_{1}

^{*}

The cosmic ray energy spectrum is calculated assuming that cosmic rays are generated by astrophysical objects provided by magnetospheres with a dipole magnetic field. With simple geometric considerations, the energy spectrum *E*^{-2.5} is obtained, independently on the particle species.

Since the discovery of the cosmic radiation in 1912 by Victor Hess, the nature of the sources has been debated. After a century, mysteries still remain. Gold [

As well known, in his original idea, Fermi considered an acceleration mechanism for cosmic rays based on the stochastic interaction of the charged particles available in space with wandering interstellar magnetic field clouds. He obtained for the accelerated particles a power law energy spectrum, and he stressed the importance to have obtained a power law, consistently with the experimental data. However, for explaining the value of the exponent in the experimental spectra, he needed to use certain values of the parameters regarding the particle energy threshold for this mechanism to operate and the energy losses in the interstellar space. We recall that the experimental differential energy spectrum is of the type

with

At least another acceleration mechanism, acting locally, has been proposed in the past years [^{1}. In this region powerful acceleration mechanisms operate on the charged particles pushing them to go in the outer space as cosmic rays.

New recent observations seem to support this mechanism:

a) The result by Reeves et al. [

b)The observation [

It is reasonable to think that if the Fermi theory applies also to the UHECR, one should expect an isotropic distribution for them, since the acceleration mechanism should operate everywhere in the intergalactic space. This is not verified by the Auger experiment.

These new experimental informations have pushed me to reconsider the magnetospheric mechanism with additional argumentations.

Let us consider a celestial body with a magnetic field and a plasma atmosphere. We make the assumption that the magnetic field be generated by a dipole M. The field at the equator at distance

(For a neutron star with a radius the magnetic moment is of the order of.)

A charged particle with electrical charge

For a particle to stay trapped it is necessary that the Larmor radius be small enough to satisfy the Alfven condition [

where the dimensionless quantity

This equation has to be interpreted in the following way: when an electrically charged particle accelerated by the electrical fields acting in the magnetosphere reaches a momentum p that violate Equation (5) (or it satisfies with the equal sign) then it escapes from the magnetosphere and goes into the outer space.

The plasma near the surface provides the source of the electrically charged particles as schematically shown in

We estimate now the energy spectrum of the escaping equatorial particles. At relativistic velocity the energy and momentum coincide. The number of particles contained in the magnetic shell between the lines of force reaching the equator at distances L and L + dL (see

where ^{2} magnetic field along the particle trajectory.

Assume a plasma source with spherical distribution. Thus the density of the particles which are accelerated is, near the star surface, independent on the latitude. Each latitude corresponds to a distance L at the equator according to the equation of the magnetic line of force

where

It remains to calculate the volume dV of the tridimensional shell delimited by the lines of forces corresponding to L and L+dL. We calculate

where the function

Combining Equations (5) (with the = sign), (6) and (7) and putting

This is the number of particles with momentum between p and p + dp which leave, with relativistic velocity, the corresponding shell, since they satisfy Equation (4) with the equal sign.

We remark on the fact that a power law spectrum, with exponent very close to the experimental ones, has been obtained, on the basis of almost purely geometrical considerations. In the case of the Fermi’s mechanism ad hoc values for the parameters involved in the initial acceleration as well in the particle loss in the interstellar space are necessary. Also we remark that the exponent

The above mechanism could be operative on small scales, like the planetary ones, or large scales as in the case of the cosmic rays proper. The maximum particle energy achievable with the above mechanism is obtained from the Equation (5)

where

In the case of the Earth, with

which compares with the highest proton energies observed in the Van Allen belt.

In the case of Jupiter, with

We note that Jupiter could give a large contribution to the cosmic rays observed on the Earth surface with neutron monitors.

In the case of cosmic ray of extra solar-system origin, for a neutron star with

This value compares with the highest cosmic ray proton energy detected so far. Higher values can be obtained for larger values of M.

The difference between the calculated value

In all cases, we may propose that the Magnetospheric Acceleration Mechanism be the base for generating, for all types of particles, the power law energy spectrum.