giant air shower

Ultra-High Energy Cosmic Rays:
Time-Dependent Fluxes and Cosmic Magnetic Fields

Charged particles of energy up to 10**(20) eV can be deflected significantly in extra-galactic magnetic fields of strength between 10**(-12) G and 10**(-9) G and/or in Galactic halo fields between 10**(-8) G and 10**(-6) G. Observable fluxes of ultra-high energy cosmic rays can therefore exhibit a significant time dependence that is determined not only by the emission history of the sources but also by deflection and delay in cosmic magnetic fields.

With Martin Lemoine (DARC, Observatoire de Paris) I have therefore developed a detailed Monte Carlo code that simulates the propagation of nucleons above 10**(19) eV in cosmic magnetic fields. In this letter co-authored with Angela Olinto and David Schramm (U of C) we discuss general signatures for the activity time scale of ultra-high energy cosmic ray sources and the time delay in large-scale magnetic fields that may be observed in the angle-time-energy images by next-generation experiments.

enery-time correlation As an example, the figure on the left shows contours of an ultra-high energy cosmic ray image projected onto the time-energy plane. The distance to the source is 60 Mpc; the r.m.s. strength of the extra-galactic magnetic field is B_rms = 2x10**(-10) G, with a white noise power spectrum and a coherence scale of l = 1 Mpc. The dotted line indicates the energy-time delay correlation t(E) proportional to E**(-2), as would be obtained in the absence of pion production losses. The dashed lines, which are not resolved here, indicate the location (arbitrarily chosen) of the observational window, of length 5 years.

enery-time correlation The figure on the left shows energy spectra that would actually be observed during a time window of a couple of years, for a continuous source (solid line), and for a burst (dashed line). The continuous source emits during 10**4 years with an average time delay of 1.3x10**3 years at 100 EeV, and the time of observation is t = 9x10**3 years, relative to propagation with the speed of light. A low energy cutoff results at the energy 40 EeV where the average delay time equals the time of observation. The dotted line shows how the spectrum would continue if the emission time scale would be much larger than 10**4 years. The case of a bursting source corresponds to a slice of the image in the time-energy plane, as indicated in the figure above by dashed lines. Both spectra assume a source distance of D = 30 Mpc, and an E**(-2) injection spectrum. The observable spectrum is given as counts per bin of width 0.05 in the logarithm of the energy, corresponding to an energy resolution typical of next generation experiments. Both spectra are normalized to a total of 50 particles detected from one source above 10 EeV, a value that could well be achieved by ,e.g., the Pierre Auger Project.

A more quantitative implementation of the effects discussed here should involve a likelihood approach. In this context, the clustering among ultra-high energy cosmic rays suggested by recent data from the Akeno Giant Air Shower Array (AGASA) is very exciting and its confirmation could have very important consequences for nature and origin of ultra-high energy cosmic rays as well as for extra-galactic and galactic magnetic fields. We have conducted such a likelihood analysis for the three pairs of ultra-high energy cosmic rays that were reported by AGASA, assuming the events in the pairs originated from a common source. Although present data are much too sparse to draw any quantitative conclusions, we observed some potentially interesting tendencies that are discussed in detail in this paper co-authored with Angela Olinto.

likelihood contours One of the pairs, for instance, turns out to be inconsistent with a burst. This is demonstrated by the figure on the left that shows contours of the logarithm of the likelihood to the base 10 in the plane of the average delay time at 100 EeV (abscissa) and the source activity time scale (ordinate). The total fluence N_0 of particles above 40 EeV that would have been observed by AGASA in the limit of inifinite integration time is constant in this figure and corresponds to the value for which the maximum of the likelihood occurs. As above, domination of the time delay by an extra-galactic magnetic field with a white noise spectrum and coherence length l=1 Mpc was assumed, for a source distance of D=30 Mpc and injection index gamma as indicated.

likelihood contours Furthermore, two of the three pairs are insensitive to the time delay. However, the pair which contains the 200 EeV event seems to significantly favor comparatively small time delays of the order of a few years, as is indicated by the likelihood marginalized with respect to the total fluence N_0 and the source activity time scale, plotted versus the average time delay at 100 EeV. This is shown on the left. The solid lines correspond to an injection index gamma=1.5, the dotted lines to gamma=2.0, and the dashed lines to gamma=2.5. All other parameters are the same as above.

This tendency can be translated into the tentative bound that the rms magnetic field should be smaller than 2x10**(-11) (l/1 Mpc)**(-0.5)(D/30 Mpc)**(-1) G. If confirmed or possibly even improved by future data, for extragalactic source distances, D > 30 Mpc, this contraint would be substantially more stringent than the existing Faraday rotation bound.

We have recently set up our code in such a way that it can directly be fed with clusters of events observed by future experiments such as the Pierre Auger Project. By simulating artificial clusters of a size typically expected from the exposure of these experiments we have employed it to quantitatively explore how much information can be reconstructed in this comprehensive paper (e-print astro-ph/9711060).

We currently start to investigate information contained in angular images, especially in the context of a strongly magnetized local environment. See also production scenarios and propagation and average fluxes of ultra-high energy cosmic rays.

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