Editing Tidal acceleration (section)

== Earth–Moon system ==
=== Discovery history of the secular acceleration ===
[[Edmond Halley]] was the first to suggest, in 1695,<ref>E Halley (1695), [https://www.jstor.org/stable/102291?seq=1#page_scan_tab_contents "Some Account of the Ancient State of the City of Palmyra, with Short Remarks upon the Inscriptions Found there"], ''Phil. Trans.'', vol.19 (1695–1697), pages 160–175; esp. at pages 174–175. (see also transcription using a modern font [https://books.google.com/books?id=b-Q_AAAAYAAJ&pg=PA65 here])</ref> that the mean motion of the Moon was apparently getting faster, by comparison with ancient [[eclipse]] observations, but he gave no data.  (It was not yet known in Halley's time that what is actually occurring includes a slowing-down of Earth's rate of rotation: see also [[Ephemeris time#History of ephemeris time (1952 standard)|Ephemeris time – History]].  When measured as a function of [[mean solar time]] rather than uniform time, the effect appears as a positive acceleration.) In 1749 [[Richard Dunthorne]] confirmed Halley's suspicion after re-examining ancient records, and produced the first quantitative estimate for the size of this apparent effect:<ref>Richard Dunthorne (1749), [http://rstl.royalsocietypublishing.org/content/46/492/162.full.pdf "A Letter from the Rev. Mr. Richard Dunthorne to the Reverend Mr. Richard Mason F. R. S. and Keeper of the Wood-Wardian Museum at Cambridge, concerning the Acceleration of the Moon"], ''Philosophical Transactions'', Vol. 46 (1749–1750) #492, pp.162–172;  also given in Philosophical Transactions (abridgements) (1809), [https://archive.org/stream/philosophicaltra09royarich#page/669/mode/2up vol.9 (for 1744–49), p669–675] as "On the Acceleration of the Moon, by the Rev. Richard Dunthorne".</ref> a centurial rate of +10″ (arcseconds) in lunar longitude, which is a surprisingly accurate result for its time, not differing greatly from values assessed later, ''e.g.'' in 1786 by de Lalande,<ref>J de Lalande (1786): [http://gallica.bnf.fr/ark:/12148/bpt6k3585j/f484.double "Sur les equations seculaires du soleil et de la lune"], Memoires de l'Academie Royale des Sciences, pp.390–397, at page 395.</ref> and to compare with values from about 10″ to nearly 13″ being derived about a century later.<ref>J D North (2008), "Cosmos: an illustrated history of astronomy and cosmology", (University of Chicago Press, 2008), chapter 14, at [https://books.google.com/books?id=qq8Luhs7rTUC&pg=PA454 page 454].</ref><ref>See also P Puiseux (1879), [http://archive.numdam.org/article/ASENS_1879_2_8__361_0.pdf "Sur l'acceleration seculaire du mouvement de la Lune"], Annales Scientifiques de l'Ecole Normale Superieure, 2nd series vol.8 (1879), pp.361–444, at pages 361–365.</ref>

[[Pierre-Simon Laplace]] produced in 1786 a theoretical analysis giving a basis on which the Moon's mean motion should accelerate in response to [[perturbation (astronomy)|perturbational]] changes in the eccentricity of the orbit of Earth around the [[Sun]]. Laplace's initial computation accounted for the whole effect, thus seeming to tie up the theory neatly with both modern and ancient observations.<ref>{{cite book|last=Britton|first=John|date=1992|title=Models and Precision: The Quality of Ptolemy's Observations and Parameters|url=https://archive.org/details/modelsprecisionq00brit|publisher=Garland Publishing Inc.|page=[https://archive.org/details/modelsprecisionq00brit/page/157 157]|isbn=978-0815302155  }}</ref>

