Gas density ρ at t = 1010 orbits (1000 orbits at the fiducial resolution, 10 at the doubled resolution used in Muley et al. 2024) with respect to the initial condition ρ₀ for our simulations with a Saturn-mass planet. The green ellipse indicates the planetary Hill radius. With the 3T scheme, the disk atmosphere shows a clear development of m = 2 spiral arms, which are absent in the β-cooling simulations. The white bands at the interior and exterior of the radiative simulation are caused by wave-damping to the initial condition.