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Improved initialization conditions and single impulsive maneuvers for \(\hbox {J}_{2}\)-invariant relative orbits

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Abstract

The determination of the initial conditions for long-term bounded relative motion under natural perturbations is an important theme in satellite cluster flight. Considering the most significant perturbation of the geopotential, namely, the \(\hbox {J}_{2}\) term, many researchers have proposed \(\hbox {J}_{2}\)-mitigating initial conditions for satellite-bounded relative motion. To improve the existing \(\hbox {J}_{2}\)-invariant conditions, two new methods for finding the correction factor are presented in this paper. In these two methods, Method 1 is obtained by minimizing the possible maximum drift in the along-track relative motion. However, Method 2 is designed by nullifying the rates of change of the bounds of the relative motion. Then the geometric character, such as the self-intersection of the \(\hbox {J}_{2}\)-invariant relative orbits, is discussed. The conditions of 0, 1 and 2 (the possible maximum number) self-intersection points are also derived. Then, using Gauss’s equations of planetary motion, an analytical optimal single-impulsive maneuver is deduced to guarantee the secular bounded relative motion under \(\hbox {J}_{2}\), too. Some numerical simulations are performed to validate the corresponding theoretical predictions. The results demonstrate that the proposed methods enhance performance for achieving the bounded relative motion under \(\hbox {J}_{2}\) effects and can be used for future satellite cluster flight missions.

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Acknowledgments

This research was supported by the National Natural Science Foundation of China (11002076) and the Graduate Student Innovation Foundation (B120101) of the National University of Defense Technology. The authors thank the anonymous reviewers and the associate editor for their invaluable advice in reviewing this paper.

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Authors and Affiliations

  1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Zhaohui Dang & Yulin Zhang

  2. School of Aerospace, Tsinghua University, Beijing, 100084, China

    Zhaokui Wang & Yulin Zhang

Authors
  1. Zhaohui Dang
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  2. Zhaokui Wang
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  3. Yulin Zhang
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Correspondence to Zhaohui Dang.

Appendix: Orbital elements’ variation under \(\hbox {J}_{2}\) perturbation

Appendix: Orbital elements’ variation under \(\hbox {J}_{2}\) perturbation

The variations of the mean elements under a \(\hbox {J}_{2}\) perturbation are as follows:

$$\begin{aligned} \frac{\partial \dot{\bar{{\Omega }}}}{\partial a}&= \frac{{7}}{{2}}\gamma a^{-\frac{{9}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{2}}\cos i, \end{aligned}$$
(122)
$$\begin{aligned} \frac{\partial \dot{\bar{{\Omega }}}}{\partial e}&= -{4}e\gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{3}}\cos i, \end{aligned}$$
(123)
$$\begin{aligned} \frac{\partial \dot{\bar{{\Omega }}}}{\partial i}&= \gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{2}}\sin i, \end{aligned}$$
(124)
$$\begin{aligned} \frac{\partial \dot{\bar{{\omega }}}}{\partial a}&= -\frac{{7}}{{2}}\gamma a^{-\frac{{9}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{2}}\left( {{2}-\frac{{5}}{{2}}\sin ^{{2}}i} \right) ,\end{aligned}$$
(125)
$$\begin{aligned} \frac{\partial \dot{\bar{{\omega }}}}{\partial e}&= {4}e\gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{3}}\left( {{2}-\frac{{5}}{{2}}\sin ^{{2}}i} \right) , \end{aligned}$$
(126)
$$\begin{aligned} \frac{\partial \dot{\bar{{\omega }}}}{\partial i}&= -{5}\gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-{2}}\sin i\cos i, \end{aligned}$$
(127)
$$\begin{aligned} \frac{\partial \dot{\bar{{M}}}}{\partial a}&= -\frac{{7}}{{2}}\gamma a^{-\frac{{9}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-\frac{{3}}{{2}}}\left( {{1}-\frac{{3}}{{2}}\sin ^{{2}}i} \right) -\frac{{3}}{{2}}\sqrt{\mu }a^{-\frac{{5}}{{2}}}, \end{aligned}$$
(128)
$$\begin{aligned} \frac{\partial \dot{\bar{{M}}}}{\partial e}&= {3}e\gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-\frac{{5}}{{2}}}\left( {{1}-\frac{{3}}{{2}}\sin ^{{2}}i} \right) ,\end{aligned}$$
(129)
$$\begin{aligned} \frac{\partial \dot{\bar{{M}}}}{\partial i}&= -{3}\gamma a^{-\frac{{7}}{{2}}}\left( {{1}-e^{{2}}} \right) ^{-\frac{{3}}{{2}}}\sin i\cos i. \end{aligned}$$
(130)

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Dang, Z., Wang, Z. & Zhang, Y. Improved initialization conditions and single impulsive maneuvers for \(\hbox {J}_{2}\)-invariant relative orbits. Celest Mech Dyn Astr 121, 301–327 (2015). https://doi.org/10.1007/s10569-014-9601-4

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  • DOI: https://doi.org/10.1007/s10569-014-9601-4

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