On the determination of lognormal flux distributions for astrophysical systems

Abstract

<jats:title>ABSTRACT</jats:title> <jats:p>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 (&amp;lt;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.</jats:p>

Study of Intra-Day Flux Distributions of Blazars Using XMM-Newton Satellite

“…The dominant shape of the resultant flux distribution is determined by relative weight of these processes (Sinha et al 2018;Bhatta & Dhital 2020). They also demonstrated that perturbation in the acceleration time-scale leads to Gaussian distribution in its photon index, whereas perturbation in the particle cooling rate produces neither of these distributions (Sinha et al 2018;Shah et al 2018Shah et al , 2020Khatoon et al 2020, 2022, andreferences therein).…”

“…The nature of variability in a long term period can be quantified or explained with the flux distribution of the variable source (Uttley et al 2005). Blazars exhibits a log-normal flux distribution over the normal distribution (Uttley et al 2005;Chakraborty 2020;Shah et al 2020) which could be an indication of the variability imprint of the accretion disk onto the jet. The flux distribution is expected to be a Gaussian for a linear stochastic process, while it is expected to be a lognormal for multiplicative processes that originate in the accretion disk (Uttley et al 2005).…”

X-Ray Studies of Blazar 1ES 1959+650 Using Swift and XMM-Newton Satellite

“…The dominant shape of the resultant flux distribution is determined by the relative weight of these processes (Sinha et al 2018;Bhatta & Dhital 2020). They also demonstrated that a perturbation in the acceleration timescale leads to a Gaussian distribution of its photon index, whereas a perturbation in the particle cooling rate produces neither of the two distributions (Shah et al 2018(Shah et al , 2020Sinha et al 2018;Khatoon et al 2020, 2022, and In order to investigate the lognormality of flux distributions in our observations, we fit histograms of the X-ray data observed by Swift-XRT within the period 2018 June to 2020 December with Gaussian and lognormal distributions. Figure 5 shows the lognormal and normal flux distribution of the blazar for the total epoch used in this analysis.…”

Multiwavelength periodicity search in a sample of $γ$-ray bright blazars

“…However, the TK algorithm is based on a gaussian-like LC generation, most likely leading to an overestimation of the statistical significance. Shah et al (2020) propose a conversion of a normal into a log-normal data series through the exponential of the former. Therefore, we also test this hypothesis with the exponential TK LCs.…”