# Modeling the resonant planetary system GJ 876

## Wilhelm Kley, Man Hoi Lee, Norman Murray, Stan Peale

## Astronomy & Astrophysics (2005), **437**, 727-742

## 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.

#### Accepted Version, March 2005
pdf-file (ca. 1.4 MB)