CHAPTER 7

MODELLING OF DIURNAL TEMPERATURE VARIATIONS

This chapter is concerned with my attempts to match the observed diurnal temperature variations on Europa, Ganymede, and Callisto with thermophysical models that consider the diurnal flow of heat in and out of the surface. Three possible surface models are considered: a homogeneous surface, one whose thermal properties vary with depth, and one whose thermal properties (and/or albedo) vary laterally across the surface. I attempt to match not only the diurnal temperature curves described in Chapter 5, but also the eclipse cooling and heating curves observed from Earth, and the IRIS spectrum slopes (Chapter 4).
The purpose of this exercise is to try to distinguish among the surface models mentioned above, and to provide quantitative descriptions of thermal properties that might constrain the surface composition and structure. Models that successfully explain the observed temperature distributions are also useful for predicting temperatures in regions not observed by Voyager, such as the morning regions of Ganymede and Europa. Such `weather forecasting' can be applied to models of temperature-dependent phenomena such as ice sublimation and ion sputtering.
I will describe the three possible surface pictures separately in the sections which follow. The details of the diurnal and eclipse thermophysical models that I use are not given here, but can be found in Appendix C. A caveat is in order: all the `best fits' given in this chapter are produced by `eyeball' only. For the eclipse models in particular the effects of varying a single parameter are complex and depend on the values of other parameters, and this makes determining a `best fit' difficult. More rigorous least-squares methods might give different and slightly better values.

1-Layer (Homogeneous) Surfaces

The only free parameters in a homogeneous surface model, once the distance from the sun, the latitude, and the rotation period are known, are the albedo and the thermal inertia. Figure 24 shows the homogeneous thermal model best fits to the observed equatorial diurnal temperature profiles on the three satellites, and the model parameters are given in Table IV. For Ganymede I fit only the Voyager 2 diurnal curve, as the Voyager 1 curve is much less `well-behaved' (Chapter 5).
These fits are poor, except for Europa where only a small portion of the diurnal curve is available. Both the Ganymede and Callisto fits suffer from temperature maxima that occur too long after midday. It is known from the work of Morrison and Cruikshank (1973), and Hansen (1973), (`MCH' henceforth) that homogeneous models cannot match the eclipse cooling and heating curves, and no laterally-homogeneous surface will give the observed IRIS spectrum slopes described in Chapter 4. I therefore move on to more sophisticated surface models.

Table IV. Best-Fit Thermal Model Parameters

These are the best-fit parameters obtained by `eyeball' techniques. The units of thermal inertia are erg cm-2 s-1/2 K-1, and the units of heat capacity per unit area are erg cm-2 K-1. Estimated errors are subjective and should be treated with caution as variations in parameters are often strongly correlated.
 
Top Layer or Component 1
Lower Layer or Component 2
Body Surface Albedo Thermal Inertia Heat Capacity
Area-1		 Areal Coverage Albedo Thermal
Inertia
Europa 1-Layer 0.72 +/-.02 5 +/-.5 X 104
2-Layer 0.70 +/-.02 1.6 +/-.2 X 104 4.3 +/-2 X 105 3 +/- 2 X 105
Ganymede 1-Layer 0.32 +/-.04 7 +/- 2 X 104
2-Layer 0.30 +/-.04 2.2 +/-.2 X 104 1.6 +/-.2 X 106 5 +/- 1 X 105
2-Component 0.10 +/-.05 1.6 +/-.6 X 104 0.5+/-.1 .50+/-.05 1 +/- .5 X 106
Callisto 1-Layer 0.20 +/-.04 5 +/-1 X 104
2-Layer .020 +/-.04 1.5 +/-.2 X 104 1.5 +/-.2 X 106 3 +/- 2 X105
Morrison and Cruikshank (1973) parameters:
Ganymede 2-Layer         - 1.4 +/-.2 X 104 1.4 +/-.3 X 106 >3 X 105
Callisto 2-Layer         - 1.0 +/-.1 X 104 1.0 +/-.1 X106 >3 X 105
Hansen (1973) parameters:
Europa 2-Layer         - 1.4 +/-.5 X 104 9.7 +/-9 X105 >3 X 105
Ganymede 2-Layer         - 1.2 +/-.3 X 104 8.3 +/-4 X106 >3 X 105

