TIDAL DISSIPATION IN THE MOON
Calculation of the dissipation of tidal energy in the moon due to imperfect elasticity, showing that heating by tidal dissipation is now insignificant
LUNAR TIDE↗
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Tidal dissipation in a viscoelastic planet
Tidal dissipation is examined using Maxwell standard liner solid (SLS), and Kelvin-Voigt models, and viscosity parameters are derived from the models that yield the amount of dissipation previously calculated for a moon model with QW = 100 in a hypothetical orbit closer to the earth. The relevance of these models is then assessed for simulating planetary tidal responses. Viscosities of 10 exp 14 and 10 ex 18 Pa s for the Kelvin-Voigt and Maxwell rheologies, respectively, are needed to match the dissipation rate calculated using the Q approach with a quality factor = 100. The SLS model requires a short time viscosity of 3 x 10 exp 17 Pa s to match the Q = 100 dissipation rate independent of the model's relaxation strength. Since Q = 100 is considered a representative value for the interiors of terrestrial planets, it is proposed that derived viscosities should characterize planetary materials. However, it is shown that neither the Kelvin-Voigt nor the SLS models simulate the behavior of real planetary materials on long time scales. The Maxwell model, by contrast, behaves realistically on both long and short time scales. The inferred Maxwell viscosity, corresponding to the time scale of days, is several times smaller than the longer time scale (greater than or equal to 10 exp 14 years) viscosity of the earth's mantle.
Ross, M.↗
Consequences of Tidal Dissipation in a Putative Venusian Ocean
The solar tide in an ancient Venusian ocean is simulated using a dedicated numerical tidal model. Simulations with varying ocean depth and rotational periods ranging from −243 to 64 sidereal Earth days are used to calculate the tidal dissipation rates and associated tidal torque. The results show that the tidal dissipation could have varied by more than 5 orders of magnitude, from 0.001 to 780 GW, depending on rotational period and ocean depth. The associated tidal torque is about 2 orders of magnitude below the present day Venusian atmospheric torque, and could change the Venusian daylength by up to 72 days per million years depending on rotation rate. Consequently, an ocean tide on ancient Venus could have had significant effects on the rotational history of the planet. These calculations have implications for the rotational periods of similarly close-in exoplanetary worlds and the location of the inner edge of the liquid water habitable zone.
planets and satellites↗
Consequences of Tidal Dissipation in a Putative Venusian Ocean
The solar tide in an ancient Venusian ocean is simulated using a dedicated numerical tidal model. Simulations with varying ocean depth and rotational periods ranging from minus 243 to 64 sidereal Earth days are used to calculate the tidal dissipation rates and associated tidal torque. The results show that the tidal dissipation could have varied by more than 5 orders of magnitude, from 0.001 to 780 gigawatts (GW), depending on rotational period and ocean depth. The associated tidal torque is about 2 orders of magnitude below the present day Venusian atmospheric torque, and could change the Venusian daylength by up to 72 days per million years depending on rotation rate. Consequently, an ocean tide on ancient Venus could have had significant effects on the rotational history of the planet. These calculations have implications for the rotational periods of similarly close-in exoplanetary worlds and the location of the inner edge of the liquid water habitable zone.
Dynamical evolution and stability↗
The Laplacian resonance and tidal dissipation
The Laplacian resonance amongst Io, Europa, and Ganymede was examined, taking into account tidal dissipation in Jupiter, Io and Europa. In equilibrium, it is not possible to neglect dissipation in Europa nor the torques of Jupiter on Europa and Ganymede. A formal calculation was made on the assumption that the tidal torques and the torques in orbit-orbit resonance reached an equilibrium such that the rates of decrease of the mean motions of the satellites are in the ratios 4:2:1. Q(j)/k(j)=167 Q(1)/k(1), Q(2)/k(2) = 0.44 Q(1)/K(1), where Q denotes the quality at the body's frequency of rotation, k its second degree Love number and the subscripts J, 1 and 2 denote Jupiter, Io, and Europa. The Q(J) found is at Jupiter's frequency of rotation. The rate of tidal dissipation in Europa comes to 1/5 that of Io. Such a low value of Q(2)/K(2) is plausible if dissipation in both Io and Europa is due to tidal friction in fluid against underlying and overlying solid layers.
Cook, A. F.↗
Gravitational tides in the outer planets. I - Implications of classical tidal theory. II - Interior calculations and estimation of the tidal dissipation factor
Classical tidal theory is applied to the atmospheres of the outer planets. The tidal geopotential due to satellites of the outer planets is discussed, and the solution of Laplace's tidal equation for Hough modes appropriate to tides on the outer planets is examined. The vertical structure of tidal modes is described, noting that only relatively high-order meridional mode numbers can propagate vertically with growing amplitude. Expected magnitudes for tides in the visible atmosphere of Jupiter are discussed. The classical theory is extended to planetary interiors taking the effects of spherically and self-gravity into account. The thermodynamic structure of Jupiter is described and the WKB theory of the vertical structure equation is presented. The regions for which inertial, gravity, and acoustic oscillations are possible are delineated. The case of a planet with a neutral interior is treated, discussing the various atmospheric boundary conditions and showing that the tidal response is small.
Ioannou, Petros J.↗
Melting of Io by tidal dissipation
The resonant structure of Io leads to forced eccentricities that are considerably larger than the free values. Although still modest by all standards, these forced eccentricities coupled with the enormous tides induced by Jupiter lead to magnitudes of tidal dissipation that are large enough to completely dominate the thermal history of Io. In the present paper, the forced eccentricities are calculated and then substituted into an expression for the total tidal dissipation. The results point to the possibility that the dissipation of tidal energy in Io may have melted a major fraction of Io's mass.
