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.
Highlights
The forces acting on an artificial satellite are responsible for its motion as well as its deviation from the desired orbit
When the exact and numerical solutions were superimposed, they exactly matched each other for the entire orbit as shown in figure 5 below. This is supported by the error analysis results which showed that the maximum difference between the exact and numerical solution for the entire orbit was 0.079%
This is the maximum relative error for all the iterations for the entire orbit. This shows that the numerical model developed closely matched the exact solution
Summary
The forces acting on an artificial satellite are responsible for its motion as well as its deviation from the desired orbit This deviation is referred to as perturbation. Various approaches have been used to develop numerical models of satellite orbit They include; Euler’s method [9, 10], Runge-Kutta method [10, 11], power series methods [12] and Lagrangian methods [13, 14] among others. The Euler, Runge-Kutta and power series methods are based on, or derived from, infinite series method It follows that their accuracy depends on the truncation, that is, the larger the number of terms the closer the method to the exact solution.
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