A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions
Round 1
Reviewer 1 Report
The study is quite significant, mathematically rich, very well developed, and an important contribution to the study of tethers. The paper is ready to be published.
Author Response
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
The study is quite significant, mathematically rich, very well developed, and an important contribution to the study of tethers. The paper is ready to be published.
Response #1
Thank you for such an appreciation of our work.
Reviewer 2 Report
The paper discusses analytical approaches to identifying stable equilibrium positions of tethered systems anchored at the L1, L2 points of the Mars-Phobos system. The paper is generally well written, however it was hard to gauge the advantages of the proposed methodology based on the discussion and literature survey provided. Some minor comments follow:
Abstract: Typo - Analitical - Analytical
Introduction:
I think this is really lacking in good discussions and state-of the art. First off, a discussion on research which design missions around L1 and L2 that have been designed assuming the CR3BP system is needed to establish context (tethered or untethered). You can discuss the challenges identified in these papers as part of your motivation. I suggest inclusion of the following papers as part of the discussion -
[1] Kim, M., & Hall, C. D. (2004). Control of a rotating variable-length tethered system. Journal of guidance, control, and dynamics, 27(5), 849-858.
[2] Singh, S., Junkins, J., Anderson, B., & Taheri, E. (2021). Eclipse-conscious transfer to lunar gateway using ephemeris-driven terminal coast arcs. Journal of Guidance, Control, and Dynamics, 44(11), 1972-1988.
[3] Niccolai, L., Caruso, A., Quarta, A. A., & Mengali, G. (2020). Artificial collinear Lagrangian point maintenance with electric solar wind sail. IEEE Transactions on Aerospace and Electronic Systems, 56(6), 4467-4477.
[4] Baião, M. F., & Stuchi, T. J. (2017). Dynamics of tethered satellites in the vicinity of the Lagrangian point L 2 of the Earth–Moon system. Astrophysics and Space Science, 362(8), 134.
[5] Wong, B., Patil, R., & Misra, A. (2008). Attitude dynamics of rigid bodies in the vicinity of the Lagrangian points. Journal of guidance, control, and dynamics, 31(1), 252-256.
Conclusion, Page 15-16:
Please do not put conclusions as a list. Shift the discussion to a paragraph form. I think comparison of some of your results with the PHLOTE mission design parameters would instill more confidence in your results and probably pave a path for further work.
Author Response
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
The paper discusses analytical approaches to identifying stable equilibrium positions of tethered systems anchored at the L1, L2 points of the Mars-Phobos system. The paper is generally well written, however it was hard to gauge the advantages of the proposed methodology based on the discussion and literature survey provided. Some minor comments follow:
Introduction: I think this is really lacking in good discussions and state-of the art. First off, a discussion on research which design missions around L1 and L2 that have been designed assuming the CR3BP system is needed to establish context (tethered or untethered). You can discuss the challenges identified in these papers as part of your motivation. I suggest inclusion of the following papers as part of the discussion –
[1] Kim, M., & Hall, C. D. (2004). Control of a rotating variable length tethered system. Journal of guidance, control, and dynamics, 27(5), 849-858.
[2] Singh, S., Junkins, J., Anderson, B., & Taheri, E. (2021). Eclipse-conscious transfer to lunar gateway using ephemerisdriven terminal coast arcs. Journal of Guidance, Control, and Dynamics, 44(11), 1972-1988.
[3] Niccolai, L., Caruso, A., Quarta, A. A., & Mengali, G. (2020). Artificial collinear Lagrangian point maintenance with electric solar wind sail. IEEE Transactions on Aerospace and Electronic Systems, 56(6), 4467-4477.
[4] Baião, M. F., & Stuchi, T. J. (2017). Dynamics of tethered satellites in the vicinity of the Lagrangian point L 2 of the Earth– Moon system. Astrophysics and Space Science, 362(8), 134.
[5] Wong, B., Patil, R., & Misra, A. (2008). Attitude dynamics of rigid bodies in the vicinity of the Lagrangian points. Journal of guidance, control, and dynamics, 31(1), 252-256.
Response #1
Thank you. All of the proposed papers have been cited in the revised version.
Comment #2
Conclusion, Page 15-16: Please do not put conclusions as a list. Shift the discussion to a paragraph form. I think comparison of some of your results with the PHLOTE mission design parameters would instill more confidence in your results and probably pave a path for further work.
Response #2
Thank you. Corrected in the paragraph form. The proposed work obtained analytical solutions that add the results of the PHLOTE study and article by Aslanov V.S. [37]. Thank you again.
