# Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid

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## Abstract

**:**

## 1. Introduction

## 2. Comparison of Numerical Schemes for Long Assessment

## 3. Stability of the Excited Spin State of the Secondary

#### 3.1. Definition of the Linearised Error Propagation Matrix $M\left(t\right)$

#### 3.2. Analysis of $M\left(t\right)$ with the Initial Angular Velocity from $1.0{\omega}_{0}$ to $1.5{\omega}_{0}$

#### 3.3. Effect of the Non-Spherical Gravitational Field of the Primary and the Shape of the Secondary on the Tumbling Motion of the Secondary

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**The internal structure of the primary: (

**a**) Uniform internal mass distribution; (

**b**) density decreases from inside to outside; (

**c**) hollow structure; and (

**d**) randomly distributed internal holes.

**Table A1.**The coefficients of the partial spherical harmonic functions of the four models corresponding to Figure A1.

Model (a) | Model (b) | Model (c) | Model (d) | |
---|---|---|---|---|

${J}_{2}$ | $1.1806016\times {10}^{-2}$ | $9.8940464\times {10}^{-3}$ | $1.6734636\times {10}^{-2}$ | $1.5034951\times {10}^{-2}$ |

${J}_{3}$ | $1.6620303\times {10}^{-3}$ | $1.5130812\times {10}^{-3}$ | $2.4893284\times {10}^{-3}$ | $1.3309408\times {10}^{-3}$ |

${J}_{4}$ | $-8.2823689\times {10}^{-3}$ | $-6.5632877\times {10}^{-3}$ | $-1.1255250\times {10}^{-3}$ | $-8.4519822\times {10}^{-3}$ |

${J}_{5}$ | $1.3132394\times {10}^{-3}$ | $1.0301008\times {10}^{-3}$ | $1.9799122\times {10}^{-3}$ | $1.7177357\times {10}^{-3}$ |

${J}_{6}$ | $5.0458902\times {10}^{-3}$ | $4.3363831\times {10}^{-3}$ | $7.6811147\times {10}^{-3}$ | $5.0889960\times {10}^{-3}$ |

${C}_{22}$ | $1.4205710\times {10}^{-3}$ | $1.2214748\times {10}^{-3}$ | $2.0914078\times {10}^{-3}$ | $1.3592346\times {10}^{-3}$ |

${S}_{22}$ | $-1.4884062\times {10}^{-17}$ | $-6.5408149\times {10}^{-17}$ | $3.4962132\times {10}^{-17}$ | $2.7487490\times {10}^{-17}$ |

${C}_{31}$ | $1.4205710\times {10}^{-3}$ | $8.2970092\times {10}^{-4}$ | $1.4993794\times {10}^{-3}$ | $1.3687619\times {10}^{-3}$ |

${C}_{32}$ | $4.5358186\times {10}^{-4}$ | $3.1217587\times {10}^{-4}$ | $5.1968086\times {10}^{-4}$ | $6.2947825\times {10}^{-4}$ |

${C}_{33}$ | $-7.3986272\times {10}^{-5}$ | $-6.7329189\times {10}^{-5}$ | $-1.3757727\times {10}^{-4}$ | $-1.1369448\times {10}^{-4}$ |

${S}_{31}$ | $-1.7122335\times {10}^{-3}$ | $4.1793971\times {10}^{-3}$ | $7.1960664\times {10}^{-3}$ | $-4.8583376\times {10}^{-3}$ |

${S}_{32}$ | $-5.5112926\times {10}^{-4}$ | $5.0319504\times {10}^{-4}$ | $8.1352188\times {10}^{-4}$ | $-5.2487380\times {10}^{-4}$ |

${S}_{33}$ | $4.3490667\times {10}^{-5}$ | $-8.4002799\times {10}^{-5}$ | $-1.5102661\times {10}^{-4}$ | $7.3761001\times {10}^{-5}$ |

${C}_{41}$ | $8.2756971\times {10}^{-4}$ | $4.8424441\times {10}^{-4}$ | $7.54612325\times {10}^{-4}$ | $8.8267016\times {10}^{-4}$ |

${C}_{42}$ | $-3.6264708\times {10}^{-5}$ | $-5.1243370\times {10}^{-6}$ | $-7.8521454\times {10}^{-5}$ | $5.8103214\times {10}^{-5}$ |

${C}_{43}$ | $-4.9772428\times {10}^{-5}$ | $-6.6087746\times {10}^{-5}$ | $-9.8008247\times {10}^{-5}$ | $-4.0274728\times {10}^{-5}$ |

