Wavelace

Horseshoe Orbit

Bodies · mechanical systems

restricted
m=0.999
m=0.001
x=-(1+δ/100)  vy=(1+δ/100)-sqrt(1/(1+δ/100))

Open in the app The dials and keys named below are the app's.

The setup

A heavy body of mass 0.999 and a planet of 0.001, about Jupiter's share of the Sun, stand one unit apart in the frame that turns with them. A massless particle starts opposite the planet, at 1 + δ/100 from the centre, moving at the speed of a circular orbit there. Outside the planet's orbit that is a little slower than the frame, so in the turning frame it drifts slowly backwards, at about 0.03 at the opening δ = 2.

Why it turns back

The particle drifts round until it comes up on the planet from ahead. The planet, now behind it, pulls it back, it loses energy and drops inside the planet's orbit. Inside, it is faster than the frame and drifts the other way, all the way round, until it meets the planet from behind. This time the pull is forward, it rises outside again, and it never passes the planet at all. At δ = 2 the two legs run at 1.023 and 0.978 from the centre, each takes about 100 seconds at Speed 1, and it turns back about 20° short of the planet.

The slider

The slider δ sets how far outside the orbit the particle starts. The further out, the faster it drifts and the shorter the legs: about 135 seconds at 1, 85 at 3. It holds its horseshoe for 15 minutes up to 3.3. From 3.4 the turn comes too close and the particle slips past the planet, at 407 seconds there and 160 at 5.

A whole cycle takes over 200 seconds, so the preset opens with a long trail, Trail 140, which keeps fewer points to reach that far back.

History

Saturn's moons Janus and Epimetheus share an orbit this way, trading places every four years, which Voyager 1 confirmed in 1980. The asteroid 2002 AA29, found in 2002, follows a horseshoe with respect to the Earth.

Try

  • Set Speed to 4: one whole horseshoe, both legs and both turns, takes under a minute.
  • Drag δ to 3.4 and wait: after 407 seconds the particle slips past the planet and the horseshoe is gone.
  • Drag δ to 1 and raise Trail to 150: the legs close in on the orbit and take longer, so the trail has to reach further back.
  • Drag Trail down to 80 and only the last 4 seconds stay on, a short arc with no shape to it.
  • Top in the deck shows the horseshoe flat, with the planet between its two ends.

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Bodies · mechanical systems