On the determination of log-normal flux distributions for astrophysical systems

TL;DR

This paper provides guidelines for testing whether astrophysical source fluxes follow Gaussian or log-normal distributions, considering factors like lightcurve length, variability, and noise, with practical applications to gamma-ray data.

Contribution

It offers empirical fits and critical values for skewness and AD tests tailored to different noise spectra, improving reliability of flux distribution analysis.

Findings

Tests are unreliable for white noise due to binning effects.

Skewness variance does not decrease for $eta geq 1.5$, limiting test applicability.

Application to Fermi data confirms a log-normal flux distribution for a blazar.

Abstract

Determining whether the flux distribution of an Astrophysical source is a Gaussian or a log-normal, provides key insight into the nature of its variability. For lightcurves of moderate length ($< 10^3$), 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 lightcurves with different lengths, variability, Gaussian measurement errors and power spectrum index $\beta$ (i.e. $P(f) \propto f^{-\beta}$), to provide a prescriptionand 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 $\beta = 0.5$ and $1.0$, which differ from the values given in the literature which are for white noise ($\beta = 0$). Moreover, we show that for white noise, for most practical situations, these tests are meaningless, since binning in time alters the flux distribution. For $\beta \gtrsim 1.5$, 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 $\beta\gtrsim 0.5$ and $\beta \lesssim 1.0$. As an example of the prescription given in this work, we reconfirm that the Fermi data of the blazar, 3FGL\,J0730.2-1141, shows that its $\gamma$ ray flux is consistent with a log-normal distribution and not with a Gaussian one.