${C}_{44}$ | $2.2303289\times {10}^{-5}$ | $1.8673576\times {10}^{-5}$ | $3.19242917\times {10}^{-5}$ | $2.40624550\times {10}^{-5}$ |

${S}_{41}$ | $1.7122335\times {10}^{-3}$ | $1.4349064\times {10}^{-3}$ | $2.4430001\times {10}^{-3}$ | $-1.1834871\times {10}^{-3}$ |

${S}_{42}$ | $1.4850440\times {10}^{-4}$ | $-1.4613356\times {10}^{-4}$ | $-2.4425241\times {10}^{-4}$ | $1.9224648\times {10}^{-4}$ |

${S}_{43}$ | $-1.4255166\times {10}^{-4}$ | $1.0462180\times {10}^{-4}$ | $1.8717446\times {10}^{-4}$ | $-1.4832237\times {10}^{-4}$ |

${S}_{44}$ | $3.7897508\times {10}^{-6}$ | $2.1443836\times {10}^{-6}$ | $-5.813172\times {10}^{-7}$ | $4.12749518\times {10}^{-6}$ |

${C}_{51}$ | $-3.1272871\times {10}^{-5}$ | $-3.4821246\times {10}^{-5}$ | $-6.8427557\times {10}^{-5}$ | $5.3334476\times {10}^{-5}$ |

${C}_{52}$ | $-1.3763895\times {10}^{-5}$ | $-9.4857243\times {10}^{-6}$ | $-1.8897903\times {10}^{-5}$ | $1.2900471\times {10}^{-5}$ |

${C}_{53}$ | $1.5176728\times {10}^{-5}$ | $1.0813339\times {10}^{-5}$ | $1.9819451\times {10}^{-5}$ | $2.3163710\times {10}^{-5}$ |

${C}_{54}$ | $-1.3104122\times {10}^{-6}$ | $-1.7654773\times {10}^{-6}$ | $-3.5584973\times {10}^{-6}$ | $-1.5303503\times {10}^{-6}$ |

${C}_{55}$ | $1.1230563\times {10}^{-7}$ | $-3.5638557\times {10}^{-8}$ | $-2.0440055\times {10}^{-8}$ | $4.6177160\times {10}^{-7}$ |

${S}_{51}$ | $-3.0555771\times {10}^{-4}$ | $2.5269240\times {10}^{-4}$ | $3.9629143\times {10}^{-4}$ | $-1.9158968\times {10}^{-4}$ |

${S}_{52}$ | $-8.3027337\times {10}^{-5}$ | $5.5164301\times {10}^{-5}$ | $1.0287761\times {10}^{-4}$ | $-6.4137630\times {10}^{-5}$ |

${S}_{53}$ | $-2.1025676\times {10}^{-5}$ | $1.9801873\times {10}^{-5}$ | $3.2486021\times {10}^{-5}$ | $-1.9990395\times {10}^{-5}$ |

${S}_{54}$ | $-8.5562671\times {10}^{-7}$ | $4.0510217\times {10}^{-7}$ | $9.9360799\times {10}^{-7}$ | $-3.4240771\times {10}^{-7}$ |

${S}_{55}$ | $-5.6674802\times {10}^{-7}$ | $7.9659473\times {10}^{-7}$ | $1.3832809\times {10}^{-6}$ | $-1.2130380\times {10}^{-6}$ |

${C}_{61}$ | $7.4598908\times {10}^{-5}$ | $1.0254846\times {10}^{-5}$ | $1.3039376\times {10}^{-4}$ | $-1.0657605\times {10}^{-4}$ |

${C}_{62}$ | $-2.2087532\times {10}^{-5}$ | $-8.6100296\times {10}^{-6}$ | $-2.6696073\times {10}^{-5}$ | $-1.5985003\times {10}^{-5}$ |

${C}_{63}$ | $-8.0428517\times {10}^{-6}$ | $-7.3253673\times {10}^{-6}$ | $-1.1413149\times {10}^{-5}$ | $-9.1612740\times {10}^{-6}$ |

${C}_{64}$ | $-4.0554689\times {10}^{-7}$ | $-4.0401878\times {10}^{-8}$ | $-1.9263216\times {10}^{-7}$ | $-2.1994592\times {10}^{-7}$ |

${C}_{65}$ | $1.2221950\times {10}^{-7}$ | $8.2412156\times {10}^{-8}$ | $1.54076076\times {10}^{-7}$ | $2.1036180\times {10}^{-7}$ |

${C}_{66}$ | $-3.6108293\times {10}^{-8}$ | $-5.009707\times {10}^{-8}$ | $-8.8044116\times {10}^{-8}$ | $-6.2420189\times {10}^{-8}$ |

