Lognormal Flux Distribution Research Articles

Axial Flux Evolution of Small-scale Magnetic Flux Ropes from 0.06 to 10 au

Small-scale magnetic flux ropes (SMFRs) fill much of the solar wind, but their origin and evolution are debated. We apply our recently developed, improved Grad–Shafranov algorithm for the detection and reconstruction of SMFRs to data from Parker Solar Probe, Solar Orbiter, Wind, and Voyager 1 and 2 to detect events from 0.06 to 10 au. We observe that the axial flux density is the same for SMFRs of all sizes at a fixed heliocentric distance but decreases with distance owing to solar wind expansion. Additionally, using the difference in speed between SMFRs, we find that the vast majority of SMFRs will make contact with others at least once during the 100 hr transit to 1 au. Such contact would allow SMFRs to undergo magnetic reconnection, allowing for processes such as merging via the coalescence instability. Furthermore, we observe that the number of SMFRs with higher axial flux increases significantly with distance from the Sun. Axial flux is conserved under solar wind expansion, but the observation can be explained by a model in which SMFRs undergo turbulent evolution by stochastically merging to produce larger SMFRs. This is supported by the observed log-normal axial flux distribution. Lastly, we derive the global number of SMFRs above 1015 Mx near the Sun to investigate whether SMFRs begin their journey as small-scale solar ejections or are continuously generated within the outer corona and solar wind.

GeV Variability Properties of TeV Blazars Detected by Fermi-LAT

Variability is a prominent observational feature of blazars. The high-energy radiation mechanism of jets has always been important but is still unclear. In this work, we performed a detailed analysis using Fermi-LAT data across 15 yr and obtained GeV light-curve information for 78 TeV blazars detected by Fermi. We provided annual GeV fluxes and corresponding spectral indices for the 78 TeV blazars and thorough monthly GeV fluxes for a subsample of 41 bright blazars. Our results suggest a strong correlation between the γ-ray photon index and logLγ for the flat spectrum radio quasars (FSRQs) and high-energy peaked BL Lacs. Fourteen sources in our sample show significant GeV outbursts/flares above the relatively stable, low-flux light curve, with six of them showing a clear sharp peak profile in their 5 day binned light curves. We quantified the variability utilizing the fractional variability parameter F var, and found that the flux of the FSRQs showed significantly stronger variability than that of the BL Lacs. The 41 bright blazars in this work are best fit by a log-normal flux distribution. We checked the spectral behavior and found 11 out of the 14 sources show a bluer-when-brighter trend, suggesting this spectral behavior for these TeV blazars at the GeV band arises from the mechanism in which the synchrotron-self Compton process dominates the GeV emission. Our research offers a systematic analysis of the GeV variability properties of TeV blazars and serves as a helpful resource for further associated blazar studies.

Probing the disc–jet coupling in S4 0954+65, PKS 0903−57, and 4C +01.02 with γ-rays

ABSTRACT We present a comprehensive variability study on three blazars, S4 0954+65, PKS 0903−57, and 4C +01.02, covering a mass range of log(M/M⊙) = 8–9, by using ∼15-yr-long γ-ray LCs from Fermi-LAT. The variability level is characterized by the fractional variability amplitude that is higher for γ-rays compared with optical/UV and X-rays emissions. A power spectral density (PSD) study and damped random walk (DRW) modelling are done to probe the characteristic time-scale. The PSD is fitted with a single power law (PL) and bending PL models and the corresponding success fraction was estimated. In the case of PKS 0903−57, We observed a break in the γ-ray PSD at 256 d, which is comparable to the viscous time-scale in the accretion disc, suggesting a possible disc–jet coupling. The non-thermal damping time-scale from the DRW modelling is compared with the thermal damping time-scale for AGNs including our three sources. Our sources lie on the best fit of the $\mathrm{\tau ^{\rm rest}_{\rm damping}}\!-\!M_{\rm BH}$ plot derived for AGN, suggesting a possible accretion disc–jet connection. If the jet’s variability is linked to the disc’s variability, we expect a log-normal flux distribution, often connected to the accretion disc’s multiplicative processes. Our study observed a double log-normal flux distribution, possibly linked to long- and short-term variability from the accretion disc and the jet. In summary, PSD and DRW modelling results for these three sources combined with blazars and AGNs studied in literature favour a disc–jet coupling scenario. However, more such studies are needed to refine this understanding.

