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Encyclopedia > Tidal acceleration
It has been suggested that Tidal friction be merged into this article or section. (Discuss)

The tidal acceleration of the Moon is an effect in the dynamics of the Earth-Moon system, that has important long-term consequences for the orbit of the Moon and the rotation of the Earth. Image File history File links Please see the file description page for further information. ... Tidal friction is a process that transferes energy between the planets. ... Bulk composition of the moons mantle and crust estimated, weight percent Oxygen 42. ... Earth (often referred to as The Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth in order of size. ...

## Contents

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. In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... The Sun is the star at the center of Earths solar system. ... The plane of the ecliptic is well seen in this picture from the 1994 lunar prospecting Clementine spacecraft. ... The equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ...

### Effects of moon's gravity

The mass of the Moon is sufficiently large and it is sufficiently close to raise tides in the Earth: the matter of the Earth, in particular the water of the oceans, bulges out to the direction of the Moon. This follows the Moon in its orbit, which takes about a month. The Earth rotates under this tidal bulge in a day. The actual matter of waters rotate with the Earth, but they rise and fall as the Moon comes overhead. 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 that is offset from the line through the centers of the Earth and Moon. Because of this offset, a portion of the gravitational pull it exerts on the Moon is perpendicular to the Earth-Moon line and hence accelerates the latter in its orbit. Conversely, the gravitational pull from the Moon on this mass exerts a torque that decelerates the rotation of the Earth. The tide is the cyclic rising and falling of Earths ocean surface caused by the tidal forces of the Moon and the Sun acting on the Earth. ... The month is a unit of time, used with calendars, which is approximately as long as some natural period related to the motion of the Moon. ... A day (symbol: d) is a unit of time equal to 24 hours. ...

### Angular momentum and energy

As in all physical processes, angular momentum and energy are conserved. So the orbital angular momentum of the Moon increases, while it moves away from the Earth. As it stays in orbit, it follows from Kepler's 3rd law that its velocity decreases: so the tidal acceleration of the Moon is an apparent deceleration of its motion across the celestial sphere. As its kinetic energy decreases, its potential energy increases. Physics (from the Greek, Ï†Ï…ÏƒÎ¹ÎºÏŒÏ‚ (physikos), natural, and Ï†ÏÏƒÎ¹Ï‚ (physis), nature) is the science of Nature. ... Gyroscope. ... Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ...

As a consequence, the rotational angular momentum of the Earth decreases: its rotation slows down, and the length of the day increases. The corresponding rotational energy dissipates through friction of the tidal waters along shallow coasts, and is lost as heat.

The Moon recedes from Earth at the rate of approximately 38 mm per year. The Earth's day lengthens by about 17 µs ever year. A millimetre (American spelling: millimeter, symbol mm) is an SI unit of length that is equal to one thousandth of a metre. ... A microsecond is an SI unit of time equal to one millionth (10-6) of a second. ...

## 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. For example, the shorter days in the Devonian period 400 million years ago meant that there were 400 days in a year (with each day being 21.8 hours long). Since some corals and shellfish exhibit both daily and annual growth rings, counting in the rings in their fossils allows one to count the number of days in a year, and the results are consistent with the calculated figure.

Tidal acceleration will continue until the rotational period of the Earth is the same as the orbital period of the Moon. At that time, the Moon will always be overhead of a single fixed place on Earth. Note that the stronger tidal forces of the Earth working on the solid Moon already have locked its rotation to its orbital period: the Moon always turns the same face to the Earth.

Tidal acceleration is one of the few examples in the dynamics of the solar system of a perturbation of an orbit that continuously increases with time and is not periodic. Up to a high order of approximation, mutual gravitational perturbations of planets only cause periodic variations in their orbits, that is, it oscillates between maximum values. The tidal effect gives rise to a quadratic term, which grows forever. The solar system comprises the Earths Sun and the retinue of celestial objects gravitationally bound to it. ... Gravity is a force of attraction that acts between bodies that have mass. ... A planet is generally considered to be a relatively large mass of accreted matter in orbit around a star. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ...

In the mathematical theories of the planetary orbits that form the basis of ephemerides, quadratic and higher order secular terms occur, but these are mostly Taylor expansions of very long time periodic terms. An ephemeris (plural: ephemerides) (from the Greek word ephemeros= daily) was, traditionally, a table providing the positions (given in a Cartesian coordinate system, or in right ascension and declination or, for astrologers, in longitude along the zodiacal ecliptic), of the Sun, the Moon, and the planets in the sky at... As the degree of the Taylor series rises, it approaches the correct function. ...

