Are Earth and the Moon "twin planets"?

Is the Moon a satellite of Earth or are Earth and the Moon a twin planet system?

I would say both. Whenever a group of bodies sharing an orbit the Sun also orbit each other, the largest one is called the planet and the rest are considered satellites. So, theoretically, if you have a double planet where one component is only 1% larger than the other, we would still call them "planet" and "satellite".

Definition of a double planet is less clear. There is no obvious lower limit to how big (or massive) a satellite has to be relative to the planet in order for the system to be considered a "double planet". Pluto and Charon (1/10 of Pluto's mass) are definitely a double planet and most people would consider Earth and the Moon (1/81 of Earth's mass) a double planet.

It is possible that once some Earth-sized extrasolar planets with moons are discovered, the term "double planet" will be defined more strictly, but so far it is completely unofficial.

January 2003, Matija Cuk (more by Matija Cuk)

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How close was the Moon to the Earth when it formed?

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