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Comments on “On the dynamics of non-rigid asteroid rotation”
Journal Pre-proof Comments on “On the dynamics of non-rigid asteroid rotation” M.Yu. Ovchinnikov, D.S. Roldugin PII:
S0094-5765(20)30030-8
DOI:
https://doi.org/10.1016/j.actaastro.2020.01.019
Reference:
AA 7843
To appear in:
Acta Astronautica
Received Date: 3 July 2019 Revised Date:
22 October 2019
Accepted Date: 12 January 2020
Please cite this article as: M.Y. Ovchinnikov, D.S. Roldugin, Comments on “On the dynamics of non-rigid asteroid rotation”, Acta Astronautica, https://doi.org/10.1016/j.actaastro.2020.01.019. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 IAA. Published by Elsevier Ltd. All rights reserved.
Comments on “On the dynamics of non-rigid asteroid rotation” M.Yu. Ovchinnikov, D.S. Roldugin Keldysh Institute of Applied Mathematics, Russia, 125047, Moscow, Miusskaya Sq., 4 Corresponding author: Dmitry Roldugin, [email protected], +7-926-15-44-983 Abstract In the commented paper the authors claim to find the solutions for the non-rigid asteroid rotation equations. The asteroid moves on an elliptic orbit and a new method is applied for the Euler equation solution. However, despite the challenging problem definition found in the introduction and highlights, a simplified approximation is covered. Namely, the asteroid motion is close to the rotation around the axis of the maximum moment of inertia. The ultimate result of the paper, that is the equations for the angular momentum component perpendicular to this axis, is correct only under specific assumptions. Small deviations from the rotation around the maximum inertia axis are analyzed. One of the angular momentum components is neglected. As a result, the omitted terms in the equations of motions have the same, or even higher, order of magnitude as the retained terms. Key words: Asteroid rotation; Angular motion; Attitude dynamics 1. Introduction Paper [1] claims to provide the solution of the Euler equations of motion for a non-rigid asteroid moving on an elliptic orbit. An elliptic orbit, stated in the Highlights, may substantially complicate the dimensionless equations of motion that utilize the derivative with respect to the argument of latitude instead of time. As the dimensionless equations are not presented in the paper, this phenomenon is not actually covered. Next, the Introduction presents a relatively complete discussion of the disturbances acting on the asteroid. However, no disturbances are accounted for in the equations of motion, which questions the necessity of half of the Introduction. Finally, the following problem statement arises: the asteroid is non-rigid (moments of inertia depend on time) and is rotating approximately around its maximum moment of inertia axis (which is the ultimate motion regime due to the natural dissipation in the body, but proper discussion of the stability is necessary). 2. Discussion of Equations 6 and 8 Section 2 provides equation manipulation that results in the expression
2 1 d ( Ki ) ∑ I dt = 0 i
(1)
where Ki are the angular momentum components and I i are the principal moments of inertia. Next, the asteroid is considered to be rotating almost around its maximum inertia axis, so Ω 2 , Ω3 = Ω1 , Ω1 ≈ const (2) where Ωi are the angular velocity components (although the paper actually uses the relation for the angular momentum components) and the first axis is considered to be the maximum inertia axis. This requires discussion, as the stability of the rotation around the maximum moment of inertia is well established for a rigid body, but not for an arbitrary non-rigid body. The paper is concerned with a non-rigid asteroid. No assumption on the nature of the moments of inertia 1
change is present. Relation (2) may be applied to the angular momentum under proper and realistic assumptions for the asteroid and corresponding stability analysis. A candidate for such an assumption is the slow and small change of the moments of inertia that induces the dissipation. Assuming that K1 is almost constant, the first term in Eq. (1) is neglected by the authors. However, all three angular momentum components are almost constant, not just the first one. The difference is that the first component oscillates near some constant rotation rate K 0 while two other components oscillate near zero. Defining the first component as K1 = K 0 + δ K1 with δ representing small (as compared to K 0 ) oscillation of Ki , Eq. (1) provides 2 2 2 2 K 0 d (δ K1 ) 1 d (δ K1 ) 1 d (δ K 2 ) 1 d (δ K 3 ) + + + =0. I1 dt I1 dt I2 dt I3 dt
(3)
