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For centuries astronomers had noticed that Jupiter seemed to be speeding up and Saturn slowing down. If that went on, the solar system would slowly come apart. Pierre-Simon Laplace showed that it would not: the drift was one half of a long, repeating cycle. This post tells his story from reliable sources, explains the idea in Class 11 terms, and lets you check the key number yourself from modern orbital data.
Who was Pierre-Simon Laplace?
The MacTutor biography records that Laplace was born in Beaumont-en-Auge, Normandy, on 23 March 1749 and died in Paris on 5 March 1827. He went to the University of Caen at 16 to study theology but discovered mathematics there. He went to Paris with a letter of introduction to d'Alembert, who found him a teaching post.
He was elected to the Academie des Sciences in 1773, after reading 13 papers to it in less than three years. In 1785, as an examiner for the Royal Artillery, he examined and passed the 16-year-old Napoleon Bonaparte. Under Napoleon he briefly served as Minister of the Interior in 1799; Napoleon later wrote that he removed him after six weeks. On the day Laplace died, the Academie cancelled its meeting as a mark of respect.
What was the problem with Jupiter and Saturn?
The Dictionary of Scientific Biography article on Laplace by C C Gillispie, hosted by MacTutor, sets out the puzzle. Old and new observations showed an acceleration of Jupiter and a larger deceleration of Saturn. Euler and Lagrange had disagreed about it. At first Laplace could not explain it with gravity between the two planets, and even wondered about the pull of comets.
The breakthrough came from a well-known near coincidence: five times Saturn's mean motion almost equals twice Jupiter's. Laplace realised that terms in the equations depending on 5n′ − 2n, normally tiny, become large when this difference is small. In memoirs written between November 1785 and April 1788, he showed that the two planets swap speed in a periodic "great inequality". His full memoir on Jupiter and Saturn gave a period of 929 years.
Historian Curtis Wilson devoted a 276-page study, The great inequality of Jupiter and Saturn: from Kepler to Laplace, to how this problem was finally cracked.
Worked example: finding the 5:2 rhythm
NASA's planetary fact sheets give sidereal orbital periods of 4,332.589 days for Jupiter and 10,755.699 days for Saturn.
Mean motion of each planet (degrees per year, using 365.25 days per year). Jupiter: 360 × 365.25 / 4332.589 = 30.35° per year. Saturn: 360 × 365.25 / 10755.699 = 12.23° per year.
Form 2n and 5n′. 2 × 30.35 = 60.70°. 5 × 12.23 = 61.13° (61.126 before rounding).
The difference. 61.126 − 60.698 = 0.428° per year. Compared with Jupiter's 30.35° per year, that is about 1/71, close to the 1/74 that Laplace worked with.
The period. For the combined angle to go once round 360° takes 360 / 0.428 ≈ 840 years.
This rough estimate gives several centuries, the same order as Laplace's 929 years. The full calculation also depends on the shapes and tilts of the orbits, which is why the exact figure differs.
Common mistakes:
Mixing up period and mean motion. The planet with the longer period has the smaller mean motion.
Rounding too early. The difference is small, so rounding each motion to whole degrees wipes it out.
Thinking 5:2 means an exact ratio. Saturn's period is about 2.48 times Jupiter's, not exactly 2.5. The small mismatch is what makes the cycle long.
For the basics of orbits and periods, see our posts on Kepler's laws and satellite problems in gravitation.
What else did Laplace do?
MacTutor records that in 1796 his Exposition du systeme du monde set out the nebular hypothesis, that the solar system formed from a cooling, rotating cloud of gas. His Traite de mecanique celeste appeared in five volumes from 1799, and his Theorie analytique des probabilites in 1812. Laplace's equation and the Laplace transform carry his name.
What can a student take from Laplace's way of working?
Take anomalies seriously. A small mismatch between theory and observation led to his central result.
Look for small denominators. A quantity close to zero in a denominator can turn a tiny effect into a large one.
Check against data. Laplace compared his formulas with observations going back to ancient oppositions of Saturn.
Orbits and their periods are easier to see than read, and the Fermi Simulation Lab has interactive simulations for ideas like this one.
Use the same method with Uranus and Neptune, whose periods are close to 1:2, and estimate how long their cycle would be.
