the length of the data set for B2 0709+37, [HB89] 1425+267, and [HB89] 1721+343 (the τ one-sixth the length of the data set in a rest frame). It was also tested that the slopes or SF breaks may be derived with insufficient accuracy due to nonregular sampling of the light curve resulting in various types of artifacts in SF shape that can be very misleading in their interpretation (Emmanoulopoulos et al. 2010). In particular, such features are visible in the SF of B2 0709+37 where the SF has a "sinusoidal" shape with peaks repeating every ∼230 days. In Figure 3 we demonstrate how the SF shape of B2 0709+37 changes when the data are evenly sampled. For this, the evenly sampled light curve is obtained by linear interpolation between consecutive data points with an interpolation interval of 8 days. The SF calculated for the evenly sampled light curve is smoothed and does not show any artificial peaks.

Zoom In Zoom Out Reset image size

Power spectral analysis uses Fourier decomposition methods, where the light curve is represented by a combination of sinusoidal signals with random phases and amplitudes, which correspond to various timescales of a source's variability in the time series (e.g., Timmer & Koenig 1995). The observed PSD shows a power-law (PL) shape $P({\nu }_{k})\propto {\nu }_{k}^{-\beta }$, where β is the slope and ν_k is the temporal frequency. As the quasar variability can be well described by a stochastic process (e.g., Kelly et al. 2009, 2011; Kozłowski 2016) the value of β represents the type of noise color. The β = 0 is an uncorrelated white-type noise, β ∼ 1 is a flicker/pink-type noise, and β ∼ 2 is a damped random-walk red-type noise. The PSD slope is related to the SF slope α as β = 4α (Kozłowski 2016), and can be used to discriminate between different variability models.

We use the power spectral response (PSRESP) method of Uttley et al. (2002) that is widely used in the community (Chatterjee et al. 2008; Max-Moerbeck et al. 2014) to derive the PSD slopes for our quasar light curves. We have previously used this method in Goyal (2020, 2021) and Goyal et al. (2022), so we direct the reader to them for details, while we briefly outline the methodology here. The discrete Fourier transform (DFT) of a time series gives the Fourier amplitude as a function of the Fourier frequency. The rms-normalized periodogram is given as the squared modulus of its DFT for the evenly sampled light curve f(t_i ), observed at discrete times t_i for a total duration T, and consists of N data points:

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}P({\nu }_{k}) & = & \displaystyle \frac{2\,T}{{\mu }^{2}\,{N}^{2}}\left\{{\left[\sum _{i=1}^{N}f({t}_{i})\,\cos (2\pi {\nu }_{k}{t}_{i})\right]}^{2}\right.\\ & & +\left.{\left[\sum _{i=1}^{N}f({t}_{i})\,\sin (2\pi {\nu }_{k}{t}_{i})\right]}^{2}\,\right\},\end{array}\end{eqnarray} \tag{ 3 }$$

where μ is the mean of the light curve and is subtracted from the flux values, f(t_i ). The light curve is evenly sampled through linear interpolation using interpolation intervals 5–10 times smaller than the mean (observed) data sampling interval. The effect of red-noise leak is minimized by multiplying the time series by the Hanning window function (Max-Moerbeck et al. 2014). Aliasing is constant and is not effective for red-noise-dominated time series (Uttley et al. 2002). The DFT is computed for evenly spaced frequencies ranging from the total duration of the light curve, T, down to the mean (observed) Nyquist sampling frequency (ν_Nyq). The "raw" periodograms, obtained using Equation (3)), provide a noisy estimate of spectral power (as it consists of independently distributed χ² variables with two degrees of freedom (DOF); Papadakis & Lawrence 1993); therefore, we average a number of them to obtain a reliable estimate, referred as log-binned periodograms. A binning factor of 1.6 is used, with the representative frequency taken as the geometric mean of each bin in our analysis. Finally, the "true" power spectrum in the log–log space is obtained from the log-binned periodograms by subtracting the expectation value of the χ² distribution with 2 DOF (=−0.25068; Vaughan 2005). The noise floor level due to measurement uncertainty is computed following Isobe et al. (2015):

$$\begin{eqnarray}&&{P}_{\mathrm{stat}}=\displaystyle \frac{2\,T}{{\mu }^{2}\,N}\,{\sigma }_{\mathrm{stat}}^{2},\end{eqnarray} \tag{ 4 }$$

where ${\sigma }_{\mathrm{stat}}^{2}={\sum }_{j=1}^{j=N}{\rm{\Delta }}f{\left({t}_{j}\right)}^{2}/N$ is the mean variance of the measurement uncertainties for the flux values, Δf(t_j ), in the observed light curve at times t_j , with N denoting the number of data points in the observed light curve.

