X-ray Variability of AGN Brandon C. Kelly, Małgorzata Sobolewska, Aneta Siemiginowska ApJ, 2011, 730, 52 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Quasar (Active Galactic Nuclei) X-ray Emission 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

AGN X-ray Variability is Aperiodic XMM Lightcurve for MRK 766 Vaughan & Fabian (2003) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

What do the random fluctuations tell us? Characteristic time scales of the fluctuations correspond to different physical mechanisms Fluctuations may probe how the accretion flow `responds’ to a perturbation Unable to do controlled perturbations, but turbulence (e.g., MHD effects) provides a constant source of chaotic perturbations May be the only observational way to probe viscosity Provides a test of GBH/SMBH connection 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Astrostatistics Group, bckelly@cfa.harvard.edu Accretion flow solutions expect simple scaling of time scale with mass and accretion rate AGN GBHs Predicted time scale vs. observed for GBHs and SMBHs (McHardy et al. 2006) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Some Example X-ray PSDs of AGN Observed PSDs are very information poor, need a better statistical technique! Updated AKN 564 PSD Markowitz et al. 2003, ApJ, 593, 96 McHardy et al., MNRAS, 2007, 382, 985 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Inadequacy of Common Methods Time Series simulated from an Autoregressive process Periodogram and SF provide poor info on variability 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Astrostatistics Group, bckelly@cfa.harvard.edu A Different Approach: Use a stochastic, generative model with the right PSD The Ornstein-Uhlenbeck (OU, autoregressive) Process, X(t) Continuous form: ω0: Characteristic angular frequency μ: Mean of X(t) σ: Amplitude of driving noise dW(t): A white noise process with unit variance Discrete form: α=exp(-ω0) ε1, … , εi : A series of standard Gaussian random variables Kelly et al. (2009, ApJ, 698, 895) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

The PSD of the OU process is a Lorentzian Flat, White Noise PSD ~ 1/ω2 Red Noise ω0 ‘Characteristic’ time scale: τ=1/ω0 Note that f = ω/2π 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

OU Process describes well the optical lightcurves of AGN Results from Kelly et al. (2009) confirmed by Kozlowski et al.(2010), and by MacLeod et al.(2010) OU process has been used a model for: Variability selection of quasars (Kozlowski et al. 2010, Butler & Bloom 2010) Reverberation mapping (Zu et al. 2010) Probably does not capture the flaring seen in sub-mm lightcurves of blazars (Strom et al., in prep) Kelly et al. (2009, ApJ, 698, 895) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

But what about X-ray lightcurves? Use a mixture of OU processes: ω1 ωM For both the OU process and mixed OU process, the likelihood function can be derived using standard techniques 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Does the Mixed OU process have any physical interpretation? Solution to the stochastic diffusion equation in a bounded medium: y(x,t) ~ Surface Density L(t) x = r1/2 See also work by Titarchuk et al. (2007) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Solution of Stochastic Diffusion Equation (Chow 2007) Denote the eigenfunctions of the diffusion operator as ek(x) and the eigenvalues as ωk Solution has the form Suppose we can express the spatial covariances of driving noise as In addition, random field W(x,t) can be expressed as {wk(t)} is a sequence of brownian motions 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Astrostatistics Group, bckelly@cfa.harvard.edu Solution (Continued) We then have the set of stochastic ODEs: This has the solution Solution is a mixture of OU processes 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Astrophysical interpretation Characteristic frequencies are the eigenvalues of the diffusion operator Mixing weights are a combination of the eigenfunctions of the diffusion operator and the projections of the spatial covariance matrix of W(x,t) onto the space spanned by the eigenfunctions Drift time scale at boundary edge Drift time scale across characteristic spatial scale of W(x,t) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

The likelihood function Mixed OU process has the state space representation: Can use Kalman recursions to derive likelihood function, efficiently calculate it y(t): Observed lightcurve at time t c: Vector of mixing weights x(t): Vector of independent OU processes at time t ε(t): Measurement errors 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Application to AGN X-ray lightcurves Characterizes the ~ 10 local Seyfert galaxies with the best X-ray lightcurves well 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Estimating Characteristic Timescales, other variability parameters Based on an MCMC sampler, available from B. Kelly 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Can also get flexible estimates of PSD MCG-6-30-15 AKN 564 Green: Best fit flexible PSD Red: Best fit assuming a bending power-law Black: Random realizations of the PSD from its probability distribution PSDs are more ‘wiggly’ than simple bending power-laws, similar to GBHs 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Trends with black hole mass X-ray Optical For optical, see Kelly et al. (2009), Collier & Peterson (2001), McHardy et al. (2007), Zhou et al. (2010), And MacLeod et al. (2010) 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Astrostatistics Group, bckelly@cfa.harvard.edu Summary X-ray Variability of AGN is well-characterized by a mixture of Ornstein-Uhlenbeck processes Enables fitting of power spectra without Fourier transforms Characteristic time scale associated with high-frequency break correlates well with MBH Rate at which variability power is injection into the lightcurve tightly anti-correlated with MBH May provide the most precise ‘cheap’ mass estimate 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu

Directions for Future Work Extend method to work with time series of photon counts, Poisson likelihood Extend methodology for analyzing multivariate lightcurves, more efficient and powerful than cross-correlation functions Add in higher order terms to the stochastic ODEs to model more complicated PSDs 5/31/11 Astrostatistics Group, bckelly@cfa.harvard.edu