Tidal acceleration is an effect of the tidal forces between an orbiting natural
satellite, and the primary planet that it orbits. The acceleration 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 the smaller first, 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. The naming is somewhat confusing,
because the speed of the satellite relative to the body it orbits is
decreased as a result of tidal acceleration, and increased as a result
of tidal deceleration. 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. 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″ 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, 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 Earth's orbital eccentricity.
Adams's 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 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. 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. Secondly there is an apparent increase
in the Moon's angular rate of orbital motion. This arises from Earth's loss of
angular momentum and the consequent increase in length of day.
= Effects of Moon's gravity = Because the Moon's mass is a
considerable fraction of that of Earth, 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 Earth lies close to the plane of Earth's orbit around the
Sun, rather than in the plane perpendicular to the axis of rotation of
Earth 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. In particular, the water of the oceans bulges out towards
and away from the Moon. The average tidal bulge is synchronized with the
Moon's orbit, and 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 Earth and the 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 Earth and the Moon. This
boosts the Moon in its orbit, and slows the rotation of Earth.
As a result of this process, 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 over time, which leads to an
increasing difference between our clock 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 occasional, irregular intervals.
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 2.1
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 is
converted to heat by frictional losses in the oceans and their interaction with
the solid Earth, and only about 1/30th is transferred to the Moon). The Moon
moves farther away from Earth, so its potential energy increases. It stays in
orbit, and from Kepler's 3rd law it follows that its angular velocity
actually decreases, so the tidal action on the Moon actually causes an angular
deceleration, i.e. a negative acceleration of its rotation around
Earth. The actual speed of the Moon also decreases. Although its kinetic energy
decreases, its potential energy increases by a larger amount.
The rotational angular momentum of 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 Earth and the
Moon as heat. If the friction and heat dissipation were not present, the Moon's
gravitational force on the tidal bulge would rapidly 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
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 are 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 Earth. 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. = 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. 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 deceleration, i.e. negative acceleration, in longitude and the rate
of change of the semimajor axis of the Earth–Moon ellipse. From the period
1970–2012, the results are: −25.82±0.03 arcsecond/century2 in
ecliptic longitude +38.08±0.04 mm/yr in the mean Earth–Moon
distance This is consistent with results from
satellite laser ranging, a similar technique applied to artificial
satellites orbiting Earth, which yields a model for the gravitational field of
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 Earth. The rotation of Earth is somewhat erratic on all time scales
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 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 than would be measured by stable clocks
calibrated to the present, longer length of the day. This is known as ΔT. Recent
values can be obtained from the International Earth Rotation and
Reference Systems Service. 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/century However, from historical records over
the past 2700 years the following average value is found:
+1.70 ± 0.05 ms/century The corresponding cumulative value is a
parabola having a coefficient of T2 of: ΔT = +31 s/century2
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. As a consequence, the polar diameter of Earth
increases, and because 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 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/century. This largely explains
the historical observations. Other cases of tidal acceleration
Most natural satellites of the planets undergo tidal acceleration to some
degree, 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's second moon Deimos, which may become an
Earth-crossing asteroid after it leaks out of Mars's grip. The effect also
arises between different components in a binary star.
Tidal deceleration This comes in two varieties:
See also Tidal locking
Tidal force Tides
Tidal heating References
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