Comprehensive Study of the Blazars from Fermi-LAT LCR: The Log-Normal Flux Distribution and Linear rms–Flux Relation

Fermi-LAT LCR provides continuous and regularly sampled gamma-ray light curves, spanning about 14 yr, for a large sample of blazars. The log-normal flux distribution and linear rms–flux relation of the light curves for a few Fermi blazars have been examined in previous studies. However, the probability that blazars exhibit the log-normal flux distribution and linear rms–flux relation in their gamma-ray light curves has not been systematically explored. In this study, we comprehensively research the distribution of γ-ray flux and the statistical characteristics on a large sample of 1414 variable blazars from the Fermi-LAT LCR catalog, including 572 FSRQs, 477 BL Lacs, and 365 BCUs, and statistically compare their flux distributions with normal and log-normal distributions. The results indicate that the probability of not rejecting log-normal is 42.05% for the large sample, and there is still a 2.05% probability of not rejecting normality, based on the joint of Kolmogorov–Smirnov, Shapiro–Wilk, and Normality tests. We further find that the probability that BL Lacs conform to the log-normal distribution is higher than that of FSRQs. Besides, after removing sources with less than 200 data points from this large sample, a sample of 549 blazars, which is still a large sample compared to the previous studies, was obtained. Based on dividing the light curves into segments every 20 points (or 40 points, or one year), we fitted the linear rms–flux relation of these three different sets and found that the Pearson correlation coefficients are all close to 1 for most blazars. This result indicates a strong linear correlation between the rms and the flux of these 549 blazars. The log-normal distribution and linear rms–flux relation indicate that the variability of the γ-ray flux for most blazars is a non-linear and multiplicative process.

Flaring activity from magnetic reconnection in BL Lacertae

ABSTRACT The evolution of the spectral energy distribution during flares constrains models of particle acceleration in blazar jets. The archetypical blazar BL Lacertae provided a unique opportunity to study spectral variations during an extended strong flaring episode from 2020 to 2021. During its brightest γ-ray state, the observed flux (0.1–300 GeV) reached up to $2.15\, \times \, 10^{-5}\, \rm {ph\, cm^{-2}\, s^{-1}}$, with sub-hour-scale variability. The synchrotron hump extended into the X-ray regime showing a minute-scale flare with an associated peak shift of inverse-Compton hump in γ-rays. In shock acceleration models, a high Doppler factor value >100 is required to explain the observed rapid variability, change of state, and γ-ray peak shift. Assuming particle acceleration in minijets produced by magnetic reconnection during flares, on the other hand, alleviates the constraint on required bulk Doppler factor. In such jet-in-jet models, observed spectral shift to higher energies (towards TeV regime) and simultaneous rapid variability arises from the accidental alignment of a magnetic plasmoid with the direction of the line of sight. We infer a magnetic field of ∼0.6 G in a reconnection region located at the edge of broad-line region (∼0.02 pc). The scenario is further supported by lognormal flux distribution arising from merging of plasmoids in reconnection region.

