Tidal acceleration

The tidal acceleration of the Moon is a peculiar 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.

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. This is apparent from the fact that 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. Hence the Earth and Moon orbit the Sun together.

Qualitative explanation

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 (and opposite to it). 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 about 2° 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.

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.

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 15 Ás every year.

This mechanism must have been working for 4.5 billion years, since oceans formed on the Earth. There is geological and paleontological evidence that the Earth rotated faster and that the month was shorter (so the Moon was closer) in the remote past.

This process will continue until in the remote future 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 in the converse situation, 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.

The Pluto-Charon system is another example of a double planet system in our solar system that went through tidal evolution of the orbit and rotation of its components. This system already has completely evolved, and both components always turn the same side to each other. See also tidal locking.

The tidal acceleration is one of the few examples in the dynamics of the solar system of a truly secular effect, i.e. 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, i.e. it oscillates between maximum values. The tidal effect gives rise to a quadratic term, which grows forever. 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.

Quantitative description

The motion of the Moon can be followed with an accuracy of a few cm. 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:

−25.858 ± 0.003 "/cy2 in ecliptic longitude (ref. [5])
+3.84 ± 0.07 m/cy in distance (ref. [1])
(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.

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 [2].

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 [3], 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 milli-seconds 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) at http://www.iers.org/iers/earth/rotation/ut1lod/table1.html . For a historical account and more comprehensive tables, see http://www.phys.uu.nl/~vgent/astro/deltatime.htm . A table of the actual length of the day in the past few centuries is available at http://www.iers.org/iers/earth/rotation/ut1lod/table3.html .

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 [2],[4], the following average value is found:

+1.70 ± 0.05 ms/cy

The corresponding cumulative value is:

ΔT = +31 s/cy²

Apparently there is another mechanism that accelerates the rotation of the Earth. Now 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.


[1] Jean O. Dickey et al. (1994): "Lunar Laser Ranging: a Continuing Legacy of the Apollo Program". Science 265, 482..490 .

[2] 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 .

[3] Jean O. Dickey (1995): "Earth Rotation Variations from Hours to Centuries". In: I. Appenzeller (ed.): Highlights of Astronomy. Vol. 10 pp.17..44 .

[4] F.R. Stephenson (1997): "Historical Eclipses and Earth's Rotation". Cambridge Univ.Press.

[5] 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) .


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