Propagation in a Strongly Magnetized Local Environment

The figure on the left shows nucleon spectra obtained from a source
at 10 Mpc distance to the observer injecting protons with
an E**(-2.4) spectrum up to 10**(22) eV. Both source and observer
sit at the center of a sheet with a Gaussian density profile of
width 10 Mpc that approximates the Local Supercluster. The magnetic
field was assumed to have a Kolmogorov spectrum up to a scale of
2 Mpc, and an r.m.s. value proportional to the density profile
with a maximum of 10**(-7) Gauss in the center. The solid blue
histogram shows the mean flux,
obtained from the average of 15 different spatial realizations of the
turbulent magnetic field, the blue dashed lines
show the upper and lower one-sided 1-sigma deviations from this mean,
and the thin red line shows the spectrum for one particular
realization.

The figure on the left shows the correlation between delay time t
and energy E of arriving nucleons in a similar situation where
the central field is 3 x 10**(-7) Gauss. The transitions from the
almost rectilinear regime where t is proportional to 1/E**2 above
200 EeV to the Bohm diffusion regime where t is proportional to 1/E,
and down to the regime where the gyro radius is smaller than the
largest turbulent eddy, below 60 EeV, can clearly be seen. In
these simulations we also found that the diffusion coefficient
in the magnetic field structure considered here
is about 10 times larger than often used analytical approximations
such as the Bohm coefficient.

Significant information on the large-scale magnetic field is not only contained in the distribution of arrival times and energies but also in angular images of sources of charged ultra-high energy cosmic rays which may become observable in relatively strong localized fields. For example, if the deflection angle at a given energy is much smaller than the ratio of the coherence scale of the field and the distance to the source, l_c/D, all cosmic rays of that energy will follow the same paths and there will be only one image. If the deflection angle is comparable to or slightly larger than l_c/D, there will be several paths possible, resulting in multiple images, somewhat reminiscent of gravitational lensing. Finally, in the limit of a very incoherent field, many images will join into one diffuse image.

By observing images at different energies, one can "scan"
through different deflection angles and, therefore, potentially
through these different limiting cases. To simulate such
situations, the code developed by Martin Lemoine
and myself records angular information as well. The
following is an example (see also **
proceedings article**):

The figure on the left is an example showing angular
distributions of nucleons from a discrete source continuously
emitting protons with an E**(-2.4) spectrum up to 10**(22) eV
at a distance of 10 Mpc for different energy
ranges, as indicated. An angular resolution of about
1 degree was assumed. Both source and observer
sit at the center of a sheet with a Gaussian density profile of
width 10 Mpc that approximates the Local Supercluster. The magnetic
field was assumed to have a Kolmogorov spectrum up to a scale of
2 Mpc, and an r.m.s. value proportional to the density profile
with a maximum of 0.05 micro Gauss in the center.
The source is at the origin and the color changes
over three orders of magnitude in intensity. A transition
from several images at lower energies to only one image at the
highest energies occurs where the linear deflection becomes
comparable to the field coherence length.

As a consequence, one could conclude in this case that the magnetic field looks coherent below a length scale which is roughly the distance times the deflection angle at energies around 10**(20) eV, which is indeed roughly 1 Mpc, the largest turbulent scale assumed in the simulation. This length scale has been "reconstructed".

As can be seen from the figures above, the angular deviations from the source direction can reach up to 20 degrees even above 200 EeV, which would explain the lack of counterpart determination so far.

However, the isotropy indicated by the newest data of the AGASA experiment now requires at least several sources. For example, Centaurus A to be the unique source of ultra high energy cosmic rays would in all likelihood require magnetic fields in excess of a micro Gauss, stretching over at least a few Mpc. This sounds unreasonable, as discussed in this paper.

Intriguingly, however, if a magnetic
field of strength > 0.5 micro Gauss permeates the
Supercluster, a diffuse source distribution following
the density in the Supergalactic
Plane can accomodate both the large scale isotropy (due
to diffusion) and the small scale clustering (due to
amplification by magnetic lensing) that was reported by
AGASA, as we
demonstrated in
**this paper (e-print astro-ph/9903124)**.

The figure on the left shows the angular distribution in Galactic
coordinates in such a
scenario with the same injection spectrum as above. The observer is
within 2 Mpc of the Supergalactic plane whose location is
indicated by the solid line, and other parameters are as indicated.
The color scale shows the intensity per solid angle,
and the distributions are averaged over 4 magnetic field realizations
with 20000 particles each.

The integral distributions with respect to the Supergalactic
plane for the case with 0.05 micro Gauss (blue histogram) and
for 0.5 micro Gauss (red histogram) are compared to a completely
isotropic distribution (dash-dotted line) on the left.

Detailed Monte Carlo simulations performed on these distributions
reveal that the anisotropy decreases with increasing magnetic field
strength due to diffusion and that small scale clustering increases
with coherence and strength of the magnetic field due to magnetic
lensing. Both anisotropy and clustering also increase with
the (unknown) source distribution radius.
Furthermore, the discriminatory power
between models with respect to anisotropy and clustering strongly
increases with exposure, as discussed in more detail in
**this paper (e-print astro-ph/9903124)**.

Finally, the corresponding solid angle integrated spectra show
negligible cosmic variance for diffuse sources and fit the data
well both for the 0.05 and for 0.5 micro Gauss as shown on the
left.

As a result, a diffuse source distribution associated with the Supergalactic plane can explain all present-day features of ultra-high energy cosmic rays at least for field strengths close to 0.5 micro Gauss. The large-scale anisotropy and the clustering predicted by this scenario will allow strong discrimination against other models with next generation experiments such as the Pierre Auger Project.

Monte Carlo simulations in highly structured fields are
quite time consuming if detailed angular images and fluctuations
around average distributions are to be calculated because most
of the calculated trajectories do not lead to the chosen observer
location. I have therefore begun to experiment with **artificial
intellicence methods** such as **Hebbian learning rules**
to find the particle emission directions which most likely lead to the
observer.

The origin of magnetic fields in the Local Supercluster and the observable predictions of more realistic implementations of this model are still in need of further study.

Much weaker large scale magnetic fields can induce
**time dependent cosmic ray fluxes** on timescale interesting
for the experiments.
See also **production scenarios** and
**propagation and average fluxes** of
ultra-high energy cosmic rays.