This appendix discusses what is known about the anisotropy, or beaming, of the thermal emission from individual regions of planetary surfaces. It also describes the measurement of beaming on the Galilean satellites, using the IRIS data.

Lunar Observations

Most of the published observations of the directionality of thermal emission have concerned the Moon (see Saari and Shorthill, 1972, and Saari et al, 1972 for a summary). It was found that:
    1) Emission from a given point peaks in the direction of zero solar phase, and decreases with increasing phase angle.
    2) The zero-phase enhancement is greater at larger solar incidence angles, so that the brightness temperature of the full Moon decreases more slowly from the center to the limb than expected on the basis of equilibrium temperatures and isotropic emission.  Unfortunately all the published lunar studies describe the 10\dmic\ atmospheric window only: there is no available data in the 20\dmic\ window to give information on the wavelength dependence of the beaming effect. Observations of beaming at a variety of wavelengths were made for both Mercury and the Moon by Murdock (1974), but these were disk-integrated measurements that are difficult to compare with the disk-resolved Voyager data.
Lunar beaming has been successfully modelled as being a consequence of local surface topography (e.g. Winter and Krupp (1971)). Sunlit depressions receive radiation (scattered solar and re-radiated thermal) from the walls around them as well as directly from the sun, and are thus warmer than nearby elevated points. An observer at low phase angles will receive the extra radiation from the warm depressions, whereas observations of the same area at high phase angles will preferentially see the cooler elevations, and shadowed regions that may be cooler still. Chapter 6 and Appendix B discuss (and partially reproduce) these models, in an attempt to explain the variations in spectrum shape with temperature that were described in Chapter 4.

Measuring Beaming on the Galilean Satellites

There are two motivations in the current work for determining the anisotropy of thermal emission from the Galilean satellites. The first is to compare with the known lunar beaming, and with theoretical modelling (Chapter 6) to see whether the beaming places any useful constraints on surface properties. The second is to see if it is possible to use the knowledge of beaming to remove the effects of changing viewing geometry from the IRIS data. Because the Voyager IRIS coverage of the Galilean satellites is inadequate to fully characterize the anisotropy of the thermal emission, this second goal also requires comparison with the Moon to fill in some of the gaps.
The Saari et al (1972) lunar data refer only to a wavelength of 11 microns, and thus give no information about the wavelength dependence of the beaming. Hansen (1977), however, using a topographic thermal model (see Chapter 6 and Appendix B), calculates (disk-integrated) beaming at a range of wavelengths. He shows that wavelength can be parameterized as the ratio $\lambda/\lambda_{max}$, where $\lambda_{max}$ is the wavelength of peak thermal emission from the planetary object at zero phase. For the Saari et al observations, he gives this ratio as about 1.2 (in his Fig. 5): for Ganymede and Callisto the wavelength corresponding to this ratio is about 26 microns. This is the appropriate wavelength of the IRIS observations to compare with the lunar data.
I measured beaming on the Galilean satellites by comparing overlapping IRIS scans of a single region taken from different viewing geometries. The temperature change between the two scans, when corrected for `real' temperature changes due to the rotation of the body, gives a measure of the directional dependence of the emission. I concentrated on measuring variations in thermal emission with phase angle, as lunar beaming depends mostly on this variable (Saari et al, 1972). To be detectable, temperature variations due to beaming must be larger than those due to pointing uncertainties. The Voyager 2 Ganymede data, having the highest resolution and best pointing accuracy (after correction) of the icy satellite data sets, were thus used for this purpose. Even here, the data is barely adequate to quantify the beaming effects. I also made one measurement on Callisto for comparison, though here the resolution is poorer still.

Table VII. Observations of Beaming on Ganymede and Callisto

26 mic m TB
Lat. Long. alpha alpha2 Local Temp. Rotional
c X 103 Observed
c X 103 Lunar 11mic m
c X 103
G1 -20 190 34 44 135 +0.13 -2.0 -2.0 -1.47 -1.5 -1.5 -1.11 -0.79
G2 -20 190 44 81 135 +0.03 -5.0 -5.0 -1.01 -3.0 -3.0 -0.60 -1.18
G3 -30 150 44 59 122 -0.67 -3.0 -3.0 -1.38 -3.0 -2.5 -1.38 -1.20
G4 -55 190 50 101 120 +0.13 -7.0 -7.0 -1.15 -6.0 -6.0 -0.98 -
G5 -60 160 48 66 107 -0.32 -4.0 -4.0 -1.81 -3.0 -2.5 -1.30 -
G6 -30 150 59 81 120 -0.70 -3.0 -3.0 -0.95 -2.0 -1.5 -0.57 -1.02
C1 +35 10 50 79 137 -1.10 -3.0 -.073

