Rigorous cosmic-ray transport equation with no restrictions on particle energy

A new transport equation for the cosmic-ray omnidirectional intensity is obtained. This equation follows exactly from the coupled pair of differential moment equations we presented earlier. It can be characterized as a nonlocal convection-diffusion equation in which the usual transport coefficients are replaced by time integral operators. The nonlocal equation is shown to reduce to the standard convection-diffusion form if the adiabatic approximation can be applied. In general, the adiabatic approximation does not apply; however, by going to the limits of infinite and zero gyroradius and, in addition, applying the adiabatic approximation, the large- and small-gyroradius transport theories due originally to Jokipii are regained. The validity of these theories as asymptotic limits and as approximate theories in the interplanetary magnetic field is discussed. It is concluded that the small-gyroradius limit cannot be considered an approximate theory for low-energy particles even when the random field strength is weak. In addition, by scaling the length and time appropriately, a low-energy scaling law is derived. This law can be used to obtain low rigidity solutions to the nonlocal convection-diffusion equation. Thus, the nonlocal convection-diffusion equation can be used to study solar-modulation or solar-particle event phenomena down to arbitrarily low rigidities, provided the particle distribution function is nearly isotropic. Subject headings: cosmic rays - hydromagnetics - interplanetary medium