Kirkwood Gap Resonant-Angle Phase Portrait (2D)

Drop asteroid at resonance

Initial conditions

Initial eccentricity 0.05

Integration speed 40 yr/s

Live state

Semi-major axis a

–

Eccentricity e

–

Resonant angle σ (3:1)

–

Behaviour (rolling window)

observing…

Peak e since reset (pumping so far)

–

How it works (2D phase portrait)

This 2D companion runs the identical restricted three-body integration as the 3D orbit view (RK4 on r̈ = −GM☉ r/|r|³ − GM♃ (r−r♃)/|r−r♃|³, Sun fixed, Jupiter on a circular orbit) but draws two things side by side: the same top-down orbit, and — genuinely distinct from a flattened 3D scene — a live resonant-angle phase portrait, plotting σ (0–360°) on the horizontal axis against osculating eccentricity e on the vertical axis every step.

σ = p·λ − q·λ♃ − (p−q)·ϖ
λ = mean longitude,  ϖ = longitude of pericentre

This σ–e plot is the actual analytical tool celestial mechanics uses to separate the two regimes: if the orbit is trapped in resonance, the trail stays inside a bounded closed loop (a libration island) around a fixed σ; if the orbit is chaotic, the trail sweeps all the way across the σ axis while e visibly random-walks upward — exactly the mechanism (Wisdom, 1982–83) that empties the Kirkwood gaps over ~10⁵–10⁶ years. The 3D orbit view alone cannot show this — the resonant angle and its libration/circulation behaviour only become visible once you fold time out onto (σ, e) axes.

Preset buttons — place the asteroid at zero radial velocity on a near-circular orbit at each named commensurability (or the stable 2.65 AU gap).
Initial eccentricity — a higher starting e widens the chaotic zone and can already sit outside a resonance's stable island at the same a.
Reset — clears both the orbit trail and the phase-portrait trail so you can watch a fresh trajectory build its own island (or smear) from scratch.
Peak e since reset — the highest osculating eccentricity reached since the last reset; at a resonance this climbs measurably above the starting value even over the short window the sim can integrate live, while the stable 2.65 AU gap stays essentially flat.

t = 0 kyr

Sun Jupiter Test asteroid Kirkwood gap radii