The two first terms in Eq. (3) are omitted in the paper leading to Eq. 6 (numbers without brackets refer to the commented paper equations) that is the base for further analysis, namely, Eq. 8. However, the second term in Eq. (3) has the same order as the third and the fourth, and the first term may even dominate the equation due to the possibly high rotation rate K 0 . Discarding the first two terms is a mistake. Also note that the same assumption of completely constant K1 is used in the Appendix. The manipulations provided in the paper may be justified under proper assumptions only. For example Eq. 8 is valid for rigid or almost rigid body (slow change in the moments of inertia). Further treatment in the paper evolves around the erroneous Eq. 6. Moreover, this treatment is also questionable. Single equation for K 2 arises. It is stated to be the Riccati equation after the change of variables. This change utilizes some new variable K 2′ that is not introduced in the paper. The equation is not solved, and only one possible problem is outlined as the basis for five self-citations. It is important to note that Eq. 8 requires direct expressions for the moments of inertia change. If such explicit functions of moments of inertia are somehow available, one can simply use the initial Euler equations of motion for the numerical integration, just as the Eq. 8. So even under proper assumptions that justify Eq. 8, its practicality is questionable. Overall, the proposed scheme can hardly be viewed as the “new method for solving Euler equations” of motion (Highlights), as no such method is present. This does not provide the “new type of solutions” (Highlights) as simply no solution is present in the paper. The paper is almost void: it contains an introduction that is almost unrelated to the problem statement, equation manipulation that results in equations that still require numerical integration, a discussion that summarizes this manipulation; a conclusion that simply summarizes everything, and an appendix not related to the paper. Conclusion The main equation presented in [1] is proved to be erroneous. The overall paper composition and discussion are found to be questionable. Bibliography [1]
S.V. Ershkov, D. Leshchenko, On the dynamics OF NON-RIGID asteroid rotation, Acta Astronaut. 161 (2019) 40–43. doi:10.1016/J.ACTAASTRO.2019.05.011. 2
• Recent paper “On the dynamics of non-rigid asteroid rotation” is discussed. • Major mistake in a core equation is outlined. • Overall paper methodology and composition are discussed.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
S0094-5765(20)30030-8
DOI:
https://doi.org/10.1016/j.actaastro.2020.01.019
Reference:
AA 7843
To appear in:
Acta Astronautica
Received Date: 3 July 2019 Revised Date:
22 October 2019
Accepted Date: 12 January 2020
Please cite this article as: M.Y. Ovchinnikov, D.S. Roldugin, Comments on “On the dynamics of non-rigid asteroid rotation”, Acta Astronautica, https://doi.org/10.1016/j.actaastro.2020.01.019. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 IAA. Published by Elsevier Ltd. All rights reserved.
Comments on “On the dynamics of non-rigid asteroid rotation” M.Yu. Ovchinnikov, D.S. Roldugin Keldysh Institute of Applied Mathematics, Russia, 125047, Moscow, Miusskaya Sq., 4 Corresponding author: Dmitry Roldugin, [email protected], +7-926-15-44-983 Abstract In the commented paper the authors claim to find the solutions for the non-rigid asteroid rotation equations. The asteroid moves on an elliptic orbit and a new method is applied for the Euler equation solution. However, despite the challenging problem definition found in the introduction and highlights, a simplified approximation is covered. Namely, the asteroid motion is close to the rotation around the axis of the maximum moment of inertia. The ultimate result of the paper, that is the equations for the angular momentum component perpendicular to this axis, is correct only under specific assumptions. Small deviations from the rotation around the maximum inertia axis are analyzed. One of the angular momentum components is neglected. As a result, the omitted terms in the equations of motions have the same, or even higher, order of magnitude as the retained terms. Key words: Asteroid rotation; Angular motion; Attitude dynamics 1. Introduction Paper [1] claims to provide the solution of the Euler equations of motion for a non-rigid asteroid moving on an elliptic orbit. An elliptic orbit, stated in the Highlights, may substantially complicate the dimensionless equations of motion that utilize the derivative with respect to the argument of latitude instead of time. As the dimensionless equations are not presented in the paper, this phenomenon is not actually covered. Next, the Introduction presents a relatively complete discussion of the disturbances acting on the asteroid. However, no disturbances are accounted for in the equations of motion, which questions the necessity of half of the Introduction. Finally, the following problem statement arises: the asteroid is non-rigid (moments of inertia depend on time) and is rotating approximately around its maximum moment of inertia axis (which is the ultimate motion regime due to the natural dissipation in the body, but proper discussion of the stability is necessary). 2. Discussion of Equations 6 and 8 Section 2 provides equation manipulation that results in the expression
