## Introduction

Diffuse reflection of light from a surface element on an atmosphereless solar system object is described by the reflection coefficient. The coefficient relates the incident flux density to the outgoing specific intensity. Alternatively, the relation can be given by using the bi-directional reflectance distribution function or the scattering law. The three alternative ways differ from each other only by certain simple trigonometric expressions that depend on the cosines of the angles of incidence and emergence as measured from the outward normal direction of the surface element.

Reflection coefficients of varying rigor have been put forward for photometric analyses of asteroid surfaces. In lightcurve inversion for asteroid rotation periods, pole orientations, and shapes, it is customary to utilize a linear combination of the Lommel–Seeliger (LS) and Lambert reflection coefficients (L; Kaasalainen et al. 2001). The former is a first-order multiple-scattering approximation of the radiative-transfer equation and thus applicable to dark particulate surfaces. The latter is typically taken as a model for bright particulate surfaces, although it does not follow from radiative transfer. For a review on asteroid lightcurve inversion, the reader is referred to Kaasalainen et al. (2002).

However, the LS–L reflection coefficient, as usually applied in lightcurve inversion, does not account for, e.g., angle-dependent shadowing effects caused by the particulate character of the surface element and the rough interface between the surface element and free space. These shadowing effects are not fully accounted for in the popular models by Lumme and Bowell (1981), Hapke (1986), and Shkuratov (1999); whereas the effects are accounted for, to varying degree of completeness, in the works by Lumme et al. (1990), Peltoniemi and Lumme (1992), Parviainen and Muinonen (2009), Muinonen et al. (2011), and Wilkman (in preparation).

The importance of the direct problem for asteroid lightcurve computation has been recognized already by, e.g., Muinonen (1998), Kaasalainen and Torppa (2001) and Kaasalainen et al. (2001). An in-depth study of the effects due to a physical reflection coefficient has, however, never been carried out. The present work sets the stage for such a study through an analysis of scattering effects in lightcurves of dark particulate media.

In the Theoretical Methods section, we describe the theoretical framework of the present study. The Numerical Methods section includes a review of the numerical methods required for the computation of lightcurve phase shifts. The Results section describes the outcome for a selection of realistic illumination and observation geometries, followed by a discussion section. Conclusions are presented in the last section.