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1.
B. Levrard 《Icarus》2008,193(2):641-643
In a recent paper, Wisdom [Wisdom, J., 2008. Icarus, 193, 637-640] derived concise expressions for the rate of tidal dissipation in a synchronously rotating body for arbitrary orbital eccentricity and obliquity. He provided numerical evidence than the derived rate is always larger than in an asymptotic nonsynchronous rotation state at any obliquity and eccentricity. Here, I present a simple mathematical proof of this conclusion and show that this result still holds for any spin-orbit resonance.  相似文献   

2.
The aim of the present paper will be to derive an equation of dissipation of energy for a rotating body of arbitrary viscosity distorted by tides, which arise from the gravitational field of its companion in a close pair of such bodies.By a transformation of the fundamental equation of energy dissipation in terms of velocity of tidal deformation (Section 2), the dissipation function is constructed for a tidally-distorted body (Section 3). From this equation, the rate of dissipation of tidal energy is formulated for a nearly-spherical rotating body distorted by second harmonic longitudinal tides (Section 4); the coefficients of viscosity (or the bulk modulus) are treated as arbitrary functions of spatial coordinates. Finally (Section 5), expressions for the total energy dissipation within the orbital cycle are given for axial rotation of the distorted body, provided its angular velocity is constant (for example, with the Keplerian angular velocity).Research financed in part by the Division of Scientific Research and Development of Ministry of Sciences and Culture of Greece.  相似文献   

3.
Several processes may produce global tectonic patterns on the surface of a planetary body. The stresses associated with distortions of biaxial figures due to despinning or reorientation were first calculated by Vening Meinesz [Vening Meinesz, F.A., 1947. Trans. Am. Geophys. Union 28 (1), 1-23]. We adopt a mathematically equivalent, but physically more meaningful treatment for distortions associated with rotation. The new approach allows us to find analytic solutions for the general case of stresses associated with distortions of biaxial or triaxial planetary figures. Distortions of biaxial figures may be driven by variations in rotation rate, rotation axis orientation, or the combination of both. Distortions of triaxial figures may be driven by the same mechanisms and/or variations in tidal axis orientation for tidally deformed satellites. While the magnitude of the resulting stresses depends on the adopted elastic and physical parameters, the expected tectonic pattern is independent of these parameters for these mechanisms. Reorientation of the rotation/tidal axis alone is expected to produce normal/thrust faulting provinces enclosing the initial rotation/tidal poles, and thrust/normal faulting provinces enclosing the final rotation/tidal poles. Reorientation of both the rotation and tidal axis results in a wide variety of tectonic patterns for different reorientation geometries. On Europa, the tidal axis reorientation which generally accompanies rotation axis reorientations may provide an alternative explanation for tectonic features that have been interpreted as evidence for non-synchronous rotation. The observed tectonic pattern on Enceladus is more easily explained by a large reorientation (∼90°) of the rotation axis, than by rotation rate variations.  相似文献   

4.
We study the tidal forcing, propagation and dissipation of linear inertial waves in a rotating fluid body. The intentionally simplified model involves a perfectly rigid core surrounded by a deep ocean consisting of a homogeneous incompressible fluid. Centrifugal effects are neglected, but the Coriolis force is considered in full, and dissipation occurs through viscous or frictional forces. The dissipation rate exhibits a complicated dependence on the tidal frequency and generally increases with the size of the core. In certain intervals of frequency, efficient dissipation is found to occur even for very small values of the coefficient of viscosity or friction. We discuss the results with reference to wave attractors, critical latitudes and other features of the propagation of inertial waves within the fluid, and comment on their relevance for tidal dissipation in planets and stars.  相似文献   

5.
Asteroids have a wide range of rotation states. While the majority spin a few times to several times each day in principal axis rotation, a small number spin so slowly that they have somehow managed to enter into a tumbling rotation state. Here we investigate whether the Yarkovsky-Radzievskii-O'Keefe-Paddack (YORP) thermal radiation effect could have produced these unusual spin states. To do this, we developed a Lie-Poisson integrator of the orbital and rotational motion of a model asteroid. Solar torques, YORP, and internal energy dissipation were included in our model. Using this code, we found that YORP can no longer drive the spin rates of bodies toward values infinitely close to zero. Instead, bodies losing too much rotation angular momentum fall into chaotic tumbling rotation states where the spin axis wanders randomly for some interval of time. Eventually, our model asteroids reach rotation states that approach regular motion of the spin axis in the body frame. An analytical model designed to describe this behavior does a good job of predicting how and when the onset of tumbling motion should take place. The question of whether a given asteroid will fall into a tumbling rotation state depends on the efficiency of its internal energy dissipation and on the precise way YORP modifies the spin rates of small bodies.  相似文献   

