A Hamiltonian theory for an elastic earth: Elastic energy of deformation

In this paper we study only the perturbation due to the deformation of the elastic mantle by a tidal body force. In a previous publication (Getino and Ferrándiz, 1989a) we defined two canonical systems of variables - we gave them the names ofelastic variables of Euler and Andoyer respectively. Next, using them, we obtained the canonical expression of rotational kinetic energy. In the present paper, using the same variables, we build up the elastic energy which is produced by the deformation of the elastic mantle. We show that the three termsm = 0, 1, 2 corresponding to the second order of the development in spherical harmonics of the perturbing potential, a tidal potential, are of the same order of magnitude. In addition, the numerical integration for a particular Earth Model (Takeuchi's Model 2) is performed, with the aim of obtaining a numerical estimate of the coefficients which intervene in both this energy and the previously mentioned kinetic energy.     

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.     

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.     

New canonical variables for orbital and rotational motions

A new canonical transformation of freedom two was found. By using this, we derived three new sets of canonical variables for the orbital motion and two for the rotational motion. New canonical variables have clear physical meanings and remain well-defined in the case when the classical sets become ill-defined, for example, when the eccentricity and/or the inclination is small for the elliptic orbital motion.     

The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

The classical Poisson equations of rotational motion are used to study the attitude motions of an Earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate Earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the Earth.A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Longterm solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.This paper represents a part of the author's Ph. D. dissertation for the Mathematics Department, Auburn University.     

Landing of Mars or Ceres on Earth (new model of our solar system)

Motivated by the recent proposals of A. Abian, we investigate in this paper the second order perturbations up to the second degree in eccentricity-inclination of a new model of our solar system after the landing of Mars or Ceres on the Earth. The theory is established through the HoriLie technique and in terms of Jacobi coordinates and Poincare canonical variables.     

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.     

The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth

Kubo (Celest Mech Dyn Astron 110:143–168, 2011) investigated the kinematical structure of the perturbation in the rotation of the elastic Earth due to the deformation caused by the outer bodies. In that paper, while the mechanism for the perturbation of the figure axis was made clear, that for the rotational axis was not shown explicitly. In the present study, following the same method, the structure of the perturbation of the rotational axis is investigated. This perturbation consists of the direct perturbation and the convective perturbation. First the direct perturbation is shown to be (A − C)/A times as large as that of the figure axis, coinciding with the analytical expressions obtained in preceding studies by other authors. As for the convective perturbation, which appears only in the perturbation of the rotational axis but not in that of the figure axis, it is shown to be (A − C)/A times the angular separation between the original figure axis and the induced figure axis produced by the elastic deformation, A and C being the principal moments of inertia of the Earth. If the perturbing bodies are motionless, the conclusion of Kubo (Celest Mech Dyn Astron 105:261–274, 2009) holds strictly, i.e. the sum of the direct and the convective perturbations of the rotational axis coincides with the perturbation of the figure axis.     

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Kinematical modeling of the Earth rotation,focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.     

Effects of the Triaxiality on the Rotation of Celestial Bodies: Application to the Earth,Mars and Eros

In this paper we discuss the influence of the triaxiality of a celestialbody on its free rotation, i.e. in absence of any external gravitationalperturbation. We compare the results obtained through two different analytical formalisms, one established from Andoyer variables by usingHamiltonian theory, the other one from Euler's variables by usingLagrangian equations. We also give a very accurate formulation of thepolar motion (polhody) in the case of a small amplitude of this motion.Then, we carry out a numerical integration of the problem, with aRunge–Kutta–Felberg algorithm, and for the two kinds of methods above, that we apply to three different celestial bodies considered as rigid : the Earth, Mars, and Eros. The reason of this choice is that each of this body corresponds to a more or less triaxial shape.In the case of the Earth and Mars we show the good agreement betweenanalytical and numerical determinations of the polar motion, and theamplitude of the effect related to the triaxial shape of the body, whichis far from being negligible, with some influence on the polhody of theorder of 10 cm for the Earth, and 1 m for Mars. In the case of Eros, weuse recent output data given by the NEAR probe, to determine in detailthe nature of its free rotational motion, characterized by the presence ofimportant oscillations for the Euler angles due to the particularly largetriaxial shape of the asteroid.     

