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

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?

• Out of curiosity, where do you see it stated that the Earth will be rotating synchronously with the Moon’s orbit at that time? The Earth’s rotation is indeed slowing as @usernumber states, due to tidal interactions with the Moon. But the rate of change is too slow to reach synchronous rotation by the time the Sun reaches the red giant phase (and will change even more slowly as the Moon continues to move away). – ELNJ Jul 13 at 12:58
• @ELNJ I corrected to "red giant or white dwarf" but usernumber said himself that both bodies may be tidally locked when the Sun is to become a red giant. – Ioannes Jul 13 at 13:01
• But it’s quite likely that the Earth will be engulfed by the Sun when the Sun becomes a red giant, so the spin evolution of the Earth beyond that time is a bit of an academic question :-) I used to think that the Earth and Moon would eventually synchronize, but if you work out the numbers, there isn’t enough time. I’ll add an answer later with the calculation. – ELNJ Jul 13 at 13:06
• @ELNJ Scientists just aren't sure whether the Earth (and Mars) will become victims of the Roche limit. They're only sure that Mercury and Venus will be swallowed while Jupiter won't. This question assumes that the Earth would survive the red giant phase. – Ioannes Jul 13 at 13:09

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.

• And how slow is the Earth predicted to rotate? – Ioannes Jul 13 at 13:00
• @Greenhorn That could probably be a new question ;) – usernumber Jul 13 at 14:07
• I'm A2A'ing you to the question then: astronomy.stackexchange.com/questions/36918/… – Ioannes Jul 13 at 14:15

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.

• If I understand you correctly, the Moon will not move away as fast as reportedly currently at about 1.6 inches per year. The distancing of the Moon would slow down and it would take more than a 100 billion years for the Moon to get to the final distance where it is to remain. – Ioannes Jul 13 at 15:37
• @Greenhorn It's hard to give confident predictions of the motions of the Earth & Moon after the Sun becomes a red giant. The orbits of all the planets will be affected because the Sun will lose around 50% of its current mass during its red giant phases. – PM 2Ring Jul 13 at 17:42
• @PM2Ring It is said that Earth-Moon, if they survive, would orbit there where Jupiter's orbit currently is, about 5 times farther from the Sun's center. Reportedly, the Sun anyway moves away from the Earth already, like the Moon reportedly from Earth does. – Ioannes Jul 13 at 17:51
• @Greenhorn No, the average distance between the Earth-Moon system & the Sun is currently fairly stable, although the distance does change over the year because our orbit is an ellipse. It ranges from about 147,095,000 km to 152,100,000 km. – PM 2Ring Jul 13 at 17:56
• @PM2Ring Allegedly Earth-Sun are slowly distancing from each other. I don't remember where I have that from, I think it's from one of the answers in the following link, but there are millions of questions. This is also where I learned some other things about the evolution of the Sun and the Moon. astronews.com/frag/index.shtml – Ioannes Jul 14 at 5:03