6.
To explain the formation of surface features on Europa, Enceladus, and other satellites, many authors have postulated the spatial localization of tidal heating within convective plumes. However, the concept that enhanced tidal heating can occur within a convective plume has not been rigorously tested. Most models of this phenomenon adopt a tidal heating with a temperature-dependence derived for an incompressible, homogeneous (zero-dimensional) Maxwell material, but it is unclear whether this formulation is relevant to the heterogeneous situation of a warm plume surrounded by cold ice. To determine whether concentrated dissipation can occur in convective plumes, we develop a two-dimensional model to compute the volumetric dissipation rate for an idealized, vertically oriented, isolated convective plume obeying a Maxwellian viscoelastic compressible rheology. We apply the model to the Europa and Enceladus ice shells, and we investigate the consequences for partial melting and resurfacing processes on these bodies. We find that the tidal heating is strongly temperature dependent in a convective ice plume and could produce elevated temperatures and local partial melting in the ice shells of Europa and Enceladus. Our calculation provides the first quantitative verification of the hypothesis by Sotin et al. [Sotin, C., Head, J.W., Tobie, G., 2002. Geophys. Res. Lett. 29. 74-1] and others that the tidal dissipation rate is a strong function of temperature inside a convective plume. On Europa, such localized heating could help allow the formation of domes and chaos terrains by convection. On Enceladus, localized tidal heating in a thermal plume could explain the concentrated activity at the south pole and its associated heat transport of 2-7 GW.  相似文献   

7.
We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.  相似文献   

8.
The tectonically and cryovolcanically resurfaced terrains of Ganymede attest to the satellite's turbulent geologic history. Yet, the ultimate cause of its geologic violence remains unknown. One plausible scenario suggests that the Galilean satellites passed through one or more Laplace-like resonances before evolving into the current Laplace resonance. Passage through such a resonance can excite Ganymede's eccentricity, leading to tidal dissipation within the ice shell. To evaluate the effects of resonance passage on Ganymede's thermal history we model the coupled orbital-thermal evolution of Ganymede both with and without passage through a Laplace-like resonance. In the absence of tidal dissipation, radiogenic heating alone is capable of creating large internal oceans within Ganymede if the ice grain size is 1 mm or greater. For larger grain sizes, oceans will exist into the present epoch. The inclusion of tidal dissipation significantly alters Ganymede's thermal history, and for some parameters (e.g. ice grain size, tidal Q of Jupiter) a thin ice shell (5 to 20 km) can be maintained throughout the period of resonance passage. The pulse of tidal heating that accompanies Laplace-like resonance capture can cause up to 2.5% volumetric expansion of the satellite and contemporaneous formation of near surface partial melt. The presence of a thin ice shell and high satellite orbital eccentricity would generate moderate diurnal tidal stresses in Ganymede's ice shell. Larger stresses result if the ice shell rotates non-synchronously. The combined effects of satellite expansion, its associated tensile stress, rapid formation of near surface partial melt, and tidal stress due to an eccentric orbit may be responsible for creating Ganymede's unique surface features.  相似文献   

9.
Since 20 years, a large population of close-in planets orbiting various classes of low-mass stars (from M-type to A-type stars) has been discovered. In such systems, the dissipation of the kinetic energy of tidal flows in the host star may modify its rotational evolution and shape the orbital architecture of the surrounding planetary system. In this context, recent observational and theoretical works demonstrated that the amplitude of this dissipation can vary over several orders of magnitude as a function of stellar mass, age and rotation. In addition, stellar spin-up occurring during the Pre-Main-Sequence (PMS) phase because of the contraction of stars and their spin-down because of the torque applied by magnetized stellar winds strongly impact angular momentum exchanges within star–planet systems. Therefore, it is now necessary to take into account the structural and rotational evolution of stars when studying the orbital evolution of close-in planets. At the same time, the presence of planets may modify the rotational dynamics of the host stars and as a consequence their evolution, magnetic activity and mixing. In this work, we present the first study of the dynamics of close-in planets of various masses orbiting low-mass stars (from \(0.6~M_\odot \) to \(1.2~M_\odot \)) where we compute the simultaneous evolution of the star’s structure, rotation and tidal dissipation in its external convective envelope. We demonstrate that tidal friction due to the stellar dynamical tide, i.e. tidal inertial waves excited in the convection zone, can be larger by several orders of magnitude than the one of the equilibrium tide currently used in Celestial Mechanics, especially during the PMS phase. Moreover, because of this stronger tidal friction in the star, the orbital migration of the planet is now more pronounced and depends more on the stellar mass, rotation and age. This would very weakly affect the planets in the habitable zone because they are located at orbital distances such that stellar tide-induced migration happens on very long timescales. We also demonstrate that the rotational evolution of host stars is only weakly affected by the presence of planets except for massive companions.  相似文献   

