Analytical expressions for a differentially moving slab

An analytical solution of the comoving frame radiative transfer equation for a differentiallymoving slab is obtained and written in terms of the wavelength-integrated extinction coefficient ψ, the spectral thickness. It is shown that ψ is much smoother than the extinction coefficient χ(λ) itself and that it can bewell approximated e.g. by a simple piecewise linear function. Compared to conventional algorithms it allows speed-ups of factors ≥ 105 in the calculation of wavelength-integrated quantities such as e.g. the total flux, the radiative force, or the energy balance with an accuracy of ≤ 1%. The total number of lines which can be taken into account is essentially unlimited. A simple expression for the total flux in the diffusion limit is derived and the dependencies on the velocity gradient are discussed.