Propagation of ultrahigh energy cosmic rays and compact sources

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The clustering of ultrahigh energy (>10²⁰ eV) cosmic rays (UHECR) suggests that they might be emitted by compact sources. Statistical analysis (Dubovsky et al., 2000) estimated the source density. We extend their analysis to give also the confidence intervals (CI) for the source density using a.) no assumptions on the relationship between clustered and unclustered events; b.) nontrivial distributions for the source luminosities and energies; c.) the energy dependence of the propagation. We also determine the probability that a proton created at a distance r with energy E arrives at earth above a threshold Ec. Using this function one can determine the observed spectrum just by one numerical integration for any injection spectrum. The observed 14 UHECR events above 10²⁰ eV with one doublet gives for the source densities 180₋₁₆₅⁺²⁷³⁰· 10⁻³ Mpc⁻³ (on the 68% confidence level).