Solution to the rotation of the elastic earth by method of rigid dynamics

Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches.     

Quaternions and the rotation of a rigid body

The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.     

On the free rotation of a rigid body

It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem.In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles.     

On the precession constant: Values and constraints on the dynamical ellipticity; link with Oppolzer terms and tilt-over-mode

The luni-solar precession, derived by theoretical considerations from the precession of the equator, is one of the most important parameters for computing not only precession but also nutations, due to its relation to the dynamical flattening. In this paper, we review the numerical values of this parameter, from the geodynamical point of view as well as the astronomical point of view, from the observational point of view as well as from the theoretical point of view. In particular, we point out a difference of about 1 percent between the global Earth dynamical flattening derived from the astronomical observations and the values derived from the different geophysical computations. The nutation amplitudes depend on the Earth dynamical flattening and this dependence is amplified by a resonance at an important normal mode, the Tilt-Over-Mode (TOM). Since the astronomical point of view as well as the geophysical one are confronted, we also take the opportunity to make the link between the TOM and the expressions of the nutations of the different axes which, in turn, are related with one another by the Oppolzer terms. Both, the Oppolzer terms and the TOM originate from a reference frame tilt effect. In writing the link between the nutational motions of the different axes, and so, in writing the Oppolzer terms, we also make the link with the precessional motion.     

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.     

Continental drift and rotation of the Earth

It is shown, based on certain simplifying assumptions, that an island the size of Greenland, drifting across the North Atlantic at 10 m/century produces a change in secular rotation T=0.01 s.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

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.     

Conservation laws and Liapounov stability of the free rotation of a rigid body

The paper derives the well known stabilities of free rotation of a rigid body about its principal axes of least and greatest moments of inertia directly from the constancy of the kinetic energy and of the square of the angular momentum. The resulting proof of Liapounov stability yields new quantitative measures of this stability. Involving only simple algebra, it depends on satisfying a weak sufficient condition that insures an unchanging sign of the main component of the angular velocity . The method cannot be used, however, to prove the well known instability of rotation about the intermediate axis.The quantitative results for the radii of the spheres in -space that occur in the Liapounov proof lead to a physical result that may be of interest. If the earth were truly a rigid body, rotating freely, the angular deviation of its instantaneous polar axis from the nearest principal axis could not increase from a given initial value by more than the factor 2.These same quantitative results for the radii of the Liapounov spheres in -space also lead to suflicient conditions for the rotational stability of almost spherical bodies of various shapes, prolate or oblate. They may be pertinent in designing spheres to be used in currently planned experiments to test general relativity by observing the rate of precession of a rotating sphere in orbit about the earth.The above results follow from restricted Liapounov stability alone. The last section contains the proof of general Liapounov stability.This paper was prepared under the sponsorship of the Electronics Research Center of the National Aeronautics and Space Administration through NASA Grant NGR 22-009-262.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

A core-mantle interaction in the rotation of the Earth

Effects of an interaction between the mantle and the core of the Earth on its rotational motion are investigated. Assuming that the Earth consists of a rigid mantle and a rigid core with a frictional coupling and a kind of inertial coupling between them, the equations of motion are derived, and they are solved in a close approximation. The solution gives the expressions for the precession, the nutation, the secular changes in the obliquity and the rotational speed, the polar motion and so on as functions of the magnitudes of these forces. A numerical estimation shows that the effect of the friction on the amplitude and phase of the nutation is small for a reasonable intensity of the friction while inertial coupling force has a decisive influence on the amplitude, and an appropriately chosen value of the latter force gives a nutation which closely agrees with observations. It is also indicated that this torque remarkably lessens the rates of the secular changes in the obliquity and the rotational speed. The possibility of a periodical change in the amplitude of the polar motion is suggested as a result of the interaction between the two consituents.     

Periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in the Kovalevskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré the existence of a periodic solution is established.     

On the planetary acceleration and the rotation of the Earth

We have developed a cosmological model for the Earth rotation and planetary acceleration that gives a good account (data) of the Earth astronomical parameters. These data can be compared with the ones obtained using space-base telescopes. The expansion of the universe has shown to have an impact on the rotation of planets, and in particular, the Earth. The expansion of the universe causes an acceleration that is exhibited by all planets.     

