Numerical Model of Perturbated Earth’s Satellite Orbit

Received: 14 October 2018     Accepted: 25 January 2019     Published: 15 February 2019
Abstract

The study aimed to develop a two-dimensional numerical model of a perturbed Earth’s satellite orbit under the influence of the Moon. The first step was to model, numerically, the Earth-satellite orbit. The interaction was assumed to be first order. The basis of the model was that for two-dimensional motion, influence in the radial direction does not affect the motion in the tangential direction and vice versa. Based on this, the satellite’s motion was decomposed into radial and tangential directions. The trajectory was segmented into time intervals and the curve swept over each interval was approximated as a straight line with the assumption that acceleration in each interval was constant. Equations of constant accelerated motion were used to describe the motion of the satellite over each interval. When the model results were compared with the exact solution, for an elliptical orbit, they matched perfectly well over the entire orbit with a maximum relative error of 0.079%. When it was tested for other orbits, circular, hyperbolic, etc., it retained all of them according to theoretical predictions. The model was then extended to incorporate the effects of the Moon by launching the satellite at quarter, half and three-quarter distance from Earth to Moon. A circular orbit was used to model the effects of the Moon. The acceleration results of the model were compared with theoretical predictions. The corresponding errors in the acceleration for the three positions of launch were 0.019% and 0.20%. This showed that this model is applicable for predicting perturbated satellite orbit and it can be applied with any extra force to describe perturbated orbit of the satellite. It can also be used to model the trajectory of projectile motion, of which the exact solution is incapable of generating. Since this model gives the speed of the satellite at any instant, it can be applied when the orbit needs to be changed as it can be used to compute the required new speed.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 1) DOI 10.11648/j.ijamtp.20190501.11 Page(s) 1-14 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2019. Published by Science Publishing Group
Keywords

Two-Dimensional, Numerical, Model, Perturbation, Three-Body, Satellite, Orbit

References
 [1] Dwidar, H. R., Prediction of satellite motion under the effects of the earth’s gravity, drag force and solar radiation pressure in terms of the ks-regularized variables. International Journal of Advanced Computer Science and Applications, 2014. 5 (5). [2] Foster, C., H. Hallam, and J. Mason Orbit Determination and Differential-Drag Control of Planet Labs Cubesat Constellations. 2015. [3] Friesen, L. J., et al., Analysis of orbital perturbations acting on objects in orbits near geosynchronous Earth orbit. Journal of Geophysical Research: Planets (1991–2012), 2012. 19 (3). [4] Prasad, P. R., et al., Mission Planning Challenges for Small Satellites International Journal of Computer Science and Information Technology and Security 2012. 2 (3): p. 553-560. [5] Sharma, B. K. Basic Mechanics of Planet-Satellite Interaction with special reference to Earth-Moon System. 2008. [6] Riaz, A., Development of an Orbit Propagator incorporating Perturbations for LEO Satellites. Journal of Space Technology, 2011. 1 (1): p. 5. [7] Diacu, F., The Solution of the n-body Problem. The Mathematical Intelligence, 1996. 18 (3): p. 66-70. [8] Senchyna, P. A (Less than Practical) Solution to the N-Body Problem. 2013. [9] Reinhardt, P. Many-Body Gravity Simulation Using Multivariable Numerical Integration. 2007. 6. [10] Krizek, M., Numerical Experience with the Three-body Problem. Journal of Computational and Applied Mathematics 1995. 63 (1995): p. 403-409. [11] Voesenek, C. J. Implementing a Fourth Order Runge-Kutta Method for Orbit Simulation. 2008. 3. [12] Broucke, R., Solution of the N-body Problem with Recurrent Power Series. Celestial Mechanics, 1971. 4 (1): p. 110-115. [13] Minesaki, Y., Lagrange Solutions to the Discrete-Time General Three-Body Problem. The Astronomical Journal, 2013. 145 (3): p. 9. [14] Abdel-Aziz, Y. A., A. M. Abdel-Hameed, and K. I. Khalil, A New Navigation Force Model for the Earth’s Albedo and Its Effects on the Orbital Motion of an Artificial Satellite. Applied Mathematics, 2011. 2: p. 801-807. [15] Serway, R. A. and J. W. Jewett, Physics for Scientists and Engineers with Modern Physics. ninth ed. 2014, Australia: Brooks/Cole CENGAGE Learning. 1484.
• APA Style

