On reverberation and cross-correlation estimates of the size of the broad-line region in active galactic nuclei

It is known that the dependence of the emission-line luminosity of a typical cloud in the active galactic nuclei (AGN) broad-line regions (BLRs) upon the incident flux of ionizing continuum can be non-linear. We study how this non-linearity can be taken into account in estimating the size of the BLR by means of the 'reverberation' methods. We show that the BLR size estimates obtained by cross-correlation of emission-line and continuum light curves can be much (up to an order of magnitude) less than the values obtained by reverberation modelling. This is demonstrated by means of numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions and by means of practical reverberation modelling of the observed optical spectral variability of NGC 4151. The time behaviour of NGC 4151 in the Hα and Hβ lines is modelled on the basis of the observational data by Kaspi et al. and the theoretical BLR model by Shevchenko. The values of the BLR parameters are estimated that allow to judge on the size and physical characteristics of the BLR. The small size of the BLR, as determined by the cross-correlation method from the data of Kaspi et al., is shown to be an artefact of this method. So, the hypothesis that the BLR size varies in time is not necessitated by the observational data.