Kirkwood Gap Eccentricity Pumping

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

observing…

How it works

A massless test asteroid orbits the Sun while Jupiter moves on a fixed circular orbit (the planar circular restricted three-body problem, with the Sun held fixed — an approximation that ignores Jupiter's own eccentricity and the Sun's small reflex motion). The equation of motion integrated every step is:

r̈ = −GM☉ r/|r|³ − GM♃ (r − r♃)/|r − r♃|³

A mean-motion resonance occurs where the asteroid's orbital frequency n is a simple ratio of Jupiter's, p·n ≈ q·n♃ (e.g. 3 asteroid orbits per 1 Jupiter orbit for the 3:1 gap at a ≈ 2.50 AU). The slow angle that controls the dynamics there is the resonant argument:

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

If σ librates (oscillates around a fixed value), the asteroid is trapped in a stable resonant island. If σ circulates (sweeps through all 360°), the orbit is chaotic — Jack Wisdom showed in 1982–83 that this chaotic zone at the 3:1, 5:2, 4:1 and 2:1 commensurabilities lets eccentricity random-walk upward over ~10⁵–10⁶ years until the asteroid starts crossing Mars's or the inner planets' orbits and is scattered or ejected. That slow eccentricity pumping — not a lack of asteroids to begin with — is what empties the Kirkwood gaps.

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 between 3:1 and 5:2).
Initial eccentricity — the width of the chaotic zone grows with e; a higher starting e can already sit outside the resonance's stable island even at the same a.
Integration speed — years of orbital motion advanced per real second; the trail keeps the most recent ~180 asteroid orbits.
Behaviour pill — flips to "circulating (chaotic)" once σ has swept through a full 360° within the trailing window, and to "librating (trapped)" once σ has stayed within one 360° span for that window.

t = 0 kyr

Sun Jupiter Test asteroid Kirkwood gap radii

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