Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite (e.g. the Moon), and the primary planet that it orbits (e.g. the Earth). The acceleration is usually negative, as it causes a gradual slowing and recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking of first the smaller, and later the larger body. The Earth–Moon system is the best studied case.
The similar process of tidal deceleration occurs for satellites that have an orbital period that is shorter than the primary's rotational period, or that orbit in a retrograde direction.
Earth–Moon system 
Discovery history of the secular acceleration 
Edmond Halley was the first to suggest, in 1695, 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 the Earth's rate of rotation: see also 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: a centurial rate of +10" (arcseconds) in lunar longitude (a surprisingly good result for its time, not far different from values assessed later, e.g. in 1786 by de Lalande, and to compare with values from about 10" to nearly 13" being derived about a century later.)
Pierre-Simon Laplace produced in 1786 a theoretical analysis giving a basis on which the Moon's mean motion should accelerate in response to perturbational changes in the eccentricity of the orbit of the 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.
However, in 1854, J C 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 the Earth's orbital eccentricity. Adams's finding provoked a sharp astronomical controversy that lasted some years, but the correctness of his result, agreed by other mathematical astronomers including C E Delaunay, was eventually accepted. 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 the Earth's rotation rate was lengthening the unit of time and causing a lunar acceleration that was only apparent.
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 the 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 the Earth and Moon. This increases the Moon's angular momentum around the Earth (and moves the Moon to a higher orbit with a slower period). 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 the Earth's loss of angular momentum and the consequent increase in length of day.
Effects of Moon's gravity 
Because the Moon's mass is a considerable fraction of that of the Earth (about 1:81), the two bodies can be regarded as a double planet system, rather than as a planet with a satellite. The plane of the Moon's orbit around the Earth lies close to the plane of the Earth's orbit around the Sun (the ecliptic), rather than in the plane perpendicular to the axis of rotation of the Earth (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 tides in the matter of the Earth. In particular, the water of the oceans bulges out towards the Moon. There is a roughly opposing bulge on the other side of the Earth that is caused by the centrifugal force of the Earth rotating about the Earth–Moon barycenter. The average tidal bulge is sychronized with the Moon's orbit, and the Earth rotates under this tidal bulge in just over a day. However, the rotation drags the position of the tidal bulge ahead of the position directly under the Moon. As a consequence, there exists a substantial amount of mass in the bulge that is offset from the line through the centers of the Earth and Moon. Because of this offset, a portion of the gravitational pull between Earth's tidal bulges and the Moon is perpendicular to the Earth–Moon line, i.e. there exists a torque between the Earth and the Moon. This boosts the Moon in its orbit, and slows the rotation of the Earth.
As a result of this process, the mean solar day, which is nominally 86400 seconds long, is actually getting longer when measured in SI seconds with stable atomic clocks. (The SI second, when adopted, was already a little shorter than the current value of the second of mean solar time.) The small difference accumulates every day, which leads to an increasing difference between our clock time (Universal Time) on the one hand, and Atomic Time and Ephemeris Time on the other hand: see ΔT. This makes it necessary to insert a leap second at occasional, irregular intervals.
In addition to the effect of the ocean tides, there is also a tidal acceleration due to flexing of the earth's crust, but this accounts for only about 4% of the total effect when expressed in terms of heat dissipation.
If other effects were ignored, tidal acceleration would continue until the rotational period of the 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 system. However, the slowdown of the Earth's rotation is not occurring fast enough for the rotation to lengthen to a month before other effects make this irrelevant: About 2.1 billion years from now, the continual increase of the Sun's radiation will cause the Earth's oceans to vaporize, 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 evolve into a red giant and likely destroy both the Earth and Moon.
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 gravitational perturbations between major or minor planets 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 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.
Angular momentum and energy 
The gravitational torque between the Moon and the tidal bulge of the Earth causes the Moon to be constantly promoted to a slightly higher orbit and the 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 the Earth to the orbital motion of the Moon (however, most of the energy lost by the Earth (-3.321 TW) 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 the Earth (+38.247±0.004 mm/y), so its potential energy (in the Earth's gravity well) increases. It stays in orbit, and from Kepler's 3rd law it follows that its angular velocity actually decreases, so the tidal action on the Moon actually causes an angular deceleration, i.e. a negative acceleration (-25.858±0.003 "/cy²) of its rotation around the Earth. Although its kinetic energy decreases, its potential energy increases by a larger amount.
The rotational angular momentum of the 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 the Earth and 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.
The dissipation of energy by tidal friction averages about 3.75 terawatts, of which 2.5 terawatts are from the principal M2 lunar component and the remainder from other components, both lunar and solar.
