A Stochastic Model for the Luminosity Fluctuations of Accreting Black Holes

Abstract

In this work, we have developed a new stochastic model for the fluctuations in light curves of accreting black holes. The model is based on a linear combination of stochastic processes and is also the solution to the linear diffusion equation perturbed by a spatially correlated noise field. This allows flexible modeling of the power spectral density (PSD), and we derive the likelihood function for the process, enabling one to estimate the parameters of the process, including break frequencies in the PSD. Our statistical technique is computationally efficient, unbiased by aliasing and red noise leak, and fully accounts for irregular sampling and measurement errors. We show that our stochastic model provides a good approximation to the X-ray light curves of galactic black holes, and the optical and X-ray light curves of active galactic nuclei (AGNs). We use the estimated timescales of our stochastic model to recover the correlation between characteristic timescale of the high-frequency X-ray fluctuations and black hole mass for AGNs, including two new "detections" of the timescale for Fairall 9 and NGC 5548. We find a tight anti-correlation between the black hole mass and the amplitude of the driving noise field, which is proportional to the amplitude of the high-frequency X-ray PSD, and we estimate that this parameter gives black hole mass estimates to within ∼0.2 dex precision, potentially the most accurate method for AGNs yet. We also find evidence that ≈13% of AGN optical PSDs fall off flatter than 1/f 2 and, similar to previous work, find that the optical fluctuations are more suppressed on short timescales compared to the X-rays, but are larger on long timescales, suggesting that the optical fluctuations are not solely due to reprocessing of X-rays.

X-ray variability of 104 active galactic nuclei

“…combinations of these datasets. The bend frequency estimate reported in the present paper is a factor ∼6 lower than that reported by McHardy et al (2007), but consistent with that from Kelly et al (2011). However, as discussed by McHardy et al (2007) the PSD of this source may be more complicated than the simple bending power law model assumes.…”

“…• Mrk 766: The PSD of Mrk 766 was studied by Vaughan & Fabian (2003), Markowitz et al (2007) and Kelly et al (2011). The main parameters (bend frequency and high frequency slope) are broadly compatible with our findings.…”

“…Kelly et al (2011) analysed the RXTE and XMM-Newton data using a different method for power spectrum estimation, and obtained a bend frequency of ∼3.7 × 10 −5 Hz. Some of the discrepancy between these results and ours may be due to the different energy bands and responses used in the different the analyses, i.e.…”

Long-term variability of AGN at hard X-rays

“…Due to the rather long time scales probed by BAT, the anticorrelations of variability with luminosity (Barr & Mushotzky 1986;Green et al 1993;Papadakis 2004) and with black hole mass (Papadakis & McHardy 1995;Zhou et al 2010;Kelly et al 2011Kelly et al , 2013Ponti et al 2012) often observed below 10 keV on short time scales are not detected in our hard X-ray study (Figs. 3 and 6).…”

The long-term millimeter activity of active galactic nuclei

“…This is a problem of importance not only in astrophysics but in many fields of physics, and various approaches towards a general explanation have been proposed (e.g., Press 1978;Pang et al 1995;Makse et al 1996). More recently, Kaulakys & Alaburda (2009) and Kelly et al (2011) proposed time series models based on superpositions of exponentially decaying outbursts occuring at random times and with random amplitudes. Both studies are able to reproduce the occurence of powerlaw power spectra.…”