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Origin and Evolution of Enceladus’s Tidal Dissipation

Enceladus possesses a subsurface ocean beneath a conductive ice shell. Based on shell thickness models, the estimated total conductive heat loss from Enceladus is 25–40 GW; the measured heat output from the South Polar Terrain (SPT) is 4–19 GW. The present-day SPT heat flux is of order 100 mWm -21 −, comparable to estimated paleo-heat fluxes for other regions of Enceladus. These regions have nominal ages of about 2 Ga, but the estimates are uncertain because the impactor flux in the Saturnian system may not resemble that elsewhere. Enceladus’s measured rate of orbital expansion implies a low dissipation factor Q p for Saturn, with Q p ≈3×10 -3 (neglecting the role of Dione). This value implies that Enceladus’s present-day equilibrium tidal heat production (roughly 50 GW, but with large uncertainties) is in approximate balance with its heat loss. If Q p is constant, Enceladus cannot be older than 1.5 Gyr (because otherwise it would have migrated more than is permissible). However, Saturn’s dissipation may be better described by the “resonance-locking” theory, in which case Enceladus’s orbit may have only evolved outwards by about 35% over the age of the Solar System. In the constant-Q p scenario, any ancient tidal heating events would have been too energetic to be consistent with the observations. Because resonance-locking makes capture into earlier mean-motion orbital resonances less likely, the inferred ancient heating episodes probably took place when the current orbital resonance was already established. In the resonance-locking scenario, tidal heating did not change significantly over time, allowing for a long-lived ocean and a relatively stable ice shell. If so, Enceladus is an attractive target for future exploration from a habitability standpoint.

Satellites of Saturn↗

Tidal dissipation in the moon

Euler's equations of motion, modified to include elasticity and solid friction, were used to study the rotation of the moon. Two heuristic models for the anelasticity were considered: Q independent of frequency and Q inversely proportional to the frequency and Q inversely proportional to the frequency of the strain oscillation. Parameters in each model were estimated by weighted least squares from 9 years of lunar laser range observations. The root mean square of the postfit range residuals was 19 cm in each case. For a strain period of 1 month, the estimates of Q obtained with the two models were similar and surprisingly low: 23 + or - 6. The range of uncertaintly, + or - 6, represents our estimate of the true standard deviation of the estimate of Q, as affected by systematic errors; it is 3 times larger than the formal, statistical, standard error.

Cappallo, R. J.↗

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.↗

Estimates of M2 Tidal Energy Dissipation from TOPEX/Poseidon Altimeter Data

Most of the tidal energy dissipation in the ocean occurs in shallow seas, as has long been recognized. However, recent work has suggested that a significant fraction of the dissipation, perhaps 1 TW or more, occurs in the deep ocean. This paper builds further evidence for that conclusion. More than 6 years of data from the TOPEX/Poseidon satellite altimeter are used to map the tidal dissipation rate throughout the world ocean. The dissipation rate is estimated as a balance between the rate of working by tidal forces and the energy flux divergence, computed using currents derived by least squares fitting of the altimeter data and the shallow water equations. Such calculations require dynamical assumptions, in particular about the nature of dissipation. To assess sensitivity of dissipation estimates to input assumptions, a large suite of tidal inversions based on a wide range of drag parameterizations and employing both real and synthetic altimeter data are compared. These experiments and Monte Carlo error fields from a generalized inverse model are used to establish error uncertainties for the dissipation estimates. Owing to the tight constraints on tidal elevation fields provided by the altimeter, area integrals of the energy balance are remarkably insensitive to required dynamical assumptions. Tidal energy dissipation is estimated for all major shallow seas (excluding individual polar seas) and compared with previous model and data-based estimates. Dissipation in the open ocean is significantly tnhanced around major bathymetric features, in a manner consistent with simple theories the generation of baroclinic tides.