10.
It is shown that the influences of the thermal and tidal effects on Mercury's libration are in equilibrium with the periods of rotation and revolution of Mercury locked in the 32 resonant state. The suggestion by Liu that the solar gravitational couple on the thermal bulges accelerates Mercury's rotation is investigated and the production of mechanical energy to balance the dissipation of the bodily tides is discussed. It is possible for Mercury to rotate with two bulges as a solar thermal engine; the tidal effect causes this engine to function and its maximum power is close to 1016 ergs per sec.  相似文献   

11.
This report is a review of Darwin’s classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.  相似文献   

12.
Iapetus' geophysics: Rotation rate, shape, and equatorial ridge   总被引：1，自引：0，他引：1  
Iapetus has preserved evidence that constrains the modeling of its geophysical history from the time of its accretion until now. The evidence is (a) its present 79.33-day rotation or spin rate, (b) its shape that corresponds to the equilibrium figure for a hydrostatic body rotating with a period of ∼16 h, and (c) its high, equatorial ridge, which is unique in the Solar System. This paper reports the results of an investigation into the coupling between Iapetus' thermal and orbital evolution for a wide range of conditions including the spatial distributions with time of composition, porosity, short-lived radioactive isotopes (SLRI), and temperature. The thermal model uses conductive heat transfer with temperature-dependent conductivity. Only models with a thick lithosphere and an interior viscosity in the range of about the water ice melting point can explain the observed shape. Short-lived radioactive isotopes provide the heat needed to decrease porosity in Iapetus' early history. This increases thermal conductivity and allows the development of the strong lithosphere that is required to preserve the 16-h rotational shape and the high vertical relief of the topography. Long-lived radioactive isotopes and SLRI raise internal temperatures high enough that significant tidal dissipation can start, and despin Iapetus to synchronous rotation. This occurred several hundred million years after Iapetus formed. The models also constrain the time when Iapetus formed because the successful models are critically dependent upon having just the right amount of heat added by SLRI decay in this early period. The amount of heat available from short-lived radioactivity is not a free parameter but is fixed by the time when Iapetus accreted, by the canonical concentration of 26Al, and, to a lesser extent, by the concentration of 60Fe. The needed amount of heat is available only if Iapetus accreted between 2.5 and 5.0 Myr after the formation of the calcium aluminum inclusions as found in meteorites. Models with these features allow us to explain Iapetus' present synchronous rotation, its fossil 16-h shape, and the context within which the equatorial ridge arose.  相似文献   

13.
Most transiting planets orbit very close to their parent star, causing strong tidal forces between the two bodies. Tidal interaction can modify the dynamics of the system through orbital alignment, circularization, synchronization and orbital decay by exchange of angular moment. Evidence for tidal circularization in close-in giant planet is well known. Here, we review the evidence for excess rotation of the parent stars due to the pull of tidal forces towards spin-orbit synchronization. We find suggestive empirical evidence for such a process in the present sample of transiting planetary systems. The corresponding angular momentum exchange would imply that some planets have spiralled towards their star by substantial amounts since the dissipation of the protoplanetary disc. We suggest that this could quantitatively account for the observed mass–period relation of close-in gas giants. We discuss how this scenario can be further tested and point out some consequences for theoretical studies of tidal interactions and for the detection and confirmation of transiting planets from radial velocity and photometric surveys.  相似文献   

14.
Each of the Galilean satellites, as well as most other satellites whose initial rotations have been substantially altered by tidal dissipation, has been widely assumed to rotate synchronously with its orbital mean motion. Such rotation would require a small permanent asymmetry in the mass distribution in order to overcome the small mean tidal torque. Since Io and Europa may be substantially fluid, they may not have the strenght to support the required permanent asymmetry. Thus, each may rotate at the unknown but slightly nonsynchronous rate that corresponds to zero mean tidal torque. This behaviour may be observable by Galileo spacecraft imaging. It may help explain the longitudinal variation of volcanism on Io and the cracking of Europa's crust.  相似文献   

