A Hamiltonian theory for an elastic earth: Canonical variables and kinetic energy

This paper is the first of a set of four, in which we shall develop the first part of a project dedicated to elaborating a Hamiltonian theory for the rotational motion of a deformable Earth. Here we study only the perturbation due to the deformation of the elastic mantle by tidal body force. In the present paper, we define two canonical systems of variables—we give these variables the names of elastic variables of Euler and Andoyer respectively. Next, using them, we obtain the canonical expression of rotational kinetic energy, which is valid for any Earth model satisfying hypotheses as general as those established in Section 2.     

A Hamiltonian theory for an elastic earth: First order analytical integration

In this article an approximate analytical integration is performed of the Hamiltonian corresponding to the rotational motion of an Earth whose elastic mantle is deformed by rotation and lunisolar attraction, using Deprit's perturbation method for the first order. Besides the usual terms, this Hamiltonian includes the perturbation of the kinetic energy and the elastic energy produced with the deformation, as well as their causes, the tidal and the centrifugal potential; these new terms have already been studied for the tidal deformation in previous articles (Getino and Ferrándiz, 1990a, 1990b). The effects of the deformation due to the centrifugal potential are studied in this article, following the same method as that used for the tidal deformation. Numerical tables of the periodic perturbations corresponding to the nutation in obliquity and longitude are obtained. As for the secular effects, a theoretical value of 457 days is obtained for Chandler's period.     

Elastic energy of a deformable earth: General expression

This work is the first in the second part of a project dedicated to elaborating a Hamiltonian theory for the rotational motion of a deformable Earth. In the four works which make up the first part the basis of this theory is laid down, studying the effects produced when the Earth's elastic mantle is deformed by lunisolar attraction. More specifically, in Getino and Ferrándiz (1991), the elastic energy which is produced on the deformation of the Earth's mantle is studied, considering solely the second order in the development in spherical harmonics of the perturbing potential (tidal potential).The present article can be considered as an amplification of the above mentioned, obtaining, under the same hypotheses, but also very general, the general expression of the said elastic energy for any order of the development of the tidal potential. Although at first this expression, in its general form, is very complicated, the final result is extremely simple, and for the case n = 2, it coincides, obviously, with that already found by the above mentioned authors.     

Kinetic energy of a non-spherical elastic earth mantle with andoyer variables

Continuing the study of the rotation of a deformable Earth begun by Getino and Ferrandiz (1990, 1991a, 1991b, 1993, 1994) for an Earth model with an elastic spherical mantle, in this paper on one hand we deal with the effect of the ellipticity, and on the other hand, we include the toroidal solution of the displacement vector. Taking an axis symmetrical, slightly ellipsoidal Earth, the modification due to the ellipticity is introduced into the solution of the displacement vector for both spheroidal and toroidal modes, and, after defining the adequate variables, we give the canonical formulation of the corresponding increase in the kinetic energy.     

Love Numbers And Elastic Energy Of Deformation For Venus

Love numbers of second order of Venus are calculated with resolving fundamental differential equations of elastic body according to the parameters of the density and the elasticity of material by means of the PVM94-01 Venus model. Meanwhile, the elastic energy of deformation of second order due to the tidal perturbation of the Sun and that due to rotational centrifugal potential are also calculated. The values of Love number provide a basis for model of internal structure of Venus. The numerical calculation of the elastic energy of deformation gives a magnitude evaluation of the perturbation terms to the Hamiltonian expressions in the study of dynamics of the elastic Venus. This revised version was published online in July 2006 with corrections to the Cover Date.     

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally, we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond.     

The interior structure of Mercury and its core sulfur content

Depth-dependent interior structure models of Mercury are calculated for several plausible chemical compositions of the core and of the mantle. For those models, we compute the associated libration amplitude, obliquity, tidal deformation, and tidal changes in the external potential. In particular we study the relation between the interior structure parameters for five different mantle mineralogies and two different temperature profiles together with two extreme crust density values. We investigate the influence of the core light element concentration, temperature, and melting law on core state and inner and outer core size. We show that a sulfur concentration above 10 wt% is unlikely if the temperature at the core-mantle boundary is above 1850 K and the silicate shell at least 240 km thick. The interior models can only have an inner core if the sulfur weight fraction is below 5 wt% for core-mantle boundary temperature in the 1850-2200 K range. Within our modeling hypotheses, we show that with the expected precision on the moment of inertia the core size can be estimated to a precision of about 50 km and the core sulfur concentration with an error of about 2 wt%. This uncertainty can only be reduced when more information on the mantle mineralogy of Mercury becomes available. However, we show that the uncertainty on the core size estimation can be greatly reduced, to about 25 km, if tidal surface displacements and tidal variations in the external potential are considered.     

