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 , \phi , librate about values > \pi ( trailing ) as opposed to < \pi ( leading ) .
We provide , for the first time , an explanation of this phenomenon , using physical , analytic , and semi-analytic arguments .
Central to our understanding is how planetary migration shifts the equilibrium points of the superposed direct and indirect potentials .
Symmetric libration , in which \phi librates about \sim \pi , precedes capture into asymmetric resonance .
As a particle transitions from symmetric to asymmetric libration , if \phi exceeds its value , \psi , at the unstable point of asymmetric resonance , then the particle is caught into trailing resonance , while if \phi < \psi , 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 \phi > \psi while in symmetric libration .
This fractional time increases with faster migration because migration not only shifts \psi to values < \pi , but also shifts the stable point of symmetric libration to values > \pi .
Smaller eccentricities prior to capture strengthen the effect of these shifts .
Large capture asymmetries appear for exponential timescales of migration , \tau , shorter than \sim 10 ^ { 7 } yr .
The observed distribution of 2:1 KBOs ( 2 trailing and 7 leading ) excludes \tau \leq 10 ^ { 6 } yr with 99.65 % confidence .