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Bridging the gap: consistent modeling of protoplanetary disk heating and gap formation by planet-induced spiral shocks

A giant planet embedded in a protoplanetary disk excites spiral density waves, which steepen into shocks as they propagate away from the planet. These shocks lead to secular disk heating and gap opening, both of which can have important implications for the evolution of solids near the planet. To date, these two effects have largely been modeled independently. In this study, we present a self-consistent model that unifies these processes by linking shock heating and angular momentum deposition through the entropy jumps across the spiral shocks. We show that this model accurately reproduces the temperature and surface density profiles around the planet's orbit, as obtained from two-dimensional hydrodynamic simulations with standard $α$ viscosity and $β$ thermal relaxation prescriptions. Furthermore, by incorporating an empirically derived scaling law for the radial distribution of the entropy jump, we construct a fully analytic model that self-consistently predicts the temperature and surface density structures of disks hosting a giant planet. This work represents a first step toward understanding how a giant planet forming in the inner disk region influences the distribution and composition of second-generation planets and planetesimals in its vicinity.