The Vulcan system is a star system located a little over sixteen light years from Sol, and containing the planet Vulcan. (ENT: "Home", "Daedalus")
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Background
Although there is no canon reference to Vulcan being located in the 40 Eridani system, it is widely considered as a fact used by Gene Roddenberry and backed by ENT: "Daedalus", in which Vulcan is mentioned as being slightly over sixteen light years from Earth, the same distance between Earth and 40 Eridani A. According to text commentary by Michael and Denise Okuda for "The Forge" on the ENT Season 4 DVD, the possibility of Vulcan being located in the 40 Eridani system originated from a suggestion by novelist James Blish. When three astronomers (Sallie Baliunas, Robert Donahue, and George Nassiopoulas) who had been studying the system at Mount Wilson Observatory published a letter stating "that [the star] 40 Eridani A could support a planet with Earth-like life" and that it would "have the Sun's brightness for a planet 50 million miles away", they theorized that Vulcan could be in orbit of this star, a theory Gene Roddenberry himself supported by signing their letter.
Recent discoveries have confirmed the likely presence of at least three planets in the Epsilon Eridanii system. [1]
40 Eridani is a trinary star system in the Eridanus constellation, located about 16.45 light years from Sol. Modern scientists have determined its second and third components are spectral classes A VII and M4e V and orbit each other at a period of about 252 years, anywhere between 21 and 49 astronomical units (AU) distant from each other. Both stars are in an orbit with an 8,000 year period about 418 AUs from 40 Eridani A. 40 Eridani B was the first white dwarf star to be recorded by Human astronomers.
References to the 40 Eridani system are mostly derived from background graphics and notations, not from any aired dialog.
Star Trek: Star Charts confirms the location. Name of the star is listed as 40 Eridani A (Omicron2 Eridani A). There are three planets and an asteroid belt in the system. Planet I is class-B. In Star Charts this is also the classification of Mercury. Planet II, Vulcan, is class-M. Planet III, T'Khul, is class-G and has a moon. In Star Charts class-G is the classification of planets similar to Delta Vega. Vulcan and T'Khul are twin planets. The stars 40 Eridani B and C have no planets. Distance between A and B is 400 AU, and between B and C 44 AU.