Characterizing Long-term Optical Variability Properties of γ-Ray-bright Blazars

Abstract Optical observations of a sample of 12 γ-ray-bright blazars from four optical data archives—American Association of Variable Star Observers, Small and Moderate Aperture Research Telescope System, Catalina, and Steward Observatory—are compiled to create densely sampled light curves spanning more than a decade. As a part of the blazar multiwavelength studies, several methods of analysis, e.g., flux distribution and rms–flux relation, are performed on the observations with the aim to compare the results with the similar ones in the γ-ray band presented in Bhatta & Dhital. It is found that, similar to the γ-ray band, blazars display significant variability in the optical band that can be characterized with lognormal flux distribution and a power-law dependence of rms on flux. It could be an indication of a possible inherent linear rms–flux relation, yet the scatter in the data does not allow to rule out other possibilities. When comparing variability properties in the two bands, the blazars in the γ-rays are found to exhibit stronger variability with a steeper possible linear rms–flux relation and a flux distribution that is more skewed toward higher fluxes. The cross-correlation study shows that except for source 3C 273, the overall optical and the γ-ray emission in the sources are highly correlated, suggesting a cospatial existence of the particles responsible for both the optical and γ-ray emission. Moreover, sources S5 0716+714, Mrk 421, Mrk 501, PKS 1424-418, and PKS 2155-304 revealed possible evidence of quasiperiodic oscillations in the optical emission with the characteristic timescales, which are comparable to those in the γ-ray band detected in our previous work.

On the determination of lognormal flux distributions for astrophysical systems

ABSTRACT Determining whether the flux distribution of an astrophysical source is a Gaussian or a lognormal, provides key insight into the nature of its variability. For light curves of moderate length (<103), a useful first analysis is to test the Gaussianity of the flux and logarithm of the flux, by estimating the skewness and applying the Anderson–Darling (AD) method. We perform extensive simulations of light curves with different lengths, variability, Gaussian measurement errors, and power spectrum index β (i.e. P(f) ∝ f−β), to provide a prescription and guidelines for reliable use of these two tests. We present empirical fits for the expected standard deviation of skewness and tabulated AD test critical values for β = 0.5 and 1.0, which differ from the values given in the literature that are for white noise (β = 0). Moreover, we show that for white noise, for most practical situations, these tests are meaningless, since binning in time alters the flux distribution. For β ≳ 1.5, the skewness variance does not decrease with length and hence the tests are not reliable. Thus, such tests can be applied only to systems with β ≳ 0.5 and β ≲ 1.0. As an example of the prescription given in this work, we reconfirm that the Fermi data of the blazar, 3FGL J0730.2−1141, show that its γ-ray flux is consistent with a lognormal distribution and not with a Gaussian one.

Studies in Astronomical Time-series Analysis. VII. An Enquiry Concerning Nonlinearity, the rms–Mean Flux Relation, and Lognormal Flux Distributions

Abstract A broad and widely used class of stationary, linear, additive time-series models can have statistical properties that many authors have asserted imply that the underlying process must be nonlinear, nonstationary, multiplicative, or inconsistent with shot noise. This result is demonstrated with exact and numerical evaluation of the model flux distribution function and dependence of flux standard deviation on mean flux (here and in the literature called the rms–flux relation). These models can (1) exhibit normal, lognormal, or other flux distributions; (2) show linear or slightly nonlinear rms–mean flux dependencies; and (3) match arbitrary second-order statistics of the time-series data. Accordingly, the above assertions cannot be made on the basis of statistical time-series analysis alone. Also discussed are ambiguities in the meaning of terms relevant to this study—linear, stationary, and multiplicative—and functions that can transform observed fluxes to a normal distribution as well as or better than the logarithm.

Fermi Large Area Telescope observations of the fast-dimming Crab Nebula in 60–600 MeV

