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Astron. Astrophys. 332, 1123-1132 (1998)

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1. Introduction

The thermal emission from atmosphereless bodies, such as the Moon , Mercury , or the asteroids, is strongly affected by the roughness and porosity of their surfaces. On a rough surface effects due to multiple scattering, shadows, and mutual heating are important. The solar radiation can penetrate and heat the porous surface material to some depth, and in addition to conduction, heat is transfered by radiation. The lack of convective heat transfer within a dense atmosphere, and the very low thermal conductivity of the surface material, results in large temperature variations on rather small scales.

Thus the emission is not that of a smooth flat surface, but rather the weighted sum of black body radiators at different temperatures, convolved with a material dependent emissivity. As a result the observed brightness temperature of the full Moon varies with the zenith angle [FORMULA] as [FORMULA], rather than [FORMULA] expected for a Lambertian surface (Saari and Shorthill, 1972). In the Standard Thermal Model of asteroids (STM) (Lebofsky and Spencer, 1989, and references therein), the opposition brightness temperature has to be increased by the empirical beaming parameter, and the slope of the phase curves are steeper than the Lambertian phase curve. Thus the tendency is to emit or "beam" more in the solar direction at the expense of the emission at larger phase angles.

Many authors have worked on various theoretical models to describe the beaming effects. Most popular are perhaps models where the surface roughness is described by spherical (Buhl et al., 1968; Winter and Krupp, 1971; Hansen, 1977; Spencer, 1990) or parabolic "craters" (Vogler et al., 1991; Johnson et al., 1993). Jämsä et al. (1993) used a stochastic surface, described by a Gaussian correlation function. The radiative transfer in the regolith has been studied by e.g. Henderson and Jakosky (1994) and Hapke (1993; 1996b). Hapke (1996a) came to the conclusion that only about 20 % of the observed beaming phenomena can be explained by effects in the porous media.

As has been pointed out by Johnson et al. (1993) there is a risk of confusing surface roughness effects and trends in the emissivity. As the beaming offers the opportunity to directly investigate the surface texture of the asteroids, it also makes the compositional studies in the thermal IR more complicated. Furthermore, as discussed in Paper I (Lagerros, 1996a), the beaming correlates with the global shape of the non-spherical asteroids. For that reason, both disk-integrated data (asteroids most of which are likely to have irregular shapes) and disk-resolved data (e.g. the Moon or Mercury) are potentially in need of detailed beaming models.

A thermophysical model of asteroids have been outlined in Papers I-III (Lagerros 1996a; 1996b; 1997). Heat conduction into the regolith, surface roughness, microwave emission, and irregular shapes have been considered. In this paper the beaming effects are followed up in more detail. The radiative transfer problem for the rough surface is formulated as a set of integral equations. Analytical solutions are derived for the roughness model using spherical craters. This makes the model easy to implement, but it is based on a highly idealised geometry. To investigate the limitations of this approach, a numerical method is applied to a stochastic surface. As a concluding example the thermophysical model is applied to IRAS data of 3 Juno .

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998