Statistics of clustering of ultra-high energy cosmic rays and the number of their sources

Observation of clustering of ultra-high energy cosmic rays (UHECR) suggests that they are emitted by compact sources. Assuming small (smaller than ~3 degrees) deflection of UHECR during the propagation, the statistical analysis of clustering allows to estimate the total number of the sources S, including those which have not yet been observed directly. When applied to astrophysical models involving extra-galactic sources, the estimate gives S ~ 400 inside the sphere of the radius ~50 Mpc. This is too large for models which associate the production of UHECR with exceptional galaxies such as AGN, powerful radio-galaxies, dead quasars, and for models based on gamma ray bursts.