${S}_{61}$ | $-2.6668643\times {10}^{-4}$ | $1.8252031\times {10}^{-4}$ | $1.7433184\times {10}^{-4}$ | $-4.6283910\times {10}^{-4}$ |

${S}_{62}$ | $9.1296564\times {10}^{-5}$ | $-8.4002799\times {10}^{-5}$ | $-1.2608227\times {10}^{-4}$ | $7.36740260\times {10}^{-5}$ |

${S}_{63}$ | $-8.4915327\times {10}^{-6}$ | $-8.4002799\times {10}^{-5}$ | $1.4218991\times {10}^{-5}$ | $-1.0994466\times {10}^{-5}$ |

${S}_{64}$ | $4.3587887\times {10}^{-7}$ | $-8.4002799\times {10}^{-7}$ | $-6.6267905\times {10}^{-7}$ | $4.28696097\times {10}^{-8}$ |

${S}_{65}$ | $-1.4568412\times {10}^{-7}$ | $-8.4002799\times {10}^{-7}$ | $1.94633108\times {10}^{-7}$ | $-1.1636263\times {10}^{-7}$ |

${S}_{66}$ | $-1.4975725\times {10}^{-8}$ | $-8.4002799\times {10}^{-8}$ | $-1.4714211\times {10}^{-8}$ | $5.34328597\times {10}^{-9}$ |

**Figure A2.**The distance error between the orbit calculated by the spherical harmonic function method and the finite element method. (

**a**–

**d**) correspond to the four models of the primary in Figure A1, and the different coloured lines indicate that different perturbation terms are considered, where $JX$ indicates the expansion to the ${X}^{\mathrm{th}}$ order of the zonal harmonic term and $TX$ indicates the expansion to the ${X}^{\mathrm{th}}$ tesseral harmonic term.

