Optical spectral index–luminosity relation for the 17 mapped Palomar–Green quasars

Abstract

In this paper, the optical spectra index–luminosity relationship is checked for the well-known 17 individually mapped quasi-stellar objects (QSOs), in order to give one more clearer conclusion on the so far conflicting dependence of the spectral index on the luminosity for an active galactic nucleus (AGN). Unlike the global relationships based on the colour difference (photometry parameters) for samples of AGNs, a more reliable relationship is determined for the multi-epoch observed individually mapped QSOs with no contamination from the host galaxies, the line variabilities and the very different central properties. The final confirmed results are as follows. (i) No strong dependence of the optical spectral index on the continuum luminosity can be found for all 17 QSOs, besides two objects (PG 0026 and PG 1613) that have some weak trends (with 3σ confidence level) for the relationship. In other words, the common expectation that ‘AGNs get bluer when they get brighter’ is not so common. (ii) There are very different damped intrinsic variability time-scales for the variability modes of the optical spectral index and the continuum emission, through the well-applied damped random walk method for the AGN variability. In other words, there are some different intrinsic mechanisms controlling the variabilities of the optical spectral index and the power-law AGN continuum emission. Therefore, the much weaker dependence of the optical spectral index on the continuum luminosity can be further confirmed.

galaxies: active, galaxies: nuclei, quasars: emission lines

1 INTRODUCTION

Variability is one of the fundamental characteristics of active galactic nuclei (AGNs; Ulrich, Maraschi & Urry 1997). Moreover, AGN variability is a powerful tool for understanding the structures of the central regions of AGNs (Rees 1984; Hawkins 1996, 2000; Kawaguchi et al. 1998; Torricelli-Ciamponi et al. 2000; Peterson et al. 2004; Hopkins et al. 2006; Kelly, Bechtold & Siemiginowska 2009; Breedt et al. 2010; Schmidt et al. 2010; Mushotzky et al. 2011; Zhang 2011, 2013a,b; Zu, Kochanek & Peterson 2011; Zu et al. 2013). However, there is so far no clear conclusion about the nature of the AGN variability, besides several well-known dependences of the AGN variability on other AGN parameters (such as the luminosity, accretion rate, black hole mass, redshift, etc.; Vanden Berk et al. 2004; Wold, Brotherton & Shang 2007; Wilhite et al. 2008). Among the dependences, the dependence of the spectral index (here we mainly consider the continuum slope α, f_λ ∝ λ^α) on the AGN luminosity (the spectral index–luminosity relation) is the one most widely and commonly studied, since the dependence was found for 3CR extragalactic galaxies (Heeschen 1960; Conway, Kellermann & Long 1963; Kellermann, Pauliny-Toth & Williams 1969; Macleod & Doherty 1972). However, the conclusion about the dependence is still uncertain.

In the previous studies, the spectral index–luminosity relationship was commonly checked by samples of AGNs with different redshifts, different luminosities, different black hole masses, different accretion rates, different contamination sources, etc. Therefore, there are still contradictory statements about the relationship. More recently, Schmidt et al. (2012) have reported that AGN colour variability (spectral index) is remarkably uniform, and independent not only of redshift but also of quasar luminosity and black hole mass, after the correction of the effects from the emission lines in the different Sloan Digital Sky Survey (SDSS) bands, using a sample of 9093 quasi-stellar objects (QSOs) in SDSS Stripe82. Zuo et al. (2012) have reported that AGN variability increases as either luminosity or Eddington ratio decreases; however, the relationship between variability and black hole mass is uncertain, as seen from a sample of 7658 QSOs from SDSS Stripe82. Certainly, there are still different results about the spectral index–luminosity relationship in the literature. Sakata et al. (2010) have shown that the spectral shape of AGN continuum emission in the optical region does not systematically change during flux variation, using a study of 11 nearby Seyfert galaxies. Using a study of 13 moderate-luminosity AGNs at z = 0.36, Woo et al. (2007) have shown that after the subtractions of the host galaxy stellar light contributions, the nuclear variability has similar amplitudes in the g and r bands within the errors. Similar results can be found in Walsh et al. (2009): there are no significant colour variations for the 13 nearby AGNs. Meanwhile, we have shown that there is one strong correlation between the spectral index and the dimensionless Eddington accretion rate, using 193 broad-line AGNs with high-quality SDSS spectra (Zhang, Dultzin & Wang 2008). Pu, Bian & Huang (2006) have reported that there was a strong global correlation between the spectral index and the continuum luminosity for the QSOs in Kaspi et al. (2000). Wilhite et al. (2005) have reported that there is a clear dependence of the spectral variability on the wavelength using one sample of about 300 variable QSOs with multi-epoch SDSS spectra. Therefore, there is still no one confirmed conclusion about the spectral index–luminosity relationship.

