Radiative transfer in protoplanetary disks under irradiation by the protostar

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative transfer in a protoplanetary accretion disk, which is irradiated by the protostar, is examined under the non-gray frequency-dependent treatment, including the anisotropic scattering effect. Due to irradiation heating, the disk is divided into an inner optically thick flat disk, a middle optically thick flared disk, and an outer optically thin flared disk. In each regime, emergent intensities as well as other radiative quantities are analytically obtained under the Eddington approximation. In the inner flat disk, where the disk is optically thick and viscous heating is dominant, we obtain solutions for the case of vertically uniform heating. In the middle flared disk, where the disk is optically thick and geometrically flared due to irradiation heating, we obtain solutions for the local thermodynamic equilibrium (LTE) case under the assumption of a linear Planck function. In the outer flared disk, where the disk is optically thin and flared due to irradiation heating, we obtain solutions for the LTE case of a vertically isothermal disk. The temperature structure is consistent with that seen in previous studies, while disk intensities are different from those in previous cases without anisotropic scattering. We found that anisotropic scattering enhances the emergent intensity in the poleward direction, when forward scattering dominates. As a result, in addition to the usual limb-darkening effect, the protoplanetary disk also becomes bright in the poleward direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .