Optical variability of quasars: a damped random walk
aa r X i v : . [ a s t r o - ph . H E ] D ec Multiwavelength AGN Surveys and StudiesProceedings IAU Symposium No. 304, 2014A. Mickaelian, F. Aharonian & D. Sanders, eds. c (cid:13) Optical variability of quasars:a damped random walk ˇZeljko Ivezi´c and Chelsea MacLeod Department of Astronomy, University of Washington,Box 351580, Seattle, WA 98195-1580, USAemail: [email protected] Department of Physics, U. S. Naval Academy,572c Holloway Rd, Annapolis, MD 21402, USAemail: [email protected]
Abstract.
A damped random walk is a stochastic process, defined by an exponential covariancematrix that behaves as a random walk for short time scales and asymptotically achieves a finitevariability amplitude at long time scales. Over the last few years, it has been demonstrated,mostly but not exclusively using SDSS data, that a damped random walk model provides asatisfactory statistical description of observed quasar variability in the optical wavelength range,for rest-frame timescales from 5 days to 2000 days. The best-fit characteristic timescale andasymptotic variability amplitude scale with the luminosity, black hole mass, and rest wavelength,and appear independent of redshift. In addition to providing insights into the physics of quasarvariability, the best-fit model parameters can be used to efficiently separate quasars from starsin imaging surveys with adequate long-term multi-epoch data, such as expected from LSST.
Keywords. surveys, galaxies: active, quasars: general, stars: variables, stars: statistics
1. Introduction
Quasars are variable sources with optical amplitudes of several tenths of a magnitudefor time scales longer than a few months (e.g., Hawkins & Veron 1995; Trevese et al.2001; Ivezi´c et al. 2004; Vanden Berk et al. 2004). Sesar et al. (2007) and Butler &Bloom (2011) showed using SDSS Stripe 82 data (a ∼
300 deg equatorial region imagedabout 60 times) that practically all quasars spectroscopically confirmed by SDSS arephotometrically variable.Quantitative statistical description of quasar variability is important both for under-standing the physics of the driving mechanism(s), and for selecting quasars in imagingsurveys. Here we describe recent progress in the analysis of quasar variability whichdemonstrated that a stochastic process called damped random walk (DRW) provides asatisfactory statistical description of quasar variability in the optical wavelength range.
2. Quantitative analysis of quasar variability
Two main methods have been utilized over the last few decades to quantitativelydescribe stochastic quasar variability: a variability structure function analysis and directmodeling of light curves. 2.1.
Structure function approach
The structure function as a function of time lag ∆ t , SF(∆ t ), is equal to the standarddeviation of the distribution of the magnitude difference m ( t ) − m ( t ) evaluated at manydifferent times t and t , such that time lag ∆ t = t − t (and divided by √ t ) = SF ∞ [1 − exp( − ∆ t/τ )] / (2.1)(for illustration see Figure 2 in the contribution by Ivezi´c et al. in these Proceedings). Atsmall time lags, SF(∆ t ) ∝ ∆ t / , and thus a DRW is equivalent to an ordinary randomwalk for ∆ t ≪ τ (the “damped” aspect manifests itself as SF(∆ t ) → SF ∞ for ∆ t ≫ τ ).The structure function is related to the autocorrelation function, which makes a Fourierpair with the power spectral density function (PSD). The PSD for a DRW is given byPSD( f ) = τ SF ∞ πf τ ) . (2.2)Therefore, a DRW is a 1 /f process at high frequencies, just as an ordinary random walk(when SF ∝ (∆ t ) γ , then PSD ∝ /f (1+2 γ ) ). The “damped” nature is seen as a flat PSDat low frequencies ( f ≪ π/τ ). A comparison of light curves drawn from a DRW andtwo other stochastic processes with similar PSDs is shown in Figure 1.2.2. Direct modeling of light curves as a damped random walk