Peale, S. J.↗
Contribution of tidal dissipation to lunar thermal history
Possible contributions of tidal heating to the thermal history of the moon are investigated, and conditions under which such heating would have been an important factor in the evolution of the moon are defined as precisely as possible within the bounds of the uncertainties. The tidal dissipation is defined as a function of position in the moon and of the earth-moon separation for a homogeneous incompressible moon. Results for the dissipation are given for eccentricities of zero and 0.055 as well as for the lunar obliquity as determined by the equilibrium Cassini states. Possible orbital resonances which interrupt the assumed simplified expansion of the lunar orbit are considered, along with radial variations in rigidity. The dissipation in a two-layer model moon with a molten inner core and an outer mantle with the moon's present rigidity is determined, and implications of the results for the role of tidal heating in the history of the moon are discussed.
Peale, S. J.↗
Tidal Dissipation in Mercury
The spatial pattern and total inventory of tidal dissipation within Mercury depends sensitively on internal structure and on orbital eccentricity. Surface heat flow from this source may exceed 3 mW/sq m, and will vary with time as the orbital eccentricity fluctuates. Additional information is contained in the original extended abstract.
Bills, B. G.↗
Spatial and Temporal Patterns of Tidal Dissipation in Synchronous Satellites
Tidal heating is an important energy source for several solar system bodies, and there is a wide-spread perception that the pattern of surface heat flow is diagnostic of internal structure. We wish to clarify that situation. Our analysis depends upon two important assumptions: First, that heat transport is dominated by conduction. Second, that the body can be modeled by a sequence of spherically symmetric layers, each with a linear visco-elastic rheology. Under these assumptions, surface heat flow patterns in tidally dominated satellites will reflect radially integrated dissipation patterns. For synchronously rotating satellites with zero obliquity, this pattern depends quite strongly on orbital eccentricity but relatively little on purely radial variations in internal structure. The total amount of heat generated within the body does depend sensitively on internal structure, but the spatial pattern is rather insensitive to structure, especially at low orbital eccentricities.
Bills, Bruce G.↗
Tidal dissipation in the earth and planets.
Earth and planets shallow seas tidal dissipation reflected in Q values, noting larger planet pressure effect
Anderson, D. L.↗
Tidal dissipation in oceans on Titan
Titan's large free eccentricity results in significant tidal dissipation. This can be used to constrain the existence and depth of hydrocarbon oceans. A hydrodynamical-numerical 2 deg ocean tide model has been constructed to investigate this connection. The model allows some simple land configurations. The results indicate that existence of such an ocean over the age of the solar system is hard to explain, as is the existence of the large eccentricity itself. If such an ocean exists, it is likely to be more than 500 m deep, ignoring the influence of land masses.
Sears, William D.↗
Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io
The deformation of Io, the tidal dissipation rate, and its interior spatial distribution are investigated by means of numerical simulations based on (1) a three-layer model (with dissipation in the mantle) or (2) a four-layer model (with dissipation in the asthenosphere). The mathematical derivation of the models is outlined; the selection of the input-parameter values is explained; the results are presented in extensive graphs and contour maps; and the constraints imposed on the models by observational data on the hot-spot distribution, tidal deformation, and gravity field are discussed in detail. It is found that both dissipation mechanisms may play a role on Io: model (2) is better able to explain the concentration of hot spots near the equator, while the presence of a large hot spot near the south pole (if confirmed by observations) would favor model (1).
Segatz, M.↗
Evolution of satellite resonances by tidal dissipation.
Analysis of a realistic model shows how satellites' gravitational interaction can halt their differential tidal evolution when resonant commensurabilities of their orbital periods are reached. The success of this study lends support to the hypothesis that orbit-orbit resonances among satellites in the solar system, including the Titan-Hyperion case, did evolve as a result of tidal energy dissipation. Consideration of the time scale for this evolution process, possible now that the capture mechanism has been revealed, can offer more sophisticated constraints on the tidal dissipation function, Q, and on past orbital conditions.
Greenberg, R.↗
Tidal dissipation in the Earth and Moon from lunar laser ranging
The evolution of the Moon's orbit which is governed by tidal dissipation in the Earth while the evolution of its spin is controlled by its own internal dissipation is discussed. Lunar laser ranging data from August 1969 through May 1982 yields the values of both of these parameters. It is suggested that if the Moon was orbited the Earth since its formation, this must be an anomalously high value presumably due to changes in dissipation in the oceans due to continental drift. The explanation that the dissipation occurs at the interface between the mantle and a liquid core of shell is preferred.
Yoder, C. F.↗
Tidal dissipation, orbital evolution, and the nature of Saturn's inner satellites
Estimates of tidal damping times of the orbital eccentricities of Saturn's inner satellites place constraints on some satellite rigidities and dissipation functions Q. These constraints favor rock-like rather than ice-like properties for Mimas and probably Dione. Photometric and other observational data are consistent with relatively higher densities for these two satellites, but require lower densities for Tethys, Enceladus, and Rhea. This leads to a nonmonotonic density distribution for Saturn's inner satellites, apparently determined by different mass fractions of rocky materials. In spite of the consequences of tidal dissipation for the orbital eccentricity decay and implications for satellite compositions, tidal heating is not an important contributor to the thermal history of any Saturnian satellite.
Peale, S. J.↗
The source of Neptune's internal heat and the value of Neptune's tidal dissipation factor
The suggestion is made that Neptune's observed internal heating is the action of tidal torques between Triton and Neptune in despinning the planet and causing the orbital decay of Triton's orbit. These result from the frictional dissipation of tides within Neptune. It is shown that the considered process implies a value of the tidal dissipation factor of approximately 170. The results of the investigation do not change the conclusion which follows from the lack of internal heating for Uranus that the interiors of Uranus and Neptune differ significantly.
Trafton, L.↗