Context. The Crab pulsar and its nebula are the origin of relativistic electrons which can be observed through their synchrotron and inverse Compton emission. The transition between synchrotron-dominated and inverse-Compton-dominated emissions takes place at ≈109 eV. Aims. The short-term (lasting for one week to months) flux variability of the synchrotron emission from the most energetic electrons is investigated with data from ten years of observations with the Fermi Large Area Telescope in the energy range from 60 MeV to 600 MeV. Methods. We reconstructed the off-pulse light curve reconstructed from phase-resolved data. The corresponding histogram of flux measurements was used to identify distributions of flux-states and the statistical significance of a lower-flux component was estimated with dedicated simulations of mock light curves. The energy spectra for different flux states were also reconstructed. Results. We confirm the presence of flaring-states which follow a log-normal flux distribution. Additionally, we discovered a low-flux state where the flux drops to as low as 18.4% of the intermediate-state average flux and remains there for several weeks. The transition time is observed to be as short as two days. The energy spectrum during the low-flux state resembles the extrapolation of the inverse-Compton spectrum measured at energies beyond several GeV energy, implying that the high-energy part of the synchrotron emission is dramatically depressed. Conclusions. The low-flux state found here and the transition time of at most ten days indicate that the bulk (>75%) of the synchrotron emission above 108 eV originates in a compact volume with apparent angular size of θ ≈ 0″​​.4 tvar/(5 d). We tentatively infer that the so-called inner knot feature is the origin of the bulk of the γ-ray emission.

Study of long-term flux and photon index distributions of blazars using RXTE observations

ABSTRACT We present a detailed study of flux and index distributions of three blazars [one flat-spectrum radio quasar (FSRQ) and two BL Lacertae objects (BL Lacs)] by using 16 yr of Rossi X-ray Timing Explorer (RXTE) archival data. The three blazars were chosen such that their flux and index distributions have sufficient number of data points (≥90) with relatively less uncertainty $\left(\overline{\sigma _{\rm err}^{2}}/\sigma ^{2} < 0.2\right)$ in light curves. Anderson–Darling (AD) test and histogram fitting show that flux distribution of FSRQ 3C 273 is lognormal, while its photon index distribution is Gaussian. This result is consistent with linear Gaussian perturbation in the particle acceleration time-scale, which produces lognormal distribution in flux. However, for two BL Lacs, viz. Mrk 501 and Mrk 421, AD test shows that their flux distributions are neither Gaussian nor lognormal, and their index distributions are non-normal. The histogram fitting of Mrk 501 and Mrk 421 suggests that their flux distributions are more likely to be a bimodal, and their index distributions are double Gaussian. Since, Sinha et al. had shown that Gaussian distribution of index produces a lognormal distribution in flux, double Gaussian distribution of index in Mrk 501 and Mrk 421 indicates that their flux distributions are probably double lognormal. Observation of double lognormal flux distribution with double Gaussian distribution in index reaffirms two flux states hypothesis. Further, the difference observed in the flux distribution of FSRQ (3C 273) and BL Lacs (Mrk 501 and Mrk 421) at X-rays suggests that the low-energy emitting electrons have a single lognormal flux distribution, while the high-energy ones have a double lognormal flux distribution.

X-Ray Reverberation Mapping and Dramatic Variability of Seyfert 1 Galaxy 1H 1934-063

A fraction of active galactic nuclei (AGN) exhibit dramatic variability, which is observed on timescales down to minutes in the X-ray band. We introduce the case study of 1H 1934-063 (z = 0.0102), a Narrow-line Seyfert 1 (NLS1) among the brightest and most variable AGN ever observed with XMM-Newton. This work includes spectral and temporal analyses of a concurrent XMM-Newton and NuSTAR 2015 observation lasting 130 kiloseconds, during which the X-ray source exhibited a steep (factor of 6) plummet and subsequent full recovery of flux level, accompanied by deviation from a single log-normal flux distribution. We rule out Compton-thin obscuration as the cause for this dramatic variability observed even at NuSTAR energies. In order to constrain coronal geometry, dynamics, and emission/absorption processes, we compare detailed spectral fitting with Fourier-based timing analysis. Similar to other well-studied, highly variable Seyfert 1s, this AGN is X-ray bright and displays strong reflection features. We find a narrower broad iron line component compared to most Seyfert 1s, and constrain black hole spin to be < 0.1, one of the lowest yet discovered for such systems. Combined spectral and timing results are consistent with a dramatic change in the continuum on timescales as short as a few kiloseconds dictating the nature of this variability. We also discover a Fe-K time lag, measuring a delay of 20 seconds between relativistically-blurred reflection off the inner accretion flow and the hard X-ray continuum emission.

