SURFACE ROUGHNESS AS A POSSIBLE EXPLANATION OF THE IRIS SPECTRUM SHAPES
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
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
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.
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
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
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
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
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.