Implications to Sources of Ultra-high-energy Cosmic Rays from their Arrival Distribution

We estimate the local number density of sources of ultra-high-energy cosmic rays (UHECRs) based on the statistical features of their arrival direction distribution. We calculate the arrival distributions of protons above $10^{19}$ eV taking into account their propagation process in the Galactic magnetic field and a structured intergalactic magnetic field, and statistically compare those with the observational result of the Pierre Auger Observatory. The anisotropy in the arrival distribution at the highest energies enables us to estimate the number density of UHECR sources as $\sim 10^{-4} {\rm Mpc}^{-3}$ assuming the persistent activity of UHECR sources. We compare the estimated number density of UHECR sources with the number densities of known astrophysical objects. This estimated number density is consistent with the number density of Fanaroff-Reily I galaxies. We also discuss the reproducability of the observed {\it isotropy} in the arrival distribution above $10^{19}$ eV. We find that the estimated source model cannot reproduce the observed isotropy. However, the observed isotropy can be reproduced with the number density of $10^{-2}$-$10^{-3} {\rm Mpc}^{-3}$. This fact indicates the existence of UHECR sources with a maximum acceleration energy of $\sim 10^{19}$ eV whose number density is an order of magnitude more than that injecting the highest energy cosmic rays.