  We run a suite of dissipative N-body simulations to determine which regions
of phase space for smooth disk migration are consistent with the GJ876 system,
an M-dwarf hosting three planets orbiting in a chaotic 4:2:1 Laplace resonance.
We adopt adaptive mesh refinement (AMR) methods which are commonly used in
hydrodynamical simulations to efficiently explore the parameter space defined
by the semi-major axis and eccentricity damping timescales. We find that there
is a large region of phase space which produces systems in the chaotic Laplace
resonance and a smaller region consistent with the observed eccentricities and
libration amplitudes for the resonant angles. Under the assumptions of Type I
migration for the outer planet, we translate these damping timescales into
constraints on the protoplanetary disk surface density and thickness. When we
strongly (weakly) damp the eccentricities of the inner two Laplace planets,
these timescales correspond to disk surface densities around ten thousand (a
few hundred) grams per square centimeter and disk aspect ratios between 1-10%.
Additionally, smooth migration produces systems with a range of chaotic
timescales, from decades and centuries to upwards of thousands of years. In
agreement with previous studies, the less chaotic regions of phase space
coincide with the system being in a low energy double apsidal corotation
resonance. Our detailed modeling of multi-planetary systems coupled with our
AMR exploration method enhances our ability to map out the parameter space of
planet formation models, and is well suited to study other resonant chain
systems such as Trappist-1, Kepler-60, and others.