## References

- Cheng, A.; Michel, P.; Jutzi, M.; Rivkin, A.; Stickle, A.; Barnouin, O.; Ernst, C.; Atchison, J.; Pravec, P.; Richardson, D.; et al. Asteroid impact & deflection assessment mission: Kinetic impactor. Planet. Space Sci.
**2016**, 121, 27–35. [Google Scholar] - Michel, P.; Cheng, A.; Küppers, M.; Pravec, P.; Blum, J.; Delbo, M.; Green, S.; Rosenblatt, P.; Tsiganis, K.; Vincent, J.B.; et al. Science case for the asteroid impact mission (AIM): A component of the asteroid impact & deflection assessment (AIDA) mission. Adv. Space Res.
**2016**, 57, 2529–2547. [Google Scholar] - Cheng, A.F.; Rivkin, A.S.; Michel, P.; Atchison, J.; Barnouin, O.; Benner, L.; Chabot, N.L.; Ernst, C.; Fahnestock, E.G.; Kueppers, M.; et al. AIDA DART asteroid deflection test: Planetary defense and science objectives. Planet. Space Sci.
**2018**, 157, 104–115. [Google Scholar] [CrossRef] - Rainey, E.S.; Stickle, A.M.; Cheng, A.F.; Rivkin, A.S.; Chabot, N.L.; Barnouin, O.S.; Ernst, C.M.; Group, A.I.S.W. Impact Modeling for the Double Asteroid Redirection Test Mission. In Proceedings of the Hypervelocity Impact Symposium, Destin, FL, USA, 16–20 April 2019; Volume 883556, p. HVIS2019-038. [Google Scholar]
- Rainey, E.S.; Stickle, A.M.; Cheng, A.F.; Rivkin, A.S.; Chabot, N.L.; Barnouin, O.S.; Ernst, C.M.; AIDA/DART Impact Simulation Working Group. Impact modeling for the Double Asteroid Redirection Test (DART) mission. Int. J. Impact Eng.
**2020**, 142, 103528. [Google Scholar] [CrossRef] - Rivkin, A.S.; Chabot, N.L.; Stickle, A.M.; Thomas, C.A.; Richardson, D.C.; Barnouin, O.; Fahnestock, E.G.; Ernst, C.M.; Cheng, A.F.; Chesley, S.; et al. The double asteroid redirection test (DART): Planetary defense investigations and requirements. Planet. Sci. J.
**2021**, 2, 173. [Google Scholar] [CrossRef] - Agrusa, H.; Richardson, D.; Barbee, B.; Bottke, W.; Cheng, A.; Eggl, S.; Ferrari, F.; Hirabayashi, M.; Karatekin, O.; McMahon, J.; et al. Predictions for the Dynamical State of the Didymos System Before and After the Planned DART Impact. LPI Contrib.
**2022**, 2678, 2447. [Google Scholar] - Agrusa, H.F.; Gkolias, I.; Tsiganis, K.; Richardson, D.C.; Meyer, A.J.; Scheeres, D.J.; Ćuk, M.; Jacobson, S.A.; Michel, P.; Karatekin, Ö.; et al. The excited spin state of Dimorphos resulting from the DART impact. Icarus
**2021**, 370, 114624. [Google Scholar] [CrossRef] - Agrusa, H.; Ballouz, R.; Meyer, A.J.; Tasev, E.; Noiset, G.; Karatekin, Ö.; Michel, P.; Richardson, D.C.; Hirabayashi, M. Rotation-induced granular motion on the secondary component of binary asteroids: Application to the DART impact on Dimorphos. Astron. Astrophys.
**2022**, 664, L3. [Google Scholar] [CrossRef] - Werner, R.A.; Scheeres, D.J. Mutual potential of homogeneous polyhedra. Celest. Mech. Dyn. Astron.
**2005**, 91, 337–349. [Google Scholar] [CrossRef] [Green Version] - Hirabayashi, M.; Scheeres, D.J. Recursive computation of mutual potential between two polyhedra. Celest. Mech. Dyn. Astron.
**2013**, 117, 245–262. [Google Scholar] [CrossRef] - Hou, X.; Scheeres, D.J.; Xin, X. Mutual potential between two rigid bodies with arbitrary shapes and mass distributions. Celest. Mech. Dyn. Astron.
**2017**, 127, 369–395. [Google Scholar] [CrossRef] - Richardson, D.C.; Quinn, T.; Stadel, J.; Lake, G. Direct large-scale N-body simulations of planetesimal dynamics. Icarus
**2000**, 143, 45–59. [Google Scholar] [CrossRef] [Green Version] - Yu, Y.; Cheng, B.; Hayabayashi, M.; Michel, P.; Baoyin, H. A finite element method for computational full two-body problem: I. The mutual potential and derivatives over bilinear tetrahedron elements. Celest. Mech. Dyn. Astron.
**2019**, 131, 51. [Google Scholar] [CrossRef] - Ruth, R.D. A canonical integration technique. IEEE Trans. Nucl. Sci.
**1983**, 30, 2669–2671. [Google Scholar] [CrossRef] - Marsden, J.E.; West, M. Discrete mechanics and variational integrators. Acta Numer.
**2001**, 10, 357–514. [Google Scholar] [CrossRef] [Green Version] - Feng, K.; Qin, M. Symplectic Geometric Algorithms for Hamiltonian Systems; Springer: Berlin/Heidelberg, Germany, 2010; Volume 449. [Google Scholar]
- Karamali, G.; Shiri, B. Numerical solution of higher index DAEs using their IAE’s structure: Trajectory-prescribed path control problem and simple pendulum. Casp. J. Math. Sci. CJMS
**2018**, 7, 1–15. [Google Scholar] - Kosmas, O.T.; Vlachos, D. Phase-fitted discrete Lagrangian integrators. Comput. Phys. Commun.
**2010**, 181, 562–568. [Google Scholar] [CrossRef] [Green Version] - Kosmas, O.T.; Vlachos, D. Local path fitting: A new approach to variational integrators. J. Comput. Appl. Math.
**2012**, 236, 2632–2642. [Google Scholar] [CrossRef] [Green Version] - Kosmas, O.; Leyendecker, S. Analysis of higher order phase fitted variational integrators. Adv. Comput. Math.
**2016**, 42, 605–619. [Google Scholar] [CrossRef] - Kosmas, O.; Leyendecker, S. Variational integrators for orbital problems using frequency estimation. Adv. Comput. Math.
**2019**, 45, 1–21. [Google Scholar] [CrossRef] [Green Version] - Kosmas, O. Energy minimization scheme for split potential systems using exponential variational integrators. Appl. Mech.
**2021**, 2, 431–441. [Google Scholar] [CrossRef] - Leimkuhler, B.; Reich, S. Simulating Hamiltonian Dynamics; Cambridge University Press: Cambridge, UK, 2004; Number 14. [Google Scholar]
- Dullweber, A.; Leimkuhler, B.; McLachlan, R. Symplectic splitting methods for rigid body molecular dynamics. J. Chem. Phys.
**1997**, 107, 5840–5851. [Google Scholar] [CrossRef] [Green Version] - Kol, A.; Laird, B.B.; Leimkuhler, B.J. A symplectic method for rigid-body molecular simulation. J. Chem. Phys.
**1997**, 107, 2580–2588. [Google Scholar] [CrossRef] [Green Version] - van Zon, R.; Schofield, J. Symplectic algorithms for simulations of rigid-body systems using the exact solution of free motion. Phys. Rev. E
**2007**, 75, 056701. [Google Scholar] [CrossRef] [PubMed] - Celledoni, E.; Fassò, F.; Säfström, N.; Zanna, A. The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput.
**2008**, 30, 2084–2112. [Google Scholar] [CrossRef] - Ćuk, M.; Jacobson, S.A.; Walsh, K.J. Barrel Instability in Binary Asteroids. Planet. Sci. J.
**2021**, 2, 231. [Google Scholar] [CrossRef] - Benner, L.A.; Margot, J.; Nolan, M.; Giorgini, J.; Brozovic, M.; Scheeres, D.; Magri, C.; Ostro, S. Radar imaging and a physical model of binary asteroid 65803 Didymos. In Proceedings of the AAS/Division for Planetary Sciences Meeting Abstracts #42, Pasadena, CA, USA, 4–8 October 2010; Volume 42, pp. 13–17. [Google Scholar]
- Hand, L.N.; Finch, J.D. Analytical Mechanics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Gao, Y.; Yu, Y.; Cheng, B.; Baoyin, H. Accelerating the finite element method for calculating the full 2-body problem with CUDA. Adv. Space Res.
**2022**, 69, 2305–2318. [Google Scholar] [CrossRef] - Naidu, S.; Benner, L.; Brozovic, M.; Nolan, M.; Ostro, S.; Margot, J.; Giorgini, J.; Hirabayashi, T.; Scheeres, D.; Pravec, P.; et al. Radar observations and a physical model of binary near-Earth asteroid 65803 Didymos, target of the DART mission. Icarus
**2020**, 348, 113777. [Google Scholar] [CrossRef] - Gao, Y.; Cheng, B.; Yu, Y. The interactive dynamics of a binary asteroid and ejecta after medium kinetic impact. Astrophys. Space Sci.
**2022**, 367, 84. [Google Scholar] [CrossRef]