The best-fit PSD slopes are derived using the PSRESP method given by Uttley et al. (2002). This method has the advantage of further mitigating the deleterious effects of Fourier transformation and, therefore, is the most robust method for estimating the spectral shape. This method is computationally expensive, as it requires Monte Carlo (MC) simulations of many light curves, mimicking the observed data. Here, an (input) PSD model is tested against the observed PSD. The estimate of the best-fit model parameters and their uncertainties is performed by varying the model parameters. To achieve this, light curves are generated with a known underlying power spectral shape using MC simulations, and DFT is produced in the same manner as the observed data (Equation (3)). We tested a simple power-law (PL) spectral shape with slopes ranging from 0.1 to 4.0 with a step of 0.1 and fixed normalization. The quality of fit is assessed by computing two quantities, ${\chi }_{\mathrm{obs}}^{2}$ and ${\chi }_{\mathrm{dist}}^{2}$, defined as

$$\begin{eqnarray}&&{\chi }_{\mathrm{obs}}^{2}=\sum _{{\nu }_{k}={\nu }_{\min }}^{{\nu }_{k}={\nu }_{\max }}\displaystyle \frac{{\left[\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}({\nu }_{k})-{\mathrm{log}}_{10}{P}_{\mathrm{obs}}({\nu }_{k})\right]}^{2}}{{\rm{\Delta }}\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}{\left({\nu }_{k}\right)}^{2}},\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{\chi }_{\mathrm{dist},\ i}^{2}=\sum _{{\nu }_{k}={\nu }_{\min }}^{{\nu }_{k}={\nu }_{\max }}\displaystyle \frac{{\left[\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}({\nu }_{k})-{\mathrm{log}}_{10}{P}_{\mathrm{sim},i}({\nu }_{k})\right]}^{2}}{{\rm{\Delta }}\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}{\left({\nu }_{k}\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

Here, ${\mathrm{log}}_{10}{P}_{\mathrm{obs}}$ and ${\mathrm{log}}_{10}{P}_{\mathrm{sim}},\ i$ are the observed and simulated log-binned periodograms, respectively; $\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}$ and ${\rm{\Delta }}\overline{{\mathrm{log}}_{10}{P}_{\mathrm{sim}}}$ are the mean and standard deviation obtained by averaging a large number of PSDs, k represents the number of frequencies in the log-binned power spectrum (ranging from ${\nu }_{k,\min }$ to ${\nu }_{k,\max }$), while i runs over the number of simulated light curves for a given β.

For a given β, ${\chi }_{\mathrm{obs}}^{2}$ is the measure of the offset of the average (input) spectral shape with the observed PSD shape, while ${\chi }_{\mathrm{dist}}^{2}$ refers to the offset when the observed data are replaced with simulations. A probability, p_β , is given by the percentile of the ${\chi }_{\mathrm{dist}}^{2}$ distribution above which ${\chi }_{\mathrm{dist}}^{2}$ is found to be greater than ${\chi }_{\mathrm{obs}}^{2}$ for a given β. A large value of p_β represents a good fit in the sense that a large fraction of random realizations of the model (input) power spectrum can recover the intrinsic PSD shape. Therefore, this analysis essentially uses the MC approach toward a frequentist estimation of the quality of the model compared to the data. For the simulation of quasar light curves, we use the method of Emmanoulopoulos et al. (2013), which preserves both the probability density function (PDF) of the flux distribution and the underlying power spectral shape, and not just the power spectral shape (e.g., Timmer & Koenig 1995). For quasar light-curve simulations, we used a log-normal PDF of the flux distribution (Kelly et al. 2009). Since the light curves are given in units of differential magnitude (Figure 2), we converted them to fluxes using the arbitrary flux scale, following Goyal (2021) for the PSD analysis. The errors in the best-fit PSD slope are obtained by fitting a Gaussian function to the probability distribution curves and are given as the standard deviation, σ, of the Gaussian function. The resulting PSD and the corresponding probability distribution curves for the quasar light curves are plotted in Figure 4 while Table 3 lists the results of the PSRESP method. The observed PSDs are well fitted to the single PL function with high confidence (p_β ≥ 0.1, except for the quasar FBQS J095206.3+235245 for which p_β = 0.094) and have slopes ∼2, over the range of spectral frequency between −3.67 and −1.42 day⁻¹, indicating a red-noise character of variability.