Observed light curves can be used to directly constrain the DRW model parameters, τ and SF ∞ (Kelly, Bechtold & Siemiginowska 2009, hereafter KBS09; Koz lowski et al.2010, MacLeod et al. 2010, 2011, 2012; Zu et al. 2012). Before summarizing the mainresults, we briefly review the statistical properties of a DRW.The CAR(1) process, as it is called in statistics literature, for a time series m ( t ) isdescribed by a stochastic differential equation which includes a damping term that pushes m ( t ) back to its mean (see KBS09). Hence, it is also known as a DRW in astronomicalliterature (another often-used name is the Ornstein–Uhlenbeck process, especially in thecontext of Brownian motion). In analogy with calling a random walk a “drunkard’s walk,”a DRW could be justifiably called a “married drunkard’s walk” – who always comes hometo his or her spouse instead of drifting away.Stochastic light curves can be modeled using the covariance matrix. For a DRW, thecovariance matrix is S ij (∆ t ij ) = σ exp( −| ∆ t ij | /τ ) , (2.3)where ∆ t ij = t i − t j , and σ and τ are model parameters; σ controls the short timescalecovariance (∆ t ij ≪ τ ), which decays exponentially on a timescale given by τ . The cor-responding autocorrelation function is ACF( t ) = exp( − t/τ ). The asymptotic value ofthe structure function, SF ∞ , is equal to 2 σ . A number of other convenient models andparametrizations for the covariance matrix are discussed in Zu et al. (2012).2.3. Tests of a damped random walk model
Both a structure function analysis and the direct modeling of light curves demonstratethat a DRW provides a good description of the optical continuum variability of quasars.For example, the time span of SDSS data from Stripe 82 is sufficiently long to constrain τ for the majority of the ∼ ptical variability of quasars Figure 1.
A comparison of simulated light curves generated using three different power spectraldensity functions (PSD), which are illustrated in the bottom right panel by lines (solid line: topleft panel; dashed line: top right panel; dotted line: bottom left panel). In all three cases, thePSD at short time scales (large frequency f ) is proportional to f − (the PSD for a randomwalk has the same index at all frequencies). The transition time scale is given by τ = 5 . f t = (2 πτ ) − (shown by the vertical dotted line in thebottom right panel). The PSD at long time scales ( f < f t ) follows f α , with α = − α = − . α = 0 (top right, similar to a DRW). The y axis in the bottom rightpanel is in arbitrary units. Observed light curves of quasars are consistent with − < α < α < − f > f − PSD at the shortest timescales found using Keplerdata (Mushotzky et al. 2011). Adapted from MacLeod et al. (2010).
The best-fit values of τ and SF ∞ are correlated with physical parameters, as discussedin the next section.MacLeod et al. (2010) have concluded that the observed light curves of quasars areconsistent with PSD ∝ f α at long timescales, with − < α <
0, while α < − − Hz up to 10 − Hz) is steeper thanthe expected f − behavior.The distribution of magnitude differences drawn from a DRW light curve should beGaussian. The number of points per observed light curve is typically too small to testthis expectation using individual objects. When using an ensemble analysis, the observeddistribution is puzzlingly closer to an exponential (Laplace) distribution than to a Gaus-sian distribution (Ivezi´c et al. 2004; MacLeod et al. 2008). Nevertheless, MacLeod et al.(2012) showed that the exponential distributions seen in the statistics of ensembles of ˇZeljko Ivezi´c et al.quasars naturally result from averaging over quasars that are individually well describedby a Gaussian DRW process.
3. Insights into the physics of quasar variability
The best-fit values of τ and SF ∞ , determined using a DRW model and SDSS Stripe82 light curves, are correlated with physical parameters, such as the luminosity, blackhole mass, and rest-frame wavelength (MacLeod et al. 2010, 2012). Their analysis showsSF ∞ to increase with decreasing luminosity and rest-frame wavelength (as was observedpreviously), and without a correlation with redshift. They found a correlation betweenSF ∞ and black hole mass with a power-law index of 0 . ± .
03, independent of the anti-correlation with luminosity. They also found that τ increases with increasing wavelengthwith a power-law index of 0.17, remains nearly constant with redshift and luminosity,and increases with increasing black hole mass with a power-law index of 0 . ± .
4. Conclusions
The last decade has seen enormous progress in both data availability and the modelingof stochastic quasar variability. The damped random walk model provides a satisfactorystatistical description for practically all the data available at this time. This progress islikely to continue thanks to new post-SDSS massive sky surveys. For example, the LargeSynoptic Survey Telescope (LSST; for a brief overview see Ivezi´c et al. 2008) will extendthe light curve baseline for ∼ ∼
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