The model obtains the temperature

where $\rho$ is density,

which represents the balance between the conductive heat flux from below, on the left-hand side of the equation, and the thermal radiation (first term on the right hand side: \ep\ is the emissivity and $\sigma$ is the Stefan-Boltzmann constant) and absorbed sunlight (second term on the right:

The boundary condition at the base of the

which implies no spatial temperature gradient at the lower boundary and therefore, from Equation 22, no temperature change with time there either. The constant temperature condition is achieved by having the lower boundary deep enough that it is below the diurnal temperature variations, which decay exponentially downwards with characteristic `skindepth'

I chose the `deep temperature'

and 23 becomes

All the physical parameters are combined into one, $\sqrt{K\rho c}$, the thermal inertia, which is the sole determiner of surface temperature for a given albedo, insolation and rotation rate and a vertically-homogeneous surface. I solved the equations in this form, with thermal inertia, length of day, albedo, and latitude (which contributes a cosine factor to

I obtained surface temperature from

which is adapted from Equation 23 in a non-obvious manner (see Carnahan

As stated above, I ran each model twice to determine the correct deep temperature. Each run started at midnight with uniform temperature with depth, and continued for 4 rotations until the diurnal temperatures stabilized and `forgot' the initial conditions. The surface temperatures on the final rotation of the second run constituted the solution to the thermal model.

1) I determined the pre-eclipse temperature profile with depth at 52 points approximately evenly spaced across a projected hemisphere of the satellite. To do this I ran a diurnal thermal model as described above for the 6 latitudes 5.7

2) For each of these 52 locations I interpolated the temperature profile, if necessary, to give slabs that were as thin as the skindepth for a time of 1.5 X 10

3) I used the interpolated temperature profile at each point as the starting point for a thermal model (using Equations 27 and 28) in which the insolation

4) I combined the 52 temperature curves with a program that weighted each curve by its projected area (which varied as the satellite rotated) as seen at zero solar phase angle, and offset them in time to simulate the passage of Jupiter's shadow across the satellite disk. See Fig. 44, which reproduces a screen output from this program, showing the temperature at each of the 52 points over the hemisphere as the satellite begins to enter Jupiter's shadow. For each timestep, I determined the disk-integrated radiance in the 10- and 20\dmic\ filter passbands given in Hansen (1972).

This model is thus very detailed, and probably suffers from overkill in several respects. Less than 52 surface regions would probably have been sufficient, for instance (experiments with 74 regions gave essentially the same results as 52). However the program results have shown that the use of a realistic pre-eclipse temperature/depth profile is very important in some circumstances, and previous models that have assumed isothermal pre-eclipse conditions have been inaccurate. See Chapter 7.

In calculating the disk-integrated emission I assumed that the Earth-based observations were taken at zero solar phase angle. The true phase angle of the Callisto event that I modelled was 8.5

Beaming (Appendix A) is ignored in my model. This may be a serious limitation: until we understand beaming fully we cannot predict how it might vary during eclipse. This is probably the largest inadequacy of the present eclipse modelling and will be very difficult to correct, though a better description of the beaming on the Galilean satellites would help in estimating the size of the problem.

The subsurface penetration of sunlight, which is likely on icy surfaces and results in something akin to an increased thermal inertia (Brown and Matson 1987), is another missing element, but one that would be easier to include than beaming.