Egbert, Gary D.↗

Significant Dissipation of Tidal Energy in the Deep Ocean Inferred from Satellite Altimeter Data

How and where the ocean tides dissipate their energy are longstanding questions that have consequences ranging from the history of the Moon to the mixing of the oceans. Historically, the principal sink of tidal energy has been thought to be bottom friction in shallow seas. There has long been suggestive however, that tidal dissipation also occurs in the open ocean through the scattering by ocean-bottom topography of surface tides into internal waves, but estimates of the magnitude of this possible sink have varied widely. Here we use satellite altimeter data from Topex/Poseidon to map empirically the tidal energy dissipation. We show that approximately 10(exp 12) watts-that is, 1 TW, representing 25-30% of the total dissipation-occurs in the deep ocean, generally near areas of rough topography. Of the estimated 2 TW of mixing energy required to maintain the large-scale thermohaline circulation of the ocean, one-half could therefore be provided by the tides, with the other half coming from action on the surface of the ocean.

Egbert, G. D.↗

Phobos Ephemeris Improvement from Recent Spacecraft Observations

Despite 127 years of observations, there is still room for improvement in the ephemeris of the Martian satellite Phobos. Early in this history, Earth-based astrometric observations of Phobos and Deimos were used to estimate the mass and oblateness of Mars. As more data accumulated, it became clear that a secular acceleration in the longitude of Phobos was occurring, and this was attributed to tidal dissipation within Mars, yielding rough estimates of the tidal quality factor, or Q. At the epoch of the earliest spacecraft observations of Phobos, from Mariner 9 and the Viking Orbiters and Landers, the gravitational field of Mars, and resulting forces on Phobos, were still not particularly well known. Thus observations of natural and artificial satellite motions continued to contribute, each in their own way, to knowledge of the mass distribution within Mars. Improvements in tracking system accuracy, and the placement of satellites, like the 1996 Mars Global Surveyor and 2001 Mars Odyssey, in circular polar orbits has lead to dramatic improvements to knowledge of the Mars gravity field. The direct gravitational influence on Phobos is no longer expected to be a limiting factor in predicting its orbital motion. Despite that progress, a variety of observations of Phobos from recent orbiters and landers suggest that the best satellite ephemeris still has along-track orbit errors which are accumulating at a rate of 1.75 kilometers per year, with Phobos gaining on the predicted positions. These recent observations alone do not span sufficient time to separately resolve the positional error into changes in mean motion and changes in secular acceleration. However, combining them with earlier observations will allow improvements in both the mean motion and its first derivative. This latter parameter is particularly interesting, as it relates to tidal dissipation, and thus uniquely constrains the internal structure of Mars. The current best estimate of the secular acceleration rate of Mars implies a surprisingly low value for the tidal dissipation factor Q = (94 plus or minus 1). For the Earth, most tidal dissipation occurs within the oceans. The terrestrial mantle tidal Q is 280. An issue of interest is then: why, where, and how does Mars dissipate tidal energy so effectively.

Bills, B. G.↗

The tides of Io

A theory is developed for the origin and evaluation of the orbital resonances between the Galilean satellites Io, Europa and Ganymede as a result of the effects of dissipative tides in Jupiter and its satellites. Following a preliminary consideration of the consequences of tidal interaction for satellite orbits and a comprehensive Hamiltonian theory of the resonance interactions which allows terms up to third order in eccentricity to be included, a dynamic model of the origin and evolution of the resonance locks is presented in which the relative expansion of the orbits by tidal torques from Jupiter together with tidal dissipation in Io lead to the rapid driving out of Io until it is captured into a 2:1 resonance with Europa and the resonance with Ganymede is achieved. Consideration of the effects of other commensurabilities on the orbital evolution of the system reveals that second-order Laplace-like resonances would act to excite free eccentricites, while the resonances associated with the two-body 3:1 commensurability could not have been encountered. Analysis of the hypothesis that the Laplace relation is primordial shows that the bounds on the tidal dissipation of Jupiter still can not be relaxed. Recent determinations of the tidal dissipation in Jupiter are discussed, and it is noted that none is sufficiently small to be consistent with the high heat flux estimates for Io. Finally, the possibility of the observational determination of the tidal dissipation in Jupiter by the measurement of the secular acceleration of Io's main motion is considered.