An averaging method to study the motion of lunar artificial satellites I: Disturbing Function

We describe a semi-analytical averaging method aimed at the computation of the motion of an artificial satellite of the Moon. In this paper, the first of the two part study, we expand the disturbing function with respect to the small parameters. In particular, a semi-analytic theory of the motion of the Moon around the Earth and the libration of the lunar equatorial plane using different reference frames are introduced. The second part of this article shows that the choice of the canonical Poincaré variables lead to equations in closed form without singularities in e = 0 or I = 0. We introduce new expressions that are sufficiently compact to be used for the study of any artificial satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

The 3:2 spin-orbit resonant motion of Mercury

Our purpose is to build a model of rotation for a rigid Mercury, involving the planetary perturbations and the non-spherical shape of the planet. The approach is purely analytical, based on Hamiltonian formalism; we start with a first-order basic averaged resonant potential (including J ₂ and C ₂₂, and the first powers of the eccentricity and the inclination of Mercury). With this kernel model, we identify the present equilibrium of Mercury; we introduce local canonical variables, describing the motion around this 3:2 resonance. We perform a canonical untangling transformation, to generate three sets of action-angle variables, and identify the three basic frequencies associated to this motion. We show how to reintroduce the short-periodic terms, lost in the averaging process, thanks to the Lie generator; we also comment about the damping effects and the planetary perturbations. At any point of the development, we use the model SONYR to compare and check our calculations.     

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.     

The motion of a symmetric gyrostat around the center of mass in a circular orbit in the presence of viscoelastic bars and disc

In the present work we consider asymmetric gyrostat which has a homogeneous viscoelastic disc and two bars attached to it. Furthermore, the gyrostat has a rotor oriented inside it such that the rotor is statically and dynamically balanced. This sytem has a rotational motion around its center of mass in a circular orbit under a central gravitational field. Bending vibrations of the bars and the disc are accompanied by dissipation of energy, which is the cause of the evolution of the system's rotational motion. Using the method of separation of motion and averaging, the approximate equations describing the evolution of rotational motion in terms of Andoyer canonical variables are obtained. The stationary motions for the system are deduced, together with the conditions of its stability.     

Satellite vibration-rotation motions studied via canonical transformations

Coupled vibration-rotation motion of a satellite is considered using a perturbation theory based on the Lie transformation method. Short-period oscillating terms are removed from the Hamiltonian function. The transformed damping forces directly affect rotational variables which were not directly influenced in the original variables. Motions and stability are more easily studied in the new variables. A dual-spin spacecraft model is used as an example; results for the usual nonresonant case are identical with the energy-sink method. Resonance cases produce a wealth of new dynamical phenomena. This canonical method extends and unifies various approximation methods in attitude dynamics.     

Study of the terms of coupling between rotational and translational motions

The equations of the translational-rotational motion of an Earth's artificial satellite, taking into account only the gravitational attraction, can be put in canonical form using the Delaunay and Andoyer variables. Terms due to the oblateness of the Earth will be considered, as well as, different moments of inertia of the satellite, when it is very close to another. Hori's Method is applied to analyse the equations and the solution is presented up to the first order.     

Hamilton's equations of motion for non-conservative systems

This paper deals with the Hamilton equations of motion and non conservative forces. The paper will show how the Hamilton formalism may be expanded so that the auxiliary equations for any problem may be found in any set of canonical variables, regardless of the nature of the forces involved. Although the expansion does not bring us closer to an analytical solution of the problem, it's simplicity makes it worth noticing.The starting point is a conservative system (for instance a satellite orbiting an oblate planet) with a known Hamiltonian (K) and canonical variables {Q, P}. This system is placed under influence of a non-conservative force (for instance drag-force). The idea is then to use, as far as possible, the same definitions used in the conservative problem, in the process of finding the auxiliary equations for the perturbed system.     

On the radial intermediaries and the time transformation in satellite theory

Taking advantage of the radial intermediaries and the regularization and linearization methods, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (, , ,R, ¡,N).The variable is eliminated in the first order of the Hamiltonian by applying Deprit's method. Then, the elimination of the perigee is carried out by another canonical transformation. As a consequence, a new radial intermediary, which contains all theJ _2n(n1) harmonics, is given. A comparison with the previous radial intermediaries of Cid and Lahulla, Deprit and Alfriend and Coffey is made.Finally, a regularizing transformation which allows us to linearize part of the radial intermediary is proposed, and an analytical study of this process is presented.     

A computer aided analysis of the Sitnikov Problem

We deal with the problem of a zero mass body oscillating perpendicular to a plane in which two heavy bodies of equal mass orbit each other on Keplerian ellipses. The zero mass body intersects the primaries plane at the systems barycenter. This problem is commonly known as theSitnikov Problem. In this work we are looking for a first integral related to the oscillatory motion of the zero mass body. This is done by first expressing the equation of motion by a second order polynomial differential equation using a Chebyshev approximation techniques. Next we search for an autonomous mapping of the canonical variables over one period of the primaries. For that we discretize the time dependent coefficient functions in a certain number of Dirac Delta Functions and we concatenate the elementary mappings related to the single Delta Function Pulses. Finally for the so obtained polynomial mapping we look for an integral also in polynomial form. The invariant curves in the two dimensional phase space of the canonical variables are investigated as function of the primaries eccentricity and their initial phase. In addition we present a detailed analysis of the linearized Sitnikov Problem which is valid for infinitesimally small oscillation amplitudes of the zero mass body. All computations are performed automatically by the FORTRAN program SALOME which has been designed for stability considerations in high energy particle accelerators.