Editing Tidal acceleration (section)

=== Quantitative description of the Earth–Moon case ===
The motion of the Moon can be followed with an accuracy of a few centimeters by [[lunar laser ranging]] (LLR).  Laser pulses are bounced off corner-cube prism retroreflectors on the surface of the Moon, emplaced during the [[Project Apollo|Apollo]] missions of 1969 to 1972 and by [[Lunokhod]] 1 in 1970 and Lunokhod 2 in 1973.<ref>Most laser pulses, 78%, are to the Apollo 15 site. See Williams, et al. (2008), p. 5.</ref><ref>A reflector emplaced by Lunokhod 1 in 1970 was lost for many years. See [http://www.space.com/scienceastronomy/060327_mystery_monday.html Lunar Lost & Found: The Search for Old Spacecraft by Leonard David]</ref><ref>{{Cite journal|last1=Murphy|first1=T. W. Jr.|last2=Adelberger|first2=E. G.|last3=Battat|first3=J. B. R.|last4=et|first4=al.|date=2011|title=Laser ranging to the lost Lunokhod 1 reflector|url=https://www.sciencedirect.com/science/article/abs/pii/S001910351000429X|journal=Icarus|language=en|volume=211|issue=2|pages=1103–1108|doi=10.1016/j.icarus.2010.11.010|issn=0019-1035|arxiv=1009.5720|bibcode=2011Icar..211.1103M |s2cid=11247676}}</ref> Measuring the return time of the pulse yields a very accurate measure of the distance.  These measurements are fitted to the equations of motion. This yields numerical values for the Moon's secular deceleration, i.e. negative acceleration, in longitude and the rate of change of the semimajor axis of the Earth–Moon ellipse. From the period 1970–2015, the results are:

:  −25.97 ± 0.05 arcsecond/century<sup>2</sup> in ecliptic longitude<ref name=":0" /><ref name=DE430>J.G. Williams, D.H. Boggs and W. M.Folkner (2013). [http://proba2.sidc.be/aux/data/spice/docs/DE430_Lunar_Ephemeris_and_Orientation.pdf DE430 Lunar Orbit, Physical Librations, and Surface Coordinates] p.10. "These derived values depend on a theory which is not accurate to the number of digits given." See also : Chapront, Chapront-Touzé, Francou (2002). [http://www.aanda.org/articles/aa/pdf/2002/20/aa2201.pdf A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements]</ref>
: +38.30 ± 0.08 mm/yr in the mean Earth–Moon distance<ref name=":0" /><ref name=DE430/>

This is consistent with results from [[satellite laser ranging]] (SLR), a similar technique applied to artificial satellites orbiting Earth, which yields a model for the gravitational field of Earth, including that of the tides. The model accurately predicts the changes in the motion of the Moon.

Finally, ancient observations of solar [[eclipse]]s give fairly accurate positions for the Moon at those moments.  Studies of these observations give results consistent with the value quoted above.<ref>{{cite journal|last1 = Stephenson|first1 = F.R.|last2 = Morrison|first2 = L.V.|year = 1995|title = Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990|url = http://rsta.royalsocietypublishing.org/content/351/1695/165.full.pdf|journal = Philosophical Transactions of the Royal Society of London, Series A|volume =  351|issue = 1695|pages =  165–202|doi = 10.1098/rsta.1995.0028|bibcode = 1995RSPTA.351..165S|s2cid = 120718607 }}</ref>

The other consequence of tidal acceleration is the deceleration of the rotation of Earth. The rotation of Earth is somewhat erratic on all time scales (from hours to centuries) due to various causes.<ref>Jean O. Dickey (1995): "Earth Rotation Variations from Hours to Centuries".  In: I. Appenzeller (ed.): ''Highlights of Astronomy''.  Vol. 10 pp.17..44.</ref> The small tidal effect cannot be observed in a short period, but the cumulative effect on Earth's rotation as measured with a stable clock (ephemeris time, International Atomic Time) of a shortfall of even a few milliseconds every day becomes readily noticeable in a few centuries. Since some event in the remote past, more days and hours have passed (as measured in full rotations of Earth) ([[Universal Time]]) than would be measured by stable clocks calibrated to the present, longer length of the day (ephemeris time). This is known as [[ΔT (timekeeping)|ΔT]]. Recent values can be obtained from the [[International Earth Rotation and Reference Systems Service]] (IERS).<ref>{{cite web|url=https://www.iers.org/nn_10910/IERS/EN/Science/EarthRotation/UT1-TAI.html|title=IERS – Observed values of UT1-TAI, 1962-1999|website=www.iers.org|access-date=2019-03-14|archive-date=2019-06-22|archive-url=https://web.archive.org/web/20190622181346/https://www.iers.org/nn_10910/IERS/EN/Science/EarthRotation/UT1-TAI.html|url-status=dead}}</ref>  A table of the actual length of the day in the past few centuries is also available.<ref>{{Cite web|url=http://www.iers.org/iers/earth/rotation/ut1lod/table3.html|archive-url=https://web.archive.org/web/20010908035636/http://www.iers.org/iers/earth/rotation/ut1lod/table3.html|url-status=dead|title=LOD|archive-date=September 8, 2001}}</ref>

