Earth’s Rotation: A Challenging Problem in Mathematics and Physics

A suitable knowledge of the orientation and motion of the Earth in space is a common need in various fields. That knowledge has been ever necessary to carry out astronomical observations, but with the advent of the space age, it became essential for making observations of satellites and predicting and determining their orbits, and for observing the Earth from space as well. Given the relevant role it plays in Space Geodesy, Earth rotation is considered as one of the three pillars of Geodesy, the other two being geometry and gravity. Besides, research on Earth rotation has fostered advances in many fields, such as Mathematics, Astronomy and Geophysics, for centuries. One remarkable feature of the problem is in the extreme requirements of accuracy that must be fulfilled in the near future, about a millimetre on the tangent plane to the planet surface, roughly speaking. That challenges all of the theories that have been devised and used to-date; the paper makes a short review of some of the most relevant methods, which can be envisaged as milestones in Earth rotation research, emphasizing the Hamiltonian approach developed by the authors. Some contemporary problems are presented, as well as the main lines of future research prospected by the International Astronomical Union/International Association of Geodesy Joint Working Group on Theory of Earth Rotation, created in 2013.

[Excerpt from the poem "Winter"], undated

One leaf containing a handwritten copy of a section of the poem "Winter" by Scottish poet James Thomson (1700-1748). The excerpt begins "'Tis done! dread Winter spreads his latest Glooms," and ends, "And one unbounded Spring encircle all."

Ueber die Bestimmung und Verbesserung der Bahnen von Himmelskörpern nach drei Beobachtungen.

Mode of access: Internet.

The Nimbus II data catalog /

v.1. 15 May through 30 June 1966.--v.2. 1 July through 31 July 1966.--v.3. 1 August through 31 August 1966 (Orbits 1035-1447)--v.4. 1 September through 30 September 1966 (Orbits 1448-1846)--v.5. 1 October through 15 November 1966 (Orbits 1847-2458)

The Nimbus 6 data catalog /

v.1. 12 June through 31 August 1975, data orbits 1 through 1082.--v.2. 1 September 1975 through 31 October 1975, data orbits 1083 through 1900.--v.3. 1 November 1975 through 31 December 1975, data orbits 1901 through 2717.--v.4. 1 Jan 1976 through 29 February 1976, data orbits 2718 through 3521.--v.5. 1 March 1976 through 30 April 1976, data orbits 3522 through 4338.--v.6. 1 May 1976 throuth 30 June 1976, data orbits 4339 through 5155.--v.7. 1 July 1976 through 31 August 1976, data orbits 5156 through 5985.--v.8. 1 September 1976 through 31 October 1976, data orbits 5986 through 6802.--v.9. 1 November 1976 through 31 December 1976, data orbits 6803 through 7619.--v.10. 1 January 1977 through 28 February 1977, data orbits 7620 through 8409.--v.11. 1 March 1977 through 30 April 1977, data orbits 8410 through 9226.--v. 12. 1 May through 30 June 1977, data orbits 9227 through 10043.

Determinación de la órbita del planeta (796) Sarita.

At head of title: Universidad Nacional de La Plata. Observatorio Astronómico.

Democedes an interlude in one act.

"Performed with unbounded applause in Annapolis, (Md.) on the 16th of August, 1827, by Messrs. Mestayer & Company."

Physical anthropology of the Lenape or Delawares, and of the eastern Indians in general,

Contributions from the Museum of the American Indian, Heye Foundation, vol. 3.

Theorie der Bewegung der Himmelskörper : welche in Kegelschnitten die Sonne umlaufen /

Added t. p. in Latin: Theoria motus corporum coelestium in sectionibus conicis solem ambientium.

Researches on the evolution of the stellar systems ...

v. 1. On the universality of the law of gravitation and on the orbits and general characteristics of binary stars.--v. 2. The capture theory of cosmical evolution, founded on dynamical principles and illustrated by phenomena observed in the spiral nebulae, the planetary system, the double and multiple stars and clusters and the star-clouds of the Milky way.

A comparison of astronomical and ballistic traditions in orbit computation /

"Air research and Development Command, Air Force Office of Scientific Research, Mechanics Division. Contract no. AF 49(638) - 498."

Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations. (c) 2006 Elsevier Ltd. All rights reserved.

Analysis of chaotic instabilities in a rotating body with internal energy dissipation

Melnikov's method is used to analytically predict the onset of chaotic instability in a rotating body with internal energy dissipation. The model has been found to exhibit chaotic instability when a harmonic disturbance torque is applied to the system for a range of forcing amplitude and frequency. Such a model may be considered to be representative of the dynamical behavior of a number of physical systems such as a spinning spacecraft. In spacecraft, disturbance torques may arise under malfunction of the control system, from an unbalanced rotor, from vibrations in appendages or from orbital variations. Chaotic instabilities arising from such disturbances could introduce uncertainties and irregularities into the motion of the multibody system and consequently could have disastrous effects on its intended operation. A comprehensive stability analysis is performed and regions of nonlinear behavior are identified. Subsequently, the closed form analytical solution for the unperturbed system is obtained in order to identify homoclinic orbits. Melnikov's method is then applied on the system once transformed into Hamiltonian form. The resulting analytical criterion for the onset of chaotic instability is obtained in terms of critical system parameters. The sufficient criterion is shown to be a useful predictor of the phenomenon via comparisons with numerical results. Finally, for the purposes of providing a complete, self-contained investigation of this fundamental system, the control of chaotic instability is demonstated using Lyapunov's method.

On the chaotic instability of a nonsliding liquid-filled top with a small spheroidal base via Melnikov-Holmes-Marsden integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.