Counter-Earth
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Counter-Earth is an Earth-like hypothetical planet, usually sharing an orbit with Earth but on the opposite side of the Sun. It was hypothesized by the Pythagorean school as Antichthon and was periodically speculated about, but its existence has been disproven by modern scientific models and observations. Nonetheless, it has appeared many times in science fiction stories, usually serving as an allegory for the real Earth.
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References in fiction
In the first episode of the radio show The Adventures of Superman, (debuting February 12, 1940), the planet Krypton was referred to as "situated on the other side of the Sun" from the Earth.
Counter-Earth was used by John Norman as the setting for his Gor novels.
It has appeared in multiple stories published by Marvel Comics. See Counter-Earth (comics).
Gerry Anderson's 1969 movie Journey to the Far Side of the Sun, (also known as Doppelgänger), features a Counter-Earth which is identical to Earth in every respect except that left and right are reversed.
In the 1974 movie The Stranger, the Counter-Earth (known as Terra by its inhabitants) is culturally and evolutionarily identical to Earth in nearly every respect, with the most obvious difference being the planet's three moons. However, it appears to have skewed significantly sometime in the last century or two. An astronaut from Earth crashes there, and discovers a strange dictatorship known as the Perfect Order. Other than the fact that everyone is left-handed, technology is about the same, although geared for such purposes as monitoring of the population to assure adherence to the Order.
The Japanese monster movie Gamera vs. Guiron had a Counter-Earth.
The series of graphic novels Les Cités Obscures by François Schuiten and Benoît Peeters features a group of city-states on a Counter-Earth.
Paul Capon wrote a series of novels, starting with "The Other Side of the Sun" (also serialised on radio by the BBC) set on Antigeos, a Counter-Earth.
Scientific analysis
If such a planet actually existed, it would be permanently hidden behind the sun, but would nevertheless be detectable from Earth because of its gravitational influence upon the other planets of the Solar System. No such influence has been detected, and indeed space probes sent to Venus, Mars and other places could not have successfully flown by or landed on their targets if a Counter-Earth existed, as it was not accounted for in navigational calculation.
Since Earth has an elliptical orbit and its orbital velocity varies slightly; a Counter-Earth would have to be in a very precisely tuned orbit to remain behind the Sun at all times. Further, a Counter-Earth's orbit would be unstable on a relatively short geological timescale, and would soon drift away from being exactly opposite Earth's location.
A Counter-Earth would be at the (unstable) Lagrange Point, L3.
The Sun-Jupiter Trojan asteroid system is an example of a stable Lagrange orbit. A look at Equation 10 in section 14 of Lectures on Celestial Mechanics by Siegel and Moser shows the relation between the masses of the bodies and the distances between them in the case of a colinear orbit. However, these linear orbits are not as stable as, say, the equilateral Lagrange orbit.