2026ApJ..1000...65P

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 $q{({h}_{{\rm{p}}}/{r}_{{\rm{p}}})}^{-1}{\left({r}_{{\rm{gap}}}/{r}_{{\rm{p}}}\right)}^{a-\frac{b}{2}-2}$, where q is the planet-to-star mass ratio, h_p/r_p is the disk aspect ratio, 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 semianalytic 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.