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 scatterings 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 to add 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 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 becomes close to or larger than 1, reddening becomes present but is 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 in changing the shape of the spectral transmission curve, with dielectric grains showing the minimum amount of reddening.