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We live in a planet that orbits 93 million miles from a G-type main-sequence star, or "yellow dwarf". That is far enough for a revolution of 365 days. Such is the case of Kepler's Third Law of Motion--the farther a body is from its parent, the longer it'd take to revolve around it.

But in the far-distant future, our sun will balloon into a red giant, a star big enough to swallow Mercury and Venus out of existence. Earth and Mars will take their place, and the habitable zone (where liquid water is possible) will be moved from between Mars and Jupiter to between Saturn and Uranus.

So, in the future, Earth will no longer be one AU from the sun, which means a far shorter year. But has anyone made any calculations as to how much shorter a Terran year will be in this future scenario?

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The year length depends on the distance between the planet's centre & the Sun's centre, not the Sun's surface. So if the Sun merely expands, Earth at 1 AU will still take a standard year to perform 1 orbit.

However, when the Sun becomes a red giant, it won't just expand. As Wikipedia mentions, red giants shed a considerable amount of mass in the form of gas and dust. It's estimated that the Sun will lose around a third of its mass over the first billion years of its red giant phase, and eventually have a mass around half of its current mass; see here for details.

It's still not clear exactly what will happen to the Earth while all this is going on. It may become engulfed, like Mercury & Venus certainly will be. But even if it's not engulfed, interacting with all that gas & dust is likely to affect its orbit.

If we ignore that complication, we can easily calculate the period of a 1 AU orbit around a Sun of reduced mass. The Newtonian form of Kepler's 3rd law says that $$T^2= \left(\frac{4\pi^2}{GM}\right)r^3$$ where $T$ is the period, $G$ is the universal gravitational constant, $M$ is the Sun's mass, and $r$ is the mean orbital radius.

If we keep $r$ at 1 AU but halve the Sun's mass, then the period is multiplied by $\sqrt{2}$, giving us a year length around 516 days. That's using the current day length, by then the rotational period of Earth will be considerably longer.

I seriously doubt that the Earth will still be at 1 AU from the Sun, though, so please don't take that figure of 516 days too seriously!

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    $\begingroup$ Right, you can't take this seriously. The consequence of non-conservative mass loss is that the orbit expands! The "complication" is that there will be tidal interactions when the Sun is an AGB star that will drag it into the envelope. $\endgroup$ – Rob Jeffries Sep 1 at 9:41
  • $\begingroup$ Thanks, @RobJeffries. So I guess it's hard to determine the combined effect of that mass-induced orbit expansion and the tidal dragging. I'd always read that the Earth will end up engulfed, and only recently have I seen speculations that it might manage to avoid that fate. In any case, Earth will be uninhabitable long before then. ;) $\endgroup$ – PM 2Ring Sep 1 at 9:54

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