I suggested in Chapter 4 that surface roughness (or topography) was a possible cause of the temperature contrasts inferred from the slopes of the IRIS spectra. The temperature of a rough surface varies with location because of the varying illumination geometry, and the trapping of heat in depressions.
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 (i = 90o), and then drop off suddenly on the night side (i > 90o). Neither Europa nor Ganymede shows such a trend, except for the steep south polar spectra of Ganymede mentioned in Chapter 4 (Fig. 10).
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.

Topographic Thermal Models

In order to determine whether the steepening of the Callisto spectra at low sun angles is due to topography (and to understand the lack of a similar effect on Ganymede and Europa), I have tried to obtain a quantitative understanding of the effects of local topography on surface temperatures. This is not a trivial problem, because the temperatures on a rough surface are determined not only by the absorption of direct sunlight, but also by the absorption of scattered sunlight and thermal radiation from nearby topography.
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:
    Model A.    Winter and Krupp (1971) calculated and published temperature profiles across hemispherical lunar craters with a           variety of solar illumination angles. I have used these profiles to calculate the expected thermal emission spectra as a function of viewing angle and solar illumination angle.
    Model B.    I have constructed my own simple (2-dimensional) topographic temperature model to investigate the effects of a wider range of input parameters (particularly albedo) than are used by Winter and Krupp. I have then calculated the expected thermal emission spectra from this model in the same way as from the Winter and Krupp temperature profiles.
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, TE, and \ep\ calculated for that spectrum using the same 2-component blackbody fit routine used on the IRIS spectra in Chapter 4.
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-1
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 D and areal coverage. In model B, (Figs. 19 and 20) the main effect of decreasing the trench depth is to decrease the variation of spectrum slope with emission angle. Otherwise, the trend of spectrum slope with solar incidence angle is seen to be relatively insensitive to trench depth. In model A (Fig. 21), the effect of trench depth is more complex.
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.


Fig. 19 confirms suspicions that the Ganymede spectrum slopes cannot be due primarily to topographic temperature contrasts. The main effect of topography on the thermal emission spectrum is to increase the spectrum slope with increasing solar incidence, a trend not observed on Ganymede. Multiple scattering of sunlight does not wash out temperature contrasts on a bright surface, as can be seen by comparing the model B results in Figs. 19 and 20. Therefore the apparently very small topographic temperature contrasts on Ganymede are not due to its high albedo, and subdued topography is a more likely explanation.
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.


Both models, with a 60% coverage of trenches (with either aspect ratio), provide a good fit to the Voyager 1 Callisto trend of spectrum slopes in the 250--500 cm-1 region (Fig. 20). The high-sun Voyager 2 spectra are somewhat steeper than the model, maybe indicating a more Ganymede-like surface in the 210o longitude region where they were taken. However the fitted temperature contrast in model B at high sun elevations is considerably too small, indicating that the model spectra, while of the right steepness in the 250--500 cm-1 region, are not of the right shape across the whole spectrum. Model A, with a 60% coverage of deep trenches, matches the fitted temperature contrasts better, but also exhibits very large variations of spectrum slope with emission angle at low solar elevation, which are not seen in the data (Fig. 21).
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.


Fig. 19 also shows the spectrum slopes for Europa. The spectrum slopes are remarkably small. It appears that the surface of Europa must be even smoother than that of Ganymede, with 15% or less coverage of thermally-significant topography. As with Ganymede there is no steepening of spectrum slopes towards the terminator, in fact if anything the near-terminator spectra are flatter than the higher-sun ones.


The variation in thermal emission from a given point with viewing geometry, seen (poorly) in the Voyager data for Ganymede and Callisto, (as I describe in Chapter 5 and Appendix A), is probably also due to local topography. It was this `beaming' effect on the Moon, rather than variations in spectrum slope, that Winter and Krupp (1971) were trying (successfully) to match by their crater temperature model. The dependence is mostly in the form of a decrease in thermal emission with increasing phase angle, and is conveniently expressed as a `thermal phase coefficient', defined in Appendix A.
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 D = 0.25 than D = 0.5 ones (D is the crater depth/width ratio in Appendix B). This is not too different from the best fit to Callisto spectrum slopes and beaming with the current models, i.e. about 60% coverage of significant topography, so Callisto probably has a `thermally significant' surface roughness comparable to the Moon's.
There is no beaming data available for Europa.

Scale of Topography

The length scale of the topography whose signature appears (or fails to appear) in the IRIS spectra is not determined by the computer model, the physics in the model being scale-independent. Limits can be placed from other considerations, though. The topography must be large enough that conduction does not remove the temperature contrasts. Conduction will be effective during a diurnal cycle only over distances of tens of centimeters on Ganymede and Callisto (the size of the skindepth; see Chapter 7), so topography giving a thermal signature is probably larger than several centimeters in scale. Apart from this limit, the relevant scale of topography will be that for which r.m.s. slopes, and thus thermal contrasts, are largest. Planetary surfaces tend to get smoother as scale increases, boulders having steeper sides than mountains. It is therefore likely, but by no means certain, that the scale of the topography probably responsible for the thermal signature on Callisto is larger than centimeter size, but not drastically larger. It is worth noting that on the scale of the Voyager images Ganymede is noticeably smoother than Callisto, as seen qualitatively in Fig. 23, which compares Voyager 1 images of Ganymede and Callisto with very similar illumination and viewing geometry, resolution, and image processing. Though the topography producing the thermal signature on Callisto is probably smaller than the kilometer-plus scales visible in this image, the Ganymede / Callisto roughness difference may persist at smaller scales too. Likewise, Europa appears even smoother than Ganymede in the thermal data discussed here and also in the Voyager images, though the spatial scale probed by the thermal data is probably smaller than that visible in the pictures.