X-Ray Variability Coherence: How to Compute It, What It Means, and How It Constrains Models of GX 339−4 and Cygnus X-1

Vaughan, Brian; Nowak, Michael A.

doi:10.1086/310430

Cited by 246 publications

(343 citation statements)

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“…The n 2 term is a bias term, which arises because the Poisson noise-level contributes to the modulussquared of the cross-spectrum 4 . Formally, the coherence indicates the fraction of variance in both bands which can be predicted via a linear transformation between the two light curves (Vaughan & Nowak 1997). A reduction in coherence can also occur due to a non-linear transformation (see the discussion in Vaughan & Nowak 1997;Nowak et al 1999).…”

Section: The Coherence and Errorsmentioning

confidence: 99%

“…the coherence attributable to the source itself, correcting for observational noise, the noise-levels should be subtracted from the PSDs in the denominator of Equation 11. Errors on the intrinsic coherence can be estimated using the approach outlined in Vaughan & Nowak (1997), but it is important to bear in mind that the errors on intrinsic coherence are difficult to correctly determine for low variability S/N . This problem does not affect the lag-error estimation however, since this is based on the raw coherence.…”

Section: The Coherence and Errorsmentioning

confidence: 99%

“…8. If desired, obtain the intrinsic coherence, following the prescription in Vaughan & Nowak (1997).…”

Section: Frequency-dependent Spectral-timing Productsmentioning

confidence: 99%

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X-ray reverberation around accreting black holes

Uttley

Cackett

Fabian³

et al. 2014

Astron Astrophys Rev

438

606

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Luminous accreting stellar mass and supermassive black holes produce power-law continuum X-ray emission from a compact central corona. Reverberation time lags occur due to light travel time-delays between changes in the direct coronal emission and corresponding variations in its reflection from the accretion flow. Reverberation is detectable using light curves made in different X-ray energy bands, since the direct and reflected components have different spectral shapes. Larger, lower frequency, lags are also seen and are identified with propagation of fluctuations through the accretion flow and associated corona. We review the evidence for X-ray reverberation in active galactic nuclei and black hole X-ray binaries, showing how it can be best measured and how it may be modelled. The timescales and energy-dependence of the high frequency reverberation lags show that much of the signal is originating from very close to the black hole in some objects, within a few gravitational radii of the event horizon. We consider how these signals can be studied in the future to carry out X-ray reverberation mapping of the regions closest to black holes.

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Section: The Coherence and Errorsmentioning

confidence: 99%

Section: The Coherence and Errorsmentioning

confidence: 99%

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X-ray reverberation around accreting black holes

Uttley

Cackett

Fabian³

et al. 2014

Astron Astrophys Rev

438

606

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show abstract

“…The coherence function, γ 2 ( f i ), a derivative of the norm of the cross spectrum, measures the Fourier-frequency dependent degree of linear correlation between two time series, in this case the light curves (Vaughan & Nowak 1997). An intuitive geometrical interpretation of γ 2 ( f i ) as a length of a vector sum in the complex plane can be found in Nowak et al (1999a).…”

