A Method to Measure the Unbiased Decorrelation Timescale of the AGN Variable Signal from Structure Functions
A simple, model-independent method to quantify the stochastic variability of active galactic nuclei (AGNs) is the structure function (SF) analysis. If the SF for the timescales shorter than the decorrelation timescale τ is a single power law and for the longer ones becomes flat (i.e., white noise), then the auto-correlation function (ACF) of the signal can have the form of the power exponential (PE). We show that the signal decorrelation timescale can be measured directly from the SF as the timescale matching the amplitude 0.795 of the flat SF part (at long timescales), and only then is the measurement independent of the ACF PE power. Typically, the timescale has been measured at an arbitrarily fixed SF amplitude, but as we prove, this approach provides biased results, because the AGN SF/power spectral density slopes, and thus the ACF shape, are not constant and depend on either the AGN luminosity and/or the black hole mass. In particular, we show that using such a method for the simulated SFs that includes a combination of empirically known dependencies between the AGN luminosity L and both the SF amplitude and the PE power, and having no intrinsic τ–L dependence, produces a fake τ∝L{sup κ} relation with 0.3≲κ≲0.6, which otherwise is expected from theoretical works (κ≡0.5). Our method provides an alternative means for analyzing AGN variability to the standard SF fitting. The caveats, for both methods, are that the light curves must be sufficiently long (with a several year rest frame) and the ensemble SF assumes AGNs to have the same underlying variability process.