  Convergent migration allows pairs of planet to become trapped into mean
motion resonances. Once in resonance, the planets' eccentricities grow to an
equilibrium value that depends on the ratio of migration time scale to the
eccentricity damping timescale, $K=\tau_a/\tau_e$, with higher values of
equilibrium eccentricity for lower values of $K$. For low equilibrium
eccentricities, $e_{eq}\propto K^{-1/2}$. The stability of a planet pair
depends on eccentricity so the system can become unstable before it reaches its
equilibrium eccentricity. Using a resonant overlap criterion that takes into
account the role of first and second order resonances and depends on
eccentricity, we find a function $K_{min}(\mu_p, j)$ that defines the lowest
value for $K$, as a function of the ratio of total planet mass to stellar mass
($\mu_p$) and the period ratio of the resonance defined as $P_1/P_2=j/(j+k)$,
that allows two convergently migrating planets to remain stable in resonance at
their equilibrium eccentricities. We scaled the functions $K_{min}$ for each
resonance of the same order into a single function $K_c$. The function $K_{c}$
for planet pairs in first order resonances is linear with increasing planet
mass and quadratic for pairs in second order resonances with a coefficient
depending on the relative migration rate and strongly on the planet to planet
mass ratio. The linear relation continues until the mass approaches a critical
mass defined by the 2/7 resonance overlap instability law and $K_c \to \infty$.
We compared our analytic boundary with an observed sample of resonant two
planet systems. All but one of the first order resonant planet pair systems
found by radial velocity measurements are well inside the stability region
estimated by this model. We calculated $K_c$ for Kepler systems without
well-constrained eccentricities and found only weak constraints on $K$.

Earth and Planetary Astrophysics