2-Layer (Vertically Inhomogeneous) Surfaces

Eclipse Thermal Models

MCH very accurately matched the thermal behavior of Ganymede and Callisto during eclipse by Jupiter with 2-layer surfaces in which a thin, very low conductivity layer overlies a substrate with the thermal inertia of solid rock or ice. Europa fails to return to its pre-eclipse temperature after re-emergence and the full eclipse curve was not successfully matched, but still seemed to require a 2-layer surface.
Their models, though successful, were very simple: Hansen assumed a flat disk, isothermal with depth before the eclipse, and simultaneous eclipse over the whole surface. Morrison and Cruikshank calculated cooling in four annuli concentric to the subsolar point, each initially in equilibrium with sunlight, and had the leading hemisphere of the satellite enter eclipse before the trailing hemisphere. However, they still started with surfaces that were isothermal with depth in each annulus.
I have used a much more complex thermal eclipse model (Appendix C), which calculates cooling and heating curves for 52 individual points on the satellite surfaces, with a realistic starting temperature profile with depth and correct timing of the eclipse at each point. This model shows that the initial isothermal-with-depth assumption is invalid for 2-layer surfaces. During the day there is a flow of heat from the surface down through the thin low-conductivity upper layer to the high thermal inertia substrate, which sets up a strong temperature gradient in the upper layer. The Morrison and Cruikshank thermal parameters for Ganymede give a subsolar pre-eclipse surface temperature of 150oK, but a temperature at the base of the upper layer of only 115oK. As a result, cooling of the surface during eclipse is much more drastic than if, as assumed by the previous work, the whole upper layer had been at 150oK initially. The difference is shown on Figure 25, which compares the observed 10- and 20-microns eclipse curves for Ganymede with the the curves derived using the Morrison and Cruikshank parameters in the improved thermal model (dashed line). Using the improved model their thermal parameters no longer give a good fit to the data, and result in excessive cooling during eclipse.
The MCH 2-layer models are thus quantitatively incorrect. Fig. 25 also shows my best fit to the Ganymede and Callisto 10- and 20\dmic\ eclipse curves using the improved 2-layer model. The thermal parameters, along with the MCH values, are given in Table IV. The lower-layer thermal inertias were chosen to give the best fit to the Voyager diurnal curves: see the next section. Compared to the previous models rather thicker upper layers with about 50% higher thermal inertia are required. The Ganymede fit is not as good as the MCH fits: it is hard with my model to get sufficient 10-micron cooling without getting excessive 20-micron cooling. It isn't clear why the improved 2-layer model should be less able to fit the data than the original simple (and incorrect) models. Perhaps the enhancement of thermal emission in a sunward (and earthward) direction due to beaming, which is not included in any of the models and would be very difficult to include, is responsible for the inadequacy of the current Ganymede model. Sub-surface penetration of sunlight, a possibly important effect that I do not consider, (Brown and Matson, 1987) might also be responsible for the poor fit. Alternatively, the 2-layer surface may not, after all, be a good model for Ganymede. The Callisto fit, in contrast, is excellent.

