Are the Parker and Focused Transport Equations Equivalent for Galactic Cosmic Ray Modulation?

arXiv:2606.09298v1 Announce Type: new Abstract: The Parker transport equation (TPE) has been the equation of choice for the past 60 years in studies of galactic cosmic ray (GCR) modulation. Conversely, the focused TPE describes the same processes on a more fundamental level than the Parker TPE by modelling an anisotropic distribution rather than an isotropic one. It is usually assumed that the Parker TPE is valid for modelling GCRs, but the two TPEs have not been tested against each other in this context. We conduct a first-of-its-kind comparison of these TPEs without particle drifts to test whether they produce the same results under identical diffusion conditions. A new model for protons during solar minimum conditions is developed to numerically solve the TPEs using stochastic differential equations. The TPEs are designed to be as consistent as possible for diffusion by normalising the pitch-angle-dependent diffusion coefficients (DCs) used in the focused TPE to the isotropic DCs used in the Parker TPE. The Parker TPE overestimates the GCR intensity at Earth's orbit for low energies by ~30%, and by ~40% over the poles. This stems from a small first-order anisotropy caused by particle fluxes over the poles. Particles gain easier access to the inner heliosphere by streaming in over the poles, where pitch-angle scattering is generally weaker, and the magnetic field is typically less wound. The focused TPE also yields nearly identical results for different pitch-angle dependencies of the DCs. The description of particle streaming and weak pitch-angle scattering as effective parallel diffusion in the Parker TPE makes it overly diffusive. This suggests that DCs derived from fitting the Parker TPE to observations are likely underestimated. Furthermore, GCR spectral and anisotropy data alone cannot distinguish between scattering theories with similar mean free paths but different pitch-angle dependencies.