The radiative transfer equations of Section 6.4 for a gray atmosphere must be substantially modified to deal with atmospheres that contain significant abundances of scattering particles, as we shall shortly see. These modifications greatly increase the computation time and thus scattering calculations are notoriously difficult and slow. There are two main ways of approaching the problem. The simplest is to ignore the curvature of the planetary atmosphere and approximate the problem by a stack of plane-parallel layers. Using this approach, a number of good approximations may be made that considerably reduce the computation time. Most calculations are made with this plane-parallel approximation. However, under certain conditions, such as for

0 50 100 150

Scattering angle

Figure 6.14. Examples of different Henyey-Greenstein phase functions depending on /, g1, g2. Here f = 1, g2 = 0, and g1 = 0.7 (solid), 0.5 (dots), 0.3 (dashes), and 0.0 (dot-dash).

0 50 100 150

Scattering angle

Figure 6.14. Examples of different Henyey-Greenstein phase functions depending on /, g1, g2. Here f = 1, g2 = 0, and g1 = 0.7 (solid), 0.5 (dots), 0.3 (dashes), and 0.0 (dot-dash).

limb observations, the plane-parallel approximation no longer applies, and thus much slower, but more general purpose techniques such as Monte Carlo calculations must be used. In this section we will outline how the scattering properties of particles, just discussed, may be used in radiative transfer models to calculate the synthetic spectra of the giant planets.

6.6.1 Plane-parallel approximation

In Section 6.4.1 we found that the thermal contribution of a thin slab of thickness dz at altitude z to the upwelling spectral radiance at wavenumber v and zenith angle 6 and azimuth angle ^ at that level in a plane-parallel atmosphere is given by dlv = kv(z)n(z)Bv(z) dz/p (Equation 6.35). Given that the optical depth of the atmosphere (which is zero at the top of the atmosphere, assumed here to be at z = œ, and increases steadily as we move to deeper levels) at altitude z is defined in Equations (6.33) and (6.34) as kv(z)n(z) dz z

the upwelling spectral radiance at altitude z (or equivalently optical depth rv) in the direction of zenith angle d and azimuth angle 0 may equivalently be expressed as dly (rs,-,0)