Section: Calculation Of X-ray Timing Quantitiesmentioning

confidence: 99%

Long term variability of Cygnus X-1

Grinberg

Pottschmidt

Böck

et al. 2014

A&A

116

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We present the most extensive analysis of Fourier-based X-ray timing properties of the black hole binary Cygnus X-1 to date, based on 12 years of bi-weekly monitoring with RXTE from 1999 to 2011. Our aim is a comprehensive study of timing behavior across all spectral states, including the elusive transitions and extreme hard and soft states. We discuss the dependence of the timing properties on spectral shape and photon energy, and study correlations between Fourier-frequency dependent coherence and time lags with features in the power spectra. Our main results follow. (a) The fractional rms in the 0.125-256 Hz range in different spectral states shows complex behavior that depends on the energy range considered. It reaches its maximum not in the hard state, but in the soft state in the Comptonized tail above 10 keV. (b) The shape of power spectra in hard and intermediate states and the normalization in the soft state are strongly energy-dependent in the 2.1-15 keV range. This emphasizes the need for an energy-dependent treatment of power spectra and a careful consideration of energy-and mass-scaling when comparing the variability of different source types, e.g., black hole binaries and AGN. PSDs during extremely hard and extremely soft states can be easily confused for energies above ∼5 keV in the 0.125-256 Hz range. (c) The coherence between energy bands drops during transitions from the intermediate into the soft state but recovers in the soft state. (d) The time lag spectra in soft and intermediate states show distinct features at frequencies related to the frequencies of the main variability components seen in the power spectra and show the same shift to higher frequencies as the source softens. Our results constitute a template for other sources and for physical models for the origin of the X-ray variability. In particular, we discuss how the timing properties of Cyg X-1 can be used to assess the evolution of variability with spectral shape in other black hole binaries. Our results suggest that none of the available theoretical models can explain the full complexity of X-ray timing behavior of Cyg X-1, although several ansatzes with different physical assumptions are promising.

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“…The variance of the coherence estimator is given by the asymptotic formula (P81; Vaughan & Nowak 1997;Nowak et al 1999;Bendat & Piersol 2011) Var[γ 2…”

Section: Smoothed/averaged Cross-spectral Estimatorsmentioning

confidence: 99%

Statistical properties of Fourier-based time-lag estimates

Epitropakis

Papadakis

2016

A&A

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Context. The study of X-ray time-lag spectra in active galactic nuclei (AGN) is currently an active research area, since it has the potential to illuminate the physics and geometry of the innermost region (i.e. close to the putative super-massive black hole) in these objects. To obtain reliable information from these studies, the statistical properties of time-lags estimated from data must be known as accurately as possible. Aims. We investigated the statistical properties of Fourier-based time-lag estimates (i.e. based on the cross-periodogram), using evenly sampled time series with no missing points. Our aim is to provide practical "guidelines" on estimating time-lags that are minimally biased (i.e. whose mean is close to their intrinsic value) and have known errors. Methods. Our investigation is based on both analytical work and extensive numerical simulations. The latter consisted of generating artificial time series with various signal-to-noise ratios and sampling patterns/durations similar to those offered by AGN observations with present and past X-ray satellites. We also considered a range of different model time-lag spectra commonly assumed in X-ray analyses of compact accreting systems. Results. Discrete sampling, binning and finite light curve duration cause the mean of the time-lag estimates to have a smaller magnitude than their intrinsic values. Smoothing (i.e. binning over consecutive frequencies) of the cross-periodogram can add extra bias at low frequencies. The use of light curves with low signal-to-noise ratio reduces the intrinsic coherence, and can introduce a bias to the sample coherence, time-lag estimates, and their predicted error. Conclusions. Our results have direct implications for X-ray time-lag studies in AGN, but can also be applied to similar studies in other research fields. We find that: a) time-lags should be estimated at frequencies lower than ≈1/2 the Nyquist frequency to minimise the effects of discrete binning of the observed time series; b) smoothing of the cross-periodogram should be avoided, as this may introduce significant bias to the time-lag estimates, which can be taken into account by assuming a model cross-spectrum (and not just a model time-lag spectrum); c) time-lags should be estimated by dividing observed time series into a number, say m, of shorter data segments and averaging the resulting cross-periodograms; d) if the data segments have a duration 20 ks, the time-lag bias is 15% of its intrinsic value for the model cross-spectra and power-spectra considered in this work. This bias should be estimated in practise (by considering possible intrinsic cross-spectra that may be applicable to the time-lag spectra at hand) to assess the reliability of any time-lag analysis; e) the effects of experimental noise can be minimised by only estimating time-lags in the frequency range where the sample coherence is larger than 1.2/(1+0.2m). In this range, the amplitude of noise variations caused by measurement errors is smaller than the amplitude of the signal's int...

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X-Ray Variability Coherence: How to Compute It, What It Means, and How It Constrains Models of GX 339−4 and Cygnus X-1

Cited by 246 publications

References 13 publications

X-ray reverberation around accreting black holes

X-ray reverberation around accreting black holes

Long term variability of Cygnus X-1

Statistical properties of Fourier-based time-lag estimates

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