Fj pn nLn dn dn623

■J 4n J 4n which, because of the normalization condition Eq. (6.17), is fj L(n') dn' = 4nffJ. (6.24)

J 4n

The integrated third term is 4kPs, which when combined with Eqs. (6.21), (6.22) and (6.24) yields

This is a general result, as valid as the equation of transfer. And it makes physical sense. For a nonabsorbing medium (k = 0) the divergence of the vector irradiance is zero: all the photons that enter a region come out.

To explain briefly why Eq. (6.15) can lead to such a bewildering array of N-stream approximations we back up just a bit. Consider first approximating an arbitrary one-dimensional integral as a finite sum:

This is called reducing an integral to quadratures; the wj are quadrature weights and the discrete set of ordinates xj are quadrature points. Simpson's rule, which you may remember from elementary calculus, is a quadrature formula, one of the simplest imaginable. Note the tremendous latitude of choices with which we are faced: N, wj, and xj, a triply-infinite set of possibilities. Thus there are many ways of replacing the integro-differential equation Eq. (6.15) with a set of N coupled ordinary differential equations for L in N discrete directions depending on the many ways of approximating the integral on the right side by a finite sum. This approach goes under the general name of discrete ordinates.

At this point, however, we are not going to launch into discussing the many ways of obtaining approximate solutions to Eq. (6.15). There are many methods and countless variations on them, most of them restricted to plane-parallel media (which don't exist). To learn more about all these methods consult the references at the end of this chapter. Instead of discussing even a restricted set of such methods, we devote Section 6.3 to only one, the Monte Carlo method, which has the peculiar characteristic that it enables us to solve equations even if we don't know what they are. By Monte Carlo methods, which are not restricted to plane-parallel media, one can solve Eq. (6.15) without knowing that it exists. Sound like magic? Read on.

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