Diurnal Thermal Models

Figure 26 shows the diurnal temperature curves for Ganymede and Callisto obtained using the current best 2-layer fit to the eclipse curve. The thermal inertia of the lower layer, which affects the eclipse curve only slightly and is thus poorly constrained by the eclipse data, is much more important for the diurnal curve (because of the deeper penetration of the diurnal wave), and was chosen to give the best fit to the diurnal temperatures. The thermal inertia of solid ice at 130oK is 2.2 X 106 c.g.s. (Hansen, 1972), and varies little with temperature, so the best-fit values for the thermal inertias of the substrate are consistent with somewhat porous or fractured ice. For Ganymede the 2-layer diurnal curve is a definite improvement over the 1-layer models. Incidently, the diurnal curves calculated using the MCH 2-layer thermal parameters (not shown) are poorer fits to the diurnal curve, being 10oK too cold after sunset. For Callisto, the fit is better than the 1-layer version of Fig. 24 during the day, being slightly more symmetrical about midday. The pre-dawn cooling rate is better matched by the 1-layer model, however, and cannot be matched by eclipse-consistent 2-layer models. Late afternoon temperatures on Callisto are much colder than either the 1- or 2-layer fits, though the 2-layer fit is better. Is seems likely, then, that Callisto's thermal properties are anomalous in this region (around 330o longitude).
Regional Thermal Inertia Variations on Ganymede. As described in Chapter 5, the Voyager 1 Ganymede temperature distribution shows `cold spots' not related to albedo, that may be due to thermal inertia variations. If this is the explanation, what magnitude of thermal inertia variations are required A precise answer is not possible without diurnal curves for each anomalous region, but the post-sunset cold spot (Fig. 13) is consistent with a regional reduction in top-layer thickness and thermal inertia of about 30%, or a reduction in lower-layer thermal inertia by a factor of about 2. The midday Voyager 1 cold spot north of the equator, with a 10oK temperature supression, is most easily achieved with a regional increase in lower-layer thermal inertia of a factor of maybe 3, approaching solid-ice values, but this may not be realistic and I have not thoroughly explored other ways of achieving the low temperatures here.
I have not attempted to match the Europa eclipse data with a thermal model. However Fig. 26 shows a good 2-layer match to the available portion of the diurnal curve. The match shown has a higher thermal inertia upper layer than Hansen's (1973) `nominal' Europa fit, obtained with the simple (and incorrect) 2-layer eclipse thermal model, but is within his very wide confidence limits. So even on Europa there is an indication that, as on Ganymede and Callisto, the best 2-layer model has higher thermal inertia in the upper layer than previously supposed.
Fig. 26 also shows the diurnal temperatures obtained by assuming instantaneous equilibrium with sunlight on the three objects. The albedos used are the same as for the 2-layer fits shown on the same figure. Midday temperatures on Europa are 15oK below equilibrium values if the 2-layer fit is realistic and, more reliably, there is a 10oK midday temperature supression on Ganymede. On Callisto, because of the long rotation period and lower fitted thermal inertias than Ganymede, midday temperatures are only about 5oK below equilibrium values. It is often said that the low surface thermal inertias on the Galilean satellites, inferred from the eclipse cooling, imply daytime surface temperatures very close to equilibrium values, but Fig. 26 shows that this statement is not very accurate.
The 2-layer model thus fits the eclipse and diurnal curves quite well. However it is still inadequate in that it does not predict local surface temperature contrasts and thus cannot explain the IRIS spectrum slopes. A rough 2-layer surface might explain the Callisto spectrum shapes in addition to the temperatures (Chapter 6). However on Ganymede, where it appears that topography cannot explain the 40oK temperature contrasts seen in the warm, near-subsolar spectra, another model is required.

2-Component (Laterally Inhomogeneous) Surfaces

This section considers surfaces in which both albedo and thermal inertia vary laterally. I will be concerned mostly with Ganymede because it is on Ganymede that there is the greatest problem explaining the IRIS spectrum slopes with topographic models. Also Ganymede, with its intermediate albedo, is the most likely object to have a large areal coverage of two materials of contrasting albedo. However, I will finish with a brief consideration of the possible contribution of lateral inhomogeneities to the Callisto spectrum shapes. See Chapter 10 for a discussion of the relevant spectroscopic evidence.
For simplicity I will only consider surfaces in which each component is vertically homogeneous. This still leaves four free parameters: the albedo and thermal inertia of each component. Their relative areal coverage is fixed if the overall albedo is assumed to be known; if it is not known then relative areal coverage (or overall albedo) is a fifth free parameter. Calculation of eclipse curves, diurnal curves, and thermal spectra for a 2-component surface is straightforward: I calculate the curves for each component, and obtain the radiance from the surface from the mean of the radiances from the two components, weighted by their relative areal coverage. Effective temperature TE of the surface at a given time, if the two components have effective temperatures TEi and fractional coverages Xi, is simply given by

