PROTON FLUX MODEL

Despite its theoretical approximation, the force field model provides a useful way to parameterize the shape of GCR differential energy spectrum.
The differential intensity J of cosmic ray (protons) at 1 AU is given by \begin{equation} J_{r = 1 AU}(E,t) = \frac{(E + M_p)^2 - M_p^2}{(E + M_p +\phi(t)) ^2 -M_p^2} J_{lis} (E+ \phi(t)),      (1) \end{equation} where E is the kinetic energy (in GeV per nucleon), \( M_p \) is the proton rest energy and \( \phi \) is the modulation potential.
\( J_{lis} \) denotes the local interstellar spectrum (LIS) of protons. Here we use a parametrization derived from the latest data from Voyager and AMS-02 [ref].
Starting from the derivation of the modulation potential developed by Usoskin in 2011 [ref], we have derived \( \phi \) for the period after 2011 assuming a quadratic relation between neutron monitor count rate \( N(t) \) and Usoskin's potential \( \phi(t) \).
This relation can be written as \begin{equation} \phi(N(t)) = A + B \cdot N(t) + C \cdot N(t)^2 \end{equation} The best-fit parameters A, B e C can be used to derive \( \phi \) for next epochs.

We can now reconstruct the proton flux J near Earth from equation (1).