Periodic Solutions of Nonlinear Relative Motion Satellites
, Elbaz I. Abouelmagd 2, Juan Luis García Guirao 3,4 and Dariusz W. Brzeziński 5,*Round 1
Reviewer 1 Report
The abstract should be written to include detailed information to clearly understand the contribution of this work.
The author should conduct more literature reviews and clearly address the existing techniques that have been used to tackle this problem.
The authors should cite the references properly. For example, no citations are included in Section 2 at all.
It is not clear what contributions were made by the authors.
The sign inside of the parenthesis in the first equation after line 193 is incorrect.
The obtained approximated solution should be compared with the existing solutions that are obtained by using other techniques.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Authors,
Research of relative motion and its simulation contributes greatly to expanding our knowledge of the world. The topic covered by your paper is rather interesting. However, I cannot recommend your article for publication in its current form due to the following reasons:
1. First of all, it is worth expanding the abstract. The abstract should briefly but succinctly describe the main results obtained in your research.
2. Many of the studies listed in the references have been published for a very long time. I recommend complementing the state-of-the-art with research recently proposed. For example, the following article is devoted to the problem you are solving: Bando M., Ichikawa A. Periodic orbits of nonlinear relative dynamics and satellite formation, DOI: 10.2514/1.41438
3. In my opinion, it is possible to shorten preliminaries by presenting the equations in a form that will then be used to describe the periodic motion.
4. From the presented results, the relevance of the proposed approach is not clear. How can the obtained results be verified?
5. Is it possible to compare the proposed method with other approaches, in particular with numerical simulation? Several studies are devoted to the problem of numerical modeling of orbital problems and the influence of numerical integration methods on the results, for example:
-Andreev et al. Six-body problem solution using symplectic integrators. DOI: 10.1109/EIConRusNW.2016.7448135
- Andreev et al. N-body problem solution with composition numerical integration methods. DOI: 10.1109/SCM.2016.7519726
please highlight this aspect in the manuscript.
Thus, my decision is major revisions.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Formation flying of multiple satellites is an important technology with many possible applications such as long baseline interferometry, synthetic aperture radar, simultaneous measurement of magnetic field at different points, and so on. To achieve these formation flying missions, a precise control of relative motion between multiple satellites is required and the equations of motions representing such a relative motion are needed. Then the studies of the orbit dynamics for the formation flying have been active. For example, the Hill-Clohessy-Wiltshire equation is suitable for the analysis of the rendezvous problem. However, in the low-Earth orbit formation flying, it is important to consider the contribution of disturbances such as the Earth flatness, the orbital eccentricity and the nonlinear effects of relative motion dynamics, and so on. The authors of the present paper consider such nonlinear effects in relative motions of satellites. Using the Lindstedt-Poincare perturbation method, the authors obtain the nonlinear periodic analytic solutions (57) which represent orbits not in a plane given in the figures 5 and 7. Such solutions would improve the plan of the formation flying of satellites. I find this work interesting but I recommend the authors to consider following points:
i) $i_y$ in the line 108 may be replaced with $i_z$.
ii) What is $\mu$ in the equation of $\dot \theta $ between the lines 126 and 127? The author may add the explanation of $\mu$.
iii) For readers' convenience, similar to the figures 2, 3 and 5, the authors may represent $\theta _0 = \pi /3, \pi /2, \pi /6$ orbits in the figure 4 by black, red and orange, respectively.
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Dear Authors,
Thank you for taking my recommendations into account.
I agree that verifying an analytical solution using numerical methods is not entirely correct. However, it is known that many problems, including some problems in relativistic mechanics, cannot be modeled using analytical models. Therefore, the results of comparison with the solutions obtained numerically can be presented. For example, you showed that the explicit Euler method does not provide target periodic oscillations. And this underlines the importance of developing analytical approaches.
I believe that the manuscript can be published as it is.
Author Response
Thank you. We agree on your comments.
