We present the distributions of geometrical covering factors of active galactic nuclei (AGNs) dusty tori (f2) using an X-ray selected complete sample of 227 AGN drawn from the Bright Ultra-hard XMM-Newton Survey. The AGN have z from 0.05 to 1.7, 2-10 keV luminosities between 10^42 and 10^46 erg/s and Compton-thin X-ray absorption. Employing data from UKIDSS, 2MASS and the Wide-field Infrared Survey Explorer in a previous work we determined the rest-frame 1-20 microns continuum emission from the torus which we model here with the clumpy torus models of Nenkova et al. Optically classified type 1 and type 2 AGN are intrinsically different, with type 2 AGN having on average tori with higher f2 than type 1 AGN. Nevertheless, ~20 per cent of type 1 AGN have tori with large covering factors while ~23-28 per cent of type 2 AGN have tori with small covering factors. Low f2 are preferred at high AGN luminosities, as postulated by simple receding torus models, although for type 2 AGN the effect is certainly small. f2 increases with the X-ray column density, which implies that dust extinction an X-ray absorption takes place in material that shares an overall geometry and most likely belongs to the same structure, the putative torus. Based on our results, the viewing angle, AGN luminosity and also f2 determine the optical appearance of an AGN and control the shape of the rest-frame ~1-20 microns nuclear continuum emission. Thus, the torus geometrical covering factor is a key ingredient of unification schemes.
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