Planetary Dances
Take two planets going round the Sun, draw a straight line between them, wait a few days and draw another. Keep going for years and the lines weave a pattern set entirely by the two orbits. The default is the Earth and Mars over 15 years: 1,050 lines, and a flower with seven petals, one for each time the Earth overtook Mars.
| Earth and Mars align every | 780 days (2.14 years) |
|---|---|
| Earth to Mars | 15 orbits to 8, within 0.3% |
| Jupiter and Saturn | 19.86 years, close to 5:2 |
| Uranus and Neptune | 171.45 years, close to 2:1 |
What you are looking at
The Sun is in the middle. The two faint rings are the real orbits of the two planets, seen from above the solar system, and every line joins where the two planets were on the same day. The colour runs from the start of the span to the end, so you can see the pattern build in the order it was drawn.
Where the planets are on the same side of the Sun the lines are short; where they are on opposite sides the lines cross the middle. The short lines pile up into petals around each alignment, the moment the inner planet overtakes the outer one, and the pattern turns because each alignment happens a little further round the orbit than the last.
The pair menu holds eight pairings, each with its own shape: the seven petals of Earth and Mars, the three lobes of Jupiter and Saturn, the slow spiral of Uranus and Neptune. Choose Custom to pair any two planets from Mercury to Neptune.
The math
Everything comes from two numbers: how long each planet takes to go round the Sun. The time between two alignments of a pair is the synodic period, one over the difference of their two orbital frequencies. For the Earth and Mars it is 780 days, about 2.14 years. For Jupiter and Saturn it is 19.86 years, and for Uranus and Neptune 171.45 years.
The shape depends on how close the ratio of the two periods is to a ratio of small whole numbers. Fifteen Earth years are within 0.3% of eight Mars years, so after seven alignments, about 15 years, the pattern nearly closes on itself. Five orbits of Jupiter take about as long as two of Saturn, which gives the three lobes, and Neptune takes 1.96 times as long as Uranus, close to two to one. The caption names the simplest such ratio, within 2.5%, for whichever pair is on screen.
No ratio is exact, so no dance ever quite closes. The small mismatch is what makes the petals step round the Sun instead of retracing the same path.
How it is computed
Positions come from the astronomy-engine library (VSOP87 planetary theory, agreeing with the Swiss Ephemeris to well under an arcminute), as heliocentric positions in the fixed J2000 ecliptic frame, so a span of centuries is not smeared by the slow turning of the equinoxes. The orbits are real ellipses, not circles, which is why some patterns lean to one side.
The span is a whole number of synodic periods, cut into between 600 and 1,500 evenly spaced lines. On Auto the length is chosen for each pair so the pattern looks complete, and the slider stretches it from one alignment up to 60. Spans are capped at 3,000 years, because the planetary theory is built for a few thousand years around the present and a dance should not be drawn from extrapolated orbits. The start date moves the whole pattern round the Sun.
Questions
Are these the real orbits?
Yes. Every line joins the true positions of the two planets on the same day, computed from the same planetary theory astronomers use, with their real oval orbits. Nothing is idealised into circles.
Why does Earth and Mars make seven petals?
Each petal forms around an alignment, when the Earth overtakes Mars. The default draws seven synodic periods, about 15 years, so there are seven alignments and seven petals. Fifteen Earth years are almost exactly eight Mars years, so the flower nearly closes.
Is this related to the Rose of Venus?
It is the same idea seen from a different place. The Rose of Venus is Venus's path seen from the Earth; a dance joins two planets with lines as seen from above the Sun. Both get their symmetry from a near whole-number ratio of orbits, eight Earth years to thirteen Venus years in that case.