Orbiting cross sections: Application to black hole scattering

The semiclassical formulas for orbiting cross sections (spiraling scattering) are derived for particles and for scalar waves. They are applied to scalar waves orbiting Schwarzschild black holes. The cross sections for scalar waves orbiting Schwarzschild black holes are also computed numerically by the method of partial wave decomposition. They are compared with the semiclassical analytic cross sections. The approximations made in the semiclassical analysis determine the scattering up to two parameters: the overall amplitude and the phase of the oscillation in the cross section. When matched to the amplitude and phase of the numerical computation there is an extremely close fit between the analytical and the numerical cross sections. Therefore we conclude that the oscillatory features observed in the numerical calculations away from forward- and backward-scattering angle are due to orbiting. Backward glory scattering has been computed analytically previously; the perfect agreement with the numerical results shows that the behavior of the cross section for a scattering angle $\theta \simeq \pi$ is due to glory scattering. Together these results interpret the numerically computed cross section for the entire range of scattering angles. The analytical calculation of the glory and orbiting cross sections is an application of the prodistribution formulation of functional integration.