Free Access
Volume 584, December 2015
Article Number A106
Number of page(s) 22
Section Extragalactic astronomy
Published online 01 December 2015

Online material

Appendix A: On the uncertainty of the excess variance

thumbnail Fig. A.1

Excess variances of all PS1 bands calculated from only the even (y-axis) and only the odd (x-axis) light curve points for all variable AGNs of the MDF04 sample. Denoted is the χ2 of together with its expectation value and standard deviation. The black line corresponds to the one-to-one relation.

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In addition to the Poisson noise error (Eq. (8)) used in this work, more uncertainties exist that are related to an excess variance measurement, and they are connected to the stochastic nature of the variations and the sampling pattern of the light curve. These error sources were studied in detail by Allevato et al. (2013). They applied Monte Carlo methods to create 5000 different light curves drawn from a power spectral density (PSD) with logarithmic slopes between −1 and −3 and measured the excess variance of these light curves adopting different sampling patterns (continuous, uniform, sparse). These investigations proved that the excess variance is a biased estimator of the intrinsic (“true”) variance, which itself arises from the underlying physical process related to variability. The bias factor associated with an individual measurement is shown to depend on the PSD logarithmic slope, the sampling pattern, and the signal-to-noise of the light curve, at least as long as S/N< 3. Since the excess variance is defined to measure the integral of the PSD over the temporal frequencies probed by a light curve, the actual value of is affected by the functional form of the PSD. The optical power spectra of AGNs are usually characterized by a “red noise” PSD, i.e. a power law with γ< −1. It is very likely that optical PSDs exhibit a break frequency, separating the low frequency part with γ ~ −1 from the high frequency part with γ ~ −2, which is similar to what was observed in many X-ray variability studies (Lawrence & Papadakis 1993; Edelson & Nandra 1999; Markowitz et al. 2003; McHardy et al. 2004). The actual value of the optical break timescale may strongly depend on the physical parameters of each source, such as the black hole mass and luminosity, and typical values between 10100 days, but even up to ~10 yr have been reported (Collier & Peterson 2001; Kelly et al. 2009). Considering the timescales encompassed by the PS1 3π and MDF light curves (shortest timescale ~1 day, longest timescale ~4 yr), it is therefore unclear whether our measurements predominantly integrate the PSD in the low or high frequency parts. Nevertheless, for both surveys, the light curve sampling pattern is closer to the sparse case than to the continuous or uniform ones. For these reasons the value of the bias factor bsparse for the measurements of this work is expected to lie somewhere between bsparse = 1.2 (for γ ~ −1), bsparse = 1.0 (for γ ~ −1.5), or bsparse = 0.6 (for γ ~ −2) according to Table 2 in Allevato et al. (2013) and is therefore negligible.

To assess the quality of the excess variance measurements for our MDF04 sample and to check whether the assumed error is reasonable according to Eq. (8), we perform a simple test by comparing the values obtained from two different realizations of each light curve by calculating the excess variance once from only the even light curve points and once from only the odd light curve points. The uncertainty corresponding to these two measurements is then contrasted to the individual errors assigned to each variability measurement. We do this by calculating (A.1)with the difference and squared error . The computed χ2(Δ), together with its expectation value and standard deviation , is quoted for all variable AGNs of the MDF04 sample in Fig. A.1. We note that the quality of the measurements is generally high for the gP1, rP1, iP1, and zP1 bands, because most values lie very close to the one to one relation. Since the χ2(Δ) values are very close to the respective expectation value for the rP1, iP1 bands and only deviate by a factor of ~1.4 for the gP1 and zP1 bands, the Poisson noise error estimate of Eq. (8) represents an appropriate measurement uncertainty of the excess variance in our light curves. Only the values of the yP1 band show significantly less accuracy than for the other PS1 bands, which is due to the fact that the yP1 band light curves contain fewer data points.

Appendix B: Method used to select the Case A photometry

thumbnail Fig. B.1

Distribution of ΔTmin for the 40 AGNs of the 3π sample and the 75 AGNs of the MDF04 sample. The distribution of ΔTrandom for one of the ten random realizations of Case C is shown for both samples in the right column.

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To identify the epochs minimizing the temporal distance of the five PS1 band light curves, we employ a combinatoric procedure. For each of our objects we have to consider five different light curves out of set B = { gP1,rP1,iP1,zP1,yP1 }, consisting of Nk magnitude values measured at times tki, with kB and i = 1,2,...,Nk. We start by taking each light curve point i of the band k = gP1 and find the four light curve points jmin with minimal temporal distance to point i, out of the j = 1,2,...,Nl points of all other bands lk. Denoting the temporal distance by Δtki,lj = | tkitlj | gives us four values Δtki,ljmin = min { Δtki,l1,...tki,lNl } for each band l = rP1,iP1,zP1,yP1. Then we compute the sum (B.1)and find its minimum value ΔTgP1,min = min { ΔTgP11,...TgP1NgP1 } out of the NgP1 points. This gives us the minimum total time interval of the different filter observations with respect to the gP1 band light curve points. However, other combinations might exist, leading to a shorter total time interval, by taking the light curve points of another band as reference values while calculating the differences | tkitlj | to the light curve points of the remaining bands. Therefore we perform the same procedure, with the reference band k running through all elements of the set B, by calculating (B.2)for each light curve point i of reference band k. For each reference band k, we then determine the minimal total time interval ΔTk,min = min { ΔTk1,...TkNk }. The set of input photometry with minimum relative temporal distance is finally obtained by selecting those five light curve points that give rise to (B.3)These five magnitude values are stored in the input catalogue for our fitting routines, together with their respective individual uncertainties err(mag).

Owing to the different sampling patterns of the 3π and MDF04 light curves, the minimized values ΔTmin differ a lot for our two samples. As shown in Fig. B.1 each of the 75 AGNs from the MDF04 sample has a value of ΔTmin< 2.5 days. In contrast, the corresponding values for the 40 AGNs of the 3π sample range between ΔTmin = 110 and 500 days, giving a very poor approximation of a snapshot SED. For comparison, Fig. B.1 also displays the histograms of the total time interval ΔTrandom for one of the ten realizations of Case C. The interval ΔTrandom is calculated after Eq. (B.2), taking the randomly chosen gP1 band point i as reference value tki in the individual addends | tkitlj |, with tlj given by the other four randomly chosen light curve points j of the remaining bands lgP1. As intended, the distributions of ΔTrandom encompass much higher values, typically between 500 and 4000 days, than the respective ΔTmin distributions for both the 3π and MDF04 samples.

Appendix C: Catalogues and light curves of variable AGNs

The catalogues of variable AGNs described in Sect. 4.2 are provided at the CDS for every PS1 band and for both the 3π

Table C.1

MDF04 survey catalogue of variable AGNs (gP1 band).

and MDF04 surveys. Part of one of these tables is shown in Table C.1. The tables (ASCII format) list basic information like the identifier number, coordinates, number of light curve points, and light curve median, as well as the variability parameters defined in Sect. 4.1. In addition, the nightly-averaged light curves for these sources, cleaned from outlier measurements as outlined in Sect. 3, are available on request for every PS1 band.

© ESO, 2015

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