Eclipse Thermal Models

Though Morrison and Cruikshank (1973) were able to set stringent upper limits on the surface coverage of high thermal inertia material on Ganymede and Callisto (5% and 1% respectively), this conclusion requires the assumption that the whole surface has the same albedo. G.H. Rieke and R.R. Howell (personal communication) have suggested that a 2-component surface with a high thermal inertia bright component and a low thermal inertia dark component might explain the eclipse curves as well as did the MCH 2-layer model. The rapid initial cooling would be due to the low thermal inertia dark component, which contributes most of the daytime flux, and the later slower cooling as the eclipse progressed would be due to the brighter, higher thermal inertia material, which would become the dominant source of flux as the dark material cooled. Figure 27 shows the best-fit 2-component eclipse curve that is also consistent with the diurnal curve (see below). The parameters are given in Table IV: they specify a 50/50 mixture of bright and dark components with albedos 0.5 and 0.1 (giving a reasonable mean albedo of 0.3). The fit is excellent except that the post-eclipse recovery at 20 microns is too slow. Other similar fits did not suffer from this problem but were poorer matches to the diurnal curve. If nothing else, this model demonstrates that Rieke and Howell were correct in their suggestion that an alternate explanation of the eclipse curves was possible.

Diurnal Thermal Models

Figure 27 also shows the diurnal curve obtained from the 2-component match to the Ganymede eclipses shown in the same figure. The fit to the Voyager 2 Ganymede data is good except for the post-sunset cooling rate, which is more constant in the data than in the model. This figure also shows the temperatures of the two components. The bright component has a thermal inertia of 1 X 106 , half that of solid ice, and an almost constant diurnal temperature, while the dark component temperature curve has large amplitude.

Spectrum Slopes

The surface temperature contrasts seen in Fig. 27 will give rise to slopes in the thermal emission spectrum. In Fig. 28, I show the fitted temperature contrast and spectrum slope (difference in TB between 20 and 40 microns) as a function of time of day, obtained by calculating the emission spectrum from the 2-component surface and subjecting the resulting spectrum to the same least-squares fitting technique used on the IRIS spectra in Chapter 4. The actual shapes of the Ganymede IRIS spectra are shown for comparison. The fit of the model spectrum shapes to the data is hardly impressive, but at least the magnitude of the daytime slopes is approximately correct. The predicted slope becomes small just before sunset, where the two component temperature curves cross (Fig. 27) and the surface is momentarily isothermal, and then increases rapidly into the evening. No such effect is seen in the data.
Possibly the 2-component surface of Figs. 27 and 28 could be made to fit the spectrum shape data if the effects of topography were included. Topography increases the surface thermal contrasts near the terminator (Chapter 6) and would increase the spectrum slope in the region where it is too small, as shown by the arrows on Fig. 28. However it is impossible to know whether this mechanism would work quantitatively without constructing a proper topographic temperature model.
The 2-component model certainly fits the spectrum shapes better than the best-fit 2-layer model which, with the assumption of a wavelength-dependent emissivity of 0.94, plots as the dashed line shown on each plot.
Though it is possible to construct 2-component surface models that fit the spectrum shapes of Fig. 28 (by giving each component a similar thermal inertia so that the dark component is always warmer and the surface is never isothermal), these models are not consistent with the eclipse curves if the two components are vertically homogeneous. However, a 2-component surface in which one or both components had two layers could probably be found that would fit all the data. Sinton and Kaminski (1987) have recently matched the thermal eclipse behavior of Io with such a model. I have not investigated models that are both laterally and vertically inhomogeneous: the number of free parameters is so large that with my `eyeball' techniques it is unlikely that I could find a unique solution. However it is quite possible that the surface of Ganymede really is like that.
An observation that may support the 2-component model for Ganymede is the shallow slope of the thermal emission spectra of the very bright region around the crater Osiris (Fig. 11, Chapter 4). If Ganymede does have a 2-component surface, and the observed macroscopic albedo patterns result from varying relative areal coverage of the two components, then a very bright region such as the Osiris ejecta blanket would be dominated by one component (the bright one). This would reduce the surface thermal contrasts and thus the spectrum slopes, as is observed.

A 2-Component Surface on Callisto?