It is clear that previous studies of the colour variability (spectral index) have mainly been based on photometry parameters of samples of AGNs. Thus, there are apparent effects on the spectral index–luminosity relation from the contributions of the host galaxies and the emission lines, and from the different AGN properties (different black hole masses, redshift, accretion rate, etc.) of the sample objects. Even though some of the above effects can be statistically corrected as discussed in the corresponding body of literature, the sample size should have apparent effects on the final conclusion about the spectral index–luminosity relationship (Tang, Zhang & Hopkins 2007). Therefore, it is necessary and interesting to re-check the spectral index–luminosity relationship using spectroscopic data of the well-known mapped objects. The advantages of using the spectroscopic parameters of the mapped objects are as follows. On one hand, the effects of the different black hole masses can be totally ignored. On the other hand, the effects of the emission lines can be clearly ignored. Moreover, in the paper, we mainly consider the 17 well-known mapped Palomar–Green (PG) QSOs. Thus the effects of the host galaxies can be totally ignored.

The paper is organized as follows. In Section 2, we present our method to determine the spectral index of the mapped PG QSOs. Section 3 gives the spectral index–luminosity relationships for the mapped objects. Finally, a discussion and conclusions are given in Section 4. In this paper, the cosmological parameters H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_Λ = 0.7 and Ω_m = 0.3 have been adopted.

2 THE SPECTRAL INDICES OF THE 17 PG QSOs

In order to totally ignore the effects of the host galaxies and/or the probable effects of the flux calibration, the 17 nearby PG QSOs with public spectra (Kaspi et al. 2000) are mainly considered, among all the reported mapped AGNs (Kaspi et al. 1996, 2000, 2005; Peterson et al. 2004; Bentz et al. 2006, 2009, 2010; Denney et al. 2006, 2009, 2010; Barth et al. 2011). The detailed description of the 17 PG QSOs can be found in Kaspi et al. (2000). Their spectra were observed using the Steward Observatory 2.3-m telescope and the Wise Observatory (WO) 1-m telescope for around 7.5 yr. Each QSO has more than 20 spectroscopic data points, which allow us to find the more reliable spectra index–luminosity relationship for the 17 individual objects without the global effects. The public spectra have been collected from the website: http://wise-obs.tau.ac.il/~shai/PG/. Moreover, the spectra have been binned into 1 Å pixel⁻¹ and are padded from 3000 to 9000 Å. Certainly, in most cases the usable part of the spectrum is the blue part around Hβ (Maoz et al. 1994; Kaspi et al. 2000). Therefore, in this paper, the spectra around Hβ are mainly considered.

Then, based on the multi-epoch observed spectra of the 17 mapped PG QSOs, the spectral index in the optical band (from 4400 to 5500 Å in the rest frame including the optical Fe ii lines) can be well determined by the following power-law function, f_λ ∝ λ^α. Moreover, due to the effects of the optical Fe ii lines, some more complicated model should be applied to determine the AGN power-law continuum, rather than to determine the continuum by the oversimplified wavelength windows (Kaspi et al. 2000; Peterson et al. 2004; Pu et al. 2006; Zhang 2013c). Due to the high noise of the spectra around the Hα, there are no further discussions about the red part of the spectra.

In this paper, the optical Fe ii lines are modelled by the more recent Fe ii template discussed by Kovacevic, Popovic & Dimitrijevic (2010). Then, the power-law function f_λ ∝ λ^α is applied for the AGN continuum component, and multi-Gaussian functions are applied to describe the emission lines around Hβ: two broad Gaussian functions for the broad component of Hβ, one narrow Gaussian function for narrow Hβ, two narrow Gaussian functions for the [O iii] λ4959, 5007 Å doublet and one broad Gaussian function for He ii λ4687 Å. In the procedure, the main objective is to determine the AGN power-law continuum; the main reason for applying two broad Gaussian functions to broad Hβ is to obtain best-fitting results for the emission lines. No further discussions are shown about the multiple components of the broad Hβ. Then, through the Levenberg–Marquardt least-squares minimization technique, the AGN power-law continuum, the optical Fe ii lines and the other emission lines can be well determined. Here, the best-fitted results for the mean spectra rather than for all the observed spectra for each object are shown in Fig. 1. It is clear that our procedure to determine the AGN power-law continuum is efficient. Finally, based on the determined power-law function and the reported redshift for each QSO, the continuum luminosity at 5100 Å and the corresponding spectral index can be well calculated in each epoch.

