Tidal acceleration is an effect of the

tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...

s between an orbiting

natural satellite A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body (or sometimes another natural satellite). Natural satellites are often colloquially referred to as ''moons'' ...

(e.g. the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

) and the primary

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

that it orbits (e.g.

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

). The acceleration causes a gradual recession of a satellite in a

prograde orbit Retrograde motion in astronomy is, in general, orbital or rotational motion of an object in the direction opposite the rotation of its primary, that is, the central object (right figure). It may also describe other motions such as precession or ...

away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body (e.g. theoretically with Earth in 50 billion years). 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. The naming is somewhat confusing, because the average speed of the satellite relative to the body it orbits is ''decreased'' as a result of tidal acceleration, and ''increased'' as a result of tidal deceleration. This conundrum occurs because a positive acceleration at one instant causes the satellite to loop farther outward during the next half orbit, decreasing its average speed. A continuing positive acceleration causes the satellite to spiral outward with a decreasing speed and angular rate, resulting in a negative acceleration of angle. A continuing negative acceleration has the opposite effect.

Earth–Moon system

Discovery history of the secular acceleration

Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, H ...

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 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 Richard Dunthorne (1711 – 3 March 1775) was an English astronomer and surveyor, who worked in Cambridge as astronomical and scientific assistant to Roger Long (master of Pembroke Hall and Lowndean Professor of Astronomy and Geometry), Library of ...

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, which is a surprisingly accurate result for its time, not differing greatly 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 Earth around the

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...

. 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, 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. Adams' finding provoked a sharp astronomical controversy that lasted some years, but the correctness of his result, agreed upon 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 Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...

) 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 William Ferrel (January 29, 1817 – September 18, 1891) was an American meteorologist who developed theories that explained the mid-latitude atmospheric circulation cell in detail, and it is after him that the Ferrel cell is named. Biograph ...

: tidal retardation of 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 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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...

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 In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...

). 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.

Effects of Moon's gravity

Because the

's mass is a considerable fraction of that of Earth (about 1:81), the two bodies can be regarded as a

double planet In astronomy, a double planet (also binary planet) is a binary satellite system where both objects are planets, or planetary-mass objects, that share an orbital axis external to both planetary bodies. Although up to a third of the star syst ...

system, rather than as a planet with a satellite. The plane of the Moon's

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...

around Earth lies close to the plane of Earth's orbit around the Sun (the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...

), 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 Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ...

s in the matter of Earth. Foremost among such matter, the

water Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...

of the

ocean The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water. An ocean can also refer to any of the large bodies of water into which the wo ...

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 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. The delay 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 In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

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 seconds with stable

atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...

s. (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 over time, which leads to an increasing difference between our clock time ( Universal Time) on the one hand, and

International Atomic Time International Atomic Time (abbreviated TAI, from its French name ) is a high-precision atomic coordinate time standard based on the notional passage of proper time on Earth's geoid. TAI is a weighted average of the time kept by over 450 ato ...

and ephemeris time on the other hand: see ΔT. This led to the introduction of the

leap second A leap second is a one- second adjustment that is occasionally applied to Coordinated Universal Time (UTC), to accommodate the difference between precise time (International Atomic Time (TAI), as measured by atomic clocks) and imprecise observ ...

in 1972 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. 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 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, 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. Tidal acceleration is one of the few examples in the dynamics of the

Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

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

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 In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...

, 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 In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

from the dynamic system in the form of

heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...

. In other words, we do not have a

Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...

here.

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

and

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) 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 mm/yr), so its potential energy, which is still negative (in Earth's

gravity well The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hil ...

), increases, i. e. becomes less negative. It stays in orbit, and from 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"/century²) of its rotation around Earth. The actual speed of the Moon also decreases. Although its

kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...

decreases, its potential energy increases by a larger amount, i. e. E_p = -2E_c ( 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 British Isles are a group of islands in the North Atlantic Ocean off the north-western coast of continental Europe, consisting of the islands of Great Britain, Ireland, the Isle of Man, the Inner and Outer Hebrides, the Northern Isles, ...