Though the thermal data for Callisto seems to be matched quite well by a rough, horizontally homogeneous surface, two observations suggest that this cannot be a complete picture.
Firstly, Figs. 20 and 21 (Chapter 6) show that the warm Callisto spectra are not completely matched by the topographic temperature models. Model A is the most successful match to the warm spectra, but while it fits the Voyager 1 data well, the high-sun Voyager 2 spectra are steeper than the model. Model B is a poor fit to the warm Callisto spectra from both encounters: standardized spectrum slopes are matched fairly well but the fitted temperature contrasts are much larger than model B predicts, for both encounters. The crudeness of the models, and the differences in the thermal emission behavior that each predicts, make it hard to draw any firm conclusions from the mismatch to the warm spectra. However, a face-value interpretation would be that there are local temperature contrasts of 10--20oK in the subsolar regions of Callisto that are in excess of what can be explained by topography alone, especially in the Voyager 2 data (longitudes 180o--240o W). Additionally, the steep nighttime spectra on Callisto probably cannot be explained by topography alone, though this would be difficult to check. A degree of surface segregation into two components would be an obvious way to produce the extra temperature contrasts.
Secondly, Chapter 10 shows that Callisto's reflectance spectrum can be best matched with a segregated, two-component surface of bright, icy material, covering about 10% of the surface, and dark, carbonaceous-like material occupying the other 90%.
I have not attempted a simultaneous match of a 2-component surface to all data sets for Callisto. However, I have looked at a two simple models, with the same surface components as the `best fit' 2-component Ganymede match (Table IV), but with a 90%/10% and an 80%/20% mix of the dark and bright components, rather than the 50%/50% mix used for Ganymede. These models have the following characteristics:
1) The 90/10 model does not fit the diurnal temperature profile, being too hot at midday, but the 80/20 model fits the Callisto temperatures as well as does the 2-layer match of Fig. 26. The mean albedo of the 80/20 model, 0.18, is also a good match to Callisto.
2) Figure 29 shows the spectrum slopes and fitted temperature contrasts for each model. When the subsolar values are added to the predictions of model B (Fig. 20) they give spectrum shapes closer to those observed on Callisto (especially the steeper Voyager 2 data) than are obtained using the topographic model alone (it is legitimate, to first order, to add slopes and temperature contrasts in this way). The slopes predicted by the 80/20 model, when added to the topographic effects, might be steeper than the observed Callisto subsolar slopes, and the 90/10 model might be a closer match, but this depends on whether topographic model A or B is preferred. Also, both 2-component models predict very steep nighttime spectra on Callisto, near the upper limit of the observed nighttime spectrum slope range (Fig. 29). The 90/10 and 80/20 2-component models thus account for the steeper nighttime spectra on Callisto than on Ganymede, which is matched quite well at night by the 50/50 2-component model (Fig. 28).
3) Unfortunately, the fit to the eclipse cooling curve is poor: both 2-component models predict much slower 20\dmic\ cooling than is actually exhibited by Callisto.
As in the case of Ganymede, it is probably possible to find a 2-component model for Callisto that is consistent with the subsolar and nighttime emission spectrum shapes, the diurnal temperature profile, the eclipse cooling, and the reflectance spectrum. I suspect that at least one of the surface components will have to be vertically inhomogeneous in order to match the eclipse behavior. The search for such a model is a major task that I have not attempted, but the above discussion suggests that the addition of up to 20% coverage of a bright component on Callisto's surface can help to explain the observed shapes of the subsolar and nighttime spectra, and is consistent with the reflectance spectrum. Because my simple 2-component surface models are such a poor fit to the eclipse behavior I am not elevating either to the status of a `best fit' to be included in Table IV, but they point the way to better models in the future.