## Quantitative description

The motion of the Moon can be followed with an accuracy of a few centimeters by lunar laser ranging (LLR). This makes use of mirrors on probes that have landed on the Moon since 1969, by bouncing off short laser pulses from them: the return time yields a very accurate measure of the distance. These measurements are fitted to the equations of motion. This yields numerical values for the parameters, among others the secular acceleration. From the period 1969–2001, the result is: The Lunar Laser Ranging Experiment from the Apollo 11 mission The ongoing Lunar Laser Ranging Experiment was first made possible by a lunar laser ranging retroreflector array planted on the Moon on July 21, 1969, by the crew of the Apollo 11. ...

−25.858 ± 0.003 "/cy² in ecliptic longitude[1]
+3.84 ± 0.07 m/cy in distance[2]
(cy is centuries; the first is a quadratic term.)

This is consistent with results from satellite laser ranging (SLR). This is a similar technique applied to artificial satellites orbiting the Earth. This yields an accurate model for the gravitational field of the Earth, including that of the tides. This can be used to predict its effect on the motion of the Moon, which yield very similar results. In satellite laser ranging (SLR) a global network of observation stations measure the round trip time of flight of ultrashort pulses of light to satellites equipped with retroreflectors. ...

Finally, ancient observations of solar eclipses give a fairly accurate position for the Moon at that moment. Studies of these give results consistent with the value quoted above.[3] The French 1999 eclipse An eclipse (Greek verb: ekleipÃ´, to vanish) is an astronomical event that occurs when one celestial object moves into the shadow of another. ...

The other consequence of the 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,[4] and the small tidal effect can not be observed in a short period. However, the cumulative effect of running behind a stable clock (ephemeris time, atomic time) a few milliseconds every day is very large, and 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 measured with 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).[5] Historical account and more comprehensive tables are availabe.[6] A table of the actual length of the day in the past few centuries is available.[7] Ephemeris Time (ET) is the time scale used in ephemerides of celestial bodies, in particular the Sun (as observed from the Earth), Moon, planets, and other members of the solar system. ... International Atomic Time (TAI, from the French name Temps Atomique International) is a high-precision atomic time standard that tracks proper time on Earths geoid. ... Universal Time (UT) is a timescale based on the rotation of the Earth. ... Ephemeris Time (ET) is the time scale used in ephemerides of celestial bodies, in particular the Sun (as observed from the Earth), Moon, planets, and other members of the solar system. ... Delta T and delta-T are ASCII substitutes for the formal ΔT, which is Terrestrial Time minus Universal Time. ... The International Earth Rotation and Reference Systems Service is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation Parameter (EOP) and International Celestial Reference System (ICRS) groups. ...

From the observed acceleration of the Moon, the corresponding change in the length of the day can be computed:

+2.3 ms/cy
(cy in centuries).

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

+1.70 ± 0.05 ms/cy[2][8]

The corresponding cumulative value is a parabola having a coefficient of T² (time in centuries squared) of:

ΔT = +31 s/cy²

Apparently there is another mechanism that accelerates the rotation of the Earth. 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 rebouncing (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 effect 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. Variations in CO2, temperature and dust from the Vostok ice core over the last 400 000 years For the animated movie, see Ice Age (movie). ...

Tidal locking makes one side of an astronomical body always face another, like the Moon facing the Earth. ... Tidal friction is a process that transferes energy between the planets. ...

## References

1. ^ J.Chapront, M.Chapront-Touzé, G.Francou: "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR". Astron.Astrophys. 387, 700..709 (2002).
2. ^ a b Jean O. Dickey et al. (1994): "Lunar Laser Ranging: a Continuing Legacy of the Apollo Program". Science 265, 482..490.
3. ^ F.R. Stephenson, L.V. Morrison (1995): Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990". Phil. Trans. Royal Soc. London Ser.A, pp.165..202.
4. ^ Jean O. Dickey (1995): "Earth Rotation Variations from Hours to Centuries". In: I. Appenzeller (ed.): Highlights of Astronomy. Vol. 10 pp.17..44.
5. ^ Observed values of UT1-TAI, 1962-1999
6. ^ Delta T: Approximate Algorithms for Historical Periods
7. ^ LOD
8. ^ F.R. Stephenson (1997): Historical Eclipses and Earth's Rotation. Cambridge Univ.Press.

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

Results from FactBites:

 Tidal force - Space Wiki - a Wikia wiki (987 words) All parts of the Earth accelerate in response to the moon's gravitational forces, but to an observer on the Earth, it appears that the Earth's center remains at rest, while water in the oceans is redistributed to form bulges on the sides near the moon and far from the moon. For a given (externally generated) gravitational field, the tidal acceleration at a point with respect to a body is obtained by vectorially subtracting the gravitational acceleration at the center of the body from the actual gravitational acceleration at the point. Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or fl holes, where they are responsible for the "spaghettification" of infalling matter.
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