Attitude Dynamics of Spinning Magnetic LEO/VLEO Satellites
Round 1
Reviewer 1 Report
This Reviewer thinks that authors have presented new development in approach with respect to investigating the rotational dynamics of satellite, which helps to describe the spinning evolution of magnetised satellite.
I think that presented work сould be a sound development in the scientific field under investigation (engineering celestial mechanics). So, this scientific work is original and results, presented herein, are novel; besides, this work corresponds enough to the scope of the journal with respect to the broad readership of the journal. My only recommendation is to cite the additional proper references (this is not mandatory, if only authors find it useful for their research):
1) Leshchenko, D., Ershkov, S., Kozachenko, T.: Rotations of a Rigid Body Close to the Lagrange Case under the Action of Nonstationary Perturbation Torque. J. Appl. Comp. Mech. 8(3), 1023-1031 (2022);
Simple algebraic manipulations are under responsibility of the authors. My recommendation: accept as is.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Authors,
My comments/suggestions are following:
- Problem statement is not well written needs to be considered again, include flowcharts graphical info where required.
- Equations are not cited in the text.
- Numerical model needs better explanation for the exact problem it is focussed towards.
- The research methodology needs to be explained clearly with the parameters and assumptions for realtime model.
- The mathematical model must be explained and comparison with other model is also required to be done.
- References must be given for each model and previous versions.
- Quality of work is impressive and alot of effort has been done.
- The paper investigates the attitude motion of arbitrarily spinning satellites in LEO and VLEO under the action of aerodynamic, gravitational, and magnetic torques. Better put the comparison part along with the relevant sections. Best wishes,
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
The goal of this manuscript is to investigate the attitude motion of satellites in LEO under the influence of aerodynamic, gravitational and magnetic torques of forces. The topic of this manuscript under review is relevant.
List of references is rather large , appropriate and interesting. However , I think that following papers would be added to the references and described in Introduction :
1. Beletsky V. V. , Grushevsky A. V. The evolution of the rotational motion of a satellite under the action of dissipative aerodynamic moment. J. Appl. Math. Mech. 58 ( 1) , 11-19 (1994)
2. Kuznetsova E.Y. , Sazonov V. V. , Chebukov S. Y.Evolution of the satellite rapid rotation under the action of gravitational and aerodynamic torques. Mech. Solids. 35 (2), 1-10 (2000)
3. Maslova A. I. , Pirozhenko A. V. Modelling of the aerodynamic moment acting upon a satellite . Cosmic Res. 48 (4), 362-370 ( 2010)
4. Akulenko L.D., Leshchenko D. D., Rachinskaya A. L. Evolution of the satellite fast rotation due to the gravitational torque in a dragging medium. Mech. Solids. 43 ( 2 ), 173-184 ( 2008)
The element of novelty is an example satellite in LEO and VLEO and numerical results for it reveal the possibility of chaos in attitude aerodynamically spinning deployable satellites.
Conclusions are consistent with arguments. The authors are focused on related problems not covered in the manuscript.
Minor revision.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
The goal of the paper is to investigate the passive attitude motion of satellites in LEO and VLEO under the action of aerodynamic, gravitational, and magnetic torques.
It is a topic of great interest that is currently being investigated by the scientific community. Although the authors have published similar works, this one presents a higher level mathematical formalism and significant contributions.
The paper presents a very serious mathematical analysis of the problem, using Lagrangian dynamics. With this analysis, non-linear solutions are obtained, that govern the dynamics of the satellite attitude under different types of conservative and non-conservative perturbations.
In the results section, the work is unrealistic. E.g. it uses a magnetic moment equal to unity. To be more realistic, it is necessary to quantify the importance of each perturbing torque. For a picosatellite in LEO orbit, the magnetic torque is much larger than the gravitational torque and the atmospheric torque.
Major Changes:
Check whether in equation (8) should be sin(2theta) or sin^2(theta). In (9), the integral of (8), cos() should appear. These errors may affect the rest of the results and should therefore be corrected and revised in the rest of the article.
These errors may affect the rest of the results and should therefore be checked and corrected in the results of the article.
To solve the problem of the relevance of the magnetic torque, I propose to the authors to assume that the principal component of the magnetic dipole can be estimated in flight and cancelled using magnetopairs. In this way they can justify that the remaining magnetic torques are of the order of the other perturbations.
Minor Changes:
PDF is attached with notes on minor corrections.
Comments for author File: Comments.pdf
Author Response
Please also see the attached file.
The authors are very grateful to the Reviewer for taking his/her invaluable time to review the current work and for his/her constructive comments. The latter is extremely helpful to enhance this manuscript. We have carefully addressed all comments in the revised version. All corrections are highlighted with blue color.
Comment #1
References 20 and 21 are not relevant.
Response #1
Thank you. Corrected.
Comment #2
Line 84: psi and phi
Response #2
Thank you. Although there is no typo here, the text has been modified in order to provide a better description of the orientation of the intermediate frames relative to the orbital frame.