Yoder, C. F.↗

Tidal Energy Dissipation from Topex/Poseidon

In a recent paper ({\it Nature, 405,} 775, 2000) we concluded that 25 to 30\% of the ocean's tidal energy dissipation, or about 1 terawatt, occurs in the deep ocean, with the remaining 2.6 TW in shallow seas. The physical mechanism for deep-ocean dissipation is apparently scattering of the surface tide into internal modes; Munk and Wunsch have suggested that this mechanism may provide half the power needed for mixing the deep-ocean. This paper builds further evidence for $1\pm 0.2$ TW of deep-ocean dissipation. The evidence is extracted from tidal elevations deduced from seven years of Topex/Poseidon satellite altimeter data. The dissipation rate Is formed as a balance between the rate of working by tidal forces and the energy flux divergence. While dynamical assumptions are required to compute fluxes, area integrals of the energy balance are, owing to the tight satellite constraints, remarkably insensitive to these assumptions. A large suite of tidal solutions based on a wide range of dynamical assumptions, on perturbations to bathymetric models, and on simulated elevation data are used to assess this sensitivity. These and Monte Carlo error fields from a generalized inverse model are used to establish error uncertainties.

Ray, Richard D.↗

Orbital evolution of the Galilean satellites

The orbital motions of the Galilean satellites exert dramatic control over their physical properties (most notably Io's) through tidal heating. In turn, tidal dissipation in the satellites, as well as in Jupiter, has governed the evolution of the orbits and, in particular, of the Laplace resonance. If the system started out of the resonance and evolved into it, forced eccentricities would have increased with time. Hence, the tidal melting of Io and the cracking of Europa'a surface may have occurred relatively recently. This theory requires that Jupiter's tidal dissipation factor be greater than about two million, a rather low value (high rate of tidal dissipation) compared with most models of Jovian interior processes. Alternatively, the system may have started even deeper in the resonance than it is today, a scenario which is consistent with larger values of the tidal dissipation factor. This model, with its correspondingly large initial forced eccentricities, would imply (1) that Io melted early and fast, and may have remained molten with only a thin solid skin until the present and (2) that the water mantles of both Europa and Ganymede remained largely molten for considerably longer than Callisto did, but later froze as their eccentricities and tidal heating decreased.

