Limitations of the force field equation to describe cosmic ray modulation

Abstract

[1] 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.

A history of solar activity over millennia

“…These uncertainties in the boundary conditions make the results of the modulation theory slightly model-dependent (see discussion in Herbst et al 2010) and require the LIS model to be explicitly cited. This approach gives results, which are at least dimensionally consistent with the full theory and can be used for long-term studies 1 (Usoskin et al 2002a;Caballero-Lopez and Moraal 2004). Differential CR intensity is described by the only time-variable parameter, called the modulation potential φ, which is mathematically interpreted as the averaged rigidity (i.e., the particle's momentum per unit of charge) loss of a CR particle in the heliosphere.…”

“…This approximation can be used only for rough estimates, since it neglects the drift effect, but it is useful for long-term studies, when the heliospheric parameters cannot be evaluated independently. Further, but still reasonable, assumptions (constant solar-wind speed, roughly power-law CR energy spectrum, slow spatial changes of the CR density) lead to the force-field approximation (Gleeson and Axford 1968;Caballero-Lopez and Moraal 2004), which can be solved analytically. The differential intensity J i of the cosmic-ray nuclei of type i with kinetic energy T at 1 AU is given in this case as…”

Modulation of galactic cosmic rays during the unusual solar minimum between cycles 23 and 24

“…The appeal of the force field approach lies in the fact that observed modulation can be described with a single-parameter termed modulation potential [Caballero-Lopez and Moraal, 2004]. The model assume an equilibrium between diffusion and adiabatic energy loss.…”

Assessment of spectral and angular characteristics of sub-GLE events using the global neutron monitor network

“…This combination provides a good balance between simplicity and realism (Kudela and Usoskin, 2004;Nevalainen et al, 2013). For the GCR spectrum we applied a parametrisation based on the force-field model (Gleeson and Axford, 1968;Caballero-Lopez and Moraal, 2004) with a solar modulation parameter calculated according to Usoskin et al (2011). For the GCRs we considered the nucleonic ratio of heavier particles including a-particles to protons in the interstellar medium as 0.3 similarly to Kovaltsov et al (2012), assuming the local interstellar spectrum according to Burger et al (2000) and Usoskin et al (2005).…”