Limitations of the force field equation to describe cosmic ray modulation

Abstract

The force field approximation to the transport equation which describes cosmic ray modulation in the heliosphere is a widely used tool. It is popular because it provides an easy to use, quasi‐analytical method to describe the level of modulation with a single parameter. A simple numerical solution of the one‐dimensional cosmic ray transport equation is used to show that this is a good approximation for galactic cosmic rays in the inner heliosphere but that its accuracy decreases toward the outer heliosphere. On the other hand, the even simpler convection‐diffusion approximation improves with radial distance. The reason for the complementary behavior of these two approximations is that energy losses are relatively important in the inner heliosphere but not in the outer heliosphere. The force field approximation is worse for anomalous cosmic rays at all radial distances due to the exponential cutoff of these spectra at high energies. The ranges of validity are quantified. Since both approximations have their limitations, a simple numerical solution of the one‐dimensional transport equation is provided for general use.