(2003Photometric modeling of Saturn’s rings I. Monte Carlo method and the effect of nonzero volume filling factor

The scattering properties of particulate rings with volume filling factors in the interval D = 0.001–0.3 are studied, with photometric Monte Carlo ray tracing simulations combining the advantages of direct (photons followed from the source) and indirect methods (brightness as seen from the observing direction). Besides vertically homogeneous models, ranging from monolayers to classical many-particle thick rings, particle distributions obtained from dynamical simulations are studied, possessing a nonuniform vertical profile and a power law distribution of particle sizes. Self-gravity is not included to assure homogeneity in planar directions. Our main goal is to check whether the moderately flattened ring models predicted by dynamical simulations (with central plane D> 0.1) are consistent with the basic photometric properties of Saturn’s rings seen in ground-based observations, including the brightening near zero phase angle (opposition effect), and the brightening of the B-ring with increasing elevation angle (tilt effect). Our photometric simulations indicate that dense rings are typically brighter in reflected light than those with D → 0, due to enhanced single scattering. For a vertically illuminated layer of identical particles this enhancement amounts at intermediate viewing elevations to roughly 1 + 2D. Increased single scattering is also obtained for low elevation illumination, further augmented at low phase angles α by the opposition brightening when D increases: the simulated opposition effect agrees very well with the Lumme and Bowell (1981, Astron. J. 86, 1694–1704) theoretical formula. For large α the total intensity may also decrease, due to reduced amount of multiple scattering. For the low (α = 13◦) and high (α = 155◦) phase angle geometries analyzed in Dones et al. (1993, Icarus 105, 184–215) the brightness change for D = 0.1 amounts to 20 % and −17%, respectively. In the case of an extended size distribution