Self-gravity in thin protoplanetary discs: 1. The smoothing-length approximation versus the exact self-gravity kernel

TL;DR

The paper tackles inaccuracies in 2D self-gravity modeling of protoplanetary discs caused by Plummer-like smoothing. It derives a closed-form, Bessel-based kernel for Gaussian-stratified, bi-fluid (gas+dust) discs, producing a 2D SG force that preserves Newtonian behavior at short and long ranges and remains compatible with FFT methods under appropriate conditions. The authors validate the kernel against analytical results for power-law and exponential discs and against 3D simulations across Gaussian disks, showing substantial improvements over smoothing-length prescriptions and highlighting potential for gravitational runaway in massive discs. This exact kernel enables a consistent, computationally efficient framework for studying SG-driven planet-formation processes in 2D simulations, with implications for gravitationalInstability analyses and future GI-focused work.

Abstract

Planet-forming discs often contain structures like spiral arms, typically linked to the disc's gravitational forces. In 2D models, an ad hoc softening prescription is commonly used for self-gravity, but this overlooks the vertical structure's impact, suppresses the Newtonian nature of gravity at short distances and doesn't respect Newton's third law. To address these issues, associated with a Plummer potential approximation, we developed an exact self-gravity kernel for thin, hydrostatically supported discs, including a dust fluid component. Our analytical framework provides a precise 2D self-gravity prescription validated by benchmarks and 2D/3D numerical tests. The derived kernel, based on modified Bessel functions, maintains Newtonian gravitation features, such as point-wise symmetry, a smooth transition from light to massive discs and a singularity at zero distance, among others. In contrast to other prescriptions found in the literature, it proves capable of leading to an additional, and previously unnoticed, source of gravitational runaway discernible only at infinitesimal distances. We finally note that our new prescription remains compatible with methods based on the fast Fourier transform, affording superior computational efficiency. Our exact kernel formulation overcomes substantial limitations inherent in the smoothing-length approach. It permits a novel, fully consistent treatment of self-gravity in Gaussian-stratified thin discs. The approach, that makes the usage of the Plummer potential obsolete, will prove useful to studying all common planet formation scenarios, which are often backed by 2D-flat numerical simulations. Accordingly, in an accompanying paper, we will investigate how the occurence of the gravitational instability is affected.

Figures (6)

• Figure 1: Normalised self-gravity kernels, $\pi d_g H_g^2 \, \text{K}$, for light (top) and massive (bottom) discs. For light discs, the prescriptions of muller_kley_2012 and 2023_rendon_restrepo are insensitive to the Toomre's parameter. This is in contrast with the Bessel prescription, which scale as $1/Q_g$ in the limit of massive discs. When a constant SL prescription is used, the kernel vanishes at short distances, whereas for a vanishing SL, the kernel diverges, suggesting an overestimation of SG. The Bessel prescription is the unique prescription that is simultaneously inversely proportional to $Q_g$ for massive discs, does not vanish at the singularity and permits a smooth transition between massive and light discs.
• Figure 2: Normalised kernels associated with dust with respect to distance for different dust-to-gas scale heights, $\eta$. The dust-dust, $K_{dd}$, and dust-gas, $K_{dg}$, kernels are insensitive to $\eta$ at large distances, an aspect also in agreement with the smoothing length prescription (see Fig. 13 of muller_kley_2012). The dust-dust kernel scales with $1/\eta$ at short range, which suggests that the SG acceleration due to dust can be comparable of the one of gas even for $Z=\Sigma_d/\Sigma_g = 0.01$. At short distances, the normalised dust-gas kernel converges quickly towards $\sqrt{2}$ (i.e., with respect to decreasing $\eta$). As a consequence this kernel adopts a nature akin to the kernel associated with a planet-disc interaction.
• Figure 3: Radial SG force of the power law disc for different scale heights. The accuracy of our kernel increases with decreasing aspect ratios. Note that for $\varpi\simeq0.07$ the relative error is beyond unity, which is a bias in the statistical measure due to a vanishing radial acceleration. We recall that in FargoCPT and Nirvana-iii we used a logarithmic and uniform radial grid, respectively.
• Figure 4: Radial SG force of the exponential disc for different scale heights. The accuracy of our kernel increases with decreasing aspect ratios and piles up at $\sim 10^{-4}$. Note that for $y=0$ the relative error is beyond unity, which is a bias in the statistical measure due to a vanishing radial acceleration.
• Figure 5: Relative difference between the 2D self-gravity prescriptions and the reference gravitational field (see Eq. \ref{['Eq: relative reference gaussian discs']}) for $H=0.4$ AU. From left to right: a) the sharp potential with $\epsilon/H=0$, b) the variable Bessel kernel, and c) the soft potential with $\epsilon/H=1.2$. The colours of the contour levels at $10^{-5}, 10^{-3}, 0.1, 0.5$ are indicated in the colorbar. Softening prescriptions exhibit errors exceeding tens of percent at disc positions, whereas the Bessel prescription maintain errors below 3%.