Saneliso Vuyo Makhanya, Wei-Hsi Liao. (2019). Numerical Model of Perturbated Earth’s Satellite Orbit. International Journal of Applied Mathematics and Theoretical Physics, 5(1), 1-14. https://doi.org/10.11648/j.ijamtp.20190501.11

ACS Style

Saneliso Vuyo Makhanya; Wei-Hsi Liao. Numerical Model of Perturbated Earth’s Satellite Orbit. Int. J. Appl. Math. Theor. Phys. 2019, 5(1), 1-14. doi: 10.11648/j.ijamtp.20190501.11

AMA Style

Saneliso Vuyo Makhanya, Wei-Hsi Liao. Numerical Model of Perturbated Earth’s Satellite Orbit. Int J Appl Math Theor Phys. 2019;5(1):1-14. doi: 10.11648/j.ijamtp.20190501.11

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title = {Numerical Model of Perturbated Earth’s Satellite Orbit},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {5},
number = {1},
pages = {1-14},
doi = {10.11648/j.ijamtp.20190501.11},
url = {https://doi.org/10.11648/j.ijamtp.20190501.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190501.11},
abstract = {The study aimed to develop a two-dimensional numerical model of a perturbed Earth’s satellite orbit under the influence of the Moon. The first step was to model, numerically, the Earth-satellite orbit. The interaction was assumed to be first order. The basis of the model was that for two-dimensional motion, influence in the radial direction does not affect the motion in the tangential direction and vice versa. Based on this, the satellite’s motion was decomposed into radial and tangential directions. The trajectory was segmented into time intervals and the curve swept over each interval was approximated as a straight line with the assumption that acceleration in each interval was constant. Equations of constant accelerated motion were used to describe the motion of the satellite over each interval. When the model results were compared with the exact solution, for an elliptical orbit, they matched perfectly well over the entire orbit with a maximum relative error of 0.079%. When it was tested for other orbits, circular, hyperbolic, etc., it retained all of them according to theoretical predictions. The model was then extended to incorporate the effects of the Moon by launching the satellite at quarter, half and three-quarter distance from Earth to Moon. A circular orbit was used to model the effects of the Moon. The acceleration results of the model were compared with theoretical predictions. The corresponding errors in the acceleration for the three positions of launch were 0.019% and 0.20%. This showed that this model is applicable for predicting perturbated satellite orbit and it can be applied with any extra force to describe perturbated orbit of the satellite. It can also be used to model the trajectory of projectile motion, of which the exact solution is incapable of generating. Since this model gives the speed of the satellite at any instant, it can be applied when the orbit needs to be changed as it can be used to compute the required new speed.},
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AB  - The study aimed to develop a two-dimensional numerical model of a perturbed Earth’s satellite orbit under the influence of the Moon. The first step was to model, numerically, the Earth-satellite orbit. The interaction was assumed to be first order. The basis of the model was that for two-dimensional motion, influence in the radial direction does not affect the motion in the tangential direction and vice versa. Based on this, the satellite’s motion was decomposed into radial and tangential directions. The trajectory was segmented into time intervals and the curve swept over each interval was approximated as a straight line with the assumption that acceleration in each interval was constant. Equations of constant accelerated motion were used to describe the motion of the satellite over each interval. When the model results were compared with the exact solution, for an elliptical orbit, they matched perfectly well over the entire orbit with a maximum relative error of 0.079%. When it was tested for other orbits, circular, hyperbolic, etc., it retained all of them according to theoretical predictions. The model was then extended to incorporate the effects of the Moon by launching the satellite at quarter, half and three-quarter distance from Earth to Moon. A circular orbit was used to model the effects of the Moon. The acceleration results of the model were compared with theoretical predictions. The corresponding errors in the acceleration for the three positions of launch were 0.019% and 0.20%. This showed that this model is applicable for predicting perturbated satellite orbit and it can be applied with any extra force to describe perturbated orbit of the satellite. It can also be used to model the trajectory of projectile motion, of which the exact solution is incapable of generating. Since this model gives the speed of the satellite at any instant, it can be applied when the orbit needs to be changed as it can be used to compute the required new speed.
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Author Information
• Faculty of Science and Engineering, University of Eswatini, P/Bag Kwaluseni, Eswatini

• Faculty of Science and Engineering, University of Eswatini, P/Bag Kwaluseni, Eswatini

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