x-ray variability of agn brandon c. kelly, małgorzata sobolewska, aneta siemiginowska apj, 2011,...

X-ray Variability of AGNX-ray Variability of AGN

Brandon C. Kelly, Małgorzata Sobolewska, Aneta Siemiginowska

ApJ, 2011, 730, 52

Astrostatistics Group, [email protected]/31/11

Quasar (Active Galactic Nuclei)Quasar (Active Galactic Nuclei)

5/31/11 Astrostatistics Group, [email protected]

X-ray Emission

AGN X-ray Variability is AperiodicAGN X-ray Variability is Aperiodic

Astrostatistics Group, [email protected]

XMM Lightcurve for MRK 766

Vaughan & Fabian (2003)

5/31/11

What do the random fluctuations tell us?

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


Accretion flow solutions expect simple scaling of time scale with mass and accretion rate

Accretion flow solutions expect simple scaling of time scale with mass and accretion rate

Predicted time scale vs. observed for GBHs and SMBHs (McHardy et al. 2006)Predicted time scale vs. observed for GBHs and SMBHs (McHardy et al. 2006)

GBHs

AGN


Some Example X-ray PSDs of AGNSome Example X-ray PSDs of AGN

Markowitz et al. 2003, ApJ, 593, 96

McHardy et al., MNRAS, 2007, 382, 985

Updated AKN 564 PSD

Observed PSDs are very information poor, need a better statistical technique!


Inadequacy of Common MethodsInadequacy of Common Methods


Time Series simulated from anAutoregressive process

Periodogram and SF providepoor info on variability

5/31/11

A Different Approach: Use a stochastic, generative model with the

right PSD

A Different Approach: Use a stochastic, generative model with the

right PSDThe Ornstein-Uhlenbeck (OU, autoregressive) Process, X(t)The Ornstein-Uhlenbeck (OU, autoregressive) Process, X(t)

Continuous form:

Discrete form:

Kelly et al. (2009, ApJ, 698, 895)

€

dX(t) = −ω0(X(t) − μ)dt +σdW (t)ω0: Characteristic angular frequencyμ: Mean of X(t)σ: Amplitude of driving noisedW(t): A white noise process with unit variance

€

x i = μ +α (x i−1 − μ) + sε iα=exp(-ω0)ε1, … , εi : A series of standard Gaussian random variables Astrostatistics Group,

[email protected]/31/11

The PSD of the OU process is a Lorentzian

The PSD of the OU process is a Lorentzian

€

PSD(ω) =σ 2

2π

1

ω 2 +ω02

ω0

Flat, White Noise

PSD ~ 1/ω2

Red Noise

‘Characteristic’ time scale:τ=1/ω0

Note that f = ω/2πAstrostatistics Group,


OU Process describes well the optical lightcurves of AGN

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)


But what about X-ray lightcurves?But what about X-ray lightcurves?Use a mixture of OU processes:Use a mixture of OU processes:

€

Y (t) = μ + c jX j (t;ω j ,σ )j=1

M

∑

PSD(ω) =c j

2σ 2

2π

1

ω j2 +ω 2

j=1

M

∑

ω1

ωM

For both the OU process and mixed OU process, the likelihoodfunction can be derived usingstandard techniques


Does the Mixed OU process have any physical interpretation?

Does the Mixed OU process have any physical interpretation?

• Solution to the stochastic diffusion equation in a bounded medium:

€

∂∂ty(x, t) = a

∂ 2

∂x 2 y(x, t) +∂

∂tW (x, t)

y(x,0) = y0(x)

y(0, t) = 0 =∂

∂ty(xmax , t)

L(t) ∝∂

∂xy(0, t)

See also work by Titarchuk et al. (2007)x = r1/2

L(t)


y(x,t) ~ Surface Density

Solution of Stochastic Diffusion Equation (Chow 2007)

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


€

y(x, t) = yk (t)ek (t)k=1

∞

∑

€

σ(x,y) =Cov[W (x, t),W (y, t)] = σ k2ek (x)ek (y),

k=1

∞

∑

σ k2 = σ (x,y)ek (x)ek (y)

0

xmax

∫ dxdy0

xmax

∫

€

W (x, t) = σ kek (x)wk (t)k=1

∞

∑ {wk(t)} is a sequence of brownian motions

5/31/11

Solution (Continued)Solution (Continued)• We then have the set of stochastic ODEs:

• This has the solution

• Solution is a mixture of OU processes


€

dyk (t) = −ω kyk (t)dt +σ kdwk (t), k =1,2,K

yk (0) = y0(x)ek (x)dx0

xmax

∫

€

y(x, t) = yk (0)ek (x)exp(−ω kt) +σ kek (x)XOU (t,ω k )[ ]k=1

∞

∑

L(t) ∝ yk (0)dek (x)

dx x=0

exp(−ω kt) +σ kdek (x)

dx x=0

XOU (t,ω k ) ⎡

⎣ ⎢

⎤

⎦ ⎥

k=1

∞

∑

5/31/11

Astrophysical interpretationAstrophysical 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 boundaryedge

Drift time scale across characteristicspatial scale of W(x,t)


The likelihood functionThe likelihood function

• Mixed OU process has the state space representation:

• Can use Kalman recursions to derive likelihood function, efficiently calculate it


€

y(t) = cTx(t) +ε(t), ε(t) ~ N(0,v(t))

x(t) =A(Δt)x(t − Δt) +w(t), w(t) ~ N(0,diag(σ 2))

y(t): Observed lightcurve at time tc: Vector of mixing weightsx(t): Vector of independent OU processes at time tε(t): Measurement errors

5/31/11

Application to AGN X-ray lightcurvesApplication to AGN X-ray lightcurves

Characterizes the ~ 10 local Seyfert galaxies with the best X-ray lightcurves well


Estimating Characteristic Timescales, other variability parameters

Estimating Characteristic Timescales, other variability parameters

Based on an MCMC sampler, availablefrom B. Kelly


Can also get flexible estimates of PSDCan also get flexible estimates of PSD

MCG-6-30-15

AKN 564

Green: Best fit flexible PSDRed: Best fit assuming a bending power-lawBlack: Random realizations of the PSD from its probability distribution PSDs are more ‘wiggly’ than simple

bending power-laws, similar to GBHs


Trends with black hole massTrends with black hole massX-

ray

Opti

cal

For optical, see Kelly et al. (2009), Collier & Peterson (2001), McHardy et al. (2007), Zhou et al. (2010),And MacLeod et al. (2010) Astrostatistics Group,


SummarySummary

• 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


Directions for Future WorkDirections 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


x-ray variability of agn brandon c. kelly, małgorzata sobolewska, aneta siemiginowska apj, 2011,...

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