Much of the dynamical structure of the Kuiper Belt can be explained if Neptune migrated over several AU, and/or if Neptune was scattered to an eccentric orbit during planetary instability. An outstanding problem with the existing formation models is that the distribution of orbital inclinations they predicted is narrower than the one inferred from observations. Here we perform numerical simulations of Kuiper Belt formation starting from an initial state with Neptune at AU and a dynamically cold outer disk extending from beyond to 30 AU. Neptune's orbit is migrated into the disk on an e-folding timescale 1 ≤ τ ≤ 100 Myr. A small fraction (~10?3) of the disk planetesimals become implanted into the Kuiper belt in the simulations. By analyzing the orbital distribution of the implanted bodies in different cases we find that the inclination constraint implies that Myr and AU. The models with Myr do not satisfy the inclination constraint, because there is not enough time for various dynamical processes to raise inclinations. The slow migration of Neptune is consistent with other Kuiper Belt constraints, and with recently developed models of planetary instability/migration. Neptune's eccentricity and inclination are never large in these models (, ), as required to avoid excessive orbital excitation in the 40 AU region, where the Cold Classicals presumably formed.
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