<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This paper introduces a different approach to obtain the exact solution of the relative equations of motion of a deputy (follower) satellite with respect to a chief (leader) satellite that both rotate about central body (Earth) in elliptic orbits by using Laplace transformation. Moreover, the paper will take the perturbation due to the oblateness of the Earth into consideration and simulate this problem with numerical example showing the effect of the perturbation on the Keplerian motion. The solution of such equations in this work is represented in terms of the eccentricity of the chief orbit and its true anomaly as the independent variable.</p></abstract>

This paper introduces a different approach to obtain the exact solution of the relative equations of motion of a deputy (follower) satellite with respect to a chief (leader) satellite that both rotate about central body (Earth) in elliptic orbits by using Laplace transformation. Moreover, the paper will take the perturbation due to the oblateness of the Earth into consideration and simulate this problem with numerical example showing the effect of the perturbation on the Keplerian motion. The solution of such equations in this work is represented in terms of the eccentricity of the chief orbit and its true anomaly as the independent variable.

Analytical solutions of the relative orbital motion in unperturbed and in <italic>J</italic><sub>2</sub> - perturbed elliptic orbits

Canadian International College for Engineering

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Analytical solutions of the relative orbital motion in unperturbed and in  - perturbed elliptic orbits

This paper introduces a different approach to obtain the exact solution of the relative equations of motion of a deputy (follower) satellite with respect to a...

No.	x(f)	$ℒ {g (f)} = G (s)$ ${\rm \mathscr{L}}\{g(f)\} =G(s)$
1	2 d (1+e cos f)	$\frac{2 d}{s} + \frac{2 e d s}{s^{2} + 1}$ $\frac{2\, d}{s} +\frac{2\, e\, d\, s}{s^{2} +1}$
2	x″	s²X(s)−sx₀ −x′₀
3	(4e cos f) x	2e [X(s − i)+X(s+i)]
4	(e cos f) x″	$\frac{e}{2} {(s - i)}^{2} X (s - i) + \frac{e}{2} {(s + i)}^{2} X (s + i) - e x_{0} s - e {x^{'}}_{0}$ $\frac{e}{2} (s-i)^{2} X(s-i)+\frac{e}{2} (s+i)^{2} X(s+i)-e\, x_{0} \, s-e\, x'_{0}$

Analytical solutions of the relative orbital motion in unperturbed and in J₂ - perturbed elliptic orbits

Published Online: Oct 03, 2017

Page range: 403 - 414

Received: Apr 04, 2017

Accepted: Oct 03, 2017

DOI: https://doi.org/10.21042/AMNS.2017.2.00032

Keywords
Relative motion of two satellites, Tschauner Hempel equations, Perturbation, Laplace transformation

© 2017 S.M. El–Shaboury, M.K. Ammar, W.M. Yousef, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Laplace transformation on the relative equation of motion of the Deputy in x-direction

Analytical solutions of the relative orbital motion in unperturbed and in J2 - perturbed elliptic orbits

Published Online: Oct 03, 2017

Page range: 403 - 414

Received: Apr 04, 2017

Accepted: Oct 03, 2017

DOI: https://doi.org/10.21042/AMNS.2017.2.00032

KeywordsRelative motion of two satellites, Tschauner Hempel equations, Perturbation, Laplace transformation

© 2017 S.M. El–Shaboury, M.K. Ammar, W.M. Yousef, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Laplace transformation on the relative equation of motion of the Deputy in x-direction

Analytical solutions of the relative orbital motion in unperturbed and in J₂ - perturbed elliptic orbits

Keywords
Relative motion of two satellites, Tschauner Hempel equations, Perturbation, Laplace transformation