Figure 1.

The best-fitting results for the mean spectra around Hβ of the 17 mapped PG QSOs. The thin dashed line and the thick solid line represent the observed spectrum and the corresponding best-fitting result, respectively. Then, the corresponding broad Hβ and probable broad He ii line (thick solid lines), narrow lines (thin solid lines), the optical Fe ii components (thin dotted line) and the power-law continuum (thick dot–dashed line) are shown under the mean spectrum.

Before the end of the section, there is one point we should note. Based on the discussions by Maoz et al. (1994) and Kaspi et al. (2000), spectrophotometric calibration has been used to provide the absolute flux calibration for each collected spectrum using the properties of the simultaneously observed comparison standard star. Moreover, the absolute flux calibration has a moderate uncertainty of ∼10 per cent, and the relative flux calibration uncertainty is about ∼3 per cent. Therefore, we do not discuss the flux calibrations any further, but we accept the value of the signal-to-noise ratio (S/N) ∼ 10 for all the collected spectra. Clearly, we can confirm that the relative flux calibrations are accurate enough and have few effects on our determined spectra index.

3 OPTICAL SPECTRAL INDEX–LUMINOSITY RELATIONSHIPS FOR THE PG QSOs

Based on the well-determined optical spectral indices and the AGN continuum luminosities at 5100 Å, the spectral index–luminosity relationship can be tested for the 17 individually mapped PG QSOs. Before proceeding further, there is one point we should note. The continuum luminosity used in the following is not the direct measured value from the observed spectrum, but the value listed by Kaspi et al. (2000), with the necessary corrections having been done for the absolute continuum flux intercalibrations at 5100 Å. Moreover, the corresponding uncertainty for the continuum flux in each epoch is the value listed in Kaspi et al. (2000).

Fig. 2 shows the optical spectral index–luminosity relationships for the 17 individual PG QSOs. The corresponding Spearman rank correlation coefficient and the number of the available observed spectra are marked for each QSO in the figure. It is clear that there are no apparently strong correlations between the optical spectral index and the continuum luminosity for the mapped QSOs, besides weak trends for the relationship (coefficient still larger than −0.5 but the two-sided significance level p_null much smaller than 0.01) for the two PG QSOs of PG 0026 and PG 1613. Besides the optical spectral index–luminosity relationship shown for the 17 mapped PG QSOs, the spectra with the lowest and the highest continuum luminosities at 5100 Å for each QSO are shown in Fig. 3, in order to more clearly show spectral variabilities. From the results in Figs 2 and 3, the basic and clear results are that there are apparent continuum variabilities but much weaker corresponding spectral index variabilities.

Figure 2.

The optical spectral index–luminosity relationships for the 17 mapped PG QSOs. In the top left-hand corner of each plot, the following information is shown in turn: the number of available spectra (first line), the corresponding Spearman rank correlation coefficient and the parameter P_null (second line), the parameters A and B (α = A + B × λL_λ) determined by the LTS method (third line) and the χ²/dof (fourth line) for the LTS best-fitting results. In each plot, the solid line represents the LTS best-fitting results, and the dashed and dotted lines are for the corresponding 68 and 99 per cent confidence bands for the LTS best-fitting results. The solid circles denote the outliers determined by the LTS method.

Figure 3.

The spectra with the lowest and the highest continuum luminosity at 5100 Å for each PG QSO. The thin and thick solid lines are for the observed spectrum and the corresponding best-fitting result, respectively. The solid line under the observed spectrum denotes the determined power-law AGN continuum.

Finally, two methods are applied to further check the relationships for the 17 individual QSOs: the commonly used bootstrap method to estimate the confidence levels of the coefficients and the more recent least trimmed squares (LTS) robust fit method (Cappellari et al. 2013) to determine the best-fitting results for the relationships.

