An initial-value method for determining the radiation field in a plane-parallel atmosphere. The vector case

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Abstract
An initial value method for determining the radiation field in a plane-parallel atmosphere is extended to include polarization. Using the first principle of invariance by Chandrasekhar we eliminate the intensity of the upward radiation from the equation of transfer thus formulating the Cauchy problem. The concomitant scattering matrix satisfies the well-known integro-differential equation and after having determined this matrix we may solve the equation of transfer itself. Next, the Chandrasekhar principle allows us to find the intensity of the upward radiation, thus completing the solution. The case with Rayleigh scattering is discussed in detail. The behaviour of neutral points in an atmosphere, where the albedo of single scattering depends exponentially on the optical depth, is studied.