Log-normal flux distribution of bright Fermi blazars

We present the results of the γ-ray flux distribution study on the brightest blazars which were observed by Fermi-LAT. We selected 50 of the brightest blazars based on the maximum number of detections reported in the Third LAT AGN Catalog. We performed standard unbinned maximum likelihood analysis on the LAT data during the period between August 2008 and December 2016, in order to obtain the average monthly flux. After quality cuts, blazars for which at least 90% of the total flux had survived were selected for further study, and this included 19 FSRQs and 19 BL Lacs. The Anderson-Darling and χ2 tests suggest that the integrated monthly flux follows a log-normal distribution for all sources, except for three FSRQs for which neither a normal nor a log-normal distribution was preferred. A double log-normal flux distribution tendency was observed in these sources, though this has to be confirmed with improved statistics. We also found that the standard deviation of the log-normal flux distribution increases with the mean spectral index of the blazar, and can be fitted with a line of slope 0.24±0.04. We repeat our study on three additional brightest unclassified blazars to identify their flux distribution properties. Based on the features of their log-normal flux distribution, we infer these unclassified blazars may be closely associated with FSRQs. We also highlight that considering the log-normal behavior of the flux distribution of blazars, averaging their long term flux on a linear scale can largely underestimate the nominal flux and this discrepancy can propagate down to the estimation of source parameters through spectral modeling.

The flux distribution of individual blazars as a key to understand the dynamics of particle acceleration

ABSTRACT The observed lognormal flux distributions in the high-energy emission from blazars have been interpreted as being due to variability stemming from non-linear multiplicative processes generated dynamically from the accretion disc. On the other hand, rapid minute scale variations in the flux point to a compact emitting region inside the jet, probably disconnected from the disc. In this work, we show that linear Gaussian variations of the intrinsic particle acceleration or escape time-scales can produce distinct non-Gaussian flux distributions, including lognormal ones. Moreover, the spectral index distributions can provide confirming evidence for the origin of the variability. Thus, modelling of the flux and index distributions can lead to quantitative identification of the micro-physical origin of the variability in these sources. As an example, we model the X-ray flux and index distribution of Mkn 421 obtained from ∼9 yr of MAXI observations and show that the variability in the X-ray emission is driven by Gaussian fluctuations of the particle acceleration process rather than that of the escape rate.

The Kepler Light Curves of AGN: A Detailed Analysis

Abstract We present a comprehensive analysis of 21 light curves of Type 1 active galactic nuclei (AGN) from the Kepler spacecraft. First, we describe the necessity and development of a customized pipeline for treating Kepler data of stochastically variable sources like AGN. We then present the light curves, power spectral density functions (PSDs), and flux histograms. The light curves display an astonishing variety of behaviors, many of which would not be detected in ground-based studies, including switching between distinct flux levels. Six objects exhibit PSD flattening at characteristic timescales that roughly correlate with black hole mass. These timescales are consistent with orbital timescales or free-fall accretion timescales. We check for correlations of variability and high-frequency PSD slope with accretion rate, black hole mass, redshift, and luminosity. We find that bolometric luminosity is anticorrelated with both variability and steepness of the PSD slope. We do not find evidence of the linear rms–flux relationships or lognormal flux distributions found in X-ray AGN light curves, indicating that reprocessing is not a significant contributor to optical variability at the 0.1%–10% level.