From the observed change in the Moon's orbit, the corresponding change in the length of the day can be computed (where "cy" means "century", d day, s second, ms millisecond, 10<sup>−3</sup> s, and ns nanosecond, 10<sup>−9</sup> s):

: +2.4 ms/d/century or +88 s/cy<sup>2</sup> or +66 ns/d<sup>2</sup>.

However, from historical records over the past 2700 years the following average value is found:

: +1.72 ± 0.03 ms/d/century<ref>{{Cite journal|doi = 10.1126/science.265.5171.482|date = 1994|last1 = Dickey|first1 = Jean O.|last2 = Bender|first2 = PL|last3 = Faller|first3 = JE|last4 = Newhall|first4 = XX|last5 = Ricklefs|first5 = RL|last6 = Ries|first6 = JG|last7 = Shelus|first7 = PJ|last8 = Veillet|first8 = C|last9 = Whipple|first9 = AL|last10 = Wiant|first10 = JR|last11 = Williams|first11 = JG|last12 = Yoder|first12 = CF|display-authors=8|title = Lunar Laser ranging: a continuing legacy of the Apollo program|url = http://www.physics.ucsd.edu/~tmurphy/apollo/doc/Dickey.pdf|journal = Science|volume = 265|issue = 5171|pages = 482–90|pmid = 17781305|bibcode=1994Sci...265..482D|s2cid = 10157934 }}</ref><ref>{{cite book|author=F. R. Stephenson|title=Historical Eclipses and Earth's Rotation|url=https://books.google.com/books?id=8RAUuAAACAAJ|year=1997|publisher=Cambridge University Press|isbn=978-0-521-46194-8}}</ref><ref>{{Cite journal|last1=Stephenson|first1=F. R.|last2=Morrison|first2=L. V.|last3=Hohenkerk|first3=C. Y.|date=2016|title=Measurement of the Earth's rotation: 720 BC to AD 2015|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=472|issue=2196|pages=20160404|doi=10.1098/rspa.2016.0404|pmc=5247521|pmid=28119545|bibcode=2016RSPSA.47260404S }}</ref><ref>{{Cite journal|last1=Morrison|first1=L. V.|last2=Stephenson|first2=F. R.|last3=Hohenkerk|first3=C. Y.|last4=Zawilski|first4=M.|date=2021|title=Addendum 2020 to 'Measurement of the Earth's rotation: 720 BC to AD 2015'|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=477|issue=2246|pages=20200776|doi=10.1098/rspa.2020.0776|bibcode=2021RSPSA.47700776M |s2cid=231938488|doi-access=free}}</ref> or +63 s/cy<sup>2</sup> or +47 ns/d<sup>2</sup>. (i.e. an accelerating cause is responsible for -0.7 ms/d/cy)

By twice integrating over the time, the corresponding cumulative value is a parabola having a coefficient of T<sup>2</sup> (time in centuries squared) of (<sup>1</sup>/<sub>2</sub>) 63 s/cy<sup>2</sup> :

: Δ''T'' = (<sup>1</sup>/<sub>2</sub>) 63 s/cy<sup>2</sup> T<sup>2</sup> = +31 s/cy<sup>2</sup> T<sup>2</sup>.

Opposing the tidal deceleration of Earth is a mechanism that is in fact accelerating the rotation. Earth is not a sphere, but rather an ellipsoid that is flattened at the poles. SLR has shown that this flattening is decreasing. The explanation is that during the [[ice age]] large masses of ice collected at the poles, and depressed the underlying rocks. The ice mass started disappearing over 10000 years ago, but Earth's crust is still not in hydrostatic equilibrium and is still rebounding (the relaxation time is estimated to be about 4000 years). As a consequence, the polar diameter of Earth increases, and the equatorial diameter decreases (Earth's volume must remain the same). This means that mass moves closer to the rotation axis of Earth, and that Earth's moment of inertia decreases. This process alone leads to an increase of the rotation rate (phenomenon of a spinning figure skater who spins ever faster as they retract their arms). From the observed change in the moment of inertia the acceleration of rotation can be computed: the average value over the historical period must have been about −0.6 ms/d/century. This largely explains the historical observations.