A comparison between two internal structure models of the Earth

In view of the elastic deformation of the Earth we performed the comparative study for the Earth's models 1066 A and PREM, calculated the static Love numbers from degree 2 to 30, and discussed the relative variations of the second degree Love numbers and their combinations due to the variation of the position of the core-mantle boundary, due to the redistribution of V _p, V _s, and in the lower mantle, and due to the possible rigidity in the outer core. From the above-mentioned discussions we recommended that the Standard Earth Model (SEM) should include two kinds of models—one is oceanless, and another has an oceanic surface. Finally, we calculated the astronomicgeodetic parameters, which are consistent with the primary constants in the IERS Standards, of the SEM.     

Thermal history of the Jovian satellite Io

The discovery of large volcanic eruptions on Io suggests that Io is one of the most geologically active planetary bodies. The energy source of this geologic activity is believed to be tidal heating induced by Jupiter. A number of thermal history calculations were done to investigate the effect of tidal heating on the thermal history of Io taking into account solid state convection and advective heat transfer. These simulations show that the total tidal heating energy in Io is almost equal to the advectively transferred heat, indicating that the observed heat flow from Io is nearly equal to the total tidal heating energy. Since total tidal heating energy is dependent on the radius of the liquid mantle and the internal dissipation factor (Q), the radius of the liquid mantle can be estimated for a given value of Q. Some reasonable thermal history models of Io were obtained using a model with Q ≈ 25–50 in which the magma source of Ionian volcanism is at a depth of 100–300 km. The models satisfy the heat flow data and the existence of a thick lithosphere. Using a model with Q = 25 and L = 300 km (thickness of the advective region) as the standard model (model II), we then studied the effect of convective heat transfer and the initial temperature distribution on the Ionian thermal history. In these calculations, the other parameters are the same as in the standard model (model II). These calculations show that although the temperature distribution in the central region reflects the difference in the efficiency of convective heat transfer and initial temperature distribution, the temperature distribution in the outer region does not changes appreciably.     

Third-order Uranus-Neptune theory

In this paper of the third order Uranus-Neptune planetary theory which is the third part of this work for the third order theory, we compute the Poisson brackets in the Lie series which is used to transform canonical variables. We apply Hori-Lie technique in this work and neglect all powers higher than the second in Poincaré variables H, K, P, Q. We restrict this work to the principal part of the disturbing function.     

Short-period fluctuation analysis of the new BIH LOD series

In this paper, we analyse the short-period tidal fluctuations in the new LOD series of ERP (1962.0 – 1982.0) reduced by Li Zhengxin, using the Marple algorithm. The Marple spectral analysis reveals sharp peaks at 13.652, 27.586 and 31.746 days. They are consistent with the tidal terms M_f, M_m and MS_m. Their amplitudes are estimated.

The elastic deformation parameter K/C is calculated at the fortnightly and monthly frequencies. A least squares solution gives 0.56 ± 0.21 and 0.89 ± 0.21 at these two frequencies. The difference between these two values is not significant.     

Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model

Improved differential equations of the rotation of the deformable Earth with the two-layer fluid core are developed. The equations describe both the precession-nutational motion and the axial rotation (i.e. variations of the Universal Time UT). Poincaré’s method of modeling the dynamical effects of the fluid core, and Sasao’s approach for calculating the tidal interaction between the core and mantle in terms of the dynamical Love number are generalized for the case of the two-layer fluid core. Some important perturbations ignored in the currently adopted theory of the Earth’s rotation are considered. In particular, these are the perturbing torques induced by redistribution of the density within the Earth due to the tidal deformations of the Earth and its core (including the effects of the dissipative cross interaction of the lunar tides with the Sun and the solar tides with the Moon). Perturbations of this kind could not be accounted for in the adopted Nutation IAU 2000, in which the tidal variations of the moments of inertia of the mantle and core are the only body tide effects taken into consideration. The equations explicitly depend on the three tidal phase lags δ, δ _c, δ _i responsible for dissipation of energy in the Earth as a whole, and in its external and inner cores, respectively. Apart from the tidal effects, the differential equations account for the non-tidal interaction between the mantle and external core near their boundary. The equations are presented in a simple close form suitable for numerical integration. Such integration has been carried out with subsequent fitting the constructed numerical theory to the VLBI-based Celestial Pole positions and variations of UT for the time span 1984–2005. Details of the fitting are given in the second part of this work presented as a separate paper (Krasinsky and Vasilyev 2006) hereafter referred to as Paper 2. The resulting Weighted Root Mean Square (WRMS) errors of the residuals dθ, sin θd for the angles of nutation θ and precession are 0.136 mas and 0.129 mas, respectively. They are significantly less than the corresponding values 0.172 and 0.165 mas for IAU 2000 theory. The WRMS error of the UT residuals is 18 ms.     

