Who discovered the planet Neptune?

Short answer: Urbain Le Verrier discovered Neptune, with the help of Johann Gottfried Galle.

Urbain Le Verrier and Johann Gottfried Galle

Long answer: A number of people were involved in this discovery, some working together, some independently.

Possibly the first recorded observation of Neptune was by Galileo Galilei in 1612, but he thought he was looking at a star, so he doesn't get credit for discovering the planet.

In the 19th century, French astronomer Alexis Bouvard noticed some strange things about the orbit of Uranus, which was the most distant known planet at the time. He suggested that this strangeness could be explained by the gravity from another, as yet unknown, celestial object.

In 1843, English mathematician John Couch Adams began working on the problem, and proposed several areas of the sky that could be home to a new planet.

In 1845, French mathematician Urbain Le Verrier began his own calculations and came up with a prediction of where the planet should be. He asked German astronomer Johann Gottfried Galle to search in that area. Galle began the search on the evening of 23 September 1846, and found Neptune within the first hour of looking.

After the discovery, there was some controversy over who should be given credit for finding Neptune. It was widely agreed that both John Couch Adams and Urbain Le Verrier should share credit for the discovery. However, in the late 20th century, scientists and historians re-examined documents from the time and concluded that Urbain Le Verrier really deserved sole credit1.

The discovery of Neptune was the first example of a celestial object being discovered using mathematics.

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[1] Sheehan W, Kollerstrom N, Waff CB. The case of the pilfered planet. Did the British steal Neptune? Sci Am. 2004 Dec;291(6):92-9. doi: 10.1038/scientificamerican1204-92. PMID: 15597985.