Reviewer 3 Report
see attached file
Comments for author File: 
Comments.pdf
Language corrections are necessary
Author Response
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
98: d is the distance between the primaries
In Eq. (1) the distance was p
Give the meaning of m1 and m2 here, not in 109-110
Response #1
Thank you. Corrected. The values of m1 and m2 have been given in lines 101-102.
Comment #2
112: The given expression of U is not the potential, but the gravitational force (multiplied by -1).
Response #2
Thank you. The CR3BP equations have been omitted in the text for brevity after editing.
Comment #3
121-123: Eq. (9) is the equation of motion, not the analytical solution
Response #3
Thank you. Corrected, this is Eq. (2) after editing.
Comment #4
127, 129: Eqs. (11) and (12) are the repetition of Eqs. (6) and (7), thus they are not necessary
Response #4
Thank you. Corrected.
Comment #5
141: What are the values of the physical parameters d, a_i, m1, m2, mu, used to obtain Fig. 2? It would give a perspective to what l=3000 m means.
There are no scales on the panels (a), (b) of Fig. 2.
Why Psi is on the axes of panel (d)? The case of Psi (ѱ) is studied in Sec. 2.2
On the other hand, Sect. 2.2 is not necessary, it does not give new informations. Fig. 3 is the same as Fig. 2, with a phase shift of Pi.
Response #5
Thank you. Corrected. The physical parameters and scales on the panels (a), (b) of Fig. 2 have been added. The angle ѱ is introduced because the dependences of the oscillation period on the length of the tether for the L1 and L2 libration points are different. This can be seen in the Eqs. (33) and (34), the Figs. (8) and (9). This is the difference between the mathematical pendulum in Earth conditions and the pendulum at the L1 libration point in the classical circular restricted three-body problem.
Comment #6
235: In Fig. 4 the numerical and analytical solutions show increasing differences. The situation is the same in Fig. 5. This should be analysed. What was the numerical integrator? How small are the deflection angles, for which the analytical solution (37) is valid? Perhaps a figure could be drawn, showing that depending on the initial deflection angle for how many oscillations do the numerical and analytical results agree within a given precision.
Response #6
We agree with this observation, and this increasing difference depends on the value of the module k, while the change in the vibration amplitude in the numerical and analytical solutions remains the same. In this case, it is important to know the amplitude values for the analysis of oscillatory motion.
Comment #7
245: Sect. 3.2 is not necessary, does not give new informations.
Response #7
Thank you. In Sect. 4 shows the differences in the variation of the oscillation period of the tether from its length at the angle φ and ѱ. Therefore, the preceding sections are important to ensure understanding.
Comment #8
281: Incomplete section title
Response #8
Thank you. Corrected.
Comment #9
283: see the above comment
Response #9
Thank you. This sentence has been deleted.
Comment #10
298-304: These results are surprising. Are they real? On the Earth, the period of a pendulum is increasing with its length.
Response #10
Thank you. We found errors in our solution of Eqs. (33) and (34) and added comparison with numerical calculations. These results are shown in Figs. (8) and (9).
Comment #11
305: Sect. 4.2 is not necessary
Response #11
Thank you. Sect. 4 shows the differences in the variation of the oscillation period of the tether from its length at the angle φ and ѱ.
Comment #12
345, 347: Eqs. (49), (50) are repetitions, they are not necessary
Response #12
Thank you. These are Eqs. (42) and (43) after editing. Eq. (42) is the distance between the primary 1 and the end mass, and Eq. (43) is the distance between the primary 2 and the end mass.
Comment #13
348: Sect. 5.2 is not necessary
Response #13
Thank you. The structure of the article has been changed. Sect. 5 includes sections for static and dynamic tether force for the angle of φ.
Comment #14
371: Sect. 6.2 is not necessary
Response #14
Thank you. The structure of the article has been changed. Sect. 6.2 was deleted after editing.
Comment #15
395: Sect. 7.2 is not necessary
Response #15
Thank you. The structure of the article has been changed. Sect. 7.2 was deleted after editing.
In the present version, all noticed typos and inaccuracies in the English writing have been corrected.
Reviewer 4 Report
Thank you for the paper. I think this is an interesting area of research worth of future studies. Below there are specific suggestions catalogued in different areas.
