This page is a compilation of two related articles:

• Tidal Force


• Tidal Acceleration

See also:

• Tidal Stream Generators


• Barrage Tidal Power

Comet Shoemaker-Levy 9 after breaking up under the influence of Jupiter's tidal forces.

The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational acceleration experienced by a large body is not constant across its diameter. One side of the body has greater acceleration than its center of mass, and the other side of the body has lesser acceleration.

Explanation

The Moon's (or Sun's) gravity differential field at the surface of the earth is known as the Tide Generating Force. This is the primary mechanism that drives tidal action and explains two tidal equipotential bulges, accounting for two high tides per day.

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. This causes strains on both bodies and may distort them or even, in extreme cases, break one or the other apart. These strains would not occur if the gravitational field is uniform, since a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.

Saturn's rings are inside the orbits of its moons. Tidal forces prevented the material in the rings from coalescing gravitationally to form moons.

The figure shows Comet Shoemaker-Levy 9 after it had broken up under the influence of Jupiter's tidal forces. The comet was falling into Jupiter, and the parts of the comet closest to Jupiter fell with a greater acceleration, due to the greater gravitational force. From the point of view of an observer riding on the comet, it would appear that the parts in front split off in the forward direction, while the parts in back split off in the backward direction. In reality, however, all parts of the comet were accelerating toward Jupiter, but at different rates.

Effects of tidal forces

In the case of an elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid, with two bulges, pointing towards and away from the other body. This is essentially what happens to the Earth's oceans. Although the Earth is not falling along a line directly toward the moon, the Earth is continuously accelerating due to the moon's gravitational forces, causing it to wobble around their common center of mass. 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.

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io.

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that in addition to variations of insolation associated with orbital forcing, harmonic beat variations in tidal forcing may contribute to climate changes.^[1]

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and the Sun.

Tidal forces are also responsible for tidal locking and tidal acceleration.

Mathematical treatment

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. Correspondingly, the term tidal force is used to describe the forces due to tidal acceleration. Note that for these purposes the only gravitational field considered is the external one; the gravitational field of the body (as shown in the graphic) is not relevant.

Graphic of tidal forces; the gravity field is generated by a body to the right. The top picture shows the gravitational forces; the bottom shows their residual once the field of the sphere is subtracted; this is the tidal force. See calculated tidal forces for a more exact version

Tidal acceleration does not require rotation or orbiting bodies; e.g. the body may be freefalling in a straight line under the influence of a gravitational field while still being influenced by (changing) tidal acceleration.

Newton's law of universal gravitation states that a particle of mass m a distance r from the center of a sphere of mass M feels a force of:

,

where is a unit vector pointing from the body M to the particle m.

Extending the description of m to a small body with spatial extent, suppose that R is the inter-object distance -- the distance from the center of M to the center of m, and let ∆r be the radius of m in the direction pointing towards M. Hence the points on the surface of m are located at distance from the centre of M. Using the above equation, and ignoring the small contribution due to m's own mass, we have the gravitational force at these points as:

Pulling out the R² term from the denominator gives:

The Maclaurin series of 1/(1 + x)² is 1 - 2 x + 3 x² - ..., which gives a series expansion of:

The first term is the traditional gravitational force; all other terms are tidal force terms. Generally, the first is much more significant than the other terms, giving:

The tidal forces can also be calculated away from the axis connecting the bodies, requiring a vector calculation of forces. In the plane perpendicular to the axis, the tidal force is directed inwards, and its magnitude is F_t / 2 in linear approximation as above (1).

See also

• Tidal resonance
• Roche limit
• Tidal locking
• Tidal acceleration
• Galactic tide

References

External links

• http://burro.astr.cwru.edu/Academics/Astr221/Gravity/tides.html

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 possibly destroy both the Earth and Moon. (Tidal acceleration 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.

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:

−25.858 ± 0.003 "/cy² in ecliptic longitude^[4]

+3.84 ± 0.07 m/cy in distance^[5]

(cy is centuries; the first is a quadratic term.)

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.^[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 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).^[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:

+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^[5][10]

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. The effect also arises between different components in a binary star.^[11]

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 (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.

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.

See also

• Tidal locking
• Tidal force
• Tides

References

1. ^ Walter Munk. "Once again: once again—tidal friction". Progress in Oceanography 40 (1997) 7-35.
2. ^ George E. Williams. "Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit". Reviews of Geophysics 38 (2000), 37-60.
3. ^ Another reflector emplaced by Lunokhod 1 in 1970 is no longer functioning.[1]
4. ^ 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).
5. ^ ^{a b} Jean O. Dickey et al. (1994): "Lunar Laser Ranging: a Continuing Legacy of the Apollo Program". Science 265, 482..490.
6. ^ 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.
7. ^ Jean O. Dickey (1995): "Earth Rotation Variations from Hours to Centuries". In: I. Appenzeller (ed.): Highlights of Astronomy. Vol. 10 pp.17..44.
8. ^ Observed values of UT1-TAI, 1962-1999
9. ^ LOD
10. ^ F.R. Stephenson (1997): Historical Eclipses and Earth's Rotation. Cambridge Univ.Press.
11. ^ Zahn, J.-P. (1977). "Tidal Friction in Close Binary Stars". Astron. Astrophys. 57: 383–394. 

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 Smith