The composition of ultra-high energy cosmic rays is an important issue in astroparticle physics research, and additional experimental results are required for further progress. Here we investigate what can be learned from the statistical correlation
factor r between the depth of shower maximum and the muon shower size, when these observables are measured simultaneously for a set of air showers. The correlation factor r contains the lowest-order moment of a two-dimensional distribution taking both observables into account, and it is independent of systematic uncertainties of the absolute scales of the two observables. We find that, assuming realistic measurement uncertainties, the value of r can provide a measure of the spread of masses in the primary beam. Particularly, one can differentiate between a well-mixed composition (i.e., a beam that contains large fractions of both light and heavy primaries) and a relatively pure composition (i.e., a beam that contains species all of a similar mass). The number of events required for a statistically significant differentiation is ~ 200. This differentiation, though diluted, is maintained to a significant extent in the presence of uncertainties in the phenomenology of high energy hadronic interactions. Testing whether the beam is pure or well-mixed is well motivated by recent measurements of the depth of shower maximum.
We introduce a method to constrain the characteristic angular size of the brightest cosmic-ray sources observed above 57 times 1018 eV. By angular size of a source, we mean the effective angular extent over which cosmic-rays from that source arrive a
t earth. The method is based on the small-scale (< 10circ) self-clustering of cosmic-ray arrival directions. The method is applicable to sparse data sets in which strong localizations of CR* directions are not yet observed. We show that useful constraints on the source size can be made in the near future and that these constraints are not strongly dependent on the assumed spatial distribution and luminosity function of the cosmic-ray sources. We suggest that an indication of the source size is quite telling. For example, an indication of the source size can be used to infer limits on the particle charge and intervening magnetic fields (not independently), both of which are not well constrained so far. This is possible because the source size is similar in scale to the magnetic deflection.
The sources of ultra-high energy cosmic rays are not yet known. However, the discovery of anisotropic cosmic rays above 57x10^18 eV by the Pierre Auger Observatory suggests that a direct source detection may soon be possible. The near-future prospect
s for such a measurement are heavily dependent on the flux of the brightest source. In this work, we show that the flux of the brightest source above 57x10^18 eV is expected to comprise 10% or more of the total flux if two general conditions are true. The conditions are: 1.) the source objects are associated with galaxies other than the Milky Way and its closest neighbors, and 2.) the cosmic ray particles are protons or heavy nuclei such as iron and the Greisen-Zatsepin-Kuzmin effect is occurring. The Pierre Auger Observatory collects approximately 23 events above 57x10^18 eV per year. Therefore, it is plausible that, over the course of several years, tens of cosmic rays from a single source will be detected.
