2n-Stream Conservative Scattering

We show how to use matrix methods of quantum mechanics to efficiently and accurately calculate axially symmetric radiation transfer in clouds, with conservative scattering of arbitrary anisotropy. Analyses of conservative scattering, where the single scattering albedo is $\tilde\omega =1$ and no energy is exchanged between the radiation and scatterers, began with work by Schwarzschild, Milne, Eddington and others on radiative transfer in stars. There the scattering is isotropic or nearly so. It has been difficult to extend traditional methods to highly anisotropic scattering, like that of sunlight in Earth's clouds. The $2n$-stream method described here is a practical way to handle highly anisotropic, conservative scattering. The basic ideas of the $2n$-stream method are an extension of Wick's seminal work on transport of thermal neutrons by isotropic scattering to scattering with arbitrary anisotropy. How to do this for finite absorption and $\tilde\omega<1$ was described in our previous paper (arXiv:2205.09713v2). But those methods fail for conservative scattering, when $\tilde \omega = 1$. Here we show that minor modifications to the fundamental $2n$-scattering theory for $\tilde\omega <1$ make it suitable for $\tilde\omega = 1$.