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All sourcesNASA NTRSDOE OSTI
NASA NTRS2015

Tidal Friction in the Earth-Moon System and Laplace Planes: Darwin Redux

The dynamical evolution of the Earth-Moon system due to tidal friction is treated here. George H. Darwin used Laplace planes (also called proper planes) in his study of tidal evolution. The Laplace plane approach is adapted here to the formalisms of W.M. Kaula and P. Goldreich. Like Darwin, the approach assumes a three-body problem: Earth, Moon, and Sun, where the Moon and Sun are point-masses. The tidal potential is written in terms of the Laplace plane angles. The resulting secular equations of motion can be easily integrated numerically assuming the Moon is in a circular orbit about the Earth and the Earth is in a circular orbit about the Sun. For Earth-Moon distances greater than ∼10 Earth radii, the Earth's approximate tidal response can be characterized with a single parameter, which is a ratio: a Love number times the sine of a lag angle divided by another such product. For low parameter values it can be shown that Darwin's low-viscosity molten Earth, M. Ross's and G. Schubert's model of an Earth near melting, and Goldreich's equal tidal lag angles must all give similar histories. For higher parameter values, as perhaps has been the case at times with the ocean tides, the Earth's obliquity may have decreased slightly instead of increased once the Moon's orbit evolved further than 50 Earth radii from the Earth, with possible implications for climate. This is contrast to the other tidal friction models mentioned, which have the obliquity always increasing with time. As for the Moon, its orbit is presently tilted to its Laplace plane by 5.2deg. The equations do not allow the Moon to evolve out of its Laplace plane by tidal friction alone, so that if it was originally in its Laplace plane, the tilt arose with the addition of other mechanisms, such as resonance passages.

Darwin Redux↗
NASA NTRS1980

Tidal-friction theory of the earth-moon system

Serious errors contained in Jeffreys' (1952, 1959, 1970, 1976) discussion of tidal friction in the earth-moon system are identified and their consequences are discussed. A direct solution of the dynamical tidal equations for the couple from the earth acting upon the moon and the couple from the earth acting upon the sun, which were left unsolved by Jeffreys, is found to be incompatible with observations and the predictions of linear or quadratic friction theory, due to his failure to take into account the possible change of the moment of inertia of the earth with time in the derivation of the dynamical equations. Consideration of this factor leads to the conclusion that the earth must be contracting at a rate of 14.7 x 10 to the -11th/year, which can be accounted for only by the Ramsey theory, in which the terrestrial core is considered as a phase change rather than a change in chemical composition. Implications of this value for the rates of changes in day length and lunar distance are also indicated.

Lyttleton, R. A.↗
NASA NTRS1973

Solar tidal friction and satellite loss

This paper treats the dynamical evolution of a satellite orbiting a primary that is being despun by solar tidal friction. The results are then applied to Mercury and Venus. It is concluded that a wide variety of satellites would not have survived the solar tidal braking of these planets. Hence, the absence of satellites around Mercury and Venus is not a compelling reason for thinking that this has always been the situation. Satellites are lost by orbital decay and impact with the primary. It is likely that Mercury would retain a record of such an event on its surface.

Ward, W. R.↗
NASA NTRS1975

Tidal friction and generalized Cassini's laws in the solar system

The tidal drift toward a generalized Cassini state of rotation of the spin axis of a planet or satellite in a precessing orbit is described. Generalized Cassini's laws are applied to several solar system objects and the location of their spin axes estimated. Of those considered only the moon definitely occupies state 2 with the spin axis near to the normal of the invariable plane. Most objects appear to occupy state 1 with the spin axis near to the orbit normal. Iapetus could occupy either state depending on its oblateness. In addition, the resonant rotation of Mercury is found to have little effect on the tidal drift of its spin axis toward state 1.

