Beyond Neptune, icy bodies cluster into distinct dynamical classes set by Neptune's gravity. A body's orbital period follows Kepler's third law, P² ∝ a³ (P in years, a in AU when GM☉ is normalized to 1):
P = a^1.5 (Neptune: a = 30.1 AU → P ≈ 165 yr)
When PKBO/PNeptune is close to a small integer ratio, the repeated gravitational tugs from Neptune can lock the orbit into mean-motion resonance. For the 3:2 resonance (the "Plutinos", including Pluto itself, near a = 39.4 AU) the relevant quantity is the resonant argument:
φ = 3λ − 2λ_N − ϖ
λ = mean longitude of the KBO, λ_N = Neptune's mean longitude
ϖ = longitude of pericenter of the KBO
Instead of circulating through 360° like a non-resonant orbit's angle, φ for a captured body librates — oscillates back and forth around a fixed center (≈180° for the 3:2) — exactly like a pendulum trapped near the bottom of a potential well. This simulator drives each resonant body's longitude directly from that pendulum solution, φ(t) = φ0 + A·sin(ω t), with a small companion oscillation in semi-major axis (a and φ are conjugate variables in the resonance, so they oscillate together) — the standard textbook "resonance pendulum" model of mean-motion resonance (Murray & Dermott, Solar System Dynamics), rather than a full N-body integration.
- Classical belt (~42–48 AU) — low-eccentricity, non-resonant bodies untouched by strong resonances; the "main" Kuiper Belt.
- Plutinos (a ≈ 39.4 AU) — trapped in Neptune's 3:2 resonance; their libration is what keeps them from ever closely approaching Neptune despite crossing its orbit's distance.
- Twotinos (a ≈ 47.8 AU) — trapped in the 2:1 resonance, φ = 2λ − λ_N − ϖ.
- Scattered disk — high-eccentricity, high-inclination bodies gravitationally scattered outward by past close encounters with Neptune.
The libration amplitude slider controls how large a swing each resonant body makes around its center — in the real belt this amplitude is set by how gently the body was captured, most famously when Neptune's orbit migrated outward early in Solar System history and swept resonances across the primordial disk (Malhotra, 1995), pulling bodies in and pumping up their eccentricities to the values still observed today.