, the Patagonian Shelf off

Argentina Argentina (), officially the Argentine Republic ( es, link=no, República Argentina), is a country in the southern half of South America. Argentina covers an area of , making it the second-largest country in South America after Brazil, th ...

, and the Bering Sea. 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 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 ocean basins as vast ''

gyre In oceanography, a gyre () is any large system of circulating ocean currents, particularly those involved with large wind movements. Gyres are caused by the Coriolis effect; planetary vorticity, horizontal friction and vertical friction dete ...

s'' 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, 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. 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 Earth, but less so at times when much or most of the water was ice. There is geological and paleontological evidence that Earth rotated faster and that the Moon was closer to Earth in the remote past. '' Tidal rhythmites'' are alternating layers of sand and silt laid down offshore from

estuaries An estuary is a partially enclosed coastal body of brackish water with one or more rivers or streams flowing into it, and with a free connection to the open sea. Estuaries form a transition zone between river environments and maritime environmen ...

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 cm/year, which is about half the present rate. The present high rate may be due to near

resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied Periodic function, periodic force (or a Fourier analysis, Fourier component of it) is equal or close to a natural frequency of the system ...

between natural ocean frequencies and tidal frequencies. Analysis of layering in fossil

mollusc shell The mollusc (or molluskOften spelled mollusk shell in the USA; the spelling "mollusc" are preferred by ) shell is typically a calcareous exoskeleton which encloses, supports and protects the soft parts of an animal in the phylum Mollusca, wh ...

s from 70 million years ago, in the

Late Cretaceous The Late Cretaceous (100.5–66 Ma) is the younger of two epochs into which the Cretaceous Period is divided in the geologic time scale. Rock strata from this epoch form the Upper Cretaceous Series. The Cretaceous is named after ''creta'', ...

period, shows that there were 372 days a year, and thus that the day was about 23.5 hours long then.

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

Apollo Apollo, grc, Ἀπόλλωνος, Apóllōnos, label=genitive , ; , grc-dor, Ἀπέλλων, Apéllōn, ; grc, Ἀπείλων, Apeílōn, label= Arcadocypriot Greek, ; grc-aeo, Ἄπλουν, Áploun, la, Apollō, la, Apollinis, label ...

missions of 1969 to 1972 and by Lunokhod 1 in 1970 and 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 1970–2015, the results are: : −25.97 ± 0.05 arcsecond/century² in ecliptic longitudeJ.G. Williams, D.H. Boggs and W. M.Folkner (2013)
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)
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 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 eclipses give fairly accurate positions for the Moon at those moments. Studies of these observations give results consistent with the value quoted above. 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. 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. 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 (where "cy" means "century"): : +2.4 ms/d/century or +88 s/cy² or +66 ns/d². However, from historical records over the past 2700 years the following average value is found: : +1.72 ± 0.03 ms/d/century or +63 s/cy² or +47 ns/d². (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 T² (time in centuries squared) of (¹/₂) 63 s/cy² : : Δ''T'' = (¹/₂) 63 s/cy² T² = +31 s/cy² T². 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 An ice age is a long period of reduction in the temperature of Earth's surface and atmosphere, resulting in the presence or expansion of continental and polar ice sheets and alpine glaciers. Earth's climate alternates between ice ages and gre ...

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/century. 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: Mercury and

are believed to have no satellites chiefly because any hypothetical satellite would have suffered deceleration long ago and crashed into the planets due to the very slow rotation speeds of both planets; in addition, Venus also has retrograde rotation.

References

External links

The Recession of the Moon and the Age of the Earth-Moon System

{{DEFAULTSORT:Tidal Acceleration

Acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...

Geodesy Orbits Orbit of the Moon