  Several properties of the Solar System, including the wide radial spacing and
orbital eccentricities of giant planets, can be explained if the early Solar
System evolved through a dynamical instability followed by migration of planets
in the planetesimal disk. Here we report the results of a statistical study, in
which we performed nearly 10^4 numerical simulations of planetary instability
starting from hundreds of different initial conditions. We found that the
dynamical evolution is typically too violent, if Jupiter and Saturn start in
the 3:2 resonance, leading to ejection of at least one ice giant from the Solar
System. Planet ejection can be avoided if the mass of the transplanetary disk
of planetesimals was large (M_disk>50 M_Earth), but we found that a massive
disk would lead to excessive dynamical damping (e.g., final e_55 < 0.01
compared to present e_55=0.044, where e_55 is the amplitude of the fifth
eccentric mode in the Jupiter's orbit), and to smooth migration that violates
constraints from the survival of the terrestrial planets. Better results were
obtained when the Solar System was assumed to have five giant planets initially
and one ice giant, with the mass comparable to that of Uranus and Neptune, was
ejected into interstellar space by Jupiter. The best results were obtained when
the ejected planet was placed into the external 3:2 or 4:3 resonance with
Saturn and M_disk ~ 20 M_Earth. The range of possible outcomes is rather broad
in this case, indicating that the present Solar System is neither a typical nor
expected result for a given initial state, and occurs, in best cases, with only
a ~5% probability (as defined by the success criteria described in the main
text). The case with six giant planets shows interesting dynamics but does
offer significant advantages relative to the five planet case.
