  (Abridged) In planetary systems with two or more giant planets, dynamical
instabilities can lead to collisions or ejections through strong planet--planet
scattering. Previous studies for simple initial configurations with two
equal-mass planets revealed some discrepancies between the results of numerical
simulations and the observed orbital elements of extrasolar planets. Here, we
show that simulations with two unequal mass planets starting on nearly circular
orbits predict a reduced frequency of collisions and a broader range of final
eccentricities. The two-planet scattering model can easily reproduce the
observed eccentricities with a plausible distribution of planet mass ratios.
Further, the two-planet scattering model predicts a maximum eccentricity of
about 0.8, independent of the distribution of planet mass ratios, provided that
both planets are initially place on nearly circular orbits. This compares
favorably with current observations and will be tested by future planet
discoveries. The combination of planet--planet scattering and tidal
circularization may be able to explain the existence of some giant planets with
very short period orbits. Orbital migration due to planet scattering could play
an important role in explaining the increased rate of giant planets with very
short period orbits. We also re-examine and discuss various possible
correlations between eccentricities and other properties of observed extrasolar
planets. We demonstrate that the observed distribution of planet masses,
orbital periods, and eccentricities can provide constraints for models of
planet formation and evolution.
