Structure of Passive Circumstellar Disks: Beyond the Two-Temperature Approximation

The structure and spectral energy distributions (SEDs) of externally irradiated circumstellar disks are often computed on the basis of the two-temperature model of Chiang & Goldreich. We refine these calculations by using a more realistic temperature profile that is continuous at all optical depths and thus goes beyond the two-temperature model. It is based on the approximate solution of the radiation transfer in the disk obtained from the frequency-integrated moment equations in the Eddington approximation. We come up with a simple procedure ("constant gz approximation") for treating the vertical structure of the disk in regions where its optical depth to stellar radiation is high. This allows us to obtain expressions for the vertical profiles of density and pressure at every point in the disk and to determine the shape of its surface. Armed with these analytical results, we calculate the full radial structure of the disk and demonstrate that it favorably agrees with the results of direct numerical calculations. We also describe a simple and efficient way of the SED calculation based on our adopted temperature profile. Resulting spectra provide very good matches (especially at short wavelengths) to the results of more detailed (but also more time-consuming) SED calculations that solve the full frequency- and angle-dependent radiation transfer within the disk.