However, in 1854, [[John Couch Adams]] caused the question to be re-opened by finding an error in Laplace's computations: it turned out that only about half of the Moon's apparent acceleration could be accounted for on Laplace's basis by the change in Earth's orbital eccentricity.<ref>{{cite journal|doi = 10.1098/rstl.1853.0017|last1 = Adams|first1 = J C|date = 1853|title = On the Secular Variation of the Moon's Mean Motion|journal = Phil. Trans. R. Soc. Lond.|volume = 143|pages = 397–406| s2cid=186213591 |doi-access =  }}</ref> Adams' finding provoked a sharp astronomical controversy that lasted some years, but the correctness of his result, agreed upon by other mathematical astronomers including [[Charles-Eugene Delaunay|C. E. Delaunay]], was eventually accepted.<ref>D. E. Cartwright (2001), [https://archive.org/details/tidesscientifich0000cart/page/144 "Tides: a scientific history"], (Cambridge University Press 2001), chapter 10, section: "Lunar acceleration, Earth retardation and tidal friction" at pages 144–146.</ref> The question depended on correct analysis of the lunar motions, and received a further complication with another discovery, around the same time, that another significant long-term perturbation that had been calculated for the Moon (supposedly due to the action of [[Venus]]) was also in error, was found on re-examination to be almost negligible, and practically had to disappear from the theory.  A part of the answer was suggested independently in the 1860s by Delaunay and by [[William Ferrel]]: tidal retardation of Earth's rotation rate was lengthening the unit of time and causing a lunar acceleration that was only apparent.<ref>{{cite journal|last1= Khalid|first1=  M.|last2=Sultana|first2= M.|last3= Zaidi|first3 =F.|date= 2014|title= Delta: Polynomial Approximation of Time Period 1620–2013|journal= Journal of Astrophysics|volume= 2014|pages=  1–4|doi= 10.1155/2014/480964|doi-access= free}}</ref>

It took some time for the astronomical community to accept the reality and the scale of tidal effects.  But eventually it became clear that three effects are involved, when measured in terms of mean solar time. Beside the effects of perturbational changes in Earth's orbital eccentricity, as found by Laplace and corrected by Adams, there are two tidal effects (a combination first suggested by [[Emmanuel Liais]]).  First there is a real retardation of the Moon's angular rate of orbital motion, due to tidal exchange of [[angular momentum]] between Earth and Moon.  This increases the Moon's angular momentum around Earth (and moves the Moon to a higher orbit with a lower [[orbital speed]]). Secondly, there is an apparent increase in the Moon's angular rate of orbital motion (when measured in terms of mean solar time). This arises from Earth's loss of angular momentum and the consequent increase in [[length of day]].<ref>F R Stephenson (2002), [http://articles.adsabs.harvard.edu/full/2003A%26G....44b..22S "Harold Jeffreys Lecture 2002: Historical eclipses and Earth's rotation"], in ''[[Astronomy & Geophysics]]'', vol.44 (2002), pp. 2.22–2.27.</ref>

=== Effects of Moon's gravity ===
[[Image:Tidal braking.svg|thumb|A diagram of the [[Earth–Moon system]] showing how the tidal bulge is pushed ahead by [[Earth]]'s rotation. This offset bulge exerts a net torque on the [[Moon]], boosting it while slowing Earth's rotation.]]

The plane of the Moon's [[orbit]] around Earth lies close to the plane of Earth's orbit around the Sun (the [[ecliptic]]), rather than in the plane of the Earth's rotation (the [[equator]]) as is usually the case with planetary satellites.  The mass of the Moon is sufficiently large, and it is sufficiently close, to raise [[tide]]s in the matter of Earth.  Foremost among such matter, the [[water]] of the [[ocean]]s bulges out both towards and away from the Moon.  If the material of the Earth responded immediately, there would be a bulge directly toward and away from the Moon. In the [[solid Earth tide]]s, there is a delayed response due to the dissipation of tidal energy. The case for the oceans is more complicated, but there is also a delay associated with the dissipation of energy since the Earth rotates at a faster rate than the Moon's orbital angular velocity. This [[lunitidal interval]] in the responses causes the tidal bulge to be carried forward. Consequently, the line through the two bulges is tilted with respect to the Earth-Moon direction exerting [[torque]] between the Earth and the Moon. This torque boosts the Moon in its orbit and slows the rotation of Earth.