15.
Hauke Hussmann  Tilman Spohn 《Icarus》2004,171(2):391-410
Coupled thermal-orbital evolution models of Europa and Io are presented. It is assumed that Io, Europa, and Ganymede evolve in the Laplace resonance and that tidal dissipation of orbital energy is an internal heat source for both Io and Europa. While dissipation in Io occurs in the mantle as in the mantle dissipation model of Segatz et al. (1988, Icarus 75, 187), two models for Europa are considered. In the first model dissipation occurs in the silicate mantle while in the second model dissipation occurs in the ice shell. In the latter model, ice shell melting and variations of the shell thickness above an ocean are explicitly included. The rheology of both the ice and the rock is cast in terms of a viscoelastic Maxwell rheology with viscosity and shear modulus depending on the average temperature of the dissipating layer. Heat transfer by convection is calculated using a parameterization for strongly temperature-dependent viscosity convection. Both models are consistent with the present orbital elements of Io, Europa, and Ganymede. It is shown that there may be phases of quasi-steady evolution with large or small dissipation rates (in comparison with radiogenic heating), phases with runaway heating or cooling and oscillatory phases during which the eccentricity and the tidal heating rate will oscillate. Europa's ice thickness varies between roughly 3 and 70 km (dissipation in the silicate layer) or 10 and 60 km (dissipation in the ice layer), suggesting that Europa's ocean existed for geological timescales. The variation in ice thickness, including both convective and purely conductive phases, may be reflected in the formation of different geological surface features on Europa. Both models suggest that at present Europa's ice thickness is several tens of km thick and is increasing, while the eccentricity decreases, implying that the satellites evolve out of resonance. Including lithospheric growth in the models makes it impossible to match the high heat flux constraint for Io. Other heat transfer processes than conduction through the lithosphere must be important for the present Io.  相似文献   

16.
The precession of the orbital plane in a close binary system can provide an important observational tool for investigating dynamical properties of the components. Tidal evolution will always tend to align the rotation axes perpendicular to the orbital plane, thereby eliminating precession. It is pointed out, however, that if observations indicate the existence of a circular orbit and synchronous rotation of the components-which is the outcome of tidal evolution-then precession may still be present, provided the interior of one of the components is, or recently has been, radiative, and is not strongly coupled to the surface layers (where tidal dissipation is greatest). The equations governing precession and nutation are derived in a concise form, and applied to the numerical study of two binary systems. The observational effects are also discussed. Finally, it is pointed out that precession may be present in a subclass of the X-ray binary systems, and its observational significance is briefly discussed.  相似文献   

17.
The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the Darwin–Kaula theory for a nonlibrating celestial object. This necessitates derivation of formulae for the tidal torque and the tidal heating rate, that are applicable under libration. We derive the tidal spectrum for longitudinal forced libration with one and two main frequencies, generalisation to more main frequencies being straightforward. (By main frequencies we understand those emerging due to the triaxiality of the librating body.) Separately, we consider a case of free libration at one frequency (once again, generalisation to more frequencies being straightforward). We also calculate the tidal torque. This torque provides correction to the triaxiality-caused physical libration. Our theory is not self-consistent: we assume that the tidal torque is much smaller than the permanent-triaxiality-caused torque, so the additional libration due to tides is much weaker than the main libration due to the permanent triaxiality. Finally, we calculate the tidal dissipation rate in a body experiencing forced libration at the main mode, or free libration at one frequency, or superimposed forced and free librations.  相似文献   

