Tidal friction

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite (i.e. a moon), and the planet (called the primary) that it orbits. It 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 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 rotation period, or that orbit in a retrograde direction.

Earth-Moon system

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, accelerating 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 Earth: 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 accelerates 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 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.

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 boil away, removing the bulk of the tidal friction and acceleration. Even without this, the slowdown to a month-long day would still not have been completed by 4.5 billion years from now when the Sun will evolve into a red giant and likely destroy both the Earth and Moon.

Tidal acceleration is one of the few examples in the dynamics of the solar system of a so-called secular perturbation of an orbit, i.e. a perturbation that continuously increases with time and is not periodic. Up to a high order of approximation, mutual gravitational perturbations between major or minor planets only cause periodic variations in their orbits, that is, parameters oscillate between maximum and minimum values. The tidal effect gives rise to a quadratic term in the equations, which leads to unbounded growth. In the mathematical theories of the planetary orbits that form the basis of ephemerides, quadratic and higher order secular terms do occur, but these are mostly Taylor expansions of very long time periodic terms. The reason that tidal effects are different is that unlike distant gravitational perturbations, friction is an essential part of tidal acceleration, and leads to permanent loss of energy in the form of heat.

Angular momentum and energy

The gravitational torque between the Moon and the tidal bulge of the Earth causes the Moon to be accelerated in its orbit, and the Earth to be decelerated in its rotation. As in any physical process, 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. 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 acceleration of the Moon causes an apparent deceleration of its motion across the celestial sphere. Although its kinetic energy decreases, its potential energy increases by a larger amount. The Moon's orbital angular momentum also 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.^[1]

A tidal bulge (called an equilibrium tide) does not really exist on Earth because the continents break up the tide when they pass under the Moon. Oceanic tides actually rotate around each ocean 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 equilibrium tide in the shape of a prolate spheroid that actually does exist for the Moon to pull on is the net result of integrating the actual undulations over all the world's oceans. Earth's net 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.^[2]

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.^[3] 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 parameters, among others the secular acceleration. From the period 1969–2001, the result is:

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.

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.^[6]

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.^[7] The small tidal effect can not be observed in a short period but the cumulative effect of the Earth's actual daily rotation occurring in less than 24 hours as measured on a stable clock (ephemeris time, atomic time) 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).^[8] A table of the actual length of the day in the past few centuries is also available.^[9]

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

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

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

Counter to the tidal deceleration of the Earth, there 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 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.

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.^[11]

Tidal deceleration

Tidal heating

Tidal heating occurs through the tidal friction processes explained above: orbital and rotational energy are dissipated as heat in the crust of the moons and planets involved. Io, a moon of Jupiter, is the most volcanically active body in the solar system, with no impact craters surviving on its surface. This is because the tidal force of Jupiter deforms Io; the eccentricity of Io's orbit (a consequence of its participation in a Laplace resonance) causes the height of Io's tidal bulge to vary significantly (by up to 100 m) over the course of an orbit; the friction from this tidal flexing then heats up its interior. A similar but weaker process is theorised to have melted the lower layers of the ice surrounding the rocky mantle of Jupiter's next large moon, Europa. Saturn's moon Enceladus is similarly thought to have a liquid water ocean beneath its icy crust. The liquid water geysers which eject material from Enceladus are thought to be powered by tidal friction of this moon's shifting ice crust.

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