  The main purpose of this paper is to obtain analytical solutions for
radiative transfer equations related to the vertical structure of accretion
discs with finite optical depth. In the non-gray atmosphere, we employ the
optical-depth dependent Eddington factor to define the relationship between the
mean intensity and radiation stress tensor. Analytical solutions are achieved
for two cases: (i) radiative equilibrium, and (ii) a disc with uniform internal
heating and both cases are assumed to be in local thermodynamical equilibrium
(LTE), too. These solutions enable us to study the probable role of scattering
and disc optical depth on the emergent intensity and other radiative
quantities. Our results show that for the first case, the surface value of mean
intensity with constant Eddington factor is three times larger than that with a
variable factor. Moreover, scattering has no role in the vertical radiative
structure 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 the
photon destruction probability decreases (which is equivalent to increasing
scattering). Furthermore, for both cases with total optical depth less than
unity, the outward intensity towards the polar direction becomes less than that
from the edges of disc which is contrary to limb-darkening. In the end, we
apply our results to find the spectrum from accretion systems, based on two
dynamical models. Consequently, we can see how the total optical depth varies
with frequency and causes remarkable changes in the emergent spectra.