As a result of this process, the mean solar day, which has to be 86,400 equal seconds, is actually getting longer when measured in [[SI]] [[second]]s with stable [[atomic clock]]s. (The SI second, when adopted, was already a little shorter than the current value of the second of mean solar time.<ref>:(1) In {{cite journal|last1 = McCarthy|first1 = D D|last2 = Hackman|first2 = C|last3 = Nelson|first3 = R A|year = 2008|title = The Physical Basis of the Leap Second|url = https://apps.dtic.mil/sti/pdfs/ADA489427.pdf|archive-url = https://web.archive.org/web/20170922113409/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA489427|url-status = live|archive-date = September 22, 2017|journal = Astronomical Journal|volume = 136|issue = 5|pages = 1906–1908|doi=10.1088/0004-6256/136/5/1906|bibcode=2008AJ....136.1906M|doi-access = free }} it is stated (page 1908), that "the SI second is equivalent to an older measure of the second of UT1, which was too small to start with and further, as the duration of the UT1 second increases, the discrepancy widens." :(2) In the late 1950s, the cesium standard was used to measure both the current mean length of the second of mean solar time (UT2) (result: 9192631830 cycles) and also the second of ephemeris time (ET) (result:9192631770±20 cycles), see [http://www.leapsecond.com/history/1968-Metrologia-v4-n4-Essen.pdf "Time Scales", by L. Essen], in Metrologia, vol.4 (1968), pp.161–165, on p.162. As is well known, the 9192631770 figure was chosen for the [[second|SI second]].  L Essen in the same 1968 article (p.162) stated that this "seemed reasonable in view of the variations in UT2".</ref>) The small difference accumulates over time, which leads to an increasing difference between our clock time ([[Universal Time]]) on the one hand, and [[International Atomic Time]] and [[ephemeris time]] on the other hand: see [[ΔT (timekeeping)|ΔT]]. This led to the introduction of the [[leap second]] in 1972 <ref>{{cite web|title=What's a Leap Second|url=http://www.timeanddate.com/time/leapseconds.html|website=Timeanddate.com}}</ref> to compensate for differences in the bases for time standardization.

In addition to the effect of the ocean tides, there is also a tidal acceleration due to flexing of Earth's crust, but this accounts for only about 4% of the total effect when expressed in terms of heat dissipation.<ref>{{cite journal|first1=Walter|last1 = Munk|year = 1997|title =  Once again: once again—tidal friction|journal = Progress in Oceanography|volume = 40|issue = 1–4|pages = 7–35|doi=10.1016/S0079-6611(97)00021-9|bibcode=1997PrOce..40....7M}}</ref>

If other effects were ignored, tidal acceleration would continue until the rotational period of Earth matched the orbital period of the Moon.  At that time, the Moon would always be overhead of a single fixed place on Earth. Such a situation already exists in the [[Pluto]]–[[Charon (moon)|Charon]] system. However, the slowdown of Earth's rotation is not occurring fast enough for the rotation to lengthen to a month before other effects make this irrelevant: about 1 to 1.5 billion years from now, the continual increase of the Sun's [[radiation]] will likely cause Earth's oceans to vaporize,<ref>Puneet Kollipara (22 January 2014), [https://www.science.org/content/article/earth-wont-die-soon-thought "Earth Won't Die as Soon as Thought"], '''Science'''.</ref> removing the bulk of the tidal friction and acceleration. Even without this, the slowdown to a month-long day would still not have been completed by 4.5 billion years from now when the Sun will probably evolve into a [[red giant]] and likely destroy both Earth and the Moon.<ref>{{cite book|last1 = Murray|first1 = C.D.|first2 = Stanley F.|last2 = Dermott|title = Solar System Dynamics|date = 1999|publisher = Cambridge University Press|isbn = 978-0-521-57295-8|page = 184 }}</ref><ref>{{cite book|last = Dickinson|first = Terence|author-link = Terence Dickinson|title = From the Big Bang to Planet X|date = 1993|publisher = [[Camden House]]|location = Camden East, Ontario|isbn = 978-0-921820-71-0|pages = 79–81 }}
</ref>

