Statistical significance of the relationship between X-ray luminosity and variability timescale in active galactic nuclei

Abstract

Recently, Barr and Mushotzky¹ have claimed that a significant correlation exists between the X-ray luminosity and the timescale of X-ray variability for Seyfert galaxies and quasars. In particular, they find that log L_x = a₁ log Δt + b₁ with a₁ = 0.9 ±0.2, b₁ = 39 ± 1 and linear correlation coefficient r = 0.7 (hereafter fit 1), as would be expected if these objects have a universal efficiency of conversion of matter into energy (η ≥ 0.005, ref. 2). Furthermore, this result indicates that the emitting plasma is close to the Eddington luminosity limit, where electron-positron pair effects may dominate the hard X-ray photon fluxes³. In view of the physical meaning implied by their linear relationship and of its possible widespread use in the literature as a way of deriving constraints on the masses of active galactic nuclei^4,5, the method of analysis and its statistical significance should be examined carefully. In particular, because in this case both variables, Lx and Δt, are affected by experimental errors, their least squares fitting procedure is inaccurate and a more general approach must be taken. Applying this more generalized fit to the same data set, we obtain a slope a = 1.5 ± 0.2, significantly different from that found by Barr and Mushotzky and more acceptable in terms of goodness of the fit. When judged on the basis of their statistical significance, both results are, in any case, too weak to be conclusive, and any physical conclusions drawn from them are therefore debatable.