How two-dimensional are planet--disc interactions? I. Locally isothermal discs

TL;DR

Planet–disc interactions are fundamentally 3D, yet 2D simulations are widely used with fitted smoothing. The authors derive physically motivated 2D force prescriptions, test them against a comprehensive set of 3D simulations in locally isothermal discs, and show that while a Bessel-type potential $\Phi_{\mathrm{B},H_p}$ best reproduces several 3D features, 2D models cannot capture the full 3D scaling of total torque with background gradients or the horseshoe vortensity striping. The work provides quantitative biases in inferred planet masses from 2D gap analyses and highlights the importance of vertical disc structure for velocity kink observations. A publicly available analysis pipeline enables broad reuse of the metrics across codes.

Abstract

Planet--disc interactions, despite being fundamentally three-dimensional, are often studied in the two-dimensional `thin-disk' approximation. The overall morphology of planet--disc interactions has ben shown to be similar in both 2D and 3D simulations, however, the ability of a 2D simulation to quantitatively match 3D results depends strongly on how the potential of the planet is handled. Typically, the 2D planetary potential is smoothed out using some `smoothing length', a free parameter, for which different values have been proposed, depending on the particular aspect of the interaction focused on. In this paper, we re-derive 2D Navier--Stokes in detail for planet--disc interactions to find better ways to represent the 2D gravitational force. We perform a large suite of 2D and 3D simulations to test these force prescriptions. We identify the parts of the interaction that are fundamentally 3D, and test how well our new force prescriptions, as well as traditional smoothed potentials, are able to match 3D simulations. Overall, we find that the optimal way to represent the planetary potential is the `Bessel-type potential', but that even in this case 2D simulations are unable to reproduce the correct scaling of the total torque with background gradients, and are at best able match the one-sided Lindblad torque and gap widths to level of 10 per cent. We find that analysis of observed gap structures based on standard 2D simulations may systematically underestimate planetary masses by a factor of two, and discuss the impacts of 3D effects on observations of velocity kinks.

Figures (21)

• Figure 1: Gravitational potential (a) and acceleration (b) for the different types of planetary potential considered in this paper, shown in units $GM_\mathrm{p} = 1$. $\Phi_{\mathrm{K}}$ is the standard Keplerian potential, $\Phi_2$ (Equation \ref{['eq:second_order_pot']}) and $\Phi_4$ (Equation \ref{['eq:fourth_order_pot']}) are the second and fourth order smoothed potentials using a smoothing length of $r_s = bH$ with $b=0.7$. $\Phi_{\mathrm{B}}$ is the Bessel-type potential (Equation \ref{['eq:besselpotl']}). All potentials were calculated for a reference disc with a constant scale height of $H = 0.05R_p$. All potentials converge to $\Phi_{\mathrm{K}}$ far from the planet, however near the planet there are significant differences, especially, in acceleration.
• Figure 2: 1D profile of the surface density evolution measured from a 3D simulation, $\langle \Sigma_0 \rangle^{-1} \, \, d \langle \Sigma \rangle/dt$, compared to the expected $\Sigma$ evolution, $d \sigma/dt$, using the theory of cordwell_early_2024, with $F_\mathrm{dep}$ measured from the simulation. This run used power law indices of $\Sigma$ and $T$ of $p=1.5$ and $q=0.5$. We also show the evolution expected if we set $F_\mathrm{dep}$ to zero within one scale height of the plane, $d \sigma/dt$ (shock only), which describes the evolution caused by the shocking of the planetary wake only (i.e. with co-rotation effects removed). The labels IG1 and OG1 refer to the locations of the inner and outer gaps closest to the planet, and are used to determine gap spacing. Despite some quantitative differences in gap depth between the measured and theoretical $\Sigma$ evolution, the theory of cordwell_early_2024 is clearly applicable to the 3D case.
• Figure 3: A sketch of fluid streamlines within the frame co-rotating with the planet. Close to the radial location of the planet the gas will undergo horseshoe turns on closed loops. The width of this region is key in determining the magnitude of the co-orbital torque brown_horseshoes_2024. The horseshoe width, $x_s$, is the radial distance between $R_\mathrm{p}$ and the most radially distant streamline that undergoes a horseshoe turn (shown in green).
• Figure 4: Vertical density slices from a $p=1.5$ and $q=0$ 3D simulation along lines of constant azimuth. Panels (a) & (b) show the full 3D structure, which is compared with the density structure derived from an equivalent 2D model in panels (c) & (d). The structure derived from 2D model is calculated as the surface density of the 3D model multiplied by $\mathrm{e}^{-z^2/2H^2}$, i.e. the initial hydrostatic vertical density (\ref{['eq:background_rho']}). In (a) and (c), where hydrostatic envelope around the planet is shown, it is clear that the 2D model misrepresents the planetary atmosphere. In (b) and (d), we compare the structure of the planet induced wake at a distance away from the planet. In the full 3D case, the wave curves slightly backwards towards the planet at higher altitudes whereas in the quasi-2D case the wake is required to be vertically straight.
• Figure 5: Left: Planet induced wake structure as a function of altitude above the midplane in the outer disc as measured from a 3D simulation ($p =1.5, q=0$). Right: Wake structure in surface density for the same 3D simulation compared to that from a 2D simulation using the potential $\Phi_2, b=0.7$. The 3D wake structure is dependent on altitude, and shocks, identified visually as when the gradient of $\delta \rho/\rho_0$ reaches its maximum, form at the midplane radially sooner (e) than at higher altitudes. The exact formation of the shock is not captured by the surface density evolution, and the literature standard 2D simulation shows slightly different shock formation and structure compared to the 3D case.