The commonly used bootstrap method is applied as follows. Before calculating the Spearman rank correlation coefficient, the values of the optical spectral index and the continuum luminosity are recalculated and randomly determined within the range from P − P_err to P + P_err, where P and P_err represent the parameter value (the optical spectral index, continuum luminosity) and the corresponding uncertainty. Then, the Spearman rank coefficients are recalculated by the new values of the optical spectral index and the continuum luminosity. The procedure is repeated 5000 times for each QSO. The probability distributions of the recalculated spearman rank correlation coefficients are shown in Fig. 4 for the 17 mapped PG QSOs. It is clear that even after the considerations of the parameter uncertainties, the previous results cannot be changed: there is no apparently strong spectral index–luminosity relationship for the QSOs (the coefficient always larger than −0.5 and smaller than 0.5).

Figure 4.

The distributions of the Spearman rank correlation coefficients for the relationships between the optical spectral index and the continuum luminosity, with the considerations of the parameter uncertainties.

The more recent LTS robust fit method is applied to find the best fits for the correlations (Cappellari et al. 2013), α = A + B × λL_λ, under the considerations of the probable intrinsic scatters of the data points with uncertainties in both coordinates. The best-fitting LTS results and the corresponding 68 and 99 per cent confidence bands for the best-fitting results are shown in Fig. 2. It is clear that besides PG 0026 and PG 1613, none of the QSOs has a slope, α, three times larger than the corresponding uncertainty of the slope. Actually, through the LTS method, some outliers (having deviations larger than 2.6σ from the expected relation; solid circles in Fig. 2) should be first ruled out, before giving the final best-fitting results, which should lead to some more apparent linear fits. The outliers are perhaps due to the bad spectra quality. Meanwhile, if the outliers were also considered, the slope should be more closer to zero.

Therefore, none of the 17 QSOs has an apparent optical spectral index–luminosity relationship with 5σ confidence levels, and only two QSOs (PG 0026 and PG 0052) have a probable and weak optical spectral index–luminosity relationship with 3σ confidence levels.

4 DISCUSSION AND CONCLUSIONS

Although there is still no confirmed conclusion about the nature of the AGN variabilities, the more recent proposed damped random walk (DRW) model (one special stochastic model) has been successfully applied to describe the AGN variabilities (Bauer et al. 2009; Kelly et al. 2009; Kozlowski et al. 2010; MacLeod et al. 2010, 2012; Schmidt et al. 2010, 2012; Meusinger, Hinze & de Hoon 2011; Bailer-Jones 2012; Andrae, Kim & Bailer-Jones 2013; Zhang 2013b; Zu et al. 2013). The basic idea of the DRW model is that the variability s(t) has one simple exponential covariance between two different epochs t_i and t_j, 〈s(t_i)s(t_j)〉 = σ² × exp( − |t_i − t_j|/τ₀). Then, through the two DRW parameters (i.e. the damped intrinsic variability time-scale τ₀ and the damped intrinsic variability amplitude σ), the AGN variabilities in both observed and unobserved epochs can be well reproduced. Moreover, the determined damped intrinsic variability time-scale τ₀ can be used to understand the origination (or the principal dependent AGN parameter) of AGN variability, as compared with the theoretical characteristic time-scales (Edelson & Nandra 1999). Therefore, the variability properties of the optical spectral index and the continuum emission are checked: very different (similar) damped intrinsic variability time-scales should lead to weak (strong) dependence of the optical spectral index on the continuum luminosity.

Based on the well-determined spectral index and the continuum emission in each epoch, the time-dependent variabilities of the optical spectral index and the continuum emission for each PG QSO can be well analysed by the well-applied DRW method. Here, we apply the DRW method discussed in Zu et al. (2011, 2013) to describe the variabilities. In the DRW method, through the Markov Chain Monte Carlo (MCMC) analysis with the uniform logarithmic priors of the DRW parameters of τ₀ and σ covering every possible corner of the parameter space (0 < τ₀ < 1e + 5 and 0 < σ < 1e + 2), the posterior distributions of the DRW parameters can be well determined and provide the final accepted parameters and the corresponding statistical confidence limits. Then, based on the posterior distributions, the exponential covariance for variability is applied to produce the corresponding variability at any epoch, i.e. the mean DRW fit and the corresponding 1σ variance. Moreover, when the DRW method is applied, one Gaussian white noise with zero mean and unit standard deviation is accepted as the model measurement noise. Fig. 5 shows the mean DRW fits for the variabilities of the optical spectral index and the continuum emission at 5100 Å and Fig. 6 shows the posterior distributions of the damped intrinsic variability time-scales. Before further discussions about the damped intrinsic variability time-scales, we find that the mean DRW fits look bad for the variations of the spectral index in several cases (such as for PG 0052 and PG 0844). We think that the bad mean DRW fits are perhaps not due to the poor DRW parameters, but more seriously due to the tiny variations of the optical spectral index and the bad time gaps. However, although the DRW parameters were poor, the observational variability modes in Fig. 5 could prove that there are different varying modes for the spectral index and the continuum emission for the several bad cases: that is, there are larger variability amplitudes for the continuum emission, but smaller amplitudes for the optical spectral index, and very different variation trends for the optical spectral index and the continuum emission.

