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

S. M. El-Shaboury

S. M. El-Shaboury

Mathematics Dept. Faculty of Science, Ain Shams UniversityCairo, Egypt
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El-Shaboury, S. M.
, 
M.K. Ammar

M.K. Ammar

Mathematics Dept. Faculty of Science, Helwan UniversityCairo, Egypt
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Ammar, M.K.
 and 
W.M. Yousef

W.M. Yousef

Basic Science Dept, Canadian International College for EngineeringCairo, Egypt
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Yousef, W.M.
  
Oct 03, 2017
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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
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© 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

ECI and LVLH frames.
ECI and LVLH frames.

Fig. 2

Perturbed and unperturbed x-component of the deputy relative position with the Chief true anomaly.
Perturbed and unperturbed x-component of the deputy relative position with the Chief true anomaly.

Fig. 3

Perturbed and unperturbed y-component of the deputy relative position vs. Chief true anomaly.
Perturbed and unperturbed y-component of the deputy relative position vs. Chief true anomaly.

Fig. 4

Perturbed and unperturbed z-component of the deputy relative position vs. Chief true anomaly.
Perturbed and unperturbed z-component of the deputy relative position vs. Chief true anomaly.

Fig. 5

Deputy 3D parametric representation.
Deputy 3D parametric representation.

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

No.x(f)ℒ{g(f)}=G(s) ${\rm \mathscr{L}}\{g(f)\} =G(s)$
12 d (1+e cos f)2ds+2edss2+1 $\frac{2\, d}{s} +\frac{2\, e\, d\, s}{s^{2} +1}$
2x″s2X(s)−sx0 −x′0
3(4e cos f) x2e [X(s − i)+X(s+i)]
4(e cos f) x″e2(s−i)2X(s−i)+e2(s+i)2X(s+i)−ex0s−ex′0 $\frac{e}{2} (s-i)^{2} X(s-i)+\frac{e}{2} (s+i)^{2} X(s+i)-e\, x_{0} \, s-e\, x'_{0}$
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