用Hamilton方法计算滞弹地球自转的潮汐变化

在日月引潮力势作用下地球产生弹性形变．地幔粘滞性子致这个形变对于引潮力滞后,成为引起地球自转长期减慢的原因之一．地幔滞弹性也使有效洛夫数k增加,并使自转变化的周期项位相滞后,即产生反常位相项．本文首先用Hamilton方法计算了地球的形变．然后考虑到地幔的滞弹性,计算了在日月引潮力作用下的地球自转长期减慢和滞弹性对周期（带谐）变化的影响．     

地球的弹性形变能

采用较完善的地球模型PREM，通过求解弹性体的基本运动方程，得到形变位移矢量，由此分别计算出因日月引潮力势和地球自转离心力势引起的地球的弹性形变能，它将为弹性地球运动的Hamilton表达式提供摄动项的量级估计，并将是理论上探讨地月系演化的一个重要方面。     

IERS1996规范中地球引力势模型和测量模型的改进

简要而系统地介绍ＩＥＲＳ１９９６规范采用的地球引力势模型和各种测量模型,着重叙述了其与ＩＥＲＳ１９９２标准相比所作的改进。规范用ＪＧＭ－３地球引力势模型取代ＧＥＭ－Ｔ３模型,在计算地球潮汐形变产生的附加势时展开到３阶,并考虑了地幔的滞弹效应。在测站位移的计算中,规范引入了３倍洛夫数,计及地幔的滞弹性,引入了计算冰后期回弹的ＩＣＥ－４Ｇ模型,列出了改正ＶＬＢＩ观测中天线形变改正的公式。关于地球自转和     

A rigorous Hamiltonian approach to the rotation of elastic bodies

When the problem of the rotation of a non-rigid body is studied, the usual procedure consists of adding perturbations to the Hamiltonian of the rigid solid. In some cases, as occurs with the centrifugal deformation, the new perturbations contains potentials which depend on the velocity, but usually one alter neither the definition of the canonical variables nor the method for obtaining the Hamiltonian. Although this procedure gives good estimates and its formulation is simpler, it is incorrect from a theoretical point of view.In this paper we rigorously develop a Hamiltonian formulation of the problem, considering potentials that depend on the velocity. Thus the differences between the two procedures are clearly shown, giving special emphasis to the case of the elastic Earth, for which we show that the differences obtained cannot be ignored within the accuracy limits at present required.     

Synchronization of perturbed non-linear Hamiltonians

We propose a new method based on Lie transformations for simplifying perturbed Hamiltonians in one degree of freedom. The method is most useful when the unperturbed part has solutions in non-elementary functions. A non-canonical Lie transformation is used to eliminate terms from the perturbation that are not of the same form as those in the main part. The system is thus transformed into a modified version of the principal part. In conjunction with a time transformation, the procedure synchronizes the motions of the perturbed system onto those of the unperturbed part.A specific algorithm is given for systems whose principal part consists of a kinetic energy plus an arbitrary potential which is polynomial in the coordinate; the perturbation applied to the principal part is a polynomial in the coordinate and possibly the momentum.We demonstrate the strategy by applying it in detail to a perturbed Duffing system. Our procedure allow us to avoid treating the system as a perturbed harmonic oscillator. In contrast to a canonical simplification, our method involves only polynomial manipulations in two variables. Only after the change of time do we start manipulating elliptic functions in an exhaustive discussion of the flows.     

The weak friction approximation and tidal evolution in close binary systems

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.     

Tidal effects in differentiated viscoelastic bodies: a numerical approach

The majority of confirmed terrestrial exoplanets orbits close to their host stars and their evolution was likely altered by tidal interaction. Nevertheless, due to their viscoelastic properties on the tidal frequencies, their response cannot be described exactly by standardly employed constant-lag models. We therefore introduce a tidal model based on the numerical evaluation of a continuum mechanics problem describing the deformation of viscoelastic (Maxwell or Andrade) planetary mantles subjected to external force. We apply the method on a model Earth-size planet orbiting a low-mass star and study the effect of the orbital eccentricity, the mantle viscosity and the chosen rheology on the tidal dissipation, the complex Love numbers and the tidal torque. The number of stable spin states (i.e., zero tidal torque) grows with increasing mantle viscosity, similarly to the analytical model of Correia et al. (Astron Astrophys 571:A50, 2014) for homogeneous bodies. This behavior is only slightly influenced by the rheology used. Similarly, the Love numbers do not distinctly depend on the considered rheological model. The increase in viscosity affects the amplitude of their variations. The tidal heating described by the Maxwell rheology attains local minima associated with low spin-orbit resonances, with depth and shape depending on both the eccentricity and the viscosity. For the Andrade rheology, the minima at low resonances are very shallow and the tidal heating for all viscosities resembles a “fluid limit.” The tidal heating is the quantity influenced the most by the rheology, having thus possible impact on the internal thermal evolution.