How Le Verrier found Neptune with mathematics

On the night of 23 September 1846, two astronomers at the Berlin Observatory pointed a telescope at a patch of sky in Aquarius that a mathematician in Paris had told them to search. Within an hour they found an object that was not on their star chart. It was a new planet, Neptune, less than a degree from the position Urbain Le Verrier had calculated.

It was the first time a planet had been found by prediction rather than by chance. Every earlier planet had been spotted by someone looking at the sky. Neptune was located first on paper, from its gravitational pull on another planet.

Uranus would not stay on its track

William Herschel found Uranus in 1781. Astronomers soon realised it had already been recorded several times as a faint star, as early as John Flamsteed in 1690, so they had a long run of positions to fit an orbit to. They could not fit them all. When Alexis Bouvard published tables of Uranus in 1821, the old and new observations disagreed, and within a few years the planet was drifting from even the new tables.

By the 1840s the error had grown to about two minutes of arc. That is a small angle, roughly one fifteenth of the width of the full Moon, but it was far larger than the uncertainty of the observations. Something was wrong. Either Newton’s law of gravitation failed at that distance from the Sun, or an unseen body was pulling Uranus off course.

Solving the problem backwards

Calculating how known planets disturb one another was routine celestial mechanics. Le Verrier set out to do the reverse: start from the disturbance and find the planet that causes it, with its mass, distance and position unknown. Encouraged by François Arago, director of the Paris Observatory, he took up the problem in 1845.

He first rebuilt the theory of Uranus, accounting carefully for the pulls of Jupiter and Saturn, to be sure the discrepancy was real. He then presented his results to the Académie des sciences in a series of papers. In June 1846 he gave the predicted position of the unknown planet in the sky. On 31 August he gave its orbit and mass.

To make the problem solvable, he assumed the new planet lay roughly where the Titius–Bode rule put the next planet out, about twice the distance of Uranus from the Sun, and solved for an orbit near that distance. That assumption turned out to be too large: Neptune orbits at about 30 times the Earth–Sun distance, not the 36 or so he used. But in the 1840s the two planets were close to each other in their orbits, and over that stretch the real and predicted planets pulled on Uranus in nearly the same way. The position was good even though the orbit was not.

The letter to Berlin

French astronomers showed little enthusiasm for a search. On 18 September 1846 Le Verrier wrote to Johann Gottfried Galle at the Berlin Observatory, giving the position and saying that the planet should show a small disc. The letter arrived on 23 September, and Galle had permission to look that night.

Heinrich d’Arrest, a student at the observatory, suggested comparing the sky with a new star chart of that region, recently printed for the Berlin Academy. Galle called out stars at the telescope while d’Arrest checked them against the chart. Shortly after midnight Galle described an eighth-magnitude star that the chart did not show. The next night it had moved against the background stars, as a planet should. Galle wrote to Le Verrier: the planet whose position you calculated really exists.

Adams, Challis and the question of credit

In Cambridge, John Couch Adams had worked on the same problem from 1843 and reached a similar position. His results were not published, and the search they prompted at Cambridge began only in July 1846. James Challis recorded Neptune twice that August without recognising it, because he had not yet compared his observations.

The announcement from Berlin set off a long dispute between French and British astronomers over who deserved the credit. Today both men are usually credited with predicting Neptune, while Le Verrier’s published prediction is the one that led directly to the observation. The name Neptune, proposed in the first weeks after the observation, was soon adopted internationally.

Neptune had also been seen much earlier, by Galileo in December 1612 and January 1613, who drew it in his notes as a star near Jupiter. It had been in plain view, and in the records, for more than two centuries.

Mercury, Vulcan and the limits of the method

Le Verrier tried the same approach on Mercury. In 1859 he showed that the point of Mercury’s closest approach to the Sun advanced slightly faster than the known planets could explain. He proposed another unseen planet inside Mercury’s orbit, which came to be called Vulcan. Despite several claimed sightings, no such planet was ever confirmed.

The anomaly was real. The explanation was not a planet but new physics: Albert Einstein’s general theory of relativity, completed in 1915, accounts for Mercury’s extra advance of about 43 seconds of arc per century. The same careful reading of data that found a planet in one case revealed a flaw in Newtonian gravity in the other.

Le Verrier’s later life

Le Verrier was born in Saint-Lô, Normandy, on 11 March 1811. He became director of the Paris Observatory in 1854, spent much of his career producing new theories of the motions of the planets, and helped build France’s network of weather observations. He died in Paris on 23 September 1877, 31 years to the day after Galle found Neptune. His name is one of the 72 engraved on the Eiffel Tower.

Why the name

Neptune was found by taking existing measurements seriously, and by asking what a small, persistent disagreement could mean. Much of astronomy still works this way. Public archives now hold far more observations than anyone has examined closely, and some of what they contain has yet to be recognised. Verrier Labs is named after Le Verrier for that reason.