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

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.