Relation between the Celestial Reference System and the Terrestrial Reference System of a rigid Earth

A relation between the Celestial Reference System (CRS) and the Terrestrial Reference System is established theoretically by solving the equations of motion of a rigid Earth under the influence of the Sun and the Moon up to the second order perturbation. The solutions include not only nutation including Oppolzer terms but also the right ascension of the dynamical departure point (DP), as well as the wobble matrix.We have found that the kinematical definition of the Non-Rotating Origin NRO (for which our term is DP) given by Capitaine, Guinot and Souchay (1987) is not entirely equivalent to that included in the solutions of the equations of motion but shows perturbation, in particular when this is taken on the instantaneous equator. Besides this serious fault, we feel little merit in taking the DP as reference: (1) Unnecessary spurious mixed secular terms appear which come from the geometrical configuration that the DP leaves far and far from the ecliptic. (2) the DP moves secularly as well as oscillating with respect to space; this literally contradicts the term NRO, or is at least misleading. (3) It does not free us from the precession uncertainty to adopt DP as reference, since we cannot avoid virtual proper motions in terms of the current CRS. (4) No terms ignored hitherto are introduced, even if we take the DP properly chosen, i.e., on the equator of the celestial ephemeris pole. The transformation is only mathematical. There is no sufficient reason to take it instead of the equinox, which is observable in principle, as reference at the cost of the labor of changing all the textbooks, ephemerides, data and computer software now existing.     

A new class of periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré, the existence of a periodic solution is established.     

Influence of the YORP effect on rotation rates of near‐Earth asteroids

The distribution of near‐Earth asteroid (NEA) rotation rates differs considerably from the similar distribution of Main Belt asteroids (MBAs) by the presence of excesses of fast and slow rotators, which are not observed or not so prominent in the distribution for MBAs. Among possible reasons for the difference, there can be influence of solar radiation on spin rate of small NEAs, the so‐called “YORP effect,” which appears due to reflection, absorption, and IR re‐emission of the sunlight by an irregularly shaped rotating asteroid. It is known that the YORP‐effect action strongly depends on the amount of solar energy obtained by the body (insolation), its size, and albedo. The analysis of observation data has shown that: (1) the mean diameter of NEAs decreases from the middle of the distribution to its ends, that is, the excesses of slow rotators (ω ≤ 2 rev day^?1) and fast rotators (ω ≥ 8 rev day^?1) are composed of smaller NEAs than in the middle of the distribution; (2) NEAs of both excesses are in the orbits where their insolation is about 8–10% larger than that of NEAs in the middle of the distribution; and (3) the objects in both excesses have a little lower albedo on average than that of objects in the middle of the distribution. All these results qualitatively agree well with the YORP‐effect action and may be considered as independent arguments in favor of it.     

Determining the pole and the siderial rotation period of a conic artificial Earth satellite from mirror flash moments

We used the mirror-cone model to derive formulas and construct an algorithm for determining the pole position and the siderial period of the rotation of an artificial Earth satellite about its center of inertia. The apex angle of cone and the precession angle are determined as well. An algorithm is also constructed for calculating a model mirror flash series at fixed satellite rotation parameters.     

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

On the solutions of the Earth rotation parameters from classical observations of time and latitude

We compare the Earth's rotation parameters claculated from various input sets and conclude 1) A reference system comparable to the BIH sustem can be set up using just a few high-precision, evenly-distributed instruments. 2) Chinese instruments for time and latitude determination play an important role in the setting up and maintenance of a global reference system. 3) There seem to be no systematic differences, of an annual or a semi-annual character between the observations by the classical methods and the newer techniques. The difference BIH (1979) – BIH (1968) is probably what has remained of the station errors when the BIH (1968) system was set up. 4) It is possible that some unknown common source of error may exist over a large geographical region, hence, to set up a good reference system, the observing insrtuments should be distributed as evenly as possible over the globe.