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I understand that the moon - Earth distance is increasing as the friction of tidal action causes energy loss. Does this mechanism happen with the Earth and other planets, is the Earth - Sun distance changing?

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Yes, the Earth is moving away from the Sun in a course of millions of years, so the moving away is about as fast as the Moon's from Earth. However don't worry because as the Sun gets older it also gets hotter so the habitable zone moves together with Earth or so. At the peak of the Sun's red giant phase the Earth is said to be either there where Mars currently is (about 50% farther from the Sun) or there where Jupiter currently is (about 5 times farther). We don't know exactly, there are several models. But we do know that the Earth and the Sun move slowly away of each other, just like the Moon from the Earth (the Moon reportedly moves away at about 1.6 inches a year on average, and with Earth-Sun it is similar). All planets' orbits are moving away from the Sun slowly.

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