Forced Nutation for Rigid Earth Model with Different Theories

Received: 13 August 2019     Accepted: 10 September 2019     Published: 23 September 2019
Abstract

Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around the Sun through the contribution of triaxial symmetry of the Earth. The formulation of the theory will be formed twice times. Firstly, deduce the tidal affect’s forces by Luni - Solar attraction coupling with the Earth’s geopotential force. Secondly, through the formulation, we will neglect the coupling between the different effects (the geopotential Earth force effect and the Luni - Solar attraction force), so, we will find the transform of the Hamiltonian for each force separately. The analytical solution for the formulated Hamiltonian will be derived for the two cases by using perturbation technique of Lie - Hori series. Once can get the analytical solution by getting the generation function, we will derive the nutation series analytically and numerically for each case and conclude over the results.

Keywords

Rotation of the Earth, Forced Nutation, Celestial Mechanics

References
 [1] Munck, W, H. and Macdonald, G. J. “The Rotation of the Earth’’. Cambridge (1960). [2] Getino, J. and ferrandiz, J. M., A Hamiltonian theory for an elastic earth: Canonical variables and kinetic energy, Celest. Mech. 49, 303 (1990). [3] Getino, J. and ferrandiz, J. M., A Hamiltonian theory for an elastic earth: Elastic energy of deformation, Celest. Mech. 51, 17 (1991a). [4] Getino, J. and ferrandiz, J. M., A Hamiltonian theory for an elastic earth: First order analytical integration, Celest. Mech. 51, 35 (1991b). [5] Getino, J. and ferrandiz, J. M., A Hamiltonian theory for an elastic earth: Secular rotational acceleration, Celest. Mech. 52, 381 (1991c). [6] Getino, J. and ferrandiz, J. M., On the effect of the mantle elasticity on the earth's rotation, Celest. Mech. 61, 117 (1995). [7] Sauchay, J. and Kinoshita, H., Comparison of new nutation series with numerical integration, Celest. Mech. 52, 45 (1991). [8] Ferrandiz, J. M.; Navarro, J. F., Escapa, A.; and Getino, J., Earth’s Rotation: A Challenging Problem in Mathematics and Physics, Pure Appl. Geophys. 172, 57–74 (2015). [9] Selim, H. H., Forced Nutation for The Rigid Earth Model At The First Order, NRIAG Journal of Astronomy and Geophysics, Special Issue, PP. 275-290 (2004). [10] Selim, H. H., Hamiltonian of A Second Order Two- Layer Earth Model, Journal of The Korean Astronomical Society, 40, 49 (2007). [11] Capitaine, N.; Mathews, P. M.; Dehant, V.; Wallace, P. T.; Lambert, S. B., On the IAU 2000/2006 precession–nutation and comparison with other models and VLBI observations, Celest Mech Dyn Astronomy, 103 (2), 179 (2009). [12] Malkin, Z., Joint analysis of celestial pole offset and free core nutation series‏, J. Geodesy, Volume 91, (7), 839 (2017). [13] Schindelegger M.; Einšpigel, D.; Salstein, D.; Böhm, J., The Global S1 Tide in Earth’s Nutation, Surv Geophys, 37 (3), pp. 643 (2016). [14] Hori, G., Theory of General Perturbation with Unspecified Canonical Variable, pub1. Astr. Soc. Japan, 18 (4), 287. (1966). [15] Kinoshita, H., Theory of the rotation of the rigid earth, Celest. Mec, h. 15, 2. (1977). [16] Getino, J., Forced nutations of a rigid mantle-liquid core Earth model, Geophysics J. Int., 122, 803-814, (2007). [17] Allen’s Astrophysical Quantities, fourth addition, Ed. Arthur N. Cox, Springer (2000).
• APA Style

Mohamed Soliman, Hadia Hassan Selim, Inal Adham Hassan. (2019). Forced Nutation for Rigid Earth Model with Different Theories. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 85-96. https://doi.org/10.11648/j.ijamtp.20190503.16

ACS Style

Mohamed Soliman; Hadia Hassan Selim; Inal Adham Hassan. Forced Nutation for Rigid Earth Model with Different Theories. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 85-96. doi: 10.11648/j.ijamtp.20190503.16

AMA Style

Mohamed Soliman, Hadia Hassan Selim, Inal Adham Hassan. Forced Nutation for Rigid Earth Model with Different Theories. Int J Appl Math Theor Phys. 2019;5(3):85-96. doi: 10.11648/j.ijamtp.20190503.16

• ```@article{10.11648/j.ijamtp.20190503.16,
author = {Mohamed Soliman and Hadia Hassan Selim and Inal Adham Hassan},
title = {Forced Nutation for Rigid Earth Model with Different Theories},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {5},
number = {3},
pages = {85-96},
doi = {10.11648/j.ijamtp.20190503.16},
url = {https://doi.org/10.11648/j.ijamtp.20190503.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.16},
abstract = {Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around the Sun through the contribution of triaxial symmetry of the Earth. The formulation of the theory will be formed twice times. Firstly, deduce the tidal affect’s forces by Luni - Solar attraction coupling with the Earth’s geopotential force. Secondly, through the formulation, we will neglect the coupling between the different effects (the geopotential Earth force effect and the Luni - Solar attraction force), so, we will find the transform of the Hamiltonian for each force separately. The analytical solution for the formulated Hamiltonian will be derived for the two cases by using perturbation technique of Lie - Hori series. Once can get the analytical solution by getting the generation function, we will derive the nutation series analytically and numerically for each case and conclude over the results.},
year = {2019}
}
```
• ```TY  - JOUR
T1  - Forced Nutation for Rigid Earth Model with Different Theories
AU  - Mohamed Soliman
Y1  - 2019/09/23
PY  - 2019
N1  - https://doi.org/10.11648/j.ijamtp.20190503.16
DO  - 10.11648/j.ijamtp.20190503.16
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 85
EP  - 96
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20190503.16
AB  - Where Earth is not strictly rigid body but can responds to any effects that tend to its rotation and shape, we will explain, in the present paper, the goal which is to define the forced nutation for a rigid Earth model using two different theories. We will formulate a first order Hamiltonian of a deformable Earth for its rotational motion around the Sun through the contribution of triaxial symmetry of the Earth. The formulation of the theory will be formed twice times. Firstly, deduce the tidal affect’s forces by Luni - Solar attraction coupling with the Earth’s geopotential force. Secondly, through the formulation, we will neglect the coupling between the different effects (the geopotential Earth force effect and the Luni - Solar attraction force), so, we will find the transform of the Hamiltonian for each force separately. The analytical solution for the formulated Hamiltonian will be derived for the two cases by using perturbation technique of Lie - Hori series. Once can get the analytical solution by getting the generation function, we will derive the nutation series analytically and numerically for each case and conclude over the results.
VL  - 5
IS  - 3
ER  - ```
Author Information
• Department of Astronomy, National Research Institute of Astronomy and Geophysics, Helwan, Egypt

• Department of Astronomy, National Research Institute of Astronomy and Geophysics, Helwan, Egypt

• Department of Astronomy and Metrology, Al-Azhar University, Cairo, Egpyt

• Sections