On the determination of lognormal flux distributions for astrophysical systems

Abstract

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.