The original Rudy model was constructed with no surface roughness,
rather using a simple smooth sphere (and parameters were derived
based on this). It is clear that surface roughness is not only
present, but affects the outgoing thermal emission, as described in a
number of papers (see e.g.,
Heiles & Drake 1963;
Hagfors & Moriello 1965;
Golden 1979;
Mitchell & de Pater 1994).
I have implemented surface roughness within the Rudy model by
modifying the emissivity of each surface facet before calculating
the emission from it. I do this by pre-calculating the emissivity,
given the roughness parameters, for the full range of outgoing
emission angles, then interpolating into these pre-calculated values
given the particular emission angle of a facet.
I allow for two ways of describing the surface roughness, either as
a Gaussian distribution or as an exponential one
(Muhleman 1964).
For either of these cases, for the range of emission angles, I do a
calculation of the average emissivity from 10000 randomly oriented
facets with a deviation from normal given by the proper distribution.
The only parameter necessary to the roughness model (aside from
choosing between a Gaussian or exponential distribution of slopes) is
the mean surface slope (very akin to the "surface rms roughness"
often used in such models).
For a more in-depth description of the mathematics, go
here.