Tidal acceleration

File:The Earth and the Moon photographed from Mars orbit.jpg

A picture of the Earth and the Moon from Mars. The presence of the moon (which has about 1/81 the mass of the Earth), is slowing Earth's rotation and lengthening the day by about 2 ms every one hundred years.

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 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' 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.

File:Tidal braking.svg

A diagram of the Earth-Moon system showing how the tidal bulge is pushed ahead by the Earth's rotation. This offset bulge exerts a net torque on the Moon, boosting it while slowing the Earth's rotation.

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 along both ends of an axis passing through the centers of the Earth and Moon. The average tidal bulge closely follows the Moon in its 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 decelerates the rotation of the Earth.

So the result is that 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 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 possibly destroy both the Earth and Moon. (Tidal acceleration and solar mass loss is also moving the Earth outward from the Sun, but it is unknown whether it will be enough to save it from destruction.)

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 dynamical 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 promoted in its 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 is converted to heat, and only about one 30th is transferred to the Moon). The Moon moves farther away from the Earth, 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 velocity actually decreases, so the tidal action on the Moon actually causes a deceleration of its motion across the celestial sphere. Although its kinetic energy decreases, its potential energy increases by a larger amount. The tidal force has a component in the direction of the Moon's motion, and therefore increases its energy, but the non-tidal part of the Earth's gravity pulls (on average) slightly backwards on the Moon (which on average has a slight outward velocity), so the net result is that the Moon slows down. The Moon's orbital angular momentum increases.

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 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:

−25.85"/cy² in ecliptic longitude

(cy is centuries, here taken to the square)

+38.14 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.

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 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 as 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). 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:

+1.70 ± 0.05 ms/cy

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' second moon Deimos, which may become an Earth-crossing asteroid after it leaks out of Mars' grip^{[citation needed]}. The effect also arises between different components in a binary star.

Tidal deceleration

This comes in two varieties:

1. 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
2. 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

• Tidal locking
• Tidal force
• Tides
• Tidal heating

External links

• The Recession of the Moon and the Age of the Earth-Moon System
• Tidal Heating as Described by University of Washington Professor Toby Smithit:Accelerazione secolare della Luna