Zoom In Zoom Out Reset image size

Table 3. Summary of the Observations and the PSD Analysis

Source Name	Duration	Tobs	Tmean	Tin	${\mathrm{log}}_{10}({{P}}_{\mathrm{stat}})$	${\mathrm{log}}_{10}({\nu }_{k})$ Range	β ± Error	pβ a
	(Start–End)	(yr)	(day)	(day)	${\left(\tfrac{\mathrm{rms}}{\mathrm{mean}}\right)}^{2}$ day	(day−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
B2 0709+37	2009 Mar 2–2021 Nov 24	12.7	12.3	0.4	−2.28	−3.66 to −1.62	3.2 ± 0.7	0.243
FBQS J095206.3+235245	2009 Mar 14–2021 May 11	12.2	19.9	0.4	−1.19	−3.64 to −1.82	2.5 ± 0.8	0.094
PG 1004+130	2009 Mar 2–2021 Nov 17	12.7	18.6	0.4	−2.56	−3.66 to −1.83	3.2 ± 0.7	0.243
[HB89] 1156+631	2009 Mar 2–2021 Sep 9	12.5	12.5	0.4	−1.86	−3.66 to −1.62	2.7 ± 0.7	0.315
[HB89] 1425+267	2009 Apr 7–2021 Sep 28	12.5	12.9	0.4	−2.02	−3.65 to −1.62	2.3 ± 0.2	0.164
[HB89] 1503+691	2009 Apr 7–2021 Sep 8	12.4	12.3	0.4	−1.46	−3.65 to −1.62	2.5 ± 0.8	0.897
[HB89] 1721+343	2009 Apr 7–2021 Nov 24	12.6	9.1	0.4	−2.63	−3.64 to −1.42	2.3 ± 0.5	0.849
4C +74.26	2009 Jan 19–2021 Nov 24	12.6	8.3	0.4	−3.13	−3.67 to −1.42	2.0 ± 0.2	0.592

Notes. Columns: (1) name of the quasar, (2) duration of monitoring (start date–end date), (3) total length of the observed light curve, (4) the mean sampling interval for the observed light curve (light-curve duration/number of data points), (5) interpolation interval, (6) the noise level in PSD due to the measurement uncertainty, (7) the temporal frequency range covered by the binned logarithmic power spectra, (8) the best-fit PL slope of the PSD along with the corresponding errors representing 68% confidence limit (see Section 4.2), (9) the corresponding p_β .

^a The PL model is considered a bad-fit if p_β ≤ 0.1 as the corresponding rejection confidence for the model is ≥90% (Section 4.2).

Download table as:  ASCIITypeset image

The results obtained using the SF and PSD methods for our light curves are notable in two aspects: (1) the slope of the SF curve is related to the PSD curve by β = 4α within the uncertainties reported for each target (Tables 2 and 3) and (2) the SF curves of quasars B2 0709+37, [HB89] 1425+267, and [HB89] 1721+343 show bends (or a plateau) around the ∼1 year timescale, indicating a presence of decorrelation timescales (Table 2), while the PSDs of the same sources show a good fit to the single PL form over the entire spectral frequency range covered with no sign of bending at lower frequencies (Table 3). The first result is trivial, as such a dependence between the SF slope (which is a measure of the rms) and the PSD slope (which is a measure of squared rms) is expected (Bauer et al. 2009). The second result, i.e., SF analysis showing a decorrelation timescale in the light curve for a few quasars, while the PSD analysis indicates no such feature, is an inconsistent result, although not completely unexpected. As discussed in detail in Emmanoulopoulos et al. (2010), the SF curves often show breaks that depend on the lengths of the data set and the underlying PSD shapes. Even for featureless PSD shapes (i.e., single PL forms), the SF breaks could be obtained on timescales corresponding to one-tenth to half of the length of the time series. For longer data sets, SF breaks are visible on longer timescales. Thus, the SF break may not reflect the intrinsic variability of the source. Furthermore, to obtain a reliable estimate of characteristic timescales, the length of the data set should be at least a few times longer than the decorrelation timescale (Kozłowski 2017). Our comparison of SF and PSD analysis methods also supports the view of Emmanoulopoulos et al. (2010) that the results of SF analysis, in particular the estimation of decorrelation timescales from the finite duration and unevenly sampled time series, should be treated with caution.

