Stanley A. Baronett, Yan-Fei Jiang, Zhaohuan Zhu, Shangjia Zhang, Philip J. Armitage

The frequency dependence of opacity is crucial for determining the thermal structure of protoplanetary disks, which in turn influences disk dynamics and planet formation. Yet many disk models adopt simplified thermodynamics, and common radiation-hydrodynamic approaches often use gray opacities, ignore scattering, and yield inaccurate results in regions with intermediate optical depth. We present a comprehensive framework that models stellar irradiation with frequency-dependent absorption and scattering across all optical depths using the Athena++ finite-volume code, extended with multigroup radiation transport and newly implemented radial rays to more accurately represent the stellar flux. To calibrate this framework, we focus exclusively on hydrostatic disk models, allowing us to isolate radiative effects and evaluate the method without additional dynamical complexity. Because dust opacity increases strongly with frequency, ultraviolet stellar irradiation heats the tenuous disk atmosphere while the optically thick midplane remains cooler. This vertical temperature gradient is captured more accurately when more frequency bands are used or when scattering is included. Our hydrostatic models achieve equilibrium temperatures that differ from Monte Carlo radiative-transfer benchmarks on average by 2--5% with 64 frequency bands and 7--11% with 3 bands. Reducing the number of bands lowers computational cost by at least an order of magnitude while increasing the maximum possible temperature deviation only from 8% to 19%. This calibration demonstrates the accuracy and efficiency of the framework and provides a solid foundation for future self-consistent studies of irradiated protoplanetary disks, including fully dynamical simulations and applications involving chemical processes and time-dependent stellar luminosity.