Abstract

Context. Optical observations cannot resolve the structure of active galactic nuclei (AGN), and a unified model for AGN was inferred mostly from indirect methods, such as spectroscopy and variability studies. Optical reverberation mapping allowed us to constrain the spatial dimension of the broad emission line region and thereby to measure the mass of supermassive black holes. Recently, reverberation was also applied to the polarized signal emerging from different AGN components. In principle, this should allow us to measure the spatial dimensions of the sub-parsec reprocessing media. Aims. We conduct numerical modeling of polarization reverberation and provide theoretical predictions for the polarization time lag induced by different AGN components. The model parameters are adjusted to the observational appearance of the Seyfert 1 galaxy NGC 4151. Methods. We modeled scattering-induced polarization and tested different geometries for the circumnuclear dust component. Our tests included the effects of clumpiness and different dust prescriptions. To further extend the model, we also explored the effects of additional ionized winds stretched along the polar direction, and of an equatorial scattering ring that is responsible for the polarization angle observed in pole-on AGN. The simulations were run using a time-dependent version of the STOKES code. Results. Our modeling confirms the previously found polarization characteristics as a function of the observer's viewing angle. When the dust adopts a flared-disk geometry, the lags reveal a clear difference between type 1 and type 2 AGN. This distinction is less clear for a torus geometry where the time lag is more sensitive to the geometry and optical depth of the inner surface layers of the funnel. The presence of a scattering equatorial ring and ionized outflows increased the recorded polarization time lags, and the polar outflows smooths out dependence on viewing angle, especially for the higher optical depth of the wind (tau = 0.3). Conclusions. Together with other AGN observables, the polarization time lag places new, independent "seismological" constraints on the inner geometry of AGN. If we conduct time-dependent spectropolarimetric observing campaigns of AGN, this method has a high potential for a census of supermassive black holes.