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16.2} Solar modulation


 The deviation from the power law, observed below 10 GeV, is a consequence of
 the influence of the solar wind called solar modulation [Gle68]. Flux intensity
 in this energy range is anti-correlated to the solar activity and follows the
 sun-spot 11-year cycle. The correlation between the solar activity and the
 modulation of the cosmic rays flux has been studied by monitoring the flux of
 atmospheric neutrons. In fact, a flux of low energy neutrons (E ~ 1.E8-1.E9 eV)
 is produced in the interaction of primary CRs with the atmosphere and it is
 mostly due to low energy primaries (1-20 GeV), due to the rapid fall of the
 primary flux intensity with energy. One assumes that far from the solar system
 there exists an unmodified flux called Local Interstellar Spectrum, which is
 modified within the solar system by the interaction with the solar wind. This
 interaction is well described by the Fokker-Planck diffusion equation.
 Describing the solar wind by a set of magnetic irregularities, and considering
 these irregularities as perfect elastic scattering centres, one obtains the
 Fokker-Planck diffusion equation. For energies above 100 MeV this equation can
 be solved using the "Force Field Approximation" [Cab04]. According to this
 approximation, at a given distance from the Sun, for example at 1 a.u., the
 population of CRs at energy E_interstellar is shifted at the energy E_0 as in
 an energy loss mechanism due to a potential V:

 E_0 = E_interstellar + Z . V_solarwind(t)

 The solar wind potential at a given distance from the Sun depends on only one
 parameter, the time: V = V(t). So it doesn't matter what the interstellar flux
 is: given a flux on the Earth at a time t, one can find the flux at another
 time just from the relative variation of the solar wind potential Phi. This
 variation can be derived from the neutron monitor counts [Bad96]. In the case
 of the fit used by FLUKA, an offline code [Bad96] makes use of an algorithm
 which takes into account a specific Phi value, or the counting rate of the
 CLIMAX neutron monitor [CLIMAX] to provide the prediction for the flux at a
 specific date or for a given value of the potential which expresses the effect
 of the interplanetary modulation of the local interstellar spectrum. Even if the
 model is not a description of the processes and of the manner in which they
 occur, it reasonably predicts the GCR modulation at Earth.

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