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

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3. Results

3.1. Implementation

A set of stochastic surfaces were generated, with the same dimensions and number of facets as in Fig. 1. Finite view factors for the triangulated surfaces were calculated by using the method described in Sect. A.1.Discrete versions (as discussed in Paper III) of the integral equations in Sect. 2.2were employed for the stochastic surfaces. The resulting matrix equations were solved by iterative methods. The matrices involved are large ([FORMULA]), but sparse and rather diagonal dominant. Therefore it is possible to find solutions in a few tens of a second on a normal workstation. The ray tracing procedures to find shadows are, however, rather costly.

The spherical craters were implemented according to the equations derived above. Heat conduction was taken into account according to Sect. 2.6. Thermal light curves were produced by using the model described in Papers I and III, and with the beaming function computed for the two types of surface roughness discussed here.

3.2. Comparing the two types of rough surfaces

The constant background approximation discussed in Sect. 2.5is both of theoretical and practical interest. If it is valid to some degree it could be a starting point to a more advanced and realistic model than the spherical craters. On the other hand it can also be used for comparing the two roughness approaches. This is because it is exact for the hemispherical segments, with 100 % crater coverage.

The approximation was applied to stochastic surfaces with r.m.s. slopes in the range [FORMULA]. The solar zenith angle was varied from [FORMULA] to [FORMULA], and the absorptivity was assumed to be [FORMULA]. Eq. (4) was solved to get the V radiosity [FORMULA]. For zenith angles smaller than [FORMULA] the mean of [FORMULA] is systematically lower than the value expected from Eq. (21). The difference increase with larger r.m.s. slope, but stays within 10 %.

The situation is, however, much worse for the background field. According to the approximation the radiation field from scattered light should be independent of the location, that is the quantity


should be a constant over the surface. Relative to its mean the standard deviation of [FORMULA] is at least 60 %. For large r.m.s. slopes and solar zenith angles the standard deviation is more than 120 %.

The conclusion is that the hemispherical segments are more efficient in "collecting" the solar radiation. In the spheres, all surface elements are participating in the scattering process. For the random surface there are valleys which to some extent behave as the spheres, but also hills with small energy exchange with the rest of the surface.

3.3. The STM beaming parameters

The STM is using an empirical correction [FORMULA] to the brightness temperature at opposition. Lebofsky et al. (1986) derived [FORMULA] from observations of 1 Ceres and 2 Pallas . This value has often been considered a standard value for main belt asteroids, and was used in for example the IRAS Minor Planet Survey (Tedesco et al., 1992). The open questions are how representative this value is for objects other than large main belt asteroids, and what is the physical explanation behind this and other values of [FORMULA]? A surface roughness model as the one presented here can be used for investigating these questions.

Model fluxes were calculated for a set of spherical non-rotating asteroids at opposition. The STM beaming parameter [FORMULA] was derived by fitting STM fluxes to the model fluxes, as shown in Figs. 2 and 3. The comparison with STM was made using both random surfaces, and surfaces with a 60 % coverage of hemispherical segments. With this crater coverage the two surface types produce beaming parameters [FORMULA] which are in rather good agreement. In the wavelength range considered here the derived beaming parameter is essentially constant. Comparing the two figures gives that increasing the albedo and the multiple scattering in the visual by lowering [FORMULA] increases the beaming correction [FORMULA] for a given roughness  [FORMULA].

[FIGURE] Fig. 2. The STM beaming parameter [FORMULA] fitted to model fluxes from spherical non-rotating asteroids, observed at opposition. The solid lines are for stochastic surfaces, and the dashed for surfaces with a 60 % coverage of hemispherical segment craters. Calculated for [FORMULA] and [FORMULA].

[FIGURE] Fig. 3. Same as Fig. 2 but using [FORMULA] instead.

In Fig. 3, for the low albedo surfaces a rather high surface roughness is required in order to achieve [FORMULA]. The [FORMULA] in Fig. 2 is probably more realistic, but might still be considered a bit too high. For example Jämsä et al. (1993) derived [FORMULA] for the Moon , when using the thermal emission data by Saari & Shorthill (1972).

It is very important, however, to remember that the [FORMULA] values derived here only includes the beaming. STM beaming parameters [FORMULA] obtained from observations depends not only on the beaming, but also on the wavelength, the shape, the heat conduction, the spin vector orientation, etc.

3.4. Example model of 3 Juno

The asteroid Juno was chosen as an example, since it is a rather well studied object. From visual light curves it has been possible to determine its spin vector and shape, as seen in Table 2. The table also gives the zero point of the light curve, by specifying the rotational phase for an epoch (see Sect. A.3.). The Bond albedo was computed from the geometrical albedo and the slope parameter. The surface roughness was varied, and the absorptivity [FORMULA] was derived by assuming that the hemispherical albedo equals the Bond albedo in Eq. (20). The longest axis of the ellipsoid was equalled to the longest stellar occultation limb profile diameter derived by Millis et al. (1981). The emissivity was assumed to equal 0.9, and the thermal inertia taken to be close to the lunar value (Spencer, 1990).


Table 2. Physical parameters for 3 Juno .

In Fig. 4, the model is compared with IRAS data. The fluxes and error estimates were taken from the IRAS Minor Planet Survey catalogue 108 (Tedesco et al., 1992). Filter colour corrected were required (M"uller, 1997). Note that the zero points of the light curves are given from the shape and spin vector solutions, and are not fitted to the IRAS data. The STM flux is also given, using a beaming parameter [FORMULA] and a phase correction of [FORMULA].

[FIGURE] Fig. 4. IRAS data at [FORMULA] of 3 Juno , compared to model thermal light curves. Curves computed for [FORMULA], and r.m.s. slopes [FORMULA], 0.4, [FORMULA], 1.0, as indicated to the right. The dashed lines are the corresponding STM fluxes.

The heat conduction was taken into account by using the detailed methods described in Sect. 2.6. The approximation discussed in Sect. 2.7and Eq. (23) were applied as well. The agreement was found to be within 1 %, for these specific circumstances. The approximation increasingly overestimates the fluxes, with larger r.m.s. slopes, and higher thermal parameters.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998