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

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4. Discussion

The beaming problem has been studied under the assumption that the small scale surface roughness is the most important factor behind the effect. The radiative heat transfer problem was formulated as a set of integral equations, which have to be solved simultaneously to give the temperature variations over the rough surface. These equations were solved for two types of surface roughness approaches. Analytical solutions were presented for a surface covered by hemispherical segment craters. A more general numerical method was devised for the stochastic surface geometries. The beaming effect was quantified by integrating the flux from the rough surfaces and compare to the flux of a similar but flat surface.

The advantage with the hemispherical segment approach is of course the analytical solutions, which makes it easy to implement. The problem is the rather idealised geometry assumed. The stochastic surfaces are in that respect probably more realistic, but much more demanding on the implementation and the computational efforts. For that reason is the comparison between these to approaches of some interest. Since the results from the two approaches are quite similar, it is likely that the spherical crater model would be the choice in most practical applications.

The beaming, as caused by the surface roughness, can probably be understood as the result of two different effects. A smooth surface dilutes the energy received according to the well known cosine factor. The rough surface will on the other hand have facets oriented towards the Sun, which will become significantly hotter than the flat comparison surface. At opposition the observer will see the hottest facets in the most favourable geometry. This is further enhanced by the second effect, the multiple scattering. A higher albedo increases the scattered radiation, which means that parts of the surface in shadow or somewhat oriented away from the Sun also gets heated. The consequences are seen in Figs. 2 and 3where both a larger surface roughness, and higher albedo increases the beaming.

The approximation discussed in Sect. 2.7and Eq. (23) is an attempt to separate the beaming from the heat conduction. By this it is possible to substantially reduce the computational efforts. The agreement in the final fluxes were found to be very good between the detailed calculations and the use of the approximation for 3 Juno in Sect. 3.4. Spencer (1990) expressed this in a somewhat different way by concluding that surface roughness adds a term to the STM beaming parameter, which is independent of the heat conduction. The approximation is probably valid in most cases, which is a major practical advantage. One should, however, keep in mind that there can be problems in situations with disc resolved data, and large thermal parameters combined with very rough surfaces.

On the theoretical side do the integral equations and their analytical solutions open possibilities for future development. The constant background approach is probably bad approximation for more general surface than the hemispherical segments. The results in Figs. 2 and 3suggests on the other hand an approach were the stochastic surface could be divided into a few different height levels. The valleys in Fig. 1 are of course similar to the hemispherical segments, while the hills can be compared to the non-cratered portion of the simpler approach. A height dependent background field is probably the next step to take in the analytical approach.

The application of the model in practise is demonstrated in Fig. 4. The introduction of surface roughness is a strong effect, and in principle it is possible to obtain surface roughness data from these kind of model-data comparisons. There are, however, several difficulties. The parameters in Table 2 is not taken from a consistent source. The ellipsoidal shape is after all an approximation, and the deviation from the true shape can produce significant effects. The relation between the albedo and the diameter is given from the theory of the HG -system (Bowell et al., 1989), and it is not clear how this relation should be corrected for non-spherical asteroids. This is clearly a problem since the H parameter varies significantly between different investigations for many objects. These and other issues will be discussed in coming papers were the model is applied to a larger data set.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998