  We examine how the late divergent migration of Jupiter and Saturn may have
perturbed the terrestrial planets. We identify six secular resonances between
the nu_5 apsidal eigenfrequency of Jupiter and Saturn and the four
eigenfrequencies of the terrestrial planets (g_{1-4}). We derive analytic upper
limits on the eccentricity and orbital migration timescale of Jupiter and
Saturn when these resonances were encountered to avoid perturbing the
eccentricities of the terrestrial planets to values larger than the observed
ones. If Jupiter and Saturn migrated with eccentricities comparable to their
present day values, smooth migration with exponential timescales characteristic
of planetesimal-driven migration (\tau~5-10 Myr) would have perturbed the
eccentricities of the terrestrial planets to values greatly exceeding the
observed ones. This excitation may be mitigated if the eccentricity of Jupiter
was small during the migration epoch, migration was very rapid (e.g. \tau<~ 0.5
Myr perhaps via planet-planet scattering or instability-driven migration) or
the observed small eccentricity amplitudes of the j=2,3 terrestrial modes
result from low probability cancellation of several large amplitude
contributions. Further, results of orbital integrations show that very short
migration timescales (\tau<0.5 Myr), characteristic of instability-driven
migration, may also perturb the terrestrial planets' eccentricities by amounts
comparable to their observed values. We discuss the implications of these
constraints for the relative timing of terrestrial planet formation, giant
planet migration, and the origin of the so-called Late Heavy Bombardment of the
Moon 3.9+/-0.1 Ga ago. We suggest that the simplest way to satisfy these
dynamical constraints may be for the bulk of any giant planet migration to be
complete in the first 30-100 Myr of solar system history.
