  A planet orbiting around a star in a binary system can be ejected if it lies
too far from its host star. We find that instability boundaries first obtained
in numerical studies can be explained by overlap between sub-resonances within
mean-motion resonances (mostly of the j:1 type). Strong secular forcing from
the companion displaces the centroids of different sub-resonances, producing
large regions of resonance overlap. Planets lying within these overlapping
regions experience chaotic diffusion, which in most cases leads to their
eventual ejection. The overlap region extends to shorter-period orbits as
either the companion's mass or its eccentricity increase. Our analytical
calculations reproduce the instability boundaries observed in numerical studies
and yield the following two additional results. Firstly, the instability
boundary as a function of eccentricity is jagged; thus, the widest stable orbit
could be reduced from previously quoted values by as much as 20%. Secondly,
very high order resonances (e.g., 50:3) do not significantly modify the
instability boundary despite the fact that these weak resonances can produce
slow chaotic diffusion which may be missed by finite-duration numerical
integrations. We present some numerical evidence for the first result. More
extensive experiments are called for to confirm these conclusions. For the
special case of circular binaries, we find that the Hill criterion (based on
the critical Jacobi integral) yields an instability boundary that is very
similar to that obtained by resonance overlap arguments, making the former both
a necessary and a sufficient condition for planet instability.
