Newton explained the orbits of the planets with a single law, but he knew it brought a problem with it. Each planet is pulled by the Sun and also, a little, by the other planets. These small attractions pile up over time. Newton suspected that, in the long run, the Solar System would fall into disorder and that divine intervention would be needed, from time to time, to set it right again.

The puzzle of Jupiter and Saturn

Astronomers had reasons to worry. Comparing ancient and modern observations, Edmond Halley had noticed in 1695 that Jupiter seemed to be moving ever faster and Saturn ever slower. If it were a permanent trend, Jupiter would end up falling into the Sun and Saturn would be lost in space. Euler and Lagrange tried to explain the phenomenon and failed.

In 1785 Laplace found the answer. It was not a trend but a very slow oscillation, with a period of almost 900 years. Astronomers had seen only a piece of it and mistaken the stretch for a straight line.

Five turns against two

The key lies in the periods. Jupiter goes around the Sun in about 11.86 years and Saturn in 29.46. That means that each year they advance

$$n_J = \frac{360^\circ}{11.86} \approx 30.35^\circ, \qquad n_S = \frac{360^\circ}{29.46} \approx 12.22^\circ.$$

Notice that two turns of Saturn last almost as long as five of Jupiter: $2 \times 29.46 = 58.9$ years and $5 \times 11.86 = 59.3$ years. We say the two planets are close to a 5 to 2 resonance. The difference that matters is

$$2n_J - 5n_S \approx 60.70^\circ - 61.10^\circ = -0.40^\circ \text{ per year}.$$

It is small, and that is what makes the perturbation slow to undo itself: a full turn of this angle takes

$$\frac{360^\circ}{0.40^\circ \text{ per year}} \approx 880 \text{ years}.$$

Conjunctions in a triangle

There is a visual way to see the same thing. The two planets line up with the Sun, in a conjunction, every

$$\frac{1}{\frac{1}{11.86} - \frac{1}{29.46}} \approx 19.86 \text{ years},$$

and each conjunction happens about $242.7^\circ$ further on than the previous one. Three consecutive conjunctions advance $3 \times 242.7^\circ = 728.1^\circ$, that is, two full turns plus $8.1^\circ$. So the conjunctions fall almost always in the same three places in the sky, at the vertices of a nearly equilateral triangle, and the triangle slowly turns. It is in those places that the tugs of one planet on the other repeat and add up:

Since the triangle looks the same again after turning $120^\circ$, the cycle lasts $\frac{120^\circ}{8.1^\circ} \approx 15$ trios of conjunctions, that is, $15 \times 3 \times 19.86 \approx 890$ years: the same period, by another road. Laplace computed that, over the cycle, Saturn runs ahead and falls behind its mean position by about $0.8^\circ$, and Jupiter by about $0.35^\circ$, always in opposite directions. The tables he built with this correction agreed with two thousand years of observations.

The Solar System corrects itself

The result on Jupiter and Saturn was part of a larger programme. As early as 1773, at 24, Laplace had shown that, to a first approximation, the semi-major axes of the orbits, which fix the size and period of each planet, do not change with time: perturbations make them oscillate but not grow without limit. In 1787 he explained another old puzzle in the same way, the apparent acceleration of the Moon, linking it to a slow change in the eccentricity of the Earth’s orbit.

The conclusion seemed complete: the irregularities of the heavens are periodic and cancel out. The Solar System would be a stable machine, needing no adjustments from outside. It is from this result that the sentence attributed to Laplace in front of Napoleon comes, that he had no need of the “hypothesis” of God to explain the sky.

Exercise: the resonance of Neptune and Pluto

Neptune goes around the Sun in about 164.8 years and Pluto in about 248.0. What simple resonance is there between the two?

Compute the ratio: $\frac{248.0}{164.8} \approx 1.505 \approx \frac{3}{2}$. While Neptune makes three turns ($3 \times 164.8 = 494.4$ years), Pluto makes two ($2 \times 248.0 = 496.0$ years). It is a 3 to 2 resonance, and it is what keeps the two from approaching each other, even though Pluto’s orbit crosses Neptune’s.

Exercise: the synodic period

Show that, if two planets have periods $T_1 < T_2$, the time between two consecutive conjunctions is $\frac{T_1 T_2}{T_2 - T_1}$, and apply it to Jupiter and Saturn.

In a time $t$, the faster planet advances $\frac{360^\circ t}{T_1}$ and the slower one $\frac{360^\circ t}{T_2}$. There is a new conjunction when the faster one gains a full turn: $\frac{t}{T_1} - \frac{t}{T_2} = 1$, that is, $t = \frac{T_1 T_2}{T_2 - T_1}$. For Jupiter and Saturn: $\frac{11.86 \times 29.46}{17.60} \approx 19.85$ years.

Sources: Pierre-Simon Laplace, “Théorie de Jupiter et de Saturne”, Mémoires de l’Académie royale des sciences (1785, published 1788) · Curtis Wilson, “The Great Inequality of Jupiter and Saturn: from Kepler to Laplace”, Archive for History of Exact Sciences 33 (1985) · Jacques Laskar, “A numerical experiment on the chaotic behaviour of the Solar System”, Nature 338 (1989) · Charles C. Gillispie, Pierre-Simon Laplace (1997).

All topics on Pierre-Simon Laplace · ← The life of Pierre-Simon Laplace · The nebular hypothesis →

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