Greenberg, R.↗

A View of the Lunar Interior Through Lunar Laser Range Analysis

Laser ranges between observatories on the Earth and retroreflectors on the Moon started in 1969 and continue to the present. Recent range accuracies are 2 cm while earliest ranges are an C, order of magnitude less certain. Four retroreflectors are ranged: three located at the Apollo 11, 14, and 15 sites and one on the Lunakhod 2 rover. Accurate analysis of the range data determines a number of lunar science parameters. The lunar interior variables include a fluid core parameter. The Lunar Laser Ranging effort is reviewed elsewhere. Many parameters are detected through their influence on rotation. Also detected are solid-body tides and accurate selenocentric reflector locations. Determined through the rotation are moment-of-inertia differences, gravitational harmonics, potential Love number, and dissipation effects due to tides and molten core. The rotation of the Moon is not at its minimum energy state; some recently active process has caused free librations. The moment differences contributed to the recent improvement of the Moon's moment of inertia from the Lunar Prospector gravity field. The Love numbers provide bulk elastic properties. Future possibilities for measurement include oblateness of the core-mantle boundary and core moment. A study of dissipation signatures in the rotation determines tidal Q vs. frequency and concludes that the Moon has a molten core. At 1 month the tidal Q is 37 and at 1 yr it is 60. The core radius is < or = 352 km for Fe and < or = 374 km for the Fe-FeS eutectic. The core detection exceeds 3x its uncertainty. The spin of the core is not aligned with the spin of the mantle and torque arises from the velocity difference at the boundary. Yoder's turbulent boundary layer theory is used to compute the radii. The present heat generation from tides and core interaction is minor compared to radiogenic heating. The heating for ancient times is more interesting. Peale and Cassen investigated lunar tidal heating while the lunar orbit expanded due to tides on Earth. Their calculations predate the measurement of Q and should be multiplied by 3.45 to match the lunar-laser-determined Love number and monthly Q. Tidal-heating computations depend on how fast the lunar orbit evolved and whether the tidal dissipation is localized. Neither is known, but plausible assumptions lead to early central region temperature increases of several hundred degrees. Most of the energy is deposited early in the Moon's history. The turbulent boundary layer theory allows a prediction of energy dissipated at the core-mantle boundary during orbit evolution. Under the assumption that the properties of the early core are the same as at present, the energy dissipated by core-mantle interaction is about the same as for tidal dissipation, but it is deposited in a smaller volume. This source of energy is capable of promoting convection in an early fluid core and driving a dynamo. This is a transient phase with a duration depending on the rate of orbit evolution. Plausible assumptions lead to a duration of a few hundred million years. Thus, the remnant magnetization of many lunar rocks is compatible with a brief global magnetic field powered by dynamical energy dissipation. Analysis of the lunar laser ranges is providing information on lunar geophysics. Future data will improve accuracies of present solution parameters, and several more interior effects should be detectable.

Williams, J. G.↗

Orbital period variability in the eclipsing pulsar binary PSR B1957+20: Evidence for a tidally powered star

Recent observations indicate that the eclipsing pulsar binary PSR B1957+20 undergoes alternating epochs of orbital period increase and decrease. We apply a model developed to explain orbital period changes of alternating sign in other binaries to the PSR B1957+20 system and find that it fits the pulsars observations well. The novel feature of the PSR B1957+20 system is that the energy flow in the companion needed to power the orbital period change mechanism can be supplied by tidal dissipation, making the companion the first identified tidally powered star. The flow of energy in the companion drives magnetic activity, which underlies the observed orbital period variations. The magnetic activity and the wind driven by the pulsar irradiation results in a torque on the spin of the companion. This torque holds the companion out of synchronous rotation, causing tidal dissipation of energy. We propose that the progenitor had a approximately 2 hr orbital period and a companion mass of 0.1-0.2 solar mass, and the system is evolving to longer orbital periods by mass and angular momentum loss on a timescale of 10(exp 8) yr.

Applegate, James H.↗

Dissipation and Bathymetric Sensitivities in an Unstructured Mesh Global Tidal Model

Abstract The mechanisms and geographic distribution of global tidal dissipation in barotropic tidal models are examined using a high resolution unstructured mesh finite element model. Mesh resolution varies between 2 and 25 km and is especially focused on inner shelves and steep bathymetric gradients. Tidal response sensitivities to bathymetric changes are examined to put into context response sensitivities to frictional processes. We confirm that the Ronne Ice Shelf dramatically affects Atlantic tides but also find that bathymetry in the Hudson Bay system is a critical control. We follow a sequential frictional parameter optimization process and use TPXO9 data‐assimilated tidal elevations as a reference solution. From simulated velocities and depths, dissipation within the global model is estimated and allows us to pinpoint dissipation at high resolution. Boundary layer dissipation is extremely focused with 1.4% of the ocean accounting for 90% of the total. Internal tide friction is much more distributed with 16.7% of the ocean accounting for 90% of the total. Often highly regional dissipation can impact basin‐scale and even ocean wide tides. Optimized boundary layer friction parameters correlate very well with the physical characteristics of the locality with high friction factors associated with energetic tidal regions, deep ocean island chains, and ice covered areas. Global complex M 2 tide errors are 1.94 cm in deep waters. Total global boundary layer and internal tide dissipation are estimated, respectively, at 1.83 and 1.49 TW. This continues the trend in the literature toward attributing more dissipation to internal tides.