2 1 d ( Ki ) ∑ I dt = 0 i
(1)
where Ki are the angular momentum components and I i are the principal moments of inertia. Next, the asteroid is considered to be rotating almost around its maximum inertia axis, so Ω 2 , Ω3 = Ω1 , Ω1 ≈ const (2) where Ωi are the angular velocity components (although the paper actually uses the relation for the angular momentum components) and the first axis is considered to be the maximum inertia axis. This requires discussion, as the stability of the rotation around the maximum moment of inertia is well established for a rigid body, but not for an arbitrary non-rigid body. The paper is concerned with a non-rigid asteroid. No assumption on the nature of the moments of inertia 1
change is present. Relation (2) may be applied to the angular momentum under proper and realistic assumptions for the asteroid and corresponding stability analysis. A candidate for such an assumption is the slow and small change of the moments of inertia that induces the dissipation. Assuming that K1 is almost constant, the first term in Eq. (1) is neglected by the authors. However, all three angular momentum components are almost constant, not just the first one. The difference is that the first component oscillates near some constant rotation rate K 0 while two other components oscillate near zero. Defining the first component as K1 = K 0 + δ K1 with δ representing small (as compared to K 0 ) oscillation of Ki , Eq. (1) provides 2 2 2 2 K 0 d (δ K1 ) 1 d (δ K1 ) 1 d (δ K 2 ) 1 d (δ K 3 ) + + + =0. I1 dt I1 dt I2 dt I3 dt
(3)
The two first terms in Eq. (3) are omitted in the paper leading to Eq. 6 (numbers without brackets refer to the commented paper equations) that is the base for further analysis, namely, Eq. 8. However, the second term in Eq. (3) has the same order as the third and the fourth, and the first term may even dominate the equation due to the possibly high rotation rate K 0 . Discarding the first two terms is a mistake. Also note that the same assumption of completely constant K1 is used in the Appendix. The manipulations provided in the paper may be justified under proper assumptions only. For example Eq. 8 is valid for rigid or almost rigid body (slow change in the moments of inertia). Further treatment in the paper evolves around the erroneous Eq. 6. Moreover, this treatment is also questionable. Single equation for K 2 arises. It is stated to be the Riccati equation after the change of variables. This change utilizes some new variable K 2′ that is not introduced in the paper. The equation is not solved, and only one possible problem is outlined as the basis for five self-citations. It is important to note that Eq. 8 requires direct expressions for the moments of inertia change. If such explicit functions of moments of inertia are somehow available, one can simply use the initial Euler equations of motion for the numerical integration, just as the Eq. 8. So even under proper assumptions that justify Eq. 8, its practicality is questionable. Overall, the proposed scheme can hardly be viewed as the “new method for solving Euler equations” of motion (Highlights), as no such method is present. This does not provide the “new type of solutions” (Highlights) as simply no solution is present in the paper. The paper is almost void: it contains an introduction that is almost unrelated to the problem statement, equation manipulation that results in equations that still require numerical integration, a discussion that summarizes this manipulation; a conclusion that simply summarizes everything, and an appendix not related to the paper. Conclusion The main equation presented in [1] is proved to be erroneous. The overall paper composition and discussion are found to be questionable. Bibliography [1]
S.V. Ershkov, D. Leshchenko, On the dynamics OF NON-RIGID asteroid rotation, Acta Astronaut. 161 (2019) 40–43. doi:10.1016/J.ACTAASTRO.2019.05.011. 2
• Recent paper “On the dynamics of non-rigid asteroid rotation” is discussed. • Major mistake in a core equation is outlined. • Overall paper methodology and composition are discussed.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.