18.
The aim of this paper is to study the dynamical problem of tidal friction in a binary system consisting of deformable components, with the restriction that the angle of lag or advance of the tidal distortion with respect to the direction of the disturbing companion is small. The fractional distortion of the bodies due to rotation and tidal interaction is also treated as a first-order small quantity, and terms up to the fourth harmonic in the tidal potential are retained. In this linear approximation, the time-dependent tidal potential can be Fourier decomposed into a spectrum of simple harmonic terms, each of which is responsible for raising a partial wave in the body; each such partial wave can then be treated independently of the others. This is the method first employed by Darwin.In Section 2, it is assumed that the phase lag in the response of the body (due to dissipation of kinetic energy of deformation) is proportional to the forcing frequency, which is justified for small amplitude oscillations of a viscous fluid or visco-elastic body. A simple expression is then obtained for the potential function for the distortion in terms of the disturbing potential and the structure of the body.In Section 3, the distortion potential function is employed in deriving the componentsR, S andW of the disturbing force which are then substituted in the Gaussian form of the equations for variation of the elements. In Section 4, the Eulerian equations for motion of deformable bodies are derived, using the so-called mean axes of the body as the rotating axes of reference. In Section 5, it is shown that the dynamical effects of rotational distortion occur on a much shorter time scale than those arising from tidal friction, which allows one to consider the two phenomena as acting independently of one another. The collected set of Gaussian (orbital) and Eulerian (body) equations is re-written in terms of dimensionless variables for the tidal friction case, and the stability of the system is examined on the basis of these equations.In Section 6, the tidal friction equations are integrated numerically for the close binary system AG Persei and for the Earth-Moon system. In the former, the integrations were started from a highly elliptical orbit and the system was found to relax into a circular orbit, with synchronous rotation perpendicular to the orbit. In the latter, the integrations were performed backwards in time from the present day, and it was found that the lunar orbit rapidly becomes highly elliptical at the time of closest approach, thus indicating a probable capture of the Moon by the Earth. This result is in agreement with that obtained by other investigators; however, it is shown that the detailed behaviour of the system at the time of capture, in particular the inclination of the lunar orbit to the ecliptic, depends critically on the chosen rate of dissipation in the Moon's interior. A simple argument is presented which allows an estimation for the mean viscosity of a fluid body from the known age of the system: for the components of AG per, the result is 2×1011 g cm–1 s–1, indicating that the stars must have possessed turbulent convective outer regions during some part of their tidal evolution, while for the Earth, the result, is 1.4×1012 g cm–1 s–1. It is shown that the angle of tidal lag in nonsynchronous close binary systems in general is expected to be extremely small, and not observationally detectable.  相似文献   

19.
Tidal dissipation has been suggested as the heat source for the south polar thermal anomaly on Enceladus. We find that under present-day conditions and assuming Maxwellian behavior, tidal dissipation is negligible in the silicate core. Dissipation may be significant in the ice shell if the shell is decoupled from the silicate core by a subsurface ocean. We have run a series of self-consistent convection and conduction models in 2D axisymmetric and 3D spherical geometry in which we include the spatially-variable tidal heat production. We find that in all cases, the shell removes more heat from the interior than can be produced in the core by radioactive decay, resulting in cooling of the interior and the freezing of any ocean. Under likely conditions, a 40-km thick ocean made of pure water would freeze solid on a ∼30 Ma timescale. An ocean containing other chemical components will have a lower freezing point, but even a water-ammonia eutectic composition will only prolong the freezing, not prevent it. If the eccentricity of Enceladus were higher (e?0.015) in the past, the increased dissipation in the ice shell may have been sufficient to maintain a liquid layer. We cannot therefore rule out the presence of a transient ocean, as a relic of an earlier era of greater heating. If the eccentricity is periodically pumped up, the ocean may have thickened and thinned on a similar timescale as the orbital evolution, provided the ocean never froze completely. We conclude that the current heat flux of Enceladus and any possible subsurface ocean is not in steady-state, and is the remnant of an epoch of higher eccentricity and tidal dissipation.  相似文献   

20.
The origin of the Martian moons, Phobos and Deimos, is still an open issue: either they are asteroids captured by Mars or they formed in situ from a circum-Mars debris disk. The capture scenario mainly relies on the remote-sensing observations of their surfaces, which suggest that the moon material is similar to outer-belt asteroid material. This scenario, however, requires high tidal dissipation rates inside the moons to account for their current orbits around Mars. Although the in situ formation scenarios have not been studied in great details, no observational constraints argue against them. Little attention has been paid to the internal structure of the moons, yet it is pertinent for explaining their origin. The low density of the moons indicates that their interior contains significant amounts of porous material and/or water ice. The porous content is estimated to be in the range of 30?C60% of the volume for both moons. This high porosity enhances the tidal dissipation rate but not sufficiently to meet the requirement of the capture scenario. On the other hand, a large porosity is a natural consequence of re-accretion of debris at Mars?? orbit, thus providing support to the in situ formation scenarios. The low density also allows for abundant water ice inside the moons, which might significantly increase the tidal dissipation rate in their interiors, possibly to a sufficient level for the capture scenario. Precise measurements of the rotation and gravity field of the moons are needed to tightly constrain their internal structure in order to help answering the question of the origin.  相似文献   

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