SIMBAD references

The spatial distribution of Kuiper Belt objects (KBOs) in 2:1 exterior resonance with Neptune constrains that planet's migration history. Numerical simulations demonstrate that fast planetary migration generates a larger population of KBOs trailing rather than leading Neptune in orbital longitude. This asymmetry corresponds to a greater proportion of objects caught into asymmetric resonance such that their resonance angles φ librate about values greater than π (trailing) as opposed to less than π (leading). We provide, for the first time, an explanation of this phenomenon, using physical, analytic, and semianalytic arguments. Central to our understanding is how planetary migration shifts the equilibrium points of the superposed direct and indirect potentials. Symmetric libration, in which φ librates about ~π, precedes capture into asymmetric resonance. As a particle transitions from symmetric to asymmetric libration, if φ exceeds the value ψ at the unstable point of asymmetric resonance, then the particle is caught into trailing resonance, while if φ<ψ, the particle is caught into leading resonance. The probability that the KBO is caught into trailing resonance is determined by the fraction of time it spends with φ>ψ while in symmetric libration. This fractional time increases with faster migration because migration not only shifts ψ to values less than π but also shifts the stable point of symmetric libration to values greater than π. Smaller eccentricities prior to capture strengthen the effect of these shifts. Large capture asymmetries appear for exponential timescales of migration τ shorter than ∼10⁷ yr. The observed distribution of 2:1 KBOs (two trailing and seven leading) excludes τ≤10⁶ yr with 99.65% confidence.