Tidal acceleration is one of the few examples in the dynamics of the [[Solar System]] of a so-called '''secular perturbation''' of an orbit, i.e. a perturbation that continuously increases with time and is not periodic.  Up to a high order of approximation, mutual [[gravity|gravitational]] perturbations between major or minor [[planet]]s only cause periodic variations in their orbits, that is, parameters oscillate between maximum and minimum values.  The tidal effect gives rise to a quadratic term in the equations, which leads to unbounded growth.  In the mathematical theories of the planetary orbits that form the basis of [[ephemerides]], quadratic and higher order secular terms do occur, but these are mostly [[Taylor series|Taylor expansions]] of very long time periodic terms. The reason that tidal effects are different is that unlike distant gravitational perturbations, friction is an essential part of tidal acceleration, and leads to permanent loss of [[energy]] from the dynamic system in the form of [[heat]]. In other words, we do not have a [[Hamiltonian system]] here.{{Citation needed|date = November 2015}}

=== Angular momentum and energy ===
The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be constantly promoted to a slightly higher orbit and Earth to be decelerated in its rotation.  As in any physical process within an isolated system, total [[energy]] and [[angular momentum]] are conserved.  Effectively, energy and angular momentum are transferred from the rotation of Earth to the orbital motion of the Moon (however, most of the energy lost by Earth (−3.78 TW)<ref name=":0">{{Cite journal|last1=Williams|first1=James G.|last2=Boggs|first2=Dale H.|date=2016|title=Secular tidal changes in lunar orbit and Earth rotation|url=http://link.springer.com/10.1007/s10569-016-9702-3|journal=Celestial Mechanics and Dynamical Astronomy|language=en|volume=126|issue=1|pages=89–129|doi=10.1007/s10569-016-9702-3|bibcode=2016CeMDA.126...89W |s2cid=124256137|issn=0923-2958|url-access=subscription}}</ref> is converted to heat by frictional losses in the oceans and their interaction with the solid Earth, and only about 1/30th (+0.121 TW) is transferred to the Moon). The Moon moves farther away from Earth (+38.30±0.08&nbsp;mm/yr), so its [[potential energy|potential energy, which is still negative]] (in Earth's [[gravity well]]), increases, i. e. becomes less negative. It stays in orbit, and from [[Laws of Kepler|Kepler's 3rd law]] it follows that its average [[angular velocity]] actually decreases, so the tidal action on the Moon actually causes an angular deceleration, i.e. a negative acceleration (−25.97±0.05"/century<sup>2</sup>) of its rotation around Earth.<ref name=":0" /> The actual speed of the Moon also decreases. Although its [[kinetic energy]] decreases, its potential energy increases by a larger amount, i. e.  E<sub>p</sub> = -2E<sub>c</sub> ([[Virial theorem|Virial Theorem]]).

The rotational angular momentum of Earth decreases and consequently the length of the day increases. The ''net'' tide raised on Earth by the Moon is dragged ahead of the Moon by Earth's much faster rotation. '''Tidal friction''' is required to drag and maintain the bulge ahead of the Moon, and it dissipates the excess energy of the exchange of rotational and orbital energy between Earth and the Moon as heat. If the friction and heat dissipation were not present, the Moon's gravitational force on the tidal bulge would rapidly (within two days) bring the tide back into synchronization with the Moon, and the Moon would no longer recede. Most of the dissipation occurs in a turbulent bottom boundary layer in shallow seas such as the [[European Shelf]] around the [[British Isles]], the [[Patagonian Shelf]] off [[Argentina]], and the [[Bering Sea]].<ref>{{Cite journal|doi = 10.1016/S0079-6611(97)00021-9|last1 = Munk|first1 = Walter|date = 1997|title = Once again: once again—tidal friction|journal = Progress in Oceanography|volume = 40|issue = 1–4|pages = 7–35|bibcode = 1997PrOce..40....7M }}</ref>

