Modeling the Thermal Structure of a Protoplanetary Disk Using Multiband Flux-Limited Diffusion Approximation

This work continues the analysis of the model for calculating the thermal structure of an axisymmetric protoplanetary disk, initiated in the paper by Pavlyuchenkov (2024). The model is based on the well-known Flux-Limited Diffusion (FLD) approximation with separate calculation of heating by direct stellar radiation (hereinafter referred to as the FLD$^{\rm s}$ method). In addition to the previously described FLD$^{\rm s}$ model with wavelength-averaged opacities, we present a multiband model mFLD$^{\rm s}$, where the spectrum of thermal radiation is divided into several frequency bands. The model is based on an implicit finite-difference scheme for the equations of thermal radiation diffusion, which reduces to a system of linear algebraic equations written in hypermatrix form. A modified Gauss method for inverting the sparse hypermatrix of the original system of linear equations is proposed. The simulation results described in the article show that the midplane radial temperature profile obtained with the mFLD$^{\rm s}$ method has a variable slope in accordance with the reference Monte Carlo radiative transfer simulations. The mFLD$^{\rm s}$ model also qualitatively reproduces the non-isothermality of the temperature distribution along the angular coordinate near the midplane, which is not provided by the FLD$^{\rm s}$ method. However, quantitative differences remain between the reference temperature values and the results of mFLD$^{\rm s}$. These differences are likely due to the diffusive nature of the FLD approximation. It is also shown that the characteristic times for the disk to reach thermal equilibrium within the mFLD$^{\rm s}$ model can be significantly shorter than in FLD$^{\rm s}$. This property should be taken into account when modeling non-stationary processes in protoplanetary disks within FLD-based models.