Orbital Resonance Lab

2D companion: the same orbit in Jupiter's rotating frame and in the inertial frame, with the osculating semi-major axis, eccentricity, Jacobi constant and resonant angle that show how resonances pump an orbit.

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Presets

Playback

Asteroid orbit

Resonances sit where the asteroid period is a simple fraction of Jupiter's 11.86 years.

Perturber and resonance

A heavier Jupiter makes the slow resonant pumping visible in seconds instead of millions of years.

Model checks

The science

Why gaps and families appear

When the orbital periods of an asteroid and Jupiter form a ratio PJ/P = p/q of small integers, the asteroid meets Jupiter at the same orbital longitudes again and again and the small pulls add up instead of averaging out. The resonant location follows from Kepler's third law: ares = aJ (q/p)2/3, so 3:1 sits at 2.50 AU, 5:2 at 2.82 AU, 7:3 at 2.96 AU and 2:1 at 3.28 AU. The gaps in the main belt (Kirkwood gaps) are the places where this repeated tugging pumps the eccentricity until the asteroid is scattered; other resonances (3:2 Hildas, 1:1 Trojans) instead trap bodies.

The resonant angle φ = p λJ - q λ - (p - q) ϖ (mean longitudes and longitude of perihelion) librates around a fixed value when the asteroid is captured and circulates when it is not.

What the simulation does

It integrates the circular restricted three-body problem in the frame that co-rotates with Jupiter: x'' - 2y' = ∂Ω/∂x, y'' + 2x' = ∂Ω/∂y, with Ω = (x2+y2)/2 + (1-μ)/r1 + μ/r2, using fourth-order Runge-Kutta. The Jacobi constant C = 2Ω - v2 is conserved exactly and is the accuracy monitor. Osculating elements are obtained by converting the state back to the inertial frame around the Sun.

Honest simplifications

  • Planar, circular Jupiter, a massless test body, no Saturn and no secular effects; real gaps take millions of years to clear, so Jupiter's mass can be boosted (the default is 20 times) to see the effect in seconds. The boost changes the strength of the resonance, not its location.
  • The Sun is drawn at the origin of the inertial panel although it wobbles around the barycentre by about 0.1 % of an orbit.
  • Close encounters with Jupiter are not softened; a body that passes within a few Hill radii is ejected in the model as it would be in reality, but the numerical step is not adaptive.
What to try
  • Load the 3:1 preset and watch the eccentricity graph climb as the resonant angle librates or circulates, then nudge a by 0.1 AU away from the resonance and compare.
  • Load the Hilda preset to see a 2:3 orbit stay locked into a triangle in the rotating frame.
  • Put the Trojan preset near L4 and watch the tadpole orbit around the green marker.
Osculating semi-major axis
Osculating eccentricity
Jacobi constant conservation
Resonant angle (libration or circulation)