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 phenom ...
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 ...
) 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 ...
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 surf ...
). The acceleration causes a gradual 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
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked b ...
, 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, Hal ...
was the first to suggest, in 1695, that the mean motion of the Moon was apparently getting faster, by comparison with ancient
eclipse
An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ce ...
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
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial t ...
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
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
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. 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
John Couch Adams (; 5 June 1819 – 21 January 1892) was a British mathematician and astronomer. He was born in Laneast, near Launceston, Cornwall, and died in Cambridge.
His most famous achievement was predicting the existence and position o ...
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: 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 sy ...
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 ma ...
). 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
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 ...
's mass is a considerable fraction of that of 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
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 a ...
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 agai ...
), rather than in the plane of the earth's rotation (the
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
) 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 can ...
s in the matter of Earth. Foremost among such matter, the
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
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 t ...
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 betw ...
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
Universal Time (UT or UT1) is a time standard based on Earth's rotation. While originally it was mean solar time at 0° longitude, precise measurements of the Sun are difficult. Therefore, UT1 is computed from a measure of the Earth's angle wi ...
) 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
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest k ...
–
Charon
In Greek mythology, Charon or Kharon (; grc, Χάρων) is a psychopomp, the ferryman of Hades, the Greek underworld. He carries the souls of those who have been given funeral rites across the rivers Acheron and Styx, which separate the wo ...
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
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, vi ...
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
A red giant is a luminous giant star of low or intermediate mass (roughly 0.3–8 solar masses ()) in a late phase of stellar evolution. The outer atmosphere is inflated and tenuous, making the radius large and the surface temperature around or ...
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
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 ...
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 v ...
, 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
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 ...
and
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 sy ...
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 Hill sp ...
), increases, i. e. becomes less negative. It stays in orbit, and from
Kepler's 3rd law it follows that its average
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
actually decreases, so the tidal action on the Moon actually causes an angular deceleration, i.e. a negative acceleration (−25.97±0.05"/century
2) 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 a ...
decreases, its potential energy increases by a larger amount, i. e. E
p = -2E
c (
Virial Theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
).
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
European, or Europeans, or Europeneans, may refer to:
In general
* ''European'', an adjective referring to something of, from, or related to Europe
** Ethnic groups in Europe
** Demographics of Europe
** European cuisine, the cuisines of Europe a ...
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 (O ...
, 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, t ...
, and the
Bering Sea
The Bering Sea (, ; rus, Бе́рингово мо́ре, r=Béringovo móre) is a marginal sea of the Northern Pacific Ocean. It forms, along with the Bering Strait, the divide between the two largest landmasses on Earth: Eurasia and The Amer ...
.
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 ''
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, 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
A rhythmite consists of layers of sediment or sedimentary rock which are laid down with an obvious periodicity and regularity. They may be created by annual processes such as seasonally varying deposits reflecting variations in the runoff cycle, b ...
'' 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 environme ...
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 force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
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, which ...
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/century
2 in ecliptic longitude
[J.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
In satellite laser ranging (SLR) a global network of observation stations measures the round trip time of flight of ultrashort pulses of light to satellites equipped with retroreflectors. This provides instantaneous range measurements of milli ...
(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
An eclipse is an astronomical event that occurs when an astronomical object or spacecraft is temporarily obscured, by passing into the shadow of another body or by having another body pass between it and the viewer. This alignment of three ce ...
s 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
Universal Time (UT or UT1) is a time standard based on Earth's rotation. While originally it was mean solar time at 0° longitude, precise measurements of the Sun are difficult. Therefore, UT1 is computed from a measure of the Earth's angle wi ...
) 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
The International Earth Rotation and Reference Systems Service (IERS), formerly the International Earth Rotation Service, is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation P ...
(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
A binary star is a system of two stars that are gravitationally bound to and in orbit around each other. Binary stars in the night sky that are seen as a single object to the naked eye are often resolved using a telescope as separate stars, in ...
.
Tidal deceleration
This comes in two varieties:
Mercury and 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 ...
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.
See also
*Tidal locking
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked b ...
*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 phenom ...
*Tides
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 tabl ...
* Tidal heating
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 ...
Geodesy
Orbits
Orbit of the Moon