# Modeling the resonant planetary system GJ 876

## Abstract:

The two planets about the star GJ 876 appear to have undergone extensive migration from their point of origin in the protoplanetary disk --- both because of their close proximity to the star (30 and 60 day orbital periods) and because of their occupying three stable orbital resonances at the 2:1 mean-motion commensurability. The resonances were most likely established by converging differential migration of the planets leading to capture into the resonances. A problem with this scenario is that continued migration of the system while it is trapped in the resonances leads to orbital eccentricities that rapidly exceed the observational upper limits of $e_1 \approx 0.31$ and $e_2 \approx 0.05$. As seen in forced 3-body simulations, these lower eccentricities would persist during migration only for an eccentricity damping rate $\dot e_2/e_2$ exceeding $\approx 40\, \dot a_2/a_2$. Previous theoretical and numerical analyses have found $\dot e/e \sim \dot a/a$ or even eccentricity growth through disk-planet interactions. In an attempt to find effects that could relax the excessive eccentricity damping requirement, we explore the evolution of the GJ 876 system using two-dimensional hydrodynamical simulations that include viscous heating and radiative cooling in some cases. Before we evolve the whole system, the disk with just the outer planet embedded is brought into equilibrium. We find that the relaxed disk remains circular in all models for low planet-to-star mass ratios $q_2$, but becomes eccentric for high mass ratios for those models with fixed temperature structure. The disk in models with full radiative thermodynamics remains circular for all $q_2$ considered due to the larger disk temperatures. Given the small stellar mass, the mass ratio for the GJ 876 system is rather high (with minimum $q_2 = 5.65 \times 10^{-3}$), and so the GJ 876 disk may have been slightly eccentric during the migration. With a range of parameter values, we find that a hydrodynamic evolution within the resonance, where only the outer planet interacts with the disk, always rapidly leads to large values of eccentricities that exceed those observed --- similar to the three-body results. The resonance corresponding to the resonant angle $\theta_1 = 2\lambda_2 - \lambda_1 - \varpi_1$ (involving the inner planet's periapse longitude, $\varpi_1$) is always captured first. There is no additional delay in capturing $\theta_2 = 2\lambda_2 - \lambda_1 - \varpi_2$ into resonance that is attributable to the secular prograde contribution to the precession of $\varpi_2$ from the interaction with the disk, but an eccentric disk can induce a large outer planet eccentricity $e_2$ before capture and thereby further delay capture of $\theta_2$ for larger planetary masses. The delay in capturing $\theta_2$ into libration, while delaying the resonance-induced growth of $e_2$, has no effect on the forced eccentricities of both planets, which are uniquely determined by the resonance conditions, once both $\theta_j$ are librating. Only if mass is removed from the disk on a time scale of the order of the migration time scale (before there has been extensive migration after capture), as might occur for photoevaporation in the late phases of planet formation, can we end up with eccentricities that are consistent with the observations.