Shadowing effect in clusters of opaque spherical particles

Abstract

The shadowing effect is studied for clusters of opaque spherical particles. The present modeling allows geometric optics computations of cluster scattering phase functions and shadowing effects with internal accuracy better than 1%. Three types of cluster structures are treated—uniform, ballistic, and hierarchical (physical fractal)—and three types of elementary surface scattering laws are examined—Lambertian, flux-isotropic, and specular. All structures investigated give rise to an opposition effect, that is, a nonlinear brightening toward zero phase angle. The amplitude and width of the opposition effect depend on the cluster parameters. For uniform clusters, the volume fraction and number of particles are the parameters that characterize the shadowing effect. The opposition effect becomes sharper with increasing number of constituent particles and with decreasing particle volume fraction. For ballistic clusters, the only parameter is the number of particles: when it increases, the opposition effect becomes sharper. For hierarchical clusters, the number of cluster structural levels plays a crucial role. With increasing number of cluster levels, the opposition behavior of brightness becomes markedly more nonlinear, mostly due to the decreasing particle volume fraction. It is notable, however, that the opposition effects of the hierarchical clusters and the uniform clusters with the same particle volume fraction differ from each other underscoring the importance of the detailed cluster structure on shadowing. It is shown that, with reasonable accuracy, the cluster scattering phase functions can be factorized as the products of the corresponding single-particle phase function and the so-called shadowing factor almost independently of the elementary surface scattering law. While the opposition effect due to shadowing is presently confirmed, it is typically wider than the opposition effect due to coherent backscattering, an interference mechanism in multiple scattering. The present work helps us to understand, e.g., the opposition effects of the Moon, asteroids, and other atmosphereless celestial bodies.