The rms-flux relation in accreting objects: not a simple “volume control”

The light curves of a diverse range of accreting objects show characteristic linear relationships between the short-term rms amplitude of variability and the flux as measured on longer time-scales. This behaviour is thought to be imprinted on the light curves by accretion rate fluctuations on different time-scales, propagating and coupling together through the accretion flow. Recently, a simple mathematical interpretation has been proposed for the rms-flux relation, where short-term variations are modulated by a single slower process. Here we show that this model was already considered and ruled out by another publication on the grounds that it did not produce the observed broad time-scale dependence of the rms-flux relation and associated lognormal flux distribution. We demonstrate the problems with the model via mathematical arguments and a case-study of Cyg X-1 data compared with numerical simulations. We also highlight another conclusion of our original work, which is that a linear rms-flux relation is easy to produce using a variety of models with positively skewed flux distributions. Observing such a relation in a non-accreting object (e.g. in solar flares) does not necessarily imply a phenomenological connection with the behaviour of accretion flows, unless the relation is seen over a similarly broad range of time-scales.

TESTING THE PROPAGATING FLUCTUATIONS MODEL WITH A LONG, GLOBAL ACCRETION DISK SIMULATION

ABSTRACT The broadband variability of many accreting systems displays characteristic structures; log-normal flux distributions, root-mean square (rms)-flux relations, and long inter-band lags. These characteristics are usually interpreted as inward propagating fluctuations of the mass accretion rate in an accretion disk driven by stochasticity of the angular momentum transport mechanism. We present the first analysis of propagating fluctuations in a long-duration, high-resolution, global three-dimensional magnetohydrodynamic (MHD) simulation of a geometrically thin (h/r ≈ 0.1) accretion disk around a black hole. While the dynamical-timescale turbulent fluctuations in the Maxwell stresses are too rapid to drive radially coherent fluctuations in the accretion rate, we find that the low-frequency quasi-periodic dynamo action introduces low-frequency fluctuations in the Maxwell stresses, which then drive the propagating fluctuations. Examining both the mass accretion rate and emission proxies, we recover log-normality, linear rms-flux relations, and radial coherence that would produce inter-band lags. Hence, we successfully relate and connect the phenomenology of propagating fluctuations to modern MHD accretion disk theory.

Rms–flux relation and fast optical variability simulations of the nova-like system MV Lyr

The stochastic variability (flickering) of the nova-like system (subclass of cataclysmic variable) MV Lyr yields a complicated power density spectrum with four break frequencies. Scaringi et al. (2012) analysed high-cadence Kepler data of MV Lyr, taken almost continuously over 600 days, giving the unique opportunity to study multicomponent Power Density Spectra (PDS) over a wide frequency range. We modelled this variability with our statistical model based on disc angular momentum transport via discrete turbulent bodies with an exponential distribution of the dimension scale. Two different models were used, a full disc (developed from the white dwarf to the outer radius of ~10^10 cm) and a radially thin disc (a ring at a distance of ~10^10 cm from the white dwarf) that imitates an outer disc rim. We succeed in explaining the two lowest observed break frequencies assuming typical values for a disc radius of 0.5 and 0.9 times the primary Roche lobe and an alpha parameter of 0.1-0.4. The highest observed break frequency was also modelled, but with a rather small accretion disc with radius of 0.3 times the primary Roche lobe and a high alpha value of 0.9 consistent with previous findings by Scaringi (2014). Furthermore, the simulated light curves exhibit the typical linear rms-flux proportionality linear relation and the typical log-normal flux distribution. As the turbulent process is generating fluctuations in mass accretion that propagate through the disc, this confirms the general knowledge that the typical rms-flux relation is mainly generated by these fluctuations. In general a higher rms is generated by a larger amount of superposed flares which is compatible with a higher mass accretion rate expressed by a larger flux.

Can self-organized critical accretion disks generate a log-normal emission variability in AGN?

