Angular broadening: Effects of nonzero, spatially varying plasma frequency between the source and observer

Angular broadening of radiation due to scattering by density irregularities is usually described using geometric optics (GO) or the parabolic wave equation (PWE) with the assumptions that the radiation frequency f greatly exceeds the local plasma frequency f(p0) or that f(p0)/f is constant along the path. These assumptions are inappropriate for many solar system radio phenomena. Here the PWE and GO formalisms are extended to treat angular broadening in plasmas with nonzero, spatially varying ratios, f(p0)(z)/f < 1. The new PWE results show that the correlation function, scattered angular spectrum, and other quantities are modified by inclusion of a denominator factor [1 - f(p0)(2)(z')/f(2)] inside the path integral over z', while the mean-square scattering angle [theta(2)] depends on both f(p0)(z)/f at the observer and the foregoing factor inside the path integral. The PWE and GO predictions for [theta(2)] are identical and involve equivalent assumptions. Previous GO and PWE results are recovered in the limits that f(p0)(z')/f is constant or zero. The new PWE and GO results will permit more accurate calculation of angular broadening for solar system and astrophysical sources. Moreover and importantly, due to the PWE and GO results for [theta(2)] being identical, previous GO analyses of [theta(2)] in solar system contexts are essentially correct, except for the neglect of or minor deficiencies in the treatment of nonzero, spatially varying f(p0)/f effects. The identical GO and PWE results for [theta(2)] and the form of the PWE equation for the correlation function raise questions as to whether diffraction is unimportant for angular broadening (under the usual PWE conditions). Future direct comparisons of the PWE predictions with angular spectra calculated using existing GO ray-tracing codes should answer these questions. Diffraction effects are probably important when the medium and turbulence are not sufficiently homogeneous transverse to the central ray path. The implications are also discussed for studies of the 2-3.5 kHz radiation observed in the outer heliosphere.