The dissipation of energy by tidal friction averages about 3.64 terawatts of the 3.78 terawatts extracted, of which 2.5 terawatts are from the principal M{{sub|2}} lunar component and the remainder from other components, both lunar and solar.<ref name=":0" /><ref name=Munk1998>{{Cite journal|author = Munk, W.|date = 1998|title = Abyssal recipes II: energetics of tidal and wind mixing|journal = Deep-Sea Research Part I|volume = 45|issue = 12|pages = 1977–2010|doi = 10.1016/S0967-0637(98)00070-3|last2 = Wunsch|first2 = C|bibcode = 1998DSRI...45.1977M }}</ref>

An ''[[equilibrium tide|equilibrium tidal]] bulge'' does not really exist on Earth because the continents do not allow this mathematical solution to take place. Oceanic tides actually rotate around the ocean basins as vast ''[[gyre]]s'' around several ''[[amphidromic point]]s'' where no tide exists. The Moon pulls on each individual undulation as Earth rotates—some undulations are ahead of the Moon, others are behind it, whereas still others are on either side. The "bulges" that actually do exist for the Moon to pull on (and which pull on the Moon) are the net result of integrating the actual undulations over all the world's oceans.

=== Historical evidence ===
This mechanism has been working for 4.5 billion years, since oceans first formed on Earth, but less so at times when much or most of the water [[Snowball Earth|was ice]]. There is geological and paleontological evidence that Earth rotated faster and that the Moon was closer to Earth in the remote past. ''[[Rhythmites|Tidal rhythmites]]'' are alternating layers of sand and silt laid down offshore from [[estuary|estuaries]] having great tidal flows.  Daily, monthly and seasonal cycles can be found in the deposits. This geological record is consistent with these conditions 620 million years ago: the day was 21.9±0.4 hours, and there were 13.1±0.1 synodic months/year and 400±7 solar days/year.  The average recession rate of the Moon between then and now has been 2.17±0.31&nbsp;cm/year, which is about half the present rate. The present high rate may be due to near [[resonance]] between natural ocean frequencies and tidal frequencies.<ref>{{Cite journal|doi = 10.1029/1999RG900016|last1 = Williams|first1 = George E.|date = 2000|title = Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit|bibcode = 2000RvGeo..38...37W|journal = Reviews of Geophysics|volume = 38|issue = 1|pages = 37–60|citeseerx = 10.1.1.597.6421 | s2cid=51948507 }}</ref>

Analysis of layering in fossil [[mollusc shell]]s from 70 million years ago, in the [[Late Cretaceous]] period, shows that there were 372 days a year, and thus that the day was about 23.5 hours long then.<ref>{{Cite web|url=https://www.sciencedaily.com/releases/2020/03/200309135410.htm|title=Ancient shell shows days were half-hour shorter 70 million years ago: Beer stein-shaped distant relative of modern clams captured snapshots of hot days in the late Cretaceous|website=ScienceDaily|language=en|access-date=2020-03-14}}</ref><ref>{{Cite journal|last1=Winter|first1=Niels J. de|last2=Goderis|first2=Steven|last3=Malderen|first3=Stijn J. M. Van|last4=Sinnesael|first4=Matthias|last5=Vansteenberge|first5=Stef|last6=Snoeck|first6=Christophe|last7=Belza|first7=Joke|last8=Vanhaecke|first8=Frank|last9=Claeys|first9=Philippe|date=2020|title=Subdaily-Scale Chemical Variability in a Torreites Sanchezi Rudist Shell: Implications for Rudist Paleobiology and the Cretaceous Day-Night Cycle|journal=Paleoceanography and Paleoclimatology|language=en|volume=35|issue=2|pages=e2019PA003723|doi=10.1029/2019PA003723|issn=2572-4525|doi-access=free|bibcode=2020PaPa...35.3723W |hdl=1854/LU-8685501|hdl-access=free}}</ref>

=== 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.