Tidal acceleration

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, and a corresponding slowdown of the primary's rotation, known as tidal braking. See supersynchronous orbit. The process eventually leads to tidal locking, usually of the smaller body 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. These satellites will have a higher and higher orbital velocity and a shorter and shorter orbital period, until a final collision with the primary. See subsynchronous orbit.
The naming is somewhat confusing, because the average 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. This conundrum occurs because a positive acceleration at one instant causes the satellite to loop farther outward during the next half orbit, decreasing its average speed. A continuing positive acceleration causes the satellite to spiral outward with a decreasing speed and angular rate, resulting in a negative acceleration of angle. A continuing negative acceleration has the opposite effect.

Earth–Moon system

Discovery history of the secular acceleration

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, 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 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

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 of the Earth's rotation 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 small difference accumulates over time, which leads to an increasing difference between our clock 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 is converted to heat by frictional losses in the oceans and their interaction with the solid Earth, and only about 1/30th. The Moon moves farther away from Earth, so its potential energy, which is still negative, 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 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, i. e. E_p = -2E_c.
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 3.64 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 are the net result of integrating the actual undulations over all the world's oceans.