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

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 $β$ (i.e. $P(f) \propto f^{-β}$), 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 $β= 0.5$ and $1.0$, which differ from the values given in the literature which 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 $β\gtrsim 1.5$, 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 $β\gtrsim 0.5$ and $β\lesssim 1.0$. 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 $γ$ ray flux is consistent with a log-normal distribution and not with a Gaussian one.