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I looked up leap second in Wikipedia. It is a second added (usually) to clocks to keep them in sync with the atomic clock. It said that 25 leap seconds were added in the last 43 years and that none were subtracted. If you assume a uniform angular deceleration, when will the Earth stop turning? The answer I got seemed low.

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    $\begingroup$ I'm not sure that the Earth will ever stop rotation - the tidal friction will end once the Moon and Earth are mutually "locked in" in rotation - see en.wikipedia.org/wiki/Tidal_locking $\endgroup$ – innisfree May 3 '15 at 18:37
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    $\begingroup$ ok, wiki says "Earth is not expected to become tidally locked to the Moon before the Sun becomes a red giant and engulfs Earth and the Moon" $\endgroup$ – innisfree May 3 '15 at 18:46
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    $\begingroup$ Extrapolation is, in general, a bad idea. A linear extrapolation when the underlying process is markedly non-linear: That's a very bad idea. A linear extrapolation over billions of years based on 43 years worth of data: That's beyond bad. $\endgroup$ – David Hammen May 3 '15 at 20:09
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    $\begingroup$ Regarding the use of leap seconds for this extrapolation: That, too, was a bad idea. Suppose the Earth's rotation rate is frozen at the current rotation rate. Despite that constant rotation rate, we will still need to add leap seconds! A day is currently about one millisecond longer than 24 hours. At that rotation rate, a leap second would need to be inserted every one thousand days or so to keep midnight at midnight. $\endgroup$ – David Hammen May 3 '15 at 20:21
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    $\begingroup$ Regarding 43 years: 100 years of data isn't enough. The day was longer 100 years ago than it is today. Ten thousand years is not enough. The right time span is several tens of millions of years, or even more. $\endgroup$ – David Hammen May 3 '15 at 20:25
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I looked up leap second in Wikipedia. It is a second added (usually) to clocks to keep them in sync with the atomic clock.

Civilian clocks use Coordinated Universal Time (UTC), sometimes erroneously called Greenwich Mean Time (which no longer exists). Atomic clocks use International Atomic Time (TAI). UTC and TAI are in sync. Civilian clocks tick at the exactly the same rate as atomic clocks.

Clocks based on the rotation of the Earth use the concept of a mean solar day, divided into exactly 86,400 mean solar seconds. This is Universal Time (UT1, and variants, sometimes also erroneously called GMT). The second used by atomic clocks (and hence civilian clocks) is not quite the same as mean solar second. Currently, a mean solar day is 86,400.001 TAI seconds long. Leap seconds are added to keep midnight UTC to within 0.9 seconds of midnight UT1.

Suppose the Earth's rotation rate remains constant at it's current value, where the mean solar day is 86,4000.001 seconds long. This constant rotation rate would not mean the end of leap seconds. A leap second would still need to be added every thousand days to keep the UTC within 0.9 seconds of UT1. This is one of many reasons that a linear extrapolation based on leap seconds over the last 43 years is invalid.

For another reason, let's see what happened over the last 43 years. The below image is from the wikipedia article Fluctuations in the Length of Day.

Image source

From the image, one can see that the day was over 2 milliseconds longer 43 years ago than it is now. The Earth appears to be speeding up rather than slowing down! Even 100 years is not enough. The length of a solar day in 1911 was longer yet, about 3.4 milliseconds longer than the current day. There are lots of complicated interactions between the Earth's atmosphere, oceans, mantle and crust, outer core, and inner core. These transfer angular momentum amongst themselves. The mantle and crust also changes shape. The Earth is still recovering from the ice that pressed down on much of the north from 110,000 to 12,000 years ago. The very slow post-glacial rebound that is still ongoing transfers rock to the poles. This speeds up the Earth's rotation rate.

All of these effects are periodic. There is one effect that is not periodic, and that is the transfer of angular momentum from the Earth's rotation to the Moon's orbit. All these periodic effects, some of which have a very long period, means using the Earth's rotation rate as a proxy for uncovering this angular momentum transfer is a bit suspect. The time frame has to be very long, hundreds of millions of years or longer.

People make mistakes even with a full treatment of the math. In their otherwise excellent article, Touma & Wisdom (Touma & Wisdom, "Evolution of the Earth-Moon system," The Astronomical Journal 108 (1994): 1943-1961) back-integrated the Earth's rotation rate and the Moon's orbit and arrived at the Moon and Earth being very close together less than 1.5 billion years ago. Their mistake was to use the current value of a certain constant that isn't constant, the $k_2$ tidal Love number. The Moon's current recession rate is anomalously high. The reason is that there are two north-south barriers to a smooth tidal flow, the Americas and Afro-Eurasia. In other times, there was a single supercontinent. In yet other times, the equatorial regions were mostly ocean. The tidal torques (and hence the Moon's recession rate) were anomalously low during those times. The current lunar recession rate is about twice the average value, averaged over the last 2.45 billion years. Using this average rate and integrating backwards yields a much saner value for the creation of the Moon.

So when will the Earth stop rotating? The answer is never. Assuming the Sun doesn't destroy the Earth and Moon when it becomes a gas giant, the Moon will either escape the Earth's gravitational field, or it will would stop receding when the Moon's sidereal period (the sidereal month) and Earth's sidereal rotation rate (the sidereal day) are the same length.

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  • $\begingroup$ You mean when the lunar month and day are the same length. Just like the sidereal day is different from the solar day. The solar day would be slightly shorter than the lunar month. $\endgroup$ – LDC3 May 4 '15 at 3:21
  • $\begingroup$ @LCD3 - That's not quite right, either. I fixed it, though. It's when the Moon's sidereal period and the Earth's rotation rate are the same. Thanks for the catch. $\endgroup$ – David Hammen May 4 '15 at 5:54

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