Technique · Geometry
Gravity orbits
Bodies pulling on each other by Newton's law, drawn as the glowing trails they leave: a figure-eight of three, a solar system seen from a planet, a falling cluster, a swarm with gaps.
Gravity orbits is drawn geometry, rebuilt as it changes. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- In 1889 Henri Poincaré won a prize offered by King Oscar II of Sweden for his work on the three-body problem; in correcting a mistake in it he found that tiny differences in the start can grow without limit, the first glimpse of what is now called chaos.
- The asteroid belt has gaps, called Kirkwood gaps after Daniel Kirkwood, who noticed them in 1866: an asteroid whose year is a simple fraction of Jupiter's is tugged in the same direction again and again until it is thrown out, which is also what makes the gaps in this swarm.
- Jacques Laskar showed in 1989 that the solar system itself is chaotic: an error of a metre in the position of a planet grows tenfold about every ten million years, so no one can say where Earth will be in a hundred million.
- Halley's comet, on one of the long thin orbits drawn here, comes back about every 76 years; Edmond Halley saw in 1705 that sightings in 1531, 1607 and 1682 were the same comet and predicted its return.
- Isaac Newton published the law of gravity in 1687; he solved the motion of two bodies completely, and a third, which makes the problem unsolvable in general, defeated him and everyone since.
- The three-body figure-eight was found on a computer by Cristopher Moore in 1993, and Alain Chenciner and Richard Montgomery proved in 2000 that it exists; in 2013 Milovan Šuvakov and Veljko Dmitrašinović found thirteen new families of periodic three-body orbits.
- As seen from Earth, the path of Venus over eight years traces a five-petalled rosette, because five meetings of the two planets fit almost exactly into eight years.
The mathematics
- Every body is pulled by every other with acceleration a = G m (x2 - x1) / (|x2 - x1|^2 + 0.000)^(3/2), the inverse-square law with a small softening so that a close pass cannot send a body flying; the figure-eight uses almost none.
- The leapfrog method moves positions and velocities alternately by half steps; it does not drift in energy over long runs the way simpler methods do, which is why it is used to follow planets for millions of years.
- Kepler's third law says the square of an orbit's period is proportional to the cube of its size, so inner planets in the picture race round while the outer ones crawl; in the rotating view the same law is what makes the paths close into rosettes.
- The figure-eight has total energy -1.287 and a period of 6.326 in the units used here, with the three bodies at the corners of a triangle that is never at rest and with no body ever alone: it is stable, but only just.
In art
- Trails like these resemble long-exposure photographs of the night sky and the drawings of a spirograph or a harmonograph, because all three trace a point that obeys a smooth law of motion.
- Ptolemy explained the rosettes that planets seem to make in the sky with circles on circles, called epicycles; the picture that follows a planet shows where they come from.
More ways to use it
Related techniques
L-system · Space colonisation · Diffusion-limited aggregation · Flow ribbons · Ridgelines · Curve families · Streamlines · Paint strokes