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Planet Vulcan, aside from being the fictional planet in Star Trek which Spock comes from, is a hypothetical planet that many astronomers in the 19th century believed may have existed. This planet was supposed to have a low mass and be closer to the Sun than Mercury, so close that the telescopes of the day could not resolve it due to the overwhelming brightness of the Sun itself. Planet Vulcan, if it existed, could have surface temperatures even hotter than that of Mercury, which peaks at 700 degrees K (801 degrees F), maybe over a thousand degrees.
The reason astronomers inferred that Planet Vulcan was likely to exist was due to discrepancies in the orbit of Mercury, detected by the French mathematician Urbain Jean Joseph Le Verrier in 1840-1843 when he was trying to predict the planet's motion based on Newton's theories. The procession of its perehelion (the point at which it is closest to the Sun) around its orbit was about 43 arcseconds per century off from what Newton's theories would predict. Considering that every other planet in the Solar System moved in ways precisely predicted by Newton's theories, this was puzzling, and Planet Vulcan was invoked as the cause of the discrepancy. In 1846, Le Verrier discovered the planet of Neptune based on the same principle, perturbations in the orbit of Uranus. This galvanized astronomers to look for the planet Le Verrier predicted.
Beginning in 1859, astronomers began to report small black spots transiting the surface of the Sun, which were thought to be Planet Vulcan. Over the next six years, about a dozen alleged observations of black dots transiting the Sun were made, but the transits always failed to rematerialize on the basis of past observations. In 1866, observations of Planet Vulcan started to drop off, but earlier observations were considered as "proof" of the planet's existence, as Le Verrier announced his discovery in 1860. In 1867, two reliable astronomers claimed to have seen a Vulcan-like planet close to the Sun during an eclipse, but in hindsight this was probably a star. Le Verrier died in 1877, still convinced that he had discovered two new planets.
In 1915, it became clear that there never was a Planet Vulcan. Einstein's new theory of relativity precisely explained the anomaly as a byproduct of the Sun's gravitational field. The new numbers his theory predicted matched observations exactly. In more modern times, astronomers have intensely searched the region around the Sun for any rocky bodies, such as Vulcanoid asteroids, which could orbit in a stable gravitational region right next to the Sun. However, observations have ruled out anything larger than about 60 km (37 mi) in width. Today, the search for Vulcanoid objects continues. Many scientists are skeptical, arguing that the Yarkovsky effect, which alters orbits based on the emission of high-energy photons on an asteroid right next to a star, would cause any Vulcanoid asteroid orbits to become unstable, either sinking into the Sun or impacting Mercury.