Acceleration at the termination shock: Anisotropies and spectra

Low-energy termination shock particle populations observed by the Voyagers upstream of the shock exhibited strong field-aligned beaming with anisotropies of the order of unity. The Parker transport equation is valid only for nearly isotropic phase space distributions and is inapplicable to these highly beamed populations. The usual approach is to revert to the more general focused transport equation retaining pitch-angle information. We developed a complimentary technique employing a three-moment expansion of the Skilling equation using Legendre polynomials. We investigate the effects of adiabatic focusing and reflection on the diffusive acceleration process at oblique shock waves. It is shown that low-energy particle intensities are discontinuous and sharply peaked at the shock, consistent with the observations. Particle spectra are not only harder than the power laws predicted from diffusive transport theory, but also exhibit spectral gaps near the low-energy acceleration threshold due to more efficient acceleration by scattering and mirroring. Our model also predicts upstream anisotropies as high as 100% for highly oblique shocks whereas downstream distributions are nearly isotropic.