Radiative Transfer in a Translucent Cloud Illuminated by an Extended
  Background Source
  We discuss the radiative transfer theory for translucent clouds illuminated
by an extended background source. First we derive a rigorous solution based on
the assumption that multiple scattering produce an isotropic flux. Then we
derive a more manageable analytic approximation showing that it nicely matches
the results of the rigorous approach. To validate our model, we compare our
predictions with accurate laboratory measurements for various types of well
characterized grains, including purely dielectric and strongly absorbing
materials representative of astronomical icy and metallic grains, respectively,
finding excellent agreement without the need of adding free parameters. We use
our model to explore the behavior of an astrophysical cloud illuminated by a
diffuse source with dust grains having parameters typical of the classic ISM
grains of Draine & Lee (1984) and protoplanetary disks, with an application to
the dark silhouette disk 114-426 in Orion Nebula. We find that the scattering
term modifies the transmitted radiation, both in terms of intensity
(extinction) and shape (reddening) of the spectral distribution. In particular,
for small optical thickness our results show that scattering makes reddening
almost negligible at visible wavelengths. Once the optical thickness increases
enough and the probability of scattering events become close to or larger than
1, reddening becomes present but appreciably modified with respect to the
standard expression for line-of-sight absorption. Moreover, variations of the
grain refractive index, in particular the amount of absorption, also play an
important role changing the shape of the spectral transmission curve, with
dielectric grain showing the minimum amount of reddening.