Comment #3
Line 85: Does it need color blue?
Response #3
Apparently, the version the editor sent to you is the revised version with our previous corrections aimed to address the comments of the other reviewers. The previous corrections are still present in this revised version, along with new added corrections, all highlighted with blue.
Comment #4
Line 100: Introduce the nomenclature
Response #4
Thank you. To clarify the nomenclature, the text before Eq. (8) has been modified, and references to the coordinate axes shown in Fig. 1 have been added.
Comment #5
Eq. (8): sin(2theta) is right? Check whether in equation (8) should be sin(2theta) or sin^2(theta).
Response #5
Yes, sin(2theta) is right. It follows directly from Eqs. (5) and (7), and some manipulations from vector algebra. Such expressions with double angle are typical for the gravitational torque. See, e.g., the pioneering work by Beletsky (Ref. [7] in the revised paper). Thank you.
Comment #6
Eq. (9): I think that this is not the integral of the previous equation. int(sin)=>-cos(). In (9), the integral of (8), cos() should appear. These errors may affect the rest of the results and should therefore be corrected and revised in the rest of the article. These errors may affect the rest of the results and should therefore be checked and corrected in the results of the article.
Response #6
Eq. (9) is correct. In order to write it in a nice symmetric form, we used a double angle formula cos(2x) = 1 - 2(sin(x))^2 after integration of Eq. (8) and then omitted constant terms. Now it is mentioned in the paper. Thank you.
Comment #7
Line 111: Lowercase e
Response #7
Thank you. Corrected.
Comment #8
Eq. (36): k=1,2 and 3?
Response #8
According to the Lagrangian mechanics, the first of Eq. (36) must be applied only to non-cyclic generalized coordinates, i.e., appearing in the Lagrangian. In our case, we only need k =1,2, corresponding to psi and theta, since the generalized coordinate phi, corresponding to k = 3, is cyclic and does not appear in the Lagrangian nor in the Routhian, as it is mentioned before Eq. (32). Thank you.
Comment #9
Line 222: Define EGG acronym
Response #9
Thank you. EGG stands for re-Entry satellite with Gossamer aeroshell and GPS/Iridium). The acronym is now defined in the paper.
Comment #10
Line 228: Define here 'both chosen altitudes'
Response #10
Thank you. The two altitudes need to be defined only in Section 4.2. Accordingly, the fragment you are referring to has been modified in order to indicate the range of applicability of the method used for calculation of the aerodynamic characteristics.
Comment #11
Line 261: Explain the condition parameter Beta and its units.
Response #11
Thank you. Beta is the dimensionless initial conjugated momentum and it is defined by Eq. (39). In order to make it clearer for the reader, some corrections have been made in the first paragraph of Section 4.2.
Comment #12
Line 268: Remove Blue.
Response #12
This blue color highlights changes in the paper and will be removed in the final version.
Comment #13
Eq. (53): Why this value? Bear in mind that for a picosatellite, the magnetic moment is much greater than gravity moments. Try to justify the value.
In the results section, the work is unrealistic. E.g. it uses a magnetic moment equal to unity. To be more realistic, it is necessary to quantify the importance of each perturbing torque. For a picosatellite in LEO orbit, the magnetic torque is much larger than the gravitational torque and the atmospheric torque. To solve the problem of the relevance of the magnetic torque, I propose to the authors to assume that the principal component of the magnetic dipole can be estimated in flight and cancelled using magnetopairs. In this way they can justify that the remaining magnetic torques are of the order of the other perturbations.
Response #13
Thank you. Equation (53) rather means that the components of the own magnetic moment are equal to each other. The absolute value of the own magnetic moment is defined by the parameter mu [Eq. (11)]. On order to justify that the remaining magnetic torques are of the order of the other perturbations, we used your kind suggestion dealing with the possibility of cancelling the principal component of the magnetic dipole using magnetopairs. The first paragraph of Section 4.3 and the text after Eq. (52) have been modified accordingly. Thank you again.
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
Dear authors,
thanks to the clarifications the article is more understandable.
I only consider it necessary to revise two aspects.
1. I still think that equation (9) contains an error. As you are using the identity sin(2x)=2sin(x)cos(x), the term /2 should disappear from equation (9).
I have done the integral by Symbolic Matlab:
int(3*sin(s)^2*sin(2*t),0,t)
ans =3*sin(s)^2*sin(t)^2
2. On the other hand, the text associated with equation (53) is still confusing, "the satellite's own magnetic moment are taken equal to each other and to unity". Equation (53) states that the direction is [1,1,1], but says nothing about the modulus of the magnetic moment. Moreover, as I told you in the previous review, I would have explained the order of magnitude of the magnetic perturbation.
Finally, congratulations on the high quality of the article.
Author Response
Please see the attachment
Author Response File: Author Response.pdf
