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The main purpose of this paper is to obtain analytical solutions forradiative transfer equations related to the vertical structure of accretiondiscs with finite optical depth. In the non-gray atmosphere, we employ theoptical-depth dependent Eddington factor to define the relationship between themean intensity and radiation stress tensor. Analytical solutions are achievedfor two cases: (i) radiative equilibrium, and (ii) a disc with uniform internalheating and both cases are assumed to be in local thermodynamical equilibrium(LTE), too. These solutions enable us to study the probable role of scatteringand disc optical depth on the emergent intensity and other radiativequantities. Our results show that for the first case, the surface value of meanintensity with constant Eddington factor is three times larger than that with avariable factor. Moreover, scattering has no role in the vertical radiativestructure of discs with the assumptions of the first case. On the other hand,for the second case, we encounter reductions in all radiative quantities as thephoton destruction probability decreases (which is equivalent to increasingscattering). Furthermore, for both cases with total optical depth less thanunity, the outward intensity towards the polar direction becomes less than thatfrom the edges of disc which is contrary to limb-darkening. In the end, weapply our results to find the spectrum from accretion systems, based on twodynamical models. Consequently, we can see how the total optical depth varieswith frequency and causes remarkable changes in the emergent spectra.

Analytical Solutions of Radiative Transfer Equations in Accretion Discs with Finite Optical Depth