General suggestions:
· A brief explanation at the beginning of the introduction of what a space tether is could be advisable
· For readability, I would suggest using capital L in place of lowercase l for the length (or \ell)
Technical suggestions:
· Uniform the variables. If end mass is M, say, then eq L105 should have M. The same applies to L108, unless the definition is sought as “general” – but the proposed case is specific so it appears redundant or not necessary
· The CR3BP equations may be omitted in the text for brevity, but they have to be cited somehow. Please add some other reference in addition to the well-known Szebehely textbook (also at the end of the article)
· Please provide a justification for the oscillation period L286, or a reference
Style suggestions:
· L38, “Paper [21]”, delete “In”, or change the sentence to “In paper [21] is showed…”
· L63-64, sentence break
· Sometimes mathematical font is different between in-text formulae and centered equations. Verify they share the same font style and font size
· L102, const should not be italicized (same applies to L122 henceforth)
· L112 parentheses should wrap the content
Questions:
· Approximate analytical solutions provide a constant trend of a procession rate with respect to numerical solutions. Is there an inherent reason for this behavior, since it is repeated for all cases (e.g. Fig 4, 5, 6, 7)?
· Could, in your opinion, the intersection point for roughly 8500s for both L1 and L2 tether be used for synchronous operations or other interesting applications?
In general, I would suggest an external check or a double check of the English. The language is ok and only minor checks are needed to make it more fluent and coherent (there are some grammatical errors and some sentence breaks, or incomplete subordinates).
Author Response
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
A brief explanation at the beginning of the introduction of what a space tether is could be advisable.
Response #1
Thank you. Corrected.
Comment #2
For readability, I would suggest using capital L in place of lowercase l for the length (or \ell).
Response #2
Thank you. Corrected.
Comment #3
Uniform the variables. If end mass is M, say, then eq L105 should have M. The same applies to L108, unless the definition is sought as “general” – but the proposed case is specific so it appears redundant or not necessary.
Response #3
Thank you. Corrected.
Comment #4
The CR3BP equations may be omitted in the text for brevity, but they have to be cited somehow. Please add some other reference in addition to the well-known Szebehely textbook (also at the end of the article).
Response #4
Thank you. Additional references [39-42] have been added.
Comment #5
Please provide a justification for the oscillation period L286, or a reference.
Response #5
Thank you. We found errors in our solution of Eqs. (33) and (34) and added comparison with numerical calculations. These results are shown in Figs. 8-9.
Comment #6
L38, “Paper [21]”, delete “In”, or change the sentence to “In paper [21] is showed…”.
Response #6
Thank you. Corrected.
Comment #7
L63-64, sentence break.
Response #7
Thank you. Corrected.
Comment #8
Sometimes mathematical font is different between in-text formulae and centered equations. Verify they share the same font style and font size.
Response #8
Thank you. Corrected.
Comment #9
L102, const should not be italicized (same applies to L122 henceforth).
Response #9
Thank you. Corrected.
Comment #10
L112 parentheses should wrap the content.
Response #10
Thank you. The CR3BP equations have been omitted in the text for brevity after editing.
Comment #11
Approximate analytical solutions provide a constant trend of a procession rate with respect to numerical solutions. Is there an inherent reason for this behavior, since it is repeated for all cases (e.g. Fig 4, 5, 6, 7)?
Response #11
Thank you. We agree with this observation, and this increasing difference depends on the value of the module k, while the change in the vibration amplitude in the numerical and analytical solutions remains the same. In this case, it is important to know the amplitude values for the analysis of oscillatory motion.
Comment #12
Could, in your opinion, the intersection point for roughly 8500s for both L1 and L2 tether be used for synchronous operations or other interesting applications?
Response #12
Thank you. The application of the L1 and L2 points for synchronous operations is a good idea, that will be considered in the future. In addition, the authors are considering the future use of a similar tether system to create a space elevator fixed at the L1/L2 libration point. In the present version, all noticed typos and inaccuracies in the English writing have been corrected.
Round 2
Reviewer 2 Report
The authors have satisfactorily responded to my comments and made the changes to the manuscript.
Author Response
Thank you very much for appreciating our work to improve the manuscript.
Reviewer 3 Report
The meaning of j&(j) in lines 118-119
and y&(y) in lines 149-150
is not clear.
Author Response
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 marked with the "Track Changes" function in MS Word.
Comment #1
The meaning of j&(j) in lines 118-119 and y&(y) in lines 149-150 is not clear.
Response #1
Thank you. Corrected. Pj is replaced by Ej in lines 118, 120 and 150, 151. Figures 2 and 3 make changes in the total energy designations.