The optical light curve of quasar 4C +74.26 for the duration 2009–2016 from our monitoring program was previously analyzed by Bhatta et al. (2018) and Zola et al. (2019). Along with the 15 GHz radio light curve from Ovens Valley Radio Observatory monitoring and the X-ray light curve from the Swift-BAT Hard X-ray Transient Monitor program of roughly similar duration, Bhatta et al. (2018) focused on cross correlation analysis of light curves between different wave bands. Statistically significant correlated variability was found between optical and radio wave bands with a time lag of 250 days (radio variation leading to optical variations). Bhatta et al. (2018) obtained an optical PSD slope of β = 1.6 ± 0.2 using the Lomb–Scargle periodogram (LSP; Lomb 1976; Scargle 1982), which is consistent with our estimates of β = 2.0 ± 0.2 within the reported uncertainties. Zola et al. (2019) analyzed the optical light curve of 4C +74.26 and the light curves of all quasars from our sample but for the duration 2009–2018 to search for possible quasiperiodic oscillations (QPOs) using the LSP and the weighted wavelet Z-transform methods (Foster 1996). No statistically significant (>99% confidence level) QPOs were reported in their analysis for any of the quasars, including 4C+74.26. We confirm the lack of QPOs in the optical light curves of our sources in longer duration data sets (2009–2021) as all of them show a good fit to the single PL form with high confidence (Table 3).

For the studied quasars, we obtained a significant anticorrelation (correlation coefficient equal to 0.86) between the projected linear size of the radio structure and the PSD slope, i.e., for larger radio sources, the PSD is flatter (see Figure 5). This relationship may indicate that the nature of the variability is related to the size of the radio source, although drawing conclusions based on the results obtained for eight objects is speculation that should be confirmed based on the study of a larger sample of lobe-dominated quasars.

Zoom In Zoom Out Reset image size

All studied quasars have large radio powers and present large-scale radio structures on the sky plane (Table 1); therefore, it would be interesting to explore the relationship between the optical variability properties (PSD slope) and the linear sizes (Figure 5). To check this, we applied Spearman's rank correlation test, which measures the statistical dependence (p-value) and the strength of the correlation (ρ-value) between the two variables (Spearman 1904). A null hypothesis of no correlation is tested against the alternate hypothesis of nonzero correlation at a certain significance level (=0.003, adopted by us). We obtained a p = 0.006 and a ρ-value = −0.86. Our results indicate that the correlation is significant at a 95%–99% confidence level and the two variables are anticorrelated. This correlation hints that the nature of the optical variability is related to the size of the radio source, although drawing conclusions based on the results obtained for eight objects is speculation that should be confirmed based on the study of a larger sample of lobe-dominated quasars.

The optical variations from quasar sources result from instabilities in the accretion disk. In such a case, one expects some characteristic/relaxation timescales beyond which the flux variations should become uncorrelated. Such a characteristic/relaxation timescale will appear as flattening in the SF and PSDs curves leading to a change of slope from ≥1 to ≃0. The natural timescales for a disk are the light-crossing, dynamical, and thermal/viscous timescales which correspond to ∼1 day, ∼104 days and 4.6 yr for a standard Shakura and Sunyaev disk (Shakura & Sunyaev 1976) with assumed Eddington ratios of 0.01–0.1, viscosity parameter 0.01, and black hole masses of 10⁸ M_⊙ (Kelly et al. 2009). Given the typical range of black hole mass for radio quasars of 10⁸–10⁹ M_⊙ (Kuźmicz & Jamrozy 2012), we expect the SF and PSDs to flatten on a timescale of around a few years, if the fluctuations are driven by thermal instabilities in the accretion disk. Our PSDs cover timescales between ∼13 yr and 2 weeks (∼2–3 times longer than the expected characteristic timescale for a 10⁸ M_⊙ SMBH) and are well fitted with single PL forms, without any sign of bending on longer timescales. Kelly et al. (2009) analyzed the 5–6 yr long light curves of 100 MACHO quasars with SMBH masses in the range 10⁶–10¹⁰ M_⊙ using the stationary Ornstein–Uhlenbeck process or a random-walk-type noise process that give a well-defined relaxation time of the process driving the variability. On timescales longer and shorter than the relaxation time, the resulting PSDs exhibit slopes equal to 0 and 2, respectively. In their analysis, relaxation timescales of ∼100–1000 days were obtained. It was interpreted that stochastic fluctuations in magnetic field intensity are dissipated in the disk, transferring energy from the magnetic field to heat in the plasma and creating thermal fluctuations in the accretion disk. This leads to stochastic fluctuations in the luminosity; however, the disk cannot react to heat content changes on timescales shorter than the thermal timescales and leads to damped (or red-noise) type variability. The disk only forgets the input heat content on timescales longer than the thermal timescales, thereby creating uncorrelated (or white-noise) type variability. Although the optical variability of our quasars is also characterized by random-walk-type processes (β ≥ 2; Table 3), we do not recover the change of slope in the PSDs, pertaining to thermal timescales in the disks of our sources. This could be due to the fact that timescales longer than ≥1000 days are poorly sampled in our data which leads to a large scatter in their estimation in our PSD modeling.