It is known that the moon is moving away from Earth 2cm a year, and in doing so makes the days longer.

I want to know how many hours will have one day, when our planet is near its end.


First off, we should take the wisdom of Jack Wisdom to heart: "Calculation of the history of the lunar orbit is fraught with difficulties." While calculating the history may well be fraught with difficulties, calculating the future is hugely problematic. For the sake of argument, I'll ignore that issue.

The OP suggests the end of the planet is about 5 billion years into the future. The end is much sooner than that. The Sun will be about 10% more luminous then than it is now about a billion years in the future. That should be enough to trigger a moist greenhouse, followed soon by a runaway greenhouse. That moist greenhouse / runaway greenhouse pretty much spells the end of the lunar recession. While some of the tidal dissipation does result from the Earth tides, that is but a tiny contributor to the lunar recession rate. Almost all of the dissipation that leads to the lunar recession results from the ocean tides. Once the oceans vanish, the tidal recession of the Moon will pretty much stop in its tracks.

Assuming the mean lunar recession rate over the next billion years is somewhere between one to four centimeters per year means the Moon will have receded an additional 10,000 km to 40,000 km beyond the current 385,000 km from the center of the Earth in one billion years. The total angular momentum of the Earth-Moon system is $$L_{\text{tot}} = \left(\frac {m_e m_m}{m_e+m_m} R^2 +I_m \right)\Omega + I_e \omega$$ where $m_e$ and $m_m$ are the masses of the Earth and the Moon, $I_m$ and $I_e$ are the moments of inertia of the Earth and Moon about their centers of mass, $R$ is the mean distance between the center of the Earth and the center of the Moon, $\Omega = \sqrt{G(m_e + m_m)/R^3}$ is the Earth-Moon orbit rate, and $\omega$ is the Earth's rotation rate.

Assuming this quantity is conserved, the length of a day in a billion years will be between 25.5 hours (1 cm/year recession rate) and 31.7 hours (4 cm/year recession rate). A recession rate of 2 cm/year will result in a day of 27.3 hours.

The above assumed a constant recession rate. While that is not a good assumption, that range of one to four centimeters per year is probably okay. The lunar recession rate has ranged from over 1.1 to under 3.9 centimeters per year over last several hundred million years. The key culprit in the variations is the shape and orientation of the continents. The current rate of 3.82 centimeters per year is anomalously high thanks to the Americas and Afro-Eurasia, which together create two huge north-south barriers to the tides. This alignment also creates for some marked resonances that increase tidal friction even more than that suggested by the barriers. At other times, the orientation and shape of the continents has impeded the flow of the tides to a much lesser extent.

Once the oceans vanish, it's only the atmosphere and solid Earth tides that can result in the Moon receding from the Earth. The atmosphere will be thicker, which will greatly magnify the atmospheric contribution to tidal recession. However, that contribution is currently extremely tiny. Greatly magnifying a tiny number results in a still tiny number. Moreover, plate tectonics will stop shortly after the oceans disappear. Mountains will disappear. Friction between the atmosphere and the Earth will be much reduced. The other factor, Earth tides, is also very small. That range of a 25.5 to 31.7 hour long day might grow a bit in the four billion years between the end of the oceans and the five billion year figure cited in the question, but not by much.

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    $\begingroup$ Hang on, won't the Sun become a red giant and engulf Earth in 5 billion years? Wouldn't that put an end to Earth as well as the length of the day? $\endgroup$ – Jim Nov 19 '14 at 15:56
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    $\begingroup$ @Jim - Different authors give different times as to when the Sun will become a red giant. Some say a bit less than five billion years from now, others over seven billion years. Whether the Sun engulfs the Earth also depends on which author you read. Both are a bit irrelevant here because the OP didn't ask about what happens after the Sun becomes a red giant. The question is about the length of a day when our planet is near the end of its life five billion years from now; i.e. before the Sun becomes a red giant. The Earth's rotation rate will be more or less constant once the oceans disappear. $\endgroup$ – David Hammen Nov 19 '14 at 16:12
  • $\begingroup$ That works for me $\endgroup$ – Jim Nov 19 '14 at 16:13

The strict answer:
By definition the day (mean solar day) has 24 hours. Ever.

The statement 'and in doing so makes the days longer' is not supported by the last 44 years or so OF official most precise satelite mesurements .

As can be seen in this official graph the Length of Day as compared to a more constant reference (86400 atomic seconds) is shortening indicated by a downward trend.

The naive mistake we observe in the media is this: One more leap second? Then the LOD is enlarging.

enter image description here

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  • $\begingroup$ As I know, yes it is supported by satellite measurements. The OP has thought on the rotation period of the Earth on the day. $\endgroup$ – peterh - Reinstate Monica Mar 28 '17 at 19:16
  • $\begingroup$ @peterh I included the official data of the actual shortening of the LOD (satellite data since 1973) $\endgroup$ – Helder Velez Mar 30 '17 at 12:00
  • $\begingroup$ Ok-ok :-) Actually, I've found your post by search for the most controversial (i.e most downvoted) users of the site. :-) You lead the list. :-) $\endgroup$ – peterh - Reinstate Monica Mar 30 '17 at 12:34
  • $\begingroup$ @peterh Yes I know and I'm honoured because it is all about the same issue: Is it that the space expands or the atom is shrinking? None was able to provide a valid argument against my viewpoint (downvotes are not arguments) . $\endgroup$ – Helder Velez Mar 30 '17 at 12:48
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    $\begingroup$ Centuries are needed, perhaps millennia. Such data exist. A few centuries by measurements of the length of a day, and quite a few millennia by where ancients reported seeing solar eclipses. $\endgroup$ – David Hammen May 21 '18 at 16:04

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