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 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, which is a surprisingly accurate result for its time, not differing greatly 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 a 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 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,
John Couch 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 Earth's orbital eccentricity. Adams' finding provoked a sharp astronomical controversy that lasted some years, but the correctness of his result, agreed upon 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 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 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 Earth and Moon. This increases the Moon's angular momentum around Earth (and moves the Moon to a higher orbit with a lower
orbital speed). 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 Earth's loss of angular momentum and the consequent increase in
length of day.
Effects of Moon's gravity showing how the tidal bulge is pushed ahead by
Earth's rotation. This offset bulge exerts a net torque on the
Moon, boosting it while slowing Earth's rotation. The plane of the Moon's
orbit around Earth lies close to the plane of Earth's orbit around the Sun (the
ecliptic), rather than in the plane of the Earth's rotation (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 Earth. Foremost among such matter, the
water of the
oceans bulges out both towards and away from the Moon. If the material of the Earth responded immediately, there would be a bulge directly toward and away from the Moon. In the
solid Earth tides, there is a delayed response due to the dissipation of tidal energy. The case for the oceans is more complicated, but there is also a delay associated with the dissipation of energy since the Earth rotates at a faster rate than the Moon's orbital angular velocity. This
lunitidal interval in the responses causes the tidal bulge to be carried forward. Consequently, the line through the two bulges is tilted with respect to the Earth-Moon direction exerting
torque between the Earth and the Moon. This torque boosts the Moon in its orbit and slows the rotation of Earth. As a result of this process, the mean solar day, which has to be 86,400 equal seconds, 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 over time, which leads to an increasing difference between our clock time (
Universal Time) on the one hand, and
International Atomic Time and
ephemeris time on the other hand: see
ΔT. This led to the introduction of the
leap second in 1972 to compensate for differences in the bases for time standardization. In addition to the effect of the ocean tides, there is also a tidal acceleration due to flexing of 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 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 Earth's rotation is not occurring fast enough for the rotation to lengthen to a month before other effects make this irrelevant: about 1 to 1.5 billion years from now, the continual increase of the Sun's
radiation will likely cause 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 probably evolve into a
red giant and likely destroy both Earth and the 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 from the dynamic 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 Earth causes the Moon to be constantly promoted to a slightly higher orbit and 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 Earth to the orbital motion of the Moon (however, most of the energy lost by Earth (−3.78 TW) is converted to heat by frictional losses in the oceans and their interaction with the solid Earth, and only about 1/30th (+0.121 TW) is transferred to the Moon). The Moon moves farther away from Earth (+38.30±0.08 mm/yr), so its
potential energy, which is still negative (in Earth's
gravity well), increases, i. e. becomes less negative. It stays in orbit, and from
Kepler's 3rd law it follows that its average
angular velocity actually decreases, so the tidal action on the Moon actually causes an angular deceleration, i.e. a negative acceleration (−25.97±0.05"/century2) of its rotation around Earth. The dissipation of energy by tidal friction averages about 3.64 terawatts of the 3.78 terawatts extracted, of which 2.5 terawatts are from the principal M 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 ocean basins 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, whereas 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.
Historical evidence This mechanism has been working for 4.5 billion years, since oceans first formed on Earth, but less so at times when much or most of the water
was ice. There is geological and paleontological evidence that Earth rotated faster and that the Moon was closer to 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 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. The present high rate may be due to near
resonance between natural ocean frequencies and tidal frequencies. Analysis of layering in fossil
mollusc shells from 70 million years ago, in the
Late Cretaceous period, shows that there were 372 days a year, and thus that the day was about 23.5 hours long then.
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 corner-cube prism retroreflectors on the surface of the Moon, emplaced during the
Apollo missions of 1969 to 1972 and by
Lunokhod 1 in 1970 and 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 deceleration, i.e. negative acceleration, in longitude and the rate of change of the semimajor axis of the Earth–Moon ellipse. From the period 1970–2015, the results are: : −25.97 ± 0.05 arcsecond/century2 in ecliptic longitude : +38.30 ± 0.08 mm/yr in the mean Earth–Moon distance The other consequence of tidal acceleration is the deceleration of the rotation of Earth. The rotation of 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 Earth's rotation as measured with a stable clock (ephemeris time, International 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 Earth) (
Universal Time) than would be measured by 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 (where "cy" means "century", d day, s second, ms millisecond, 10−3 s, and ns nanosecond, 10−9 s): : +2.4 ms/d/century or +88 s/cy2 or +66 ns/d2. However, from historical records over the past 2700 years the following average value is found: : +1.72 ± 0.03 ms/d/century or +63 s/cy2 or +47 ns/d2. (i.e. an accelerating cause is responsible for -0.7 ms/d/cy) By twice integrating over the time, the corresponding cumulative value is a parabola having a coefficient of T2 (time in centuries squared) of (1/2) 63 s/cy2 : : Δ
T = (1/2) 63 s/cy2 T2 = +31 s/cy2 T2. Opposing the tidal deceleration of Earth is a mechanism that is in fact accelerating the rotation. 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 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 Earth increases, and the equatorial diameter decreases (Earth's volume must remain the same). This means that mass moves closer to the rotation axis of Earth, and that Earth's moment of inertia decreases. This process alone leads to an increase of the rotation rate (phenomenon of a spinning figure skater who spins ever faster as they retract their 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/d/century. This largely explains the historical observations. == Other cases of tidal acceleration ==