Presets
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Jupiter
Mass multiplier compresses millions of years into thousands.
Asteroid belt
Extent and starting eccentricity of the belt.
Layers
Model checks
The science
Resonances with Jupiter
An asteroid whose orbital period is a simple fraction of Jupiter's is tugged at the same place in its orbit again and again. For a period ratio PJ : Past = p : q the asteroid sits at
ares = aJ (q/p)2/3 (Kepler III), 3:1 → 2.50 AU, 5:2 → 2.82 AU, 7:3 → 2.96 AU, 2:1 → 3.28 AUThe repeated kicks pump the eccentricity of the orbit (and, near 3:1, chaotically) until its perihelion crosses the orbit of Mars or Earth, where close encounters remove the body from the belt. The result is the set of Kirkwood gaps in the real main belt, found by Daniel Kirkwood in 1866.
The model
The page integrates the planar restricted three-body problem: the Sun and Jupiter follow a Kepler ellipse (Jupiter's eccentricity is a slider), and each asteroid is a massless test particle feeling both bodies. In heliocentric coordinates the acceleration of an asteroid is
d2r/dt2 = −G M☉ r/r3 + G MJ [ (rJ − r)/|rJ − r|3 − rJ/rJ3 ]where the last term is the indirect term from the Sun's own motion. A leapfrog step of 0.04 yr advances everything. Osculating elements follow from the energy ε = v2/2 − GM/r (a = −GM/2ε) and the eccentricity vector. An asteroid is removed when its perihelion falls below 1 AU or it escapes.
Honest simplifications
- Time compression. Real gaps need millions of years. The page multiplies Jupiter's mass by a factor (10 by default, up to 60) so the same effects appear within a few thousand years; this also widens the resonances, so the gaps are drawn too wide. The "Real Jupiter mass" preset is accurate but slow.
- Planar motion only, no Saturn, no Mars, no planetary secular resonances (which matter for the 3:1 gap's chaotic part) and no Yarkovsky drift. The removal criterion (perihelion below 1 AU) is a stand-in for real planetary encounters.
- The belt is a uniform random sample of a and e with random phases, not the real distribution.
Checks run in the page
(1) A massless particle started at the Kepler-III 3:1 distance is measured to complete exactly 3 turns per Jupiter orbit. (2) With Jupiter switched off a particle keeps its energy and angular momentum. (3) The integrated period of Jupiter equals 2π√(a3/(G(M+m))).
What to try
- Run Jupiter x40 at 300 yr/s: the survival bars (graph 1) dip first at the 3:1 line, then at 5:2 and 7:3, with the red dots of removed asteroids in graph 2.
- Compare No Jupiter: nothing happens, every asteroid keeps its orbit.
- Try Eccentric Jupiter: more resonances and secular effects strip a wider region.
- Choose Outer belt and Hildas: the 2:1 gap and the stable 3:2 clump near 3.97 AU.
- Watch graph 3: the orange curve (3:1 zone) falls much faster than the all-belt curve.