On Eccentric Protoplanetary Disks I -- How Eccentric are Planet-Perturbed Disks?

TL;DR

This work addresses how eccentric protoplanetary disks become when a planet opens a deep gap. It uses 2D hydrodynamical simulations with embedded planets, extracting an $m=1$ disk eccentricity measure $e_1(r)$ and a global diagnostic $\,\mathcal{L}_1$, and derives a semi-analytic profile $e_{sa}$ that matches simulations to about 30% accuracy. The key finding is that the steady-state outer-disk eccentricity results from a balance between eccentric Lindblad resonance excitation at the 1:3 resonance and damping by gas pressure, with $e_1$ scaling roughly as $e_1 \propto Q = q(h(r_{gap})/r_{gap})^{-1}(r_{1:3}/r_{gap})^{-a}$ and declining as $r^{-(b+3/2)}$; the edge eccentricity $e_{gap}$ is nearly independent of $q$ but sensitive to gap width and viscosity. The paper further defines applicable parameter space, $30 \lesssim K' \lesssim 700$, and provides a framework to interpret observed eccentric disks (e.g., MWC 758, HD 142527, IRS 48, CI Tau), linking disk structure to planetary properties through measurable diagnostics like $\\mathcal{L}_1$ and the predicted $e_{gap}$.

Abstract

Protoplanetary disks can become eccentric when planets open deep gaps within, but how eccentric are they? We answer this question by analyzing two-dimensional hydrodynamical simulations of planet-disk interaction. The steady state eccentricity of the outer disk (outside of the planet's orbit) is described as a balance between eccentricity excitation by the 1:3 eccentric Lindblad resonance and eccentricity damping by gas pressure. This eccentricity scales with $q(\frac{h_p}{r_p})^{(-1)}(\frac{r_{gap}}{r_p})^{(a-\frac{b}{2}-2)}$, where $q$ is the planet-to-star mass ratio, $\frac{h_p}{r_p}$ is the disk aspect ratio, $\frac{r_{gap}}{r_p}$ is the radial position of the outer gap edge divided by the planet's position, and $a$ and $b$ are the negative exponents in the disk's surface density and temperature power law profiles, respectively. We derive a semi-analytic eccentricity profile that agrees with numerical simulations to within 30%. Our result is a first step to quantitatively interpret observations of eccentric protoplanetary disks, such as MWC 758, HD 142527, IRS 48, and CI Tau.

Figures (7)

• Figure 1: Azimuthally averaged eccentricity (left) and surface density (right) at different resolutions. The parameters for these simulations are $\{q,h/r,a,b\}=\{0.004,0.05,1,1\}$. Our simulations converge well at our choice of resolution, $1200 (r)\times2160(\theta)$, especially in the outer disk ($r>r_{\rm p}$) where we focus our analysis.
• Figure 2: A gallery of our simulations displaying how disk morphology changes with the planet-to-star mass ratio $q$ and the surface temperature profile's (eq. \ref{['disk_temp_struct']}) parameter $b$. Other parameters are fixed at $\{h_{\rm p}/r_{\rm p}, a, \alpha_{\rm p}\}=\{0.05, 1, 10^{-3}\}$. The image color gradient is linear scale from $0.0-0.5 \space \Sigma_p$. Planets' locations and orbits are de-marked with a red dot and a white dotted line. Green dashed circles references the locations of $r_{\rm gap}$ (eq. \ref{['eq:rgap']}), the outer gap edges if they were circular. The true gap edges are eccentric, and are represented by the blue ellipses that have semi-major axis $r_{\rm gap}$ and eccentricity $e_{\rm gap}$ (eq. \ref{['eq:e_gap']}).
• Figure 3: $\mathcal{L}_1$, normalized by $Q$ (eq. \ref{['eq:Q']}), versus $a+b$. Two strong correlations emerge in this plot: 1) $\mathcal{L}_1$ is mostly proportional to $Q$, the ratio that describes eccentricity excitation vs damping; and 2) $\mathcal{L}_1$ is mostly a function of $a+b$, rather than $a$ and $b$ separately. $\mathcal{L}_1/Q$ appears to follow an exponential scaling with $a+b$ similar to the dashed black line, which is described by eq. \ref{['eq:L1_fit']}. These scalings enable us to predict disk eccentricity from the values of $Q$ and $a+b$.
• Figure 4: Profiles of $\ell_1$ from different simulations, all normalized by the scaling factor $Q$ (eq. \ref{['eq:Q']}). The four plots are separated by values of $a+b$: profiles with $a+b=1$ are shown in the upper left, $a+b=1.5$ in the upper right, $a+b=2$ in the lower left, and $a+b=2.5$ in the lower right. Blue lines indicate $\{q,h_{\rm p}/r_{\rm p}\}=\{0.004,0.05\}$; green lines indicate $\{q,h_{\rm p}/r_{\rm p}\}=\{0.004,0.035\}$; brown ones have $\{q,h_{\rm p}/r_{\rm p}\}=\{0.008,0.1\}$; red ones have $\{q,h_{\rm p}/r_{\rm p}\}=\{0.008,0.05\}$, and orange ones have $\{q,h_{\rm p}/r_{\rm p}\}=\{0.008,0.035\}$. Dotted, dot-dashed, dashed, and solid lines correspond to $b=$ 0, 0.5, 0.75, and 1, respectively. The grey lines are power laws that scale with $r^{-(a+b)}$. We find that $\ell_1$ closely follows the $r^{-(a+b)}$ profile in the outer disk, roughly between $2r_{\rm p}$ and $7r_{\rm p}$. Eccentricity is artificially damped near the outer boundary by our circular boundary condition.
• Figure 5: Plots of disk eccentricity $e_1$ divided by our semi-analytic expression $e_{\rm sa}$ (eq. \ref{['eq:e1_appr2']}). The $a$ values for these curves vary between $0.5$ and $2.0$. The vertical gray line marks the location of the 1:3 ELR, and the horizontal dashed gray lines bracket a $30\%$ difference between $e_1$ and $e_{\rm sa}$, which captures most of our models outward of the 1:3 ELR. We discuss the exceptional case of $\{q,h_{\rm p}/r_{\rm p}\}=\{0.008,0.035\}$ in sec. \ref{['sec:wide_gap']}.