Lengthscales

I now briefly discuss the physical distance over which heat can be conducted during the course of a day on the icy Galilean satellites. This is an important parameter because it provides a crude lower limit to the size of thermally-significant topography (Chapter 6) and the scale of possible lateral segregation of the surface (this chapter and Chapter 8).
The characteristic lengthscale for thermal conduction is the skindepth, given by

where K is thermal conductivity, $\rho$ is density, c is specific heat capacity, $\sqrt{K\rho c}$ is the thermal inertia, and tD is the length of a day (Appendix C). This equation shows that s cannot be determined uniquely from the thermal inertia listed in Table IV without assuming a value for K or $\rho c$ individually.
c for water ice at 120 K is 107 erg oK-1 g-1 (CRC Handbook), and \rho will vary from 0.9 g cm-3 for solid ice to perhaps 0.3 g cm-3 if the ice is porous. \sqrt tD varies from 550 s-1/2 for Europa to 1200 s-1/2 for Callisto. So for porous ice with a thermal inertia of 2 X 104  c.g.s., s is about 7 cm, while at the other extreme almost solid ice with a thermal inertia of  1 X 106  will give s around 100 cm. These distances will apply to horizontal as well as vertical conduction, if there are large horizontal temperature gradients as might occur at the boundary between the light and dark components on the 2-component model.

Questions not Addressed

This thermophysical modelling of the IRIS data has been far from exhaustive. I have not covered several interesting aspects of the data, such as: Why are the south polar Ganymede spectra so steep How different are the thermal properties of the dark cratered and bright grooved terrain on Ganymede Are the diurnal temperature curves at high latitudes consistent with the thermal parameters derived from fitting the equatorial data Are the global temperature distributions determined from the data and the modelling consistent with the disk-integrated out-of-eclipse thermal emission observed from earth (Beaming effects complicate this last question). These questions must wait for further analysis.

Table V. Surface Thermal Models vs. The Data: A Summary

The table shows how well the various pictures of the surfaces of the icy Galilean satellites considered in Chapters 6 and 7 match the thermal infrared evidence. I indicate the consistency of each surface with the IRIS diurnal temperature profile, the ground-based eclipse curves, and the IRIS spectrum slopes. `Y' indicates a good fit, `M' a moderately good fit, and `N' a poor fit. `p' means the data is probably consistent but that my modelling is insufficient to be sure. I assume that surface roughness affects only the spectrum slopes and does not otherwise alter the thermal behavior of the surface. The 2-component models are those discussed in this chapter, which assume that each component is vertically homogeneous.
 
Satellite
Surface
Topography
Diurnal
Eclipse
Spectrum Slopes
Europa
1-Layer
Smooth
Y
N
M
   
Rough
Y
N
P
 
2-Layer
Smooth
Y
P
M
   
Rough
Y
P
P
Ganymede
1-Layer
Smooth
N
N
N
   
Rough
N
N
N
 
2-Layer
Smooth
Y
M
N
   
Rough
Y
M
N
 
2-Component
Smooth
Y
M
N
   
Rough
Y
M
P
Callisto
1-Layer
Smooth
M
N
N
   
Rough
M
N
P
 
2-Layer
Smooth
M
Y
N
   
Rough
M
Y
P
 
2-Component
Smooth
Y
N
N
   
Rough
Y
N
P

Summary of the Results of the Thermal Modelling

Table V summarizes the qualitative results of this chapter. My judgements of the goodness of fit of particular model are subjective, and can generally be checked by reference to the relevant sections and figures of this chapter. I also draw on the results of Chapter 6 in judging the rough-surface models. I assume that the addition of topography will not affect the diurnal and eclipse models substantially, and will significantly affect only the spectrum slopes. Refer to Table IV for the quantitative details of the various fits. The Europa data seems most consistent with a surface that is smooth, because of the small near-terminator spectrum slopes, and 2-layer, because of the eclipse results.
Ganymede probably requires a 2-component surface (about equal surface coverage of dark, low thermal inertia and bright, high thermal inertia meterial) to match the three data sets, but the near-terminator spectrum slopes are much steeper than predicted by such a model if the surface is smooth. It is not yet known whether the addition of topography to the model could correct this discrepancy and produce the observed almost-constant slopes. A 2-component surface model in which one or both components is vertically inhomogeneous is another possibility that needs to be checked.
Callisto is matched quite well by a 2-layer, rough surface, though the fit to the diurnal curve is not perfect. Spectrum slopes are probably due mostly to topography. There could be a small amount of cold bright material on the surface: it might improve the fits to the subsolar spectrum slopes and might explain the steepness of the nighttime spectra, if the bright material had high thermal inertia.