Ward, W. R.↗
NASA NTRS1978

Tidal friction in the solid earth

The earth's imperfectly elastic response to body and loading tidal forces is discussed using complex Love numbers and complex mass loading coefficients. Exact analytical expressions were derived relating the energy dissipation within an inhomogeneous, compressible solid earth to the surface values of these complex characteristic numbers, thus relating the global dissipation function Q to the phase shifts in the potential, gravity, tilt, strain and displacement tides. Integration of a global ocean tidal model shows that energy dissipated in the solid earth due to ocean loading is at least 10% of that dissipated in the body tides.

Zschau, J.↗
NASA NTRS1975

Tidal friction and the early history of the moon's orbit

The present work investigates the consequences implied by various rheological models of the early earth for the orbital history of the moon subsequent to its formation. Models of the earth that yield small tidal angles, such as low-viscosity models, imply that the moon never orbited in the earth's equatorial plane, thereby ruling out an equatorial origin for the moon. A high-viscosity model is shown to permit the moon to originate in the equatorial plane and still account for the present-day characteristics of the moon's orbit.

Rubincam, D. P.↗
NASA NTRS1987

Could Ariel have been heated by tidal friction?

For significant dissipation in Ariel, the past existence of an orbital resonance to force and maintain a substantial eccentricity in the orbit is hypothesized. The absolute maximum dissipation rate would be reduced both by the time necessary to establish the resonance in the first place and by a consequently larger minimum Q. The actual heating rate was much less than this maximum so even the existence of a 2/1 resonance may not be sufficient to account for Ariel's smooth sufrace.

Peale, S. J.↗
NASA NTRS1993

The connection between Venus' free obliquity and its CMB oblateness

The most striking feature of Venus rotational state is its slow retrograde rotation which is apparently maintained by a balance between solid tidal friction and thermal tidal torques. Solid tides tend to drive the spin toward synchronous rotation while thermal tides drive it away. A balance is achieved at a specific rate because of the inverse frequency dependence of the thermal tide to the semi-diurnal heating. Atmospheric models have been constructed to estimate the thermal tidal torque based on ground heating. The solid friction dissipation factor Q approximately equal to 50 can be deduced assuming rotation has achieved steady state. The most perplexing feature of Venus orientation is its non-zero free obliquity epsilon approximately equal to 1.5 deg relative to its orbit. Although solid tides and perhaps atmospheric tides tend to increase the free obliquity on a time 1/K(sub t) approximately equal to 1 x 10(exp 8) yr, viscous friction (CMF) at a core-mantle boundary (CMB) resulting from the differential angular orientation Delta-epsilon of the core and mantle spin axes should have damped the free obliquity on a time scale as short as 10(exp 6) yr. One means of achieving a balance similar to that controlling rotation is to introduce a comparatively large CMB ellipticity e(sub c) to reduce Delta-epsilon such that there is a balance between solid-thermal tides and CMF. The balance depends not so much on the potential frequency dependence on the tides as on the quadratic dependence of CMF on Delta-epsilon if the layer is turbulent.

Yoder, C. F.↗
NASA NTRS1982

Tidal rigidity of Phobos

A novel scenario is proposed in which the orbital eccentricity of Phobos results from several gravitational resonance excitations within the past one billion years, assuming that tidal friction in Phobos has had only a small effect on its orbit. Both the primordial eccentricity and inclination may have been much smaller than presently observed. Constraints imposed on tidal friction in Phobos by the apparent age of its surface, which is greater than one billion years, in addition to those of the gravitational resonance excitation scenario, can be satisfied only if the product of the rigidity and the Q factor is greater than 10 to the 12th dynes/sq cm. In view of the fact that the Q factor is only about 100, so that the rigidity must be greater than 10 to the 10th dynes/sq cm, Phobos should have substantial internal strength.

Yoder, C. F.↗
NASA NTRS1973

The early history of the lunar inclination

The effect of tidal friction on the inclination of the lunar orbit to the earth's equator for earth-moon distances of less than 10 earth radii is examined. The results obtained bear on a conclusion drawn by Gerstenkorn and others which has been raised as a fatal objection to the fission hypothesis of lunar origin, namely, that the present nonzero inclination of the moon's orbit to the ecliptic implies a steep inclination of the moon's orbit to the earth's equatorial plane in the early history of the earth-moon system. This conclusion is shown to be valid only for particular rheological models of the earth. The earth is assumed to behave like a highly viscous fluid in response to tides raised in it by the moon. The moon is assumed to be tideless and in a circular orbit about the earth. The equations of tidal friction are integrated numerically to give inclination of the lunar orbit as a function of earth-moon distance.

