where m is the cosine of the observer's zenith angle; and k is a constant between 0 and 1, which may be fitted to the experimental data. For a Lambert reflector k = 1. Once measured, the observed limb-darkening curves may be fitted with scattering models to determine the vertical distribution of particles and their scattering properties.
Calculated thermal emission spectra of the giant planets are shown in Figure 6.16 for nadir-viewing geometry, and for the case of zero cloud opacity for Jupiter and Saturn, and for the case of a deep thick cloud for Uranus and Neptune with an optical depth of unity at bar for all wavelengths. These cloud structures were chosen to reflect the observation that for Jupiter and Saturn the clouds are mostly above or below the line-forming region, while for Uranus and Neptune near-IR observations detect a substantial cloud (presumably composed of H2S) in the middle of the line-forming region at 2 bar to 3 bar. Atmospheric compositions were set to the current estimates of the gaseous composition outlined in Tables 4.6-4.9. For reference, the Planck function for a number of temperatures has also been plotted. Thermal emission diminishes rapidly with temperature and wavenumber and thus the emission of the ice giants is significantly smaller than for gas giants, as can be seen by the vertical scales of Figure 6.16 and by the steady diminishment of 5 ^m and mid-IR radiance as we go from Jupiter to Neptune. Hence, it can be immediately seen that it is very much more difficult to measure the thermal emission spectra of the ice giants than the gas giants since the radiance levels are so much lower. An alternative way of
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