On the radiative equilibrium of a stellar atmosphere. X.

In this paper a detailed theory of the radiative equilibrium of an atmosphere in which the Thomson scattering by free electrons governs the transfer of radiation is developed. In particular, allowance has been made for the polarization of the scattered radiation; and the equations of transfer for the intensities Il and Ir referring, respectively, to the two states of polarization in which the electric vector vibrates in the meridian plane and at right angles to it, are separately formulated. The equations of transfer are found to be μdIl/dτ=Il-3/8{2∫+1-1Il(τ, μ')(1-μ'2)dμ'+μ2∫+1-1Il(τ, μ')(3μ'2-2)dμ'+μ2∫+1-1Iτ(τ, μ')dμ'} and μdIr/dτ=Ir-3/8{∫+1-1Ir(τ, μ')dμ'+∫+1-1Il(τ, μ')μ'2dμ'}. These equations have been solved in a general nth approximation, and their explicit numerical forms have been found in the second and the third approximations. It is found that the theory predicts different laws of darkening for the two states of polarization distinguished by Il and Ir. The emergent radiation is therefore polarized, and it is further predicted that the degree of polarization must vary from zero at the center of the disk to 11 per cent at the limb