X-ray Variability of AGN Brandon C. Kelly, Małgorzata Sobolewska, Aneta Siemiginowska ApJ, 2011,...
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Transcript of 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
Astrostatistics Group, [email protected]/31/11
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
Astrostatistics Group, [email protected]/31/11
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!
Astrostatistics Group, [email protected]/31/11
Inadequacy of Common MethodsInadequacy of Common Methods
Astrostatistics Group, [email protected]
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,
[email protected]/31/11
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)
5/31/11 Astrostatistics Group, [email protected]
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
Astrostatistics Group, [email protected]/31/11
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)
5/31/11 Astrostatistics Group, [email protected]
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
Astrostatistics Group, [email protected]
€
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
Astrostatistics Group, [email protected]
€
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)
Astrostatistics Group, [email protected]/31/11
The likelihood functionThe likelihood function
• Mixed OU process has the state space representation:
• Can use Kalman recursions to derive likelihood function, efficiently calculate it
Astrostatistics Group, [email protected]
€
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
Astrostatistics Group, [email protected]/31/11
Estimating Characteristic Timescales, other variability parameters
Estimating Characteristic Timescales, other variability parameters
Based on an MCMC sampler, availablefrom B. Kelly
5/31/11 Astrostatistics Group, [email protected]
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
5/31/11 Astrostatistics Group, [email protected]
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,
[email protected]/31/11
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
Astrostatistics Group, [email protected]/31/11
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
5/31/11 Astrostatistics Group, [email protected]