How can the Earth and the Moon be in synchronous rotation if the Moon won't be in geostationary orbit?

Question

It is said that the Moon moves away from Earth and that during the Sun's red giant or white dwarf phase the Moon will be about 40% farther than now and in a synchronous rotation ("hantle rotation") with the Earth, meaning it will always be above the same location on Earth, like Pluto-Charon. But how would this be possible if the Moon wouldn't be in a geostationary orbit which is 35,786 kilometers (22,236 miles) above the Earth's surface?

Answer 1

Although the Earth's rotation is slowing due to tidal interactions with the Moon, the timescale for the Earth to reach synchronous rotation with the Moon's orbit is quite long, even by astronomical standards, and certainly won't happen during the red giant phase.

The current change in Earth's rotation is 1.7 ms/century over the past 100 years, and 2.3 ms/century since the 8th century BCE. (See Wikipedia for references.) To be conservative (i.e. to give the Earth the best chance to get to synchronous rotation), let's take the larger rate of spindown.

Converting units, this is

$$ \frac{2.3 \times 10^{-3}\ \mathrm{sec}}{10^2\ \mathrm{yr}} \times \frac{1\ \mathrm{day}}{86400\ \mathrm{sec}}=\frac{2.66\times 10^{-10}\ \mathrm{day}}{\mathrm{yr}}=\frac{1\ \mathrm{day}}{3.76\times 10^{9}\ \mathrm{yr}}$$

So at the current rate of change, the day length increases by one (current) day every 3.8 billion years. So by the time the Sun reaches the red giant phase in 5–7 billion years, the day will be at most 2 days longer, i.e. about 3 of our current days. The orbit of the Moon will be longer than its current 27.3 days (it is spiraling outward due to the same interactions), so we won't be anywhere near synchronous rotation at that point.

Assuming the Earth survives being engulfed by the Sun during the red giant phase, how long will it take? At the rate calculated above, changing from the current 1 day to the Moon's current orbital period of 27.3 days requires a 26.3 day change, which takes about 100 billion years (about 7 times longer than the current age of the universe) at the current rate of change. We still aren't synchronized at that point, since the Moon will have moved outward at that point, and thus (by Kepler's third law) will have a longer orbital period. (To calculate the right period to compare to, you would have to work out what orbital period gives synchronous rotation given the current amount of angular momentum in the Earth-Moon system.)

There are two major reasons why the above is an underestimate of how long this would take:

1. As already noted, as the Moon gains orbital angular momentum from the Earth's rotation, the Moon moves into a larger orbit. This weakens the tidal interactions between Earth and Moon, which makes the rate of change slow down over time.

2. During the red giant phase (or well before that), Earth's oceans will evaporate. Without a liquid ocean, the tidal bulge raised on the Earth by the Moon will be much smaller. Thus the exchange of angular momentum, and resulting rate of change of day length, is again slower.

There are various other effects you could speculate about (i.e. will the Earth even have a Moon, assuming it survives the Sun's red giant phase?), but this gives a sense of the long timescales you'd have to consider for this specific orbital evolution.

Answer 2

The tidal acceleration of the Moon on the Earth slows the rotation speed of the Earth down.

The altitude of the geostationary orbit depends on how fast the Earth spins. If the Earth would spin faster, the geostationary orbit would be lower. On the contrary, as the Earth's rotation speed decreases, the geostationary orbit will become higher and higher.

By the time the Sun becomes a red giant, the rotation speed of the Earth will have slowed down so much that the Moon and the Earth can be tidally locked.