Orbital resonance - 2D belt and Kirkwood gaps

Equations

Units: G M_Sun = 1, planet distance 1. Each particle obeys d²r/dt² = -(r - r_S)/|r - r_S|³ - μ(r - r_P)/|r - r_P|³ (restricted three-body problem: the particles do not pull back), integrated with a leapfrog scheme while the Sun and planet follow their exact two-body orbit. A particle with period ratio p:q to the planet (p orbits per q planet orbits) sits at a = (q/p)^2/3: 3:1 at 0.481, 5:2 at 0.543, 7:3 at 0.568, 2:1 at 0.630, 3:2 at 0.763. At these places the planet's pulls repeat at the same orbital phase, so they add up instead of averaging out; eccentricity grows until the particle crosses another orbit or is scattered. For a circular planet the Jacobi constant C = 2nL_z - 2E is exactly conserved (used as a check).

Honest simplifications: a flat (planar) system, a planet mass of 3 to 6 times Jupiter's by default so gaps open within minutes (the real value is a preset but needs thousands of orbits), no collisions or particle-particle forces, and particles close to the planet or Sun are simply removed. The real Kirkwood gaps also owe part of their depth to secular and 3D effects not modelled here.

• Run the 'Strong planet' preset for a minute and read the gaps in the histogram against the yellow lines.
• Switch on the corotating frame to see resonant particles trace closed curves around the planet.
• Click a particle near a ring and watch its a and e in the third graph.