The Orbital Harmonograph
This is the figure two swinging pendulums draw when their rhythms are set to the orbits of two planets. One swings at the outer planet's pace, the other at the inner planet's, and as friction slowly stops them the pen spirals inward. Neptune and Pluto, whose orbits are locked in a 2 to 3 rhythm, draw a five-pointed star.
| Venus : Earth | 8 : 13, off by 0.03% |
|---|---|
| Jupiter : Saturn | 2 : 5, off by 0.67% |
| Neptune : Pluto | 2 : 3, a true resonance |
| Uranus : Neptune | Near 1 : 2, off by 1.95% |
What you are looking at
A harmonograph is a Victorian drawing machine: pendulums move a pen across paper, and because a pendulum slowly loses energy, the figure shrinks as it goes, leaving many nested copies of itself. This one has two pendulums, each free to swing in any direction, so each traces a slowly shrinking circle. The slow one keeps the outer planet's time and the fast one the inner planet's. The line fades as the pendulums run down, and its colour runs from the first swing to the last.
The title under the figure gives the two planets and their orbital periods as a simple ratio. Choose another pair to hear, in effect, another interval: Jupiter and Saturn draw a seven-lobed flower, Uranus and Neptune three sweeping wings, Venus and the Earth a 21-pointed star.
The math
The pen's position is the sum of the two pendulums: x(t) = sin(2πt) e^(-d t) + a sin(2πrt) e^(-1.6 d t), and y(t) the same with the swings turned a quarter turn, so each pendulum circles. Time is counted in swings of the slow pendulum, r is the ratio of the two orbital periods, and d is the damping. At a phase of 90 degrees the two pendulums circle in opposite senses, and a ratio of p to q draws a figure with p + q points. At 270 degrees they circle the same way and it has p minus q lobes, so Venus and the Earth, 13 to 8, draw five, the same five-fold pattern as the Rose of Venus.
Every pair here sits near a simple ratio. Venus and the Earth are 8 to 13, off by 0.03 percent. The Earth and Mars are 8 to 15, off by 0.31 percent. Jupiter and Saturn are 2 to 5, off by 0.67 percent, the near-resonance behind their famous Great Inequality, explained by Laplace in 1785. Neptune and Pluto are 2 to 3, a true resonance that keeps Pluto from ever meeting Neptune. Uranus and Neptune are near 1 to 2 and Mercury and Venus near 2 to 5, but both miss by about 2 percent.
How it is computed
The ratios come from the planets' orbital periods in days, the same values the rest of the gallery uses, and nothing is rounded by hand. The simple ratio for each pair is the first fraction in the continued fraction of the real ratio that comes within 2.25 percent of it, which is the best small-number fraction for its size.
The Real ratio slider moves the pendulums from that exact fraction to the real orbital ratio. At the fraction the figure closes perfectly and stays symmetrical as it shrinks. At the real ratio it slowly turns as it shrinks, by exactly the amount the two orbits miss the fraction, so Venus and the Earth barely move while Mercury and Venus twist visibly. The fast pendulum loses energy 1.6 times as fast as the slow one, which is what turns the outer star into a softer spiral toward the middle.
Questions
What is a harmonograph?
A drawing machine from the 19th century in which swinging pendulums move a pen, or the paper under it. Because the pendulums slowly lose energy, the figure spirals inward, and frequencies in simple ratios, like musical intervals, draw the most beautiful patterns.
What is an orbital resonance?
Two orbits whose periods are in a simple ratio, so the two bodies line up in the same pattern again and again. Pluto goes round the Sun twice for every three times Neptune does, and that rhythm steers Pluto away from Neptune even though their orbits cross on paper.
Is this the real path of the planets?
No. It is a picture of their rhythm, not their positions: the two orbital periods set the two pendulums. For the paths the planets really draw around each other, see Planetary Dances.