An 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 oceans basin as vast gyres around several amphidromic points 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, while 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. Earth's net (or equivalent) equilibrium tide has an amplitude of only 3.23 cm, which is totally swamped by oceanic tides that can exceed one metre.
Historical evidence 
This mechanism has been working for 4.5 billion years, since oceans first formed on the Earth. There is geological and paleontological evidence that the Earth rotated faster and that the Moon was closer to the Earth in the remote past. Tidal rhythmites are alternating layers of sand and silt laid down offshore from 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 length of the year has remained virtually unchanged during this period because no evidence exists that the constant of gravitation has changed. The average recession rate of the Moon between then and now has been 2.17±0.31 cm/year, which is about half the present rate.
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 mirrors on the surface of the moon, emplaced during the Apollo missions of 1969 to 1972 and by Lunokhod 2 in 1973. 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 1970–2007, the results are:
- −25.858±0.003 arc minute/century² in ecliptic longitude
- +38.247±0.004 mm/yr in the mean Earth–Moon distance
This is consistent with results from satellite laser ranging (SLR), a similar technique applied to artificial satellites orbiting the Earth, which yields a model for the gravitational field of the Earth, including that of the tides. The model accurately predicts the changes in the motion of the Moon.
The other consequence of tidal acceleration is the deceleration of the rotation of the Earth. The rotation of the Earth is somewhat erratic on all time scales (from hours to centuries) due to various causes. The small tidal effect cannot be observed in a short period, but the cumulative effect on the Earth's rotation as measured with a stable clock (ephemeris time, 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 the 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. Recent values can be obtained from the International Earth Rotation and Reference Systems Service (IERS). A table of the actual length of the day in the past few centuries is also available.
From the observed change in the Moon's orbit, the corresponding change in the length of the day can be computed:
- +2.3 ms/cy
- (cy is centuries).
However, from historical records over the past 2700 years the following average value is found:
The corresponding cumulative value is a parabola having a coefficient of T² (time in centuries squared) of:
- ΔT = +31 s/cy²
Opposing the tidal deceleration of the Earth is a mechanism that is in fact accelerating the rotation. The 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 the 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 the Earth increases, and since the mass and density remain the same, the volume remains the same; therefore the equatorial diameter is decreasing. As a consequence, mass moves closer to the rotation axis of the Earth. This means that its moment of inertia is decreasing. Because its total angular momentum remains the same during this process, the rotation rate increases. This is the well-known phenomenon of a spinning figure skater who spins ever faster as she retracts her 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/cy. This largely explains the historical observations.
Other cases of tidal acceleration 
Most natural satellites of the planets undergo tidal acceleration to some degree (usually small), except for the two classes of tidally decelerated bodies. In most cases, however, the effect is small enough that even after billions of years most satellites will not actually be lost. The effect is probably most pronounced for Mars's second moon Deimos, which may become an Earth-crossing asteroid after it leaks out of Mars's grip. The effect also arises between different components in a binary star.
Tidal deceleration 
This comes in two varieties:
- Fast satellites: Some inner moons of the gas giant planets and Phobos orbit within the synchronous orbit radius so that their orbital period is shorter than their planet's rotation. In this case the tidal bulges raised by the moon on their planet lag behind the moon, and act to decelerate it in its orbit. The net effect is a decay of that moon's orbit as it gradually spirals towards the planet. The planet's rotation also speeds up slightly in the process. In the distant future these moons will impact the planet or cross within their Roche limit and be tidally disrupted into fragments. However, all such moons in the Solar System are very small bodies and the tidal bulges raised by them on the planet are also small, so the effect is usually weak and the orbit decays slowly. The moons affected are:
- Around Mars: Phobos
- Around Jupiter: Metis and Adrastea
- Around Saturn: none, except for the ring particles (like Jupiter, Saturn is a very rapid rotator but has no satellites close enough)
- Around Uranus: Cordelia, Ophelia, Bianca, Cressida, Desdemona, Juliet, Portia, Rosalind, Cupid, Belinda, and Perdita
- Around Neptune: Naiad, Thalassa, Despina, Galatea and Larissa
- Retrograde satellites: All retrograde satellites experience tidal deceleration to some degree because the moon's orbital motion and the planet's rotation are in opposite directions, causing restoring forces from their tidal bulges. A difference to the previous "fast satellite" case here is that the planet's rotation is also slowed down rather than sped up (angular momentum is still conserved because in such a case the values for the planet's rotation and the moon's revolution have opposite signs). The only satellite in the Solar System for which this effect is non-negligible is Neptune's moon Triton. All the other retrograde satellites are on distant orbits and tidal forces between them and the planet are negligible. The planet Venus is believed to have no satellites chiefly because any hypothetical satellites would have suffered deceleration long ago, from either cause; Venus has a very slow and retrograde rotation.