Active Galactic Nuclei (AGN), such as Seyfert galaxies, quasars, etc., show light variations in all wavelength bands, with various amplitude and in many time scales. The variations usually look erratic, not periodic nor purely random. Many of these objects also show lognormal flux distribution and RMS - flux relation and power law frequency distribution. So far, the lognormal flux distribution of black hole objects is only observational facts without satisfactory explanation about the physical mechanism producing such distribution in the accretion disk. One of the most promising models based on cellular automaton mechanism has been successful in reproducing PSD (Power Spectral Density) of the observed objects but could not reproduce lognormal flux distribution. Such distribution requires the existence of underlying multiplicative process while the existing SOC models are based on additive processes. A modified SOC model based on cellular automaton mechanism for producing lognormal flux distribution is presented in this paper. The idea is that the energy released in the avalanche and diffusion in the accretion disk is not entirely emitted instantaneously as in the original cellular automaton model. Some part of the energy is kept in the disk and thus increase its energy content so that the next avalanche will be in higher energy condition and will release more energy. The later an avalanche occurs, the more amount of energy is emitted to the observers. This can provide multiplicative effects to the flux and produces lognormal flux distribution.

Superorbital variability of X-ray and radio emission of Cyg X-1 - II. Dependence of the orbital modulation and spectral hardness on the superorbital phase

We discover a pronounced dependence of the strength of the X-ray orbital modulation and the hardness in Cyg X-1 in the hard state on its superorbital phase. Our results can be well modelled as a combination of two effects: the precession of the accretion disc (which causes the superorbital flux modulation) and the orbital-phase dependent X-ray absorption in an accretion bulge, located at the accretion disc edge close to the supergiant companion but displaced from the line connecting the stars by about 25^o. Our findings are supported by the distribution of the X-ray dips showing concentration towards zero superorbital phase, which corresponds to the bulge passing through the line of sight. We Fourier analyse our model, and find it explains the previous finding of asymmetric beat (between the orbital and superorbital modulations) frequencies in the observed power spectrum, provided the disc precession is prograde. We find no statistically significant changes of the orbital modulation with the superorbital phase in the 15-GHz radio data. This absence is consistent with the radio being emitted by a jet in the system, in which case the orbital modulation is caused by wind absorption far away from the disc. We also find that both the X-ray and radio fluxes of Cyg X-1 in the hard state on time scales >10^4-s have lognormal distributions, which complements a previous finding of a lognormal flux distribution in the hard state on 1-s time scales. We point out that the lognormal character of the flux distribution requires that flux logarithms rather than fluxes themselves should be used for averaging and error analysis. We also provide a correct formula for the uncertainty of rms of a light curve for the case when the uncertainty is higher than the measurement.

Non-linear X-ray variability in X-ray binaries and active galaxies

We show that the rms-flux relation recently discovered in the X-ray light curves of Active Galactic Nuclei (AGN) and X-ray binaries (XRBs) implies that the light curves have a formally non-linear, exponential form, provided the rms-flux relation applies to variations on all time-scales (as it appears to). This phenomenological model implies that stationary data will have a log normal flux distribution. We confirm this result using an observation of Cyg X-1, and further demonstrate that our model predicts the existence of the powerful millisecond flares observed in Cyg X-1 in the low/hardstate, and explains the general shape and amplitude of the bicoherence spectrum in that source. Our model predicts that the most variable light curves will show the most extreme non-linearity. This result can naturally explain the apparent non-linear variability observed in some highly variable Narrow Line Seyfert 1 (NLS1) galaxies, as well as the low states observed on long time-scales in the NLS1 NGC 4051, as being nothing more than extreme manifestations of the same variability process that is observed in XRBs and less variable AGN. That variability process must be multiplicative (with variations coupled together on all time-scales) and cannot be additive (such as shot-noise), or related to self-organised criticality, or result from completely independent variations in many separate emitting regions. Successful models for variability must reproduce the observed rms-flux relation and non-linear behaviour, which are more fundamental characteristics of the variability process than the power spectrum or spectral-timing properties. Models where X-ray variability is driven by accretion rate variations produced at different radii remain the most promising.