It is shown that the widely used Force-Field (FF) approximation to the cosmic ray transport equation (TPE) does not work well in the outer heliopshere, in particular for anomalous cosmic rays (ACRs). The even simpler ConvectionDiffusion (CD) approximation produces better results in the outer heliosphere. Using a simple 1D solution to the TPE for comparison, we show that the widely used FF and CD approximations are poor in commonly occurring situations. The FF fares badly in the outer heliosphere and for ACRs in general, while the CD is only appropriate in the outer heliosphere. A full 1D solution is provided in the hope that it may be more commonly used instead of these approximations.

The well established cosmic ray transport equation describes the physics of cosmic ray modulation in the heliosphere and is thought to be complete, its solution for anomalous cosmic rays has special characteristics due to the local acceleration of these particles at the solar wind termination shock. Some of these characteristics are demonstrated here, namely effects by the strength of the shock, shock drift, the cutoff in the spectrum at the shock, species scaling, drift, and ionization.

New insights regarding the form of the accelerated anomalous cosmic ray spectrum at the solar wind termination shock, obtained from solutions of the cosmic ray transport equation, are presented. A simple analytical expression for the spectrum on the shock is derived, and its dependence on the acceleration parameters is shown. Particular attention is paid to the high‐energy cutoff of this spectrum. This expression and its parameterization should be applicable for the acceleration of charged particles in any spherical shock.

The anomalous cosmic ray component is a much more sensitive probe of modulation/acceleration in the heliosphere than galactic cosmic rays. Based on one-dimensional, no-drift solutions of the transport equation, we formulate several properties that govern their acceleration and modulation.

The companion paper of Fisk (1999) in this volume convincingly argues that the theory of cosmic ray transport in the heliosphere has probably been complete for almost three decades. The outstanding problem is to find the magnitude, spatial, and energy dependence of the convection, diffusion, drift, adiabatic cooling, and acceleration mechanisms. This problem can be addressed in two complementary ways: the relevant parameters can be calculated from theory, or the observed particle spectra can be used to deduce them. This paper emphasises the second approach, i.e., we review to what extent one is able to fit cosmic ray observations self-consistently with a single set of modulation parameters. For this purpose we mainly concentrate on model fits to the intensity spectra of cosmic ray protons, helium, and oxygen as observed during the solar minimum periods of 1987 and 1997. The main results are that: (1) the scattering parameters deduced from the particle spectra broadly fall in the same range as the newest values derived from theory, but (2) the separate sets of scattering parameters needed to simulate the 1987 and 1997 solar minimum spectra differ so much from one another that it is unlikely that the final solution has been found.

The global processes that determine cosmic ray modulation are reviewed. The essential elements of the theory which describes cosmic ray behavior in the heliosphere are summarized, and a series of discussions is presented which compare the expectations of this theory with observations of the spatial and temporal behavior of both galactic cosmic rays and the anomalous component; the behavior of cosmic ray electrons and ions; and the 26-day variations in cosmic rays as a function of heliographic latitude. The general conclusion is that the current theory is essentially correct. There is clear evidence, in solar minimum conditions, that the cosmic rays and the anomalous component behave as is expected from theory, with strong effects of gradient and curvature drifts. There is strong evidence of considerable latitude transport of the cosmic rays, at all energies, but the mechanism by which this occurs is unclear. Despite the apparent success of the theory, there is no single choice for the parameters which describe cosmic ray behavior, which can account for all of the observed temporal and spatial variations, spectra, and electron vs. ion behavior.

The global processes that determine cosmic ray modulation are reviewed. The essential elements of the theory which describes cosmic ray behavior in the heliosphere are summarized, and a series of discussions is presented which compare the expectations of this theory with observations of the spatial and temporal behavior of both galactic cosmic rays and the anomalous component; the behavior of cosmic ray electrons and ions; and the 26-day variations in cosmic rays as a function of heliographic latitude. The general conclusion is that the current theory is essentially correct. There is clear evidence, in solar minimum conditions, that the cosmic rays and the anomalous component behave as is expected from theory, with strong effects of gradient and curvature drifts. There is strong evidence of considerable latitude transport of the cosmic rays, at all energies, but the mechanism by which this occurs is unclear. Despite the apparent success of the theory, there is no single choice for the parameters which describe cosmic ray behavior, which can account for all of the observed temporal and spatial variations, spectra, and electron vs, ion behavior.

We review the observed properties of anomalous cosmic rays and the present status of our knowledge of the processes by which they originate. We compiled a comprehensive set of ACR energy spectral data from various spacecraft throughout the heliosphere during the passes of Ulysses over the poles of the Sun and present first results of a detailed modeling effort. In several contributions, we discuss the questions of injection and possible pre-acceleration of pickup ions, summarize new observations on the ionic charge composition, and present new results on the composition of minor ions in ACRs.

We review the observed properties of anomalous cosmic rays and the present status of our knowledge of the processes by which they originate. We compiled a comprehensive set of ACR energy spectral data from various spacecraft throughout the heliosphere during the passes of Ulysses over the poles of the Sun and present first results of a detailed modeling effort. In several contributions, we discuss the questions of injection and possible pre-acceleration of pickup ions, summarize new observations on the ionic charge composition, and present new results on the composition of minor ions in ACRs.

We have done a numerical solution of the cosmic ray transport equation to simulate a heliosphere containing a discontinuous solar wind termination shock in the outer heliosphere. Our previous studies demonstrated how hydrogen spectra could be explained by a no-drift modulation model, including an estimate of the size of the relative contribution of the anomalous H component. In this contribution we concentrate on a simplified diffusion model which allows us to better study the effects of particle drifts on anomalous cosmic rays.

The first results from a different approach in the search for solutions of the transport equation to describe particle spectra measured during consecutive solar minimum periods, are presented. A simple two-dimensional no-drift model with ϰrr independent of r and θ satisfactorily decribes the proton data measured at different radial distances during consecutive solar minimum periods. The negative latitudinal gradient between Voyagers 1 and 2 in 1987 is successfully modelled due to a latitudinally dependent solar wind speed.

A modulation model that includes shock acceleration of cosmic rays at the solar wind termination shock: is used to study the characteristics of anomalous cosmic ray hydrogen, guided by the 1987 solar minimum spectra as observed by IMP 8, Voyager 2, and Pioneer 10. This first attempt at a comprehensive data fit excludes the effects of drifts and must therefore be considered as preliminary. It is found that an anomalous component dominates hydrogen spectra in the outer heliosphere but, due to large radial gradients, the anomalous contribution in the inner heliosphere is negligible. Similar model fits are shown for anomalous cosmic ray He and O to demonstrate the degree of internal consistency of the model.