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

TL;DR

This study presents a self-consistent framework unifying planet-induced spiral shock heating and gap opening in protoplanetary disks through entropy jumps across shocks. By deriving the shock-induced radial velocity from the entropy jump and balancing it with viscous transport, the authors obtain a 1D model for the gap’s temperature and surface density, validated against 2D simulations with $\alpha$ viscosity and $\beta$ thermal relaxation. They introduce an empirical scaling for the entropy jumps $\delta(r)$ and develop both semi-analytic and full analytic gap models that reproduce the simulated structures and offer competitive predictions compared to the KanagawaTanaka17 gap model, with implications for inner disk heating and solid-body distribution near forming planets. The framework advances a pathway to study how giant planets influence the distribution and composition of solids in the inner disk, while noting limitations related to global steady-state behavior, relaxation physics, and contributions from secondary shocks. Overall, the work provides a coherent, testable bridge between disk thermodynamics and gap morphology in planet-disk interactions, particularly relevant for the inner $\sim 10$ au where cooling is slow and heating from shocks can dominate.

Abstract

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.

Figures (10)

• Figure 1: Schematic illustration of the dual role of planet-induced spiral shocks Rafikov16. Entropy generation at the shock fronts leads to secular, irreversible heating of the disk, while angular momentum deposition by the dissipating shocks drives gap opening. The entropy jumps across the shocks, $\delta$, determine the shock heating rate $Q_{\rm shock}$ (equation \ref{['eq:Qshock']}; see also Ono25). The connection between shock-induced heating and angular momentum transfer (equation \ref{['eq:Lambda_shock']}, GoodmanRafikov01Rafikov16ArzamasskiyRafikov18) also relates $\delta$ to the gap-opening flow velocity $v_{r,\rm shock}$ (equation \ref{['eq:vr_shock']}).
• Figure 2: 2D profiles of $\Sigma$ and ${\@fontswitch\mathcal{S}}/{\@fontswitch\mathcal{S}}_{\rm rmin} - 1$ (upper and lower panels, respectively) from all simulation runs. Here, ${\@fontswitch\mathcal{S}}_{\rm rmin}(r)$ denotes the minimum value of the exponential entropy ${\@fontswitch\mathcal{S}}$ at radial distance $r$. The planet is located at $(r,\phi) = (r_{\rm p}, 0)$. The letters "P" and "S" label the primary and secondary spiral arms. Dashed and dotted lines in the lower panels mark the locations of the primary and secondary shocks, respectively, with $\delta > 5\times 10^{-5}$. Gray-shaded areas in the lower panels are excluded from streamline analysis.
• Figure 3: Radial structure of the simulated disk models. Top two rows: azimuthally averaged temperature and surface density. Dotted lines show the initial profiles. Third row: dimensionless specific entropy jumps, $\delta$ (equation \ref{['eq:delta']}), across the primary and secondary shocks, obtained from streaming analysis. Bottom row: shock- and viscosity-induced radial gas velocity components, $v_{r,{\rm shock}}$ and $v_{r,{\rm visc}}$, estimated from equations \ref{['eq:vr_shock']} and \ref{['eq:vr_visc']} (bottom row) for all simulation runs. Shaded regions around the planet's orbit are excluded from streamline analysis. Dotted lines in the first, second, and bottom rows represent the temperature, surface density, and radial velocity profiles before the planet's insertion. Dotted lines in the third row show the empirical scaling laws for the primary shocks' entropy jumps, equation \ref{['eq:delta_formula']} with $({\@fontswitch\mathcal{A}}, {\@fontswitch\mathcal{B}}) = (0.5, 0.5)$, as presented in subsection \ref{['sec:simulationresults']}.
• Figure 4: Comparison between simulations and the semi-analytic model (section \ref{['sec:semi']}). The upper and lower panels show the radial temperature and surface density profiles, respectively. Solid lines represent azimuthally averaged profiles from four 2D simulations, Dashed lines show semi-analytic predictions based on equations \ref{['eq:Sigma_semi_in']}--\ref{['eq:T_predict']}, using numerical $\delta$ profiles for primary and secondary spiral shocks obtained directly from streamline analysis. Shaded regions around the planet's orbit are excluded from streamline analysis. Dotted lines represent the profiles before the planet's insertion.
• Figure 5: Comparison between simulations and the full-analytic model (section \ref{['sec:analytic']}). The upper and lower panels show the radial temperature and surface density profiles, respectively. Solid lines represent azimuthally averaged profiles from four 2D simulations. Dashed lines show full-analytic predictions ($T$ from equations \ref{['eq:delta_formula']} and \ref{['eq:T_predict']}, and $\Sigma$ from equations \ref{['eq:Sigma_analytic_in']}--\ref{['eq:Kanagawa']}) based on the empirical scaling law for the primary shocks' $\delta$ profiles, equation \ref{['eq:delta_formula']} with $({\@fontswitch\mathcal{A}}, {\@fontswitch\mathcal{B}}) = (0.5, 0.5)$. Dotted lines represent the profiles before the planet's insertion, $T = T_{\rm init}$ and $\Sigma = \Sigma_{\rm init}$.