See also 
- E Halley (1695), "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.
- Richard Dunthorne (1749), "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 (1683–1775), Vol. 46 (1749–1750) #492, pp.162–172; also given in Philosophical Transactions (abridgements) (1809), vol.9 (for 1744–49), p669–675 as "On the Acceleration of the Moon, by the Rev. Richard Dunthorne".
- J de Lalande (1786): "Sur les equations seculaires du soleil et de la lune", Memoires de l'Academie Royale des Sciences, pp.390–397, at page 395.
- J D North (2008), "Cosmos: an illustrated history of astronomy and cosmology", (University of Chicago Press, 2008), chapter 14, at page 454.
- See also P Puiseux (1879), "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.
- Adams, J C (1853). "On the Secular Variation of the Moon's Mean Motion" (PDF). Phil. Trans. R. Soc. Lond. 143: 397–406. doi:10.1098/rstl.1853.0017.
- D E Cartwright (2001), "Tides: a scientific history", (Cambridge University Press 2001), chapter 10, section: "Lunar acceleration, earth retardation and tidal friction" at pages 144–146.
- F R Stephenson (2002), "Harold Jeffreys Lecture 2002: Historical eclipses and Earth's rotation", in Astronomy & Geophysics, vol.44 (2002), pp. 2.22–2.27.
- :(1) In "The Physical Basis of the Leap Second", by D D McCarthy, C Hackman and R A Nelson, in Astronomical Journal, vol.136 (2008), pages 1906–1908, 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 "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 SI second. L Essen in the same 1968 article (p.162) stated that this "seemed reasonable in view of the variations in UT2".
- Munk, Progress in Oceanography 40 (1997) 7; http://www.sciencedirect.com/science/article/pii/S0079661197000219
- Munk, Walter (1997). "Once again: once again—tidal friction". Progress in Oceanography 40 (1–4): 7–35. Bibcode:1997PrOce..40....7M. doi:10.1016/S0079-6611(97)00021-9.
- Munk, W.; Wunsch, C (1998). "Abyssal recipes II: energetics of tidal and wind mixing". Deep Sea Research Part I Oceanographic Research Papers 45 (12): 1977. Bibcode:1998DSRI...45.1977M. doi:10.1016/S0967-0637(98)00070-3
- Williams, George E. (2000). "Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit". Reviews of Geophysics 38 (1): 37–60. Bibcode:2000RvGeo..38...37W. doi:10.1029/1999RG900016.
- Most laser pulses, 78%, are to the Apollo 15 site. See Williams, et al. (2008), p. 5.
- Another reflector emplaced by Lunokhod 1 in 1970 is no longer functioning. See Lunar Lost & Found: The Search for Old Spacecraft by Leonard David
- J. G. Williams, D. H. Boggs and W. M. Folkner (2008). DE421 Lunar orbit, physical librations, and surface coordinates p. 7. "These derived values depend on a theory which is not accurate to the number of digits given." See also : Chapront, Chapront-Touzé, Francou (2002). A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements
- F.R. Stephenson, L.V. Morrison (1995): "Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990". Philosophical Transactions of the Royal Society of London Series A, pp.165–202. doi:10.1098/rsta.1995.0028
- Jean O. Dickey (1995): "Earth Rotation Variations from Hours to Centuries". In: I. Appenzeller (ed.): Highlights of Astronomy. Vol. 10 pp.17..44.
- Dickey, Jean O.; Bender, PL; Faller, JE; Newhall, XX; Ricklefs, RL; Ries, JG; Shelus, PJ; Veillet, C et al. (1994). "Lunar Laser ranging: a continuing legacy of the Apollo program". Science 265 (5171): 482–90. Bibcode:1994Sci...265..482D. doi:10.1126/science.265.5171.482. PMID 17781305.
- F.R. Stephenson (1997): Historical Eclipses and Earth's Rotation. Cambridge Univ.Press.
- Zahn, J.-P. (1977). "Tidal Friction in Close Binary Stars". Astron. Astrophys. 57: 383–394. Bibcode:1977A&A....57..383Z.
- Schröder, K.-P.; Smith, R.C. (2008). "Distant future of the Sun and Earth revisited". Monthly Notices of the Royal Astronomical Society 386 (1): 155. arXiv:0801.4031. Bibcode:2008MNRAS.386..155S. doi:10.1111/j.1365-2966.2008.13022.x. See also Palmer, J. (2008). "Hope dims that Earth will survive Sun's death". New Scientist. Retrieved 2008-03-24.