54 ENVIRONMENTAL SCIENCES↗

Tidal heating and mass loss in neutron star binaries - Implications for gamma-ray burst models

A neutron star in a close binary orbit around another neutron star (or stellar-mass black hole) spirals inward owing to gravitational radiation. We discuss the effects of tidal dissipation during this process. Tidal energy dissipated in the neutron star's core escapes mainly as neutrinos, but heating of the crust, and outward diffusion of photons, blows off the outer layers of the star. This photon-driven mass loss precedes the final coalescence. The presence of this eject material impedes the escape of gamma-rays created via neutrino interactions. If an e(+) - e(-) fireball, created in the late stages of coalescence, were loaded with (or surrounded by) material with the mean column density of the ejecta, it could not be an efficient source of gamma-rays. Models for cosmologically distant gamma-rays burst that involve neutron stars must therefore be anisotropic, so that the fireball expands preferentially in directions where the column density of previously blown-off material is far below the spherically averaged value which we have calculated. Some possible 'scenarios' along these lines are briefly discussed.

Meszaros, P.↗

New estimates of oceanic tidal energy dissipation from satellite altimetry

As a novel approach to computing the flux of tidal power into shelf areas, tidal maps of an oceanic area near the Patagonian Shelf are derived from 11 months of altimetry records from the GEOSAT satellite. Power fluxes are computed from the maps through Laplace's tidal equations. Flux vectors for M2 clearly show a convergence on to the southern portion of the shelf sea and their total is nearly twice the loss computed by Miller for that area. A decrease of 'quality factor' with frequency from M2 to S2 is in keeping with Webb's hypothesis of shelf resonances at frequencies a little higher than the tidal band.

Cartwright, David E.↗

Decaying Orbit of the Hot Jupiter WASP-12b: Confirmation with TESS Observations

Theory suggests that the orbits of some close-in giant planets should decay due to tidal interactions with their host stars. To date, WASP-12b is the only hot Jupiter reported to have a decaying orbit, at a rate of 29 ± 2 ms yr{sup −1}. We analyzed data from NASA’s Transiting Exoplanet Survey Satellite (TESS) to verify that WASP-12b’s orbit is indeed changing. We find that the TESS transit and occultation data are consistent with a decaying orbit with an updated period of 1.091420090 ± 0.000000041 days and a decay rate of 32.53 ± 1.62 ms yr{sup −1}. We find an orbital decay timescale of τ=P/| P-dot |=2.90±0.14 Myr. If the observed decay results from tidal dissipation, the modified tidal quality factor is Q{sub ⋆}{sup ′}=1.39±0.15×10{sup 5}, which falls at the lower end of values derived for binary star systems and hot Jupiters. Our result highlights the power of space-based photometry for investigating the orbital evolution of short-period exoplanets.

47 OTHER INSTRUMENTATION↗

Physical Oceanography in the Solar System and Beyond

A key controller of a planet’s rotational evolution, and hence habitability, is tidal dissipation, which on Earth occurs primarily in the oceans. As the discovery of habitable exoplanets is a primary objective of exoplanet research, it is imperative that we understand how “exo-oceans” behave. Despite this importance, little research has investigated the physical oceanography of worlds other than Earth. This oversight has occurred even though the Earth science community has studied tidal flows in Earth’s oceans for over a century and developed sophisticated models that exquisitely match satellite altimetry data, e.g. Here, we present a) models of tidal effects on exoplanets to motivate the problem, b) the application of a physical oceanography model to a putative ancient Venus ocean, and c) the application of that model to an ensemble of “alternative Earths” with a range of continental configurations and seafloor properties. We find that oceanic tidal dissipation can span 5 orders of magnitude, revealing that simulating exo-oceans with Earth science tools will provide fundamental insight into exoplanet evolution and habitability

Physical Oceanography↗