Rubincam, D. P.↗
NASA NTRS1981

The free librations of a dissipative moon

It is noted that dissipation in the moon produces a small offset (approximately 0.23 arcsec) of the moon's rotation axis from the plane defined by the ecliptic and lunar orbit normals. Both solid body tidal friction and viscous fluid friction at a core-mantle interface are thought to be plausible mechanisms. The merits of both are discussed, and it is found that solid friction requires a low lunar tidal Q (approximately 28), whereas turbulent fluid friction requires a core with a radius of approximately 330 km to cause the signature observed by lunar laser ranging. Large (approximately 0.4-8.0 arcsec) free librations of the lunar figure have also been detected through laser ranging. Both a very recent impact on the moon and fluid turbulence in the lunar core are considered plausible mechanisms for generating these librations.

Yoder, C. F.↗
NASA NTRS1977

Are striations on Phobos evidence for tidal stress

A series of striations or grooves over a great part of the surface of Phobos, the inner satellite of Mars, can be observed on Viking orbiter photographs. It is suggested that some of these features are related to a readjustment of the satellite's figure with increasing tidal stress as the orbit evolves inwards under the action of tidal friction. If the width of the grooves is in fact due to tidal readjustments of the figure of Phobos, then the older craters should be systematically deformed from their initial nearly circular shapes.

Soter, S.↗
NASA NTRS1972

Dynamical characteristics of Phobos and Deimos.

The orbital properties of the two small Martian satellites, Phobos and Deimos, are discussed, as well as those dynamical constants of Mars that can be determined from the satellite orbits. The secular acceleration of the mean motion of Phobos is shown to be very small. Of mechanisms that could cause any such acceleration, only tidal friction appears to be important. From the orbital evolution of the Martian satellites under tidal forces, Phobos and Deimos seem to have originated in nearly circular orbits of low inclination fairly close to the distance at which a satellite's orbital period is the planet's rotation period. It is proposed that the Martian satellites were born at the same time as Mars from equatorial dust clouds. The satellites are predicted to be locked in synchronous rotation, with their axes of minimum moment of inertia pointing on the average toward Mars, whereas their maximum axes are approximately normal to their orbit planes.

Burns, J. A.↗
NASA NTRS1994

Evolution of the Earth-Moon system

The tidal evolution of the Earth-Moon system is reexamined. Several models of tidal friction are first compared in an averaged Hamiltonian formulation of the dynamics. With one of these models, full integrations of the tidally evolving Earth-Moon system are carried out in the complete, fully interacting, and chaotically evolving planetary system. Classic results on the history of the lunar orbit are confirmed by our more general model. A detailed history of the obliquity of the Earth which takes into account the evolving lunar orbit is presented.

Touma, Jihad↗
NASA NTRS1984

The Moon's orbit history and inferences on its origin

A frequency dependent model of tidal friction was used to determine the evolution of the Earth-Moon system. The analysis considers the lunar orbit eccentricity and inclination, the solar tide on the Earth, Earth oblateness, and higher order terms in the tidal potential. A solution of the equations governing the precession of the Earth's rotational angular momentum and the lunar ascending node is found. The history is consistent with a capture origin for the Moon. It rules out the origin of the Moon by fission. Results are shown for a range of assumed values for the lunar tidal dissipation. Tidal dissipation within the Moon, during what would be the immediate postcapture period, is shown to be capable of significantly heating the Moon. The immediate postcapture orbit has a periapsis within the Earth's Roche limit. Capture into resonance with the Earth's gravitational field as this orbit tidally evolves is suggested to be a mechanism to prevent so close, an approach. It is shown that the probability of such capture is negligibly small and alternative hypotheses for the survival of the Roche limit passage is offered.

Conway, B. A.↗