They calculated the temperature at each point in an obliquely-illuminated crater of albedo

Note that in this appendix,

Figures 6 and 7 of Winter and Krupp (1971) show calculated temperature profiles along the meridian (the line of symmetry aligned with the direction of the sun) of spherical-section `craters'. Profiles are given for a hemispherical crater (depth/diameter ratio,

The temperature information provided by Winter and Krupp pertains only to the crater meridian: they do not give temperatures elsewhere in the crater so a calculation of the thermal emission from the crater in three dimensions is not possible. I therefore used the temperature profile in a two-dimensional calculation, applying it to an infinite trench with the same circular-arc profile as the crater cross-section. This introduces an error: the geometry of the trench is different and the actual temperature profile within it will be somewhat different from that within a three-dimensional crater. However, this simplification allows the calculation of results that should be at least qualitatively valid. Figure 42 illustrates the model.

The radiated flux from each surface element is then reduced by the factor \ep\ at all wavelengths in Equation 18 below.

Simple (but tedious) geometry determines what portions of the trench interior are visible from a given emission angle. The visible opening of the trench is divided into ten sections of equal projected area and the location of the center of each within the crater is determined. The temperature at that point is read from the digitized and scaled temperature profile, interpolating as necessary. This gives ten temperatures

The thermal emission spectrum $R(\lambda)$ of a surface with fractional trench coverage

where

The spectrum $R(\lambda)$ can be compared directly with the IRIS spectra, or can be fitted with a 2-component blackbody in an identical way to the IRIS spectra so that the fit parameters can be compared.

I ran this model for all combinations of the following values of the input parameters: both available values of

The model first calculates the size of each surface element, a function of

A major (and invalid) assumption implicit in Equation 19 is that conditions are the same over the whole of each surface element, i.e.\ the flux

All

Finally, the temperature

where

Finally, the thermal emission spectrum from the trench and its surroundings is calculated using Equation 18 as in model A.

The resulting spectrum is treated as in model A, being compared directly with the IRIS spectra or first fitted with a 2-component blackbody.

I ran model B with the same range of parameters as model A, with the additional variable