 See text for details. alpha1 and alpha2 are the two phase angles at which temperatures were compared to obtain the thermal phase coefficients c.
In the southern hemisphere of the Voyager 2 region of Ganymede, I found six instances of overlapping coverage of the same region at a pair of phase angles. Results for these instances (G1--G6) are listed in Table VII, along with the Callisto measurement (C1). For each instance, using plots showing brightness temperature for each spectrum as a function of geographical position, I drew isotherms by hand for each phase angle. I then recorded the offset between the two sets of isotherms, corresponding to the temperature change between the pair of observations. In each case the offset was in the expected sense, with higher phase angle observations being cooler, providing some confidence that I was observing a real effect. I determined the offset for both 26\dmic\ TB, for comparison with the 11\dmic\ lunar data, and for the wavelength-integrated TE.
For Ganymede, I corrected for the `real' cooling of the surface between the observations, due to the satellite rotation (all the data comes from afternoon regions), by reference to theoretical diurnal temperature profiles. Direct inference of the afternoon cooling rate, from the observed temperature distribution as a function of local time, is probably unreliable on Ganymede, because of the quite strong perturbations due to local albedo variations. At the time that I was doing this analysis, my best fit to the Ganymede diurnal temperature profile was a single-layer thermal model with a thermal inertia of 105 erg-cgs and an albedo of 0.3. I ran this model for each latitude in Table VII to determine the expected surface cooling during the time interval between the overlapping scans.
A problem with this approach is that this 1-layer model predicts a diurnal temperature peak that is significantly longer after midday than is observed on Ganymede. This means that the temperature is predicted to be still rising slightly for locations G1, G2, and G3, while in reality there was probably a slight cooling during the observations. Fortunately, as Table VII shows, the rotational corrections are much smaller than the observed phase-related reductions in brightness temperature so the effect of this error is small. In the three cases where the thermal model predicted a temperature rise, I applied no rotational correction (constant surface temperature was assumed).
Subsequent thermal modelling (Chapter 7) gives an improved fit to the Ganymede diurnal temperature profile, but I have not repeated the rotational corrections because their small magnitude does not justify an improved calculation.
On Callisto, where albedo variations are much smaller, I used the actual observed afternoon cooling rate, deduced from the isotherm spacing, to determine the expected surface cooling between the pair of observations.
I define a `thermal phase coefficient', c, as the fractional change in temperature per degree of phase angle, so that for two values of TE (or TB) at two phase angles

c is tabulated in Table VII. It varies by almost a factor of two between the various cases. Table VII compares the 26\dmic\ c values for cases G1, G2, G3, and G6, which are quite close to the thermal meridian, with those for corresponding geometries at 11 microns on the Moon, derived from both the observations of Saari et al Fig. 13, and the theoretical model of Winter and Krupp, Fig. 9. It can be seen that there is no good correspondence in the variations of c with viewing geometry between the lunar and Ganymede data. This suggests, if the beaming on Ganymede is similar to that on the Moon, that the twofold variability in c seen on Ganymede is a result of observational error rather than a true dependence on viewing geometry. Such large errors are not suprising considering the pointing uncertainties and resolution of the data. The single value of c for Callisto is probably equally or more inaccurate. However, the Ganymede and Callisto thermal phase coefficients are of the same magnitude (and sign!) as those observed on the Moon, and are therefore probably real.

Possibility of Corrections for Beaming Effects

Because of the probable factor of two (at least) uncertainty in the c values, and the limited range of geometries for which they are available, I decided that an attempt to correct the Ganymede and Callisto effective temperatures for viewing geometry would not be worthwhile. There would be too much danger of overcorrecting and obtaining `corrected' temperatures that were less accurate than the original values.
However the values of c that I obtain, close to 10-3, allow an estimate of the temperature uncertainties introduced by the varying viewing geometry. Saari et al (1972) Fig. 11 shows, for the Moon, that `accurate' 11\dmic\ brightness temperatures (equal to the surface equilibrium temperature) are obtained at phase angles between 45o and 60o on the thermal meridian. From equation 15, using a c value of 10-3, and an `accurate' phase angle of 50o, a 120oK surface would appear to be at 122.4oK at 30o solar phase angle and 115.3oK at 90o phase. The plots of surface temperature distributions presented in Chapter 5 will include uncertainties of this magnitude due to viewing geometry variations.