Motivated by the population of multi-planet systems with orbital period
ratios 1<P2/P1<2, we study the long-term stability of packed two planet
systems. The Hamiltonian for two massive planets on nearly circular and nearly
coplanar orbits near a first order mean motion resonance can be reduced to a
one degree of freedom problem (Sessin & Ferraz Mello (1984), Wisdom (1986),
Henrard et al. (1986)). Using this analytically tractable Hamiltonian, we apply
the resonance overlap criterion to predict the onset of large scale chaotic
motion in close two planet systems. The reduced Hamiltonian has only a weak
dependence on the planetary mass ratio, and hence the overlap criterion is
independent of the planetary mass ratio at lowest order. Numerical integrations
confirm that the planetary mass ratio has little effect on the structure of the
chaotic phase space for close orbits in the low eccentricity (e <~0.1) regime.
We show numerically that orbits in the chaotic web produced primarily by first
order resonance overlap eventually experience large scale erratic variation in
semimajor axes and are Lagrange unstable. This is also true of the orbits in
this overlap region which are Hill stable. As a result, we can use the first
order resonance overlap criterion as an effective stability criterion for pairs
of observed planets. We show that for low mass (<~10 M_Earth) planetary systems
with initially circular orbits the period ratio at which complete overlap
occurs and widespread chaos results lies in a region of parameter space which
is Hill stable. Our work indicates that a resonance overlap criterion which
would apply for initially eccentric orbits needs to take into account second
order resonances. Finally, we address the connection found in previous work
between the Hill stability criterion and numerically determined Lagrange
instability boundaries in the context of resonance overlap.Comment: Accepted for publication in Ap
