Figure 18 plots spectrum slope against solar incidence angle for all the on-planet icy satellite IRIS spectra. It shows, in a slightly different form from Fig. 9 in Chapter 4, the different trends of spectrum slope observed on Europa, Ganymede, and Callisto. The Callisto spectrum slopes increase dramatically towards the terminator (

The Callisto trend is what would be expected, intuitively, from a rough surface. Spectrum slopes due to roughness should increase with increasing solar incidence angle, due to increasing temperature contrast between lengthening shadows and warm slopes tilted towards the sun. It therefore seems likely that this is the chief explanation for the steep near-terminator slopes seen on Callisto in Fig. 18. Once the sun is below the horizon temperature contrasts and spectrum slopes should decrease, also in accord with the Callisto trend in Fig. 18.

The temperature distribution across, and the resulting angular distribution of thermal emission from, an obliquely-illuminated rough surface has been modelled by Winter and Krupp (1971) and Hansen (1977). However both papers are concerned with different problems than mine, and neither gives results in a form that I can use directly to determine the shapes of thermal emission spectra from a rough surface. I have therefore adopted two approaches:

Both approaches are described in detail in Appendix B.

The possible input variables for both models are the depth/width ratio and fractional areal coverage of the trenches (the rest of the surface being smooth and horizontal), the surface emissivity (not treated rigorously), and the solar incidence and viewing emission angles, and for model B only, the surface albedo. For each combination of input parameters the model produces a thermal emission spectrum, or the fit parameters \dt,

Figures 19, 20 and 21 show, for each model, the effects of some of these variables on the calculated thermal emission spectra, and compare the model trends to those observed on Ganymede and Callisto. The slopes of the spectra are shown in two ways, as the temperature contrast \dt\ of the 2-component blackbody fit to the spectrum, and more directly as the difference in brightness temperature between 500 and 250 cm

The dependence of spectrum slope on viewing (emission) angle is shown by the width of the ribbons representing each combination of trench depth/width ratio

Fig. 19 shows model B albedos roughly appropriate for Ganymede, and Fig. 20 shows the same for Callisto, both with the actual data superimposed. The model albedos are `single scattering' albedos, (see Appendix B) so the chosen values are higher than the actual satellite albedos, though comparison of the Figs. 19 and 20 shows that the effect of model albedo on the results is small. The model A albedo is fixed at a lunar value (0.08) most appropriate for Callisto, so only Callisto data is shown on Fig. 21.

The topographic model can be used to place very crude upper limits on the roughness of Ganymede's surface (see the end of this chapter for a discussion of the length scale of the roughness). Whatever else is producing the spectrum slopes on Ganymede, it can only add to the slopes due to the inevitable topographic temperature contrasts. From Fig. 19, the low-sun Ganymede spectra are consistent with a surface with not much more than about 15% coverage of topography rough enough to give significant thermal contrasts. A rougher surface would give steeper low-sun spectra than are observed. However, this is not a firm quantitative conclusion because of the crudeness of the topographic model. The small number of Ganymede spectra in Fig. 19 that show steep slopes at high solar incidence are all from very high southern latitudes, as was mentioned previously. Maybe the surface is rougher here, perhaps due to reduced ice mobility.

So neither model fits the Callisto trend perfectly, but as both models are rather crude this is not suprising. The differences between them at least give some idea of the range of effects that topography can have on thermal emission spectra. Because both models can match the increase in spectrum slope with decreasing sun elevation, with reasonable topography, and because of the sudden reduction in spectrum slope from day to night across the terminator (Fig. 18), I am satisfied that the Callisto spectrum slopes are due largely to topography. In Chapter 7 I briefly consider the ways in which the fit to the Callisto slopes might be improved further by the introduction of a small amount of lateral inhomogeneity on the surface.

A more complete topographic model, with three dimensions and arbitrarily shaped topography, (essentially a reconstruction and extension of the Winter and Krupp, and Hansen, models) could be developed from model B and would be a logical next step in the analysis of the Callisto data.

Models A and B should be able to match the observed beaming as well as the spectrum slope variations. I used both models to calculate thermal phase coefficients as a function of solar incidence angle, emission angle, and the other variables (trench fractional coverage, albedo, trench depth) available in each. The model results are shown in Fig. 22, along with the thermal phase coefficients observed for Ganymede and Callisto, from Appendix A.

Both models give about the correct beaming if a 60% trench coverage is assumed. Deeper trenches give considerably higher thermal phase coefficients: it seems that trench depth has more effect on phase coefficient than on spectrum slope. The problem, of course, is that the Ganymede phase coefficients suggest a high value for trench coverage whereas the Ganymede spectrum slopes suggest a smoother surface. The discrepancy is probably a result of the inadequacy of the current topographic thermal models, as well as the imprecision of the measurements. For Callisto the single thermal phase coefficient derived from the data is consistent with the surface roughness suggested by the spectrum slopes.

It is worth noting that Winter and Krupp (1971), using the original version of model A, concluded that the beaming observed on the Moon could be matched best by a surface with about 2/3 coverage of sharp and subdued craters, with rather more

There is no beaming data available for Europa.