From Academic Kids

Counter-Earth is a non-existent Earth-like hypothetical planet on Earth's orbit but on the other side of the Sun. It was hypothesized by the Pythagorean school and used by John Norman as the setting for his Gor novels, as well as appearing in multiple Marvel Comics.

A similar concept was used in Gerry Anderson's 1969 movie Journey to the Far Side of the Sun, (also known as Doppelganger), in which the Counter-Earth is identical to Earth in every respect except that left and right are reversed. (But if that were so, then surely the planet would also orbit the sun in the opposite direction and collide with the Earth....)

The concept was also used in the 1974 movie The Stranger, in which the Counter-Earth, known as Terra by its inhabitants, is culturally and evolutionarily identical to Earth in nearly every respect, however, it appears to have skewed significantly sometime in the last century or two. An astronaut from Earth, whose three-man spaceship crashes there, 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 concept of the movie begs explanation as to how the astronaut's craft could mistake their position by not properly charting Venus' and Mars' positions relative to Earth and realize that the Earth-like orb is a previously unknown planet.

The Japanese monster movie Gamera vs. Guiron also had such a counter-Earth.

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. Furthermore, 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.

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. This would eventually result in the two planets either colliding or experiencing a near-miss that would eject both of them from their current orbit.

A counter earth cannot be at the Lagrange point L3. The Sun-Jupiter-Trojan Asteriods system is a stable Lagrange orbit forming an equilateral triangle. 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. Also, Lagrange orbits are by definition circular, or at least nearly so in the real world, and Earth's orbit is too elliptical to be part of a Lagrangian system.

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