We use the current orbital structure of large (>50 km) asteroids in the main asteroid belt to constrain the evolution of the giant planets when they migrated from their primordial orbits to their current ones. Minton & Malhotra showed that the orbital distribution of large asteroids in the main belt can be reproduced by an exponentially decaying migration of the giant planets on a timescale of τ ~ 0.5 Myr. However, self-consistent numerical simulations show that the planetesimal-driven migration of the giant planets is inconsistent with an exponential change in their semi-major axes on such a short timescale. In fact, the typical timescale is τ ≥ 5 Myr. When giant planet migration on this timescale is applied to the asteroid belt, the resulting orbital distribution is incompatible with the observed one. However, the planet migration can be significantly sped up by planet-planet encounters. Consider an evolution where both Jupiter and Saturn have close encounters with a Neptune-mass planet (presumably Uranus or Neptune itself) and where this third planet, after being scattered inward by Saturn, is scattered outward by Jupiter. This scenario leads to a very rapid increase in the orbital separation between Jupiter and Saturn which we show here to have only mild effects on the structure of the asteroid belt. This type of evolution is called a "jumping-Jupiter" case. Our results suggest that the total mass and dynamical excitation of the asteroid belt before migration were comparable to those currently observed. Moreover, they imply that, before migration, the orbits of Jupiter and Saturn were much less eccentric than their current ones.
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