This appendix describes the numerical thermal models used in Chapter 7 to match the observed diurnal temperature distributions on the Galilean satellites, and also the cooling and heating behavior during eclipse by Jupiter. The diurnal models are described first.

Diurnal Thermal Models

Numerical models of diurnal conductive heat flow have been used to determine planetary surface temperatures for many years. They have been applied to such tasks as the radiometric determination of asteroid and satellite diameters (Lebofsky et al, 1986), determination of rock abundance on Mars (Christensen 1986), and water ice stability on the Galilean satellites (Squyres, 1980). My model is similar to most others but differs in detail, and I describe it here for completeness.
The model obtains the temperature T as a function of depth below the surface x and time t by numerically solving the one-dimensional conductive heat flow equation
  where $\rho$ is density, c is specific heat capacity, and K is thermal conductivity. $\rho$, c, and K may vary with depth, but at Galilean satellite temperatures they do not vary significantly with temperature. The equation is evaluated over the x interval from 0, at the surface, to some depth d.

Upper and Lower Boundary Conditions

The boundary condition at the surface is

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: A is the bolometric albedo and FS the time-dependent flux of sunlight). I ignore the possible variation of A with solar incidence angle (Squyres and Veverka, 1982), and also the possible subsurface penetration of sunlight (Brown and Matson, 1987).
The boundary condition at the base of the x interval is that there should be no heat flux through the boundary, i.e.

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' s=\sqrt{(K t_D) / (\rho c)}$, where tD here is the length of a day. I generally used d=3.2 for homogeneous surface models.
I chose the `deep temperature' T(d,t) by using the fact that for c and K independent of T, the `true' deep temperature (the value for which there is no heat flow through the base of the model) is equal to the diurnally-averaged surface temperature. I ran every model twice, first with an estimated deep temperature and second with a deep temperature equal to the mean surface temperature from the first run, with a correction factor based on the imbalance of radiated thermal and absorbed solar radiation at the surface in the first run (too high a deep temperature raises the mean surface temperature and thus gives an imbalance in absorbed and re-radiated energy at the surface). As a result, thermal balance at the surface on the second run was always excellent, equivalent to an error in mean surface temperature of less than 0.05oK.


If x is replaced with X, defined as x/s, (i.e. if depth is measured as the number of skindepths below the surface), and K, $\rho$ and c are assumed constant with x, then Equation 22 simplifies to

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 FS) as the input parameters

Numerical Solution

For the vertically-homogeneous case, I divided the subsurface into 32 `slabs', each with an X-thickness \delta X of 0.1 (to give 3.2 skindepths total), and calculated the heat flow from each slab to the next over successive time increments $\delta t$ using the `discrete' equivalent of equation 25, i.e.

I obtained surface temperature from

which is adapted from Equation 23 in a non-obvious manner (see Carnahan et al (1969) for details). I chose the fewest number of timesteps that gave a stable numerical solution: typically this was around 600 timesteps per rotation ($\delta t = tD /600$). Using smaller values of $\delta X$ and $\delta t$ did not change the solution significantly.
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.

2 Layer Models

The above discussion refers to vertically-homogeneous models, but I also ran models with a low thermal inertia layer overlying a higher thermal inertia substrate. For a 2-layer surface there are three physical parameters rather than one: the thermal inertias of the two layers and the thickness of the top layer, expressed as a heat capacity per unit area ($\rho c xt, where xt is the top-layer thickness). Calculation of surface temperatures is similar to that for the homogeneous case, except that the top layer is divided into a fixed number of slabs, usually 6, while the lower layer temperatures are calculated with $\delta X=0.1$ as before. For most 2-layer runs this gave top-layer slabs much thinner than for the homogeneous case, necessitating up to 30,000 timesteps per rotation for stability in some cases.

Eclipse Thermal Models

The physics of the eclipse modelling is identical to that for the diurnal modelling described above, except that the insolation FS(t), instead of varying smoothly during the day, drops rapidly to zero from some `normal' daytime value, stays there awhile, and then sharply recovers to something like its pre-eclipse value, as the sun is eclipsed by Jupiter. Most of the complexities of the eclipse modelling involve calculating the timing of the eclipses, and combining temperatures to get a disk-integrated flux as a function of time. The procedure I employed was as follows:
    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 5.7o, 17.5o, 30.0o,     44.4o, 58.2o, and 71.8o (sin-1 0.1, 0.3, 0.5, 0.7, 0.85, and 0.95 respectively). I recorded the temperature profile at various times during each diurnal run: specifically 12, 12, 10, 8, 6, and 4 times for successively higher-latitude bands. The times chosen were symmetric about the noon meridian and approximately evenly spaced across the projected hemisphere (Fig. 44).
    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 10-4  satellite days. Such thin slabs were necessary in order to accurately track the response to the change in sunlight at the beginning and end of the eclipse, as in a central eclipse the sunlight fades and returns at each point over a timespan of 2.9 X 10-4  satellite days, due to the finite angular diameter of the sun.
    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 FS(t) immediately began to fade, remained at zero for the duration of the eclipse, and then returned. I generally ran this eclipse model with higher time resolution than the diurnal model, with a timestep $\delta t$ no larger than 6 X10-5 satellite days. I modelled the fading and return of the sunlight on the assumption that the sun was a uniform-brightness disk being occulted by a straight edge: sunlight scattered in the Jovian atmosphere was ignored, as justified by the photometric eclipse observations of Price and Hall (1971). I continued the model after the eclipse for a time equal to the eclipse duration. The 52 surface temperature vs. time curves were stored.
    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.

Limitations of the Eclipse Model

The current program cannot model non-central eclipses, where Jupiter's shadow crosses the satellite obliquely. This would require consideration of both hemispheres of the satellite. In the 20\dmic\ Ganymede eclipse matched in Figure 25 of Chapter 7, for instance, Jupiter's shadow crossed Ganymede at an angle 52o from Ganymede's spin axis, and the Callisto eclipse the corresponding angle was 68o. I simulated an oblique eclipse by increasing the satellite diameter and penumbra width so that the time taken for the satellite to cross the penumbra was correct even though I was assuming parallelism of the shadow and spin axis.
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 8.5o, with the morning hemisphere being preferentially in view. However, the Ganymede data I attempted to fit came from 3 separate events with different phase angles, and the scatter of the Ganymede data (Figure 25), is small enough that phase effects do not seem to be too important.
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.