Resonances – the numerical ratios of the heavenly bodies
Io, Europa and Ganymede orbit Jupiter in the ratio 1 : 2 : 4, the planets of TRAPPIST-1 form a chain of fifths and fourths. The harmony of the spheres in today's astronomy.
The Pythagoreans believed the heavenly bodies were ordered by musical ratios (see Harmony of the spheres). Kepler searched for this order all his life. Today’s astronomy does indeed know heavenly bodies whose orbital periods stand in simple whole-number ratios. This is called orbital resonance.
The moons of Jupiter
The three inner large moons of Jupiter take 1.77 days (Io), 3.55 days (Europa) and 7.16 days (Ganymede) for one orbit. That is almost exactly 1 : 2 : 4. While Ganymede circles Jupiter once, Europa circles twice and Io four times. This threefold resonance was described by Pierre-Simon Laplace in the 18th century and is therefore called the Laplace resonance. It has a remarkable consequence: the three moons can never all line up on one side of Jupiter at the same time.
Orbits in the correct proportions (by Kepler's third law). 'Listen' plays each orbit as a tone whose pitch matches the orbital frequency. For Jupiter's moons the result is octaves.
The resonance is no accident. At each encounter the moons pull on one another a little, and over millions of years their orbits have settled into this stable ratio. Heard as tones, the three orbits sound three octaves stacked on each other, the purest harmony the Pythagoreans knew, 1 : 2.
TRAPPIST-1: a chain of intervals
In 2017 seven Earth-sized planets were discovered around the small star TRAPPIST-1, about 40 light years away. Their orbital periods form a chain of nearly whole-number ratios from neighbour to neighbour:
| Planets | Periods (days) | Ratio | musically |
|---|---|---|---|
| b – c | 1.51 : 2.42 | ≈ 8 : 5 | minor sixth |
| c – d | 2.42 : 4.05 | ≈ 5 : 3 | major sixth |
| d – e | 4.05 : 6.10 | ≈ 3 : 2 | fifth |
| e – f | 6.10 : 9.21 | ≈ 3 : 2 | fifth |
| f – g | 9.21 : 12.35 | ≈ 4 : 3 | fourth |
| g – h | 12.35 : 18.77 | ≈ 3 : 2 | fifth |
Astronomers were even able to predict the period of the outermost planet h from the resonances before it was measured. On the current view the planets migrated inwards as they formed and “locked” into these ratios one after another.
Neptune, Pluto and the gaps in the asteroid belt
Our own solar system has such ratios too. Pluto circles the sun twice while Neptune does so three times, 2 : 3, a fifth. That is why the two never come close, although Pluto’s orbit crosses Neptune’s.
Resonances also clear things out. In the asteroid belt between Mars and Jupiter there are almost empty bands, the Kirkwood gaps, named after Daniel Kirkwood, who explained them in 1866. They lie exactly where an asteroid would circle the sun three, five or seven times while Jupiter does so once, twice or three times:
| Gap | Ratio to Jupiter | Distance from the sun |
|---|---|---|
| 1 | 3 : 1 | 2.50 AU |
| 2 | 5 : 2 | 2.82 AU |
| 3 | 7 : 3 | 2.95 AU |
| 4 | 2 : 1 | 3.27 AU |
There every asteroid is nudged by Jupiter again and again in the same rhythm until it leaves its orbit. The same produces the Cassini division in Saturn’s rings, through a 2 : 1 resonance with the moon Mimas.
What to make of it
The planets of the solar system as a whole follow no simple scale, and Kepler’s attempt to explain their distances by solids and intervals did not work out. But the principle the Pythagoreans trusted is real: where bodies act on one another long enough, they settle into simple ratios, 1 : 2, 2 : 3, 3 : 4, exactly the ratios of the tetractys (see Number and sound). In this sense the harmony of the spheres is no fantasy but the result of gravity and time.