Figure 5.

The mean DRW fits for the variabilities of the optical spectral index and the continuum emission for the 17 mapped PG QSOs using the DRW method (Zu et al. 2011, 2013) with the DRW parameters determined by the MCMC analysis. The circles denote the observational data points, and the solid line and the two dotted lines represent the determined mean DRW fit and the corresponding 1σ variance, respectively. The left-hand panels show the results for the optical spectral index, and the right-hand panels are for the continuum emission.

Figure 5

– continued

Figure 6.

The posterior distributions of the DRW determined damped intrinsic variability time-scales for the variabilities of the optical spectral index and the continuum emission, using the MCMC analysis. The solid line represents the distribution for the optical spectral index variabilities, and the dotted line is for the continuum variability.

It is clear that the varying modes are very different for the optical spectral index and for the continuum emission: the damped intrinsic variability time-scales for the optical spectral index are commonly much smaller than those for the continuum emission (besides the results for PG 2130). This strongly indicates that there are very different intrinsic mechanisms controlling the variabilities of the optical spectral index and the continuum emission. Thus, we cannot expect one strong dependence of the optical spectral index on the continuum luminosity. Therefore, it is clear that except for the global effects of the black hole masses, the host galaxy contamination, the narrow-line contamination, etc., no strong dependence of the optical spectral index on the continuum luminosity can be confirmed for the 17 mapped individual PG QSOs. In other words, there is no intrinsic dependence of the optical spectral index on the continuum luminosity.

Finally, there are two points we should note. On one hand, we should note that Pu et al. (2006) have discussed the optical spectral index–luminosity relationships for the 17 PG QSOs, and reported that most of the 17 objects show apparent correlations between the spectral slope and the continuum flux (five of them have much stronger correlations with coefficients larger than 0.5), which is in contrast to our results. The different results from Pu et al. (2006) are mainly due to the following two main reasons. The first main reason is the process for the continuum flux calibration, because of different instruments used for the spectra of the PG QSOs. In Pu et al. (2006), no further process is considered for the flux calibration; the values direct from the observed spectra were used. The second main reason is the effects of the optical Fe ii lines and the probable broad He ii λ4687 Å line on the determined spectral slope (Bian et al. 2010; Zhang 2013c). Thus, very different results from the results in Pu et al. (2006) can be found. On the other hand, in the paper, we first show the direct evidence for the very different intrinsic mechanisms for the variabilities of the optical spectral index and the continuum emission by the well-applied DRW method. Therefore, no strong relationship between the optical spectral index and the continuum luminosity can be confirmed for the individual AGNs. The very different damped intrinsic variability time-scales are enough to support our final results: the common expectation that ‘AGNs get bluer when they get brighter’ is not so common.

The final conclusions are as follows.

• In order to ignore the global effects of the host stellar lights, the emission lines, the central region parameters on the relationship between the optical spectral index and the continuum luminosity, we carefully analyse the multi-epoch spectra of the 17 well-known individually mapped PG QSOs from Kaspi et al. (2000). We have found that there is no confirmed strong dependence of the optical spectral index on the continuum luminosity, even after considering the parameter uncertainties.

• The well-accepted DRW method is applied for the variabilities of the optical spectral index and the continuum emission for the 17 PG QSOs; the damped intrinsic variability time-scales are very different for the optical spectral index and the continuum emission, which strongly indicates there are very different mechanisms controlling the optical spectral index variability and the continuum emission variability.

Z-XG is very grateful to C. A. L. Bailer-Jones from MPIA for providing constructive comments and suggestions that have greatly improved our paper. Z-XG gratefully acknowledges the kind support provided by the Chinese grants NSFC-11003043 and NSFC-11178003, and is grateful to Dr S. Kaspi who has conveniently collected the public spectra of the 17 mapped PG QSOs (http://wise-obs.tau.ac.il/~shai/PG/).