**Figure 1.**Variation of the relative error of the conserved quantities over ten years for simulations of the full two-body problem using the RK4 integrator.

**Figure 2.**Variation of the relative error of the conserved quantities over ten years for simulations of the full two-body problem using the RK78 integrator.

**Figure 3.**Variation of the relative error of the conserved quantities over ten years for simulations of the full two-body problem using the SI integrator.

**Figure 4.**Diagram of the 1-2-3 Euler angles of the secondary: (

**a**) diagram of the Didymos system; and (

**b**) the simplified model.

**Figure 5.**Variation of the three Euler angles of the secondary over time: (

**a**) initial spin angular velocity = $1.0{\omega}_{0}$ and (

**b**) initial spin angular velocity = $1.5{\omega}_{0}$.

**Figure 6.**The distribution of the eigenvalues of $M\left(t\right)$ at $t=$ 8000 s, 110,000 s, 623,000 s, and 990,000 s, where the initial angular velocity of the secondary is $1.0{\omega}_{0}$.

**Figure 7.**The distribution of the eigenvalues of $M\left(t\right)$ in ${10}^{6}$ s, with initial angular velocity of the secondary of $1.0{\omega}_{0}$.

**Figure 8.**The maximum eigenvalue of $M\left(\mathbf{t}\right)$ and the corresponding eigenvector component in the $\varphi $-direction, where the initial angular velocity of the secondary is $1.0{\omega}_{0}$: (

**a**) Variation of the modulus of the maximum eigenvalue over ${10}^{6}$ s seconds; and (

**b**) the component of the eigenvector in the $\varphi $-direction corresponding to the maximum eigenvalue.

**Figure 9.**Components of the eigenvector in the x-direction and y-direction corresponding to the maximum eigenvalue.

**Figure 10.**Distribution of the eigenvalues of $M\left(t\right)$ over ${10}^{6}$ s seconds. (

**a**,

**b**) The initial angular velocity of the secondary is $1.1{\omega}_{0}$. (

**c**,

**d**) The initial angular velocity of the secondary is $1.2{\omega}_{0}$. (

**e**,

**f**) The initial angular velocity of the secondary is $1.3{\omega}_{0}$.

**Figure 11.**The variation of the modulus of the eigenvalues of $M\left(t\right)$ and the components of the eigenvector in the $\varphi $- and $\theta $-directions corresponding to the eigenvalues over ${10}^{6}$ s, where the initial angular velocity of the secondary is $1.3{\omega}_{0}$. (

**a**) Modulus of the 1st eigenvalue. (

**b**) Modulus of the 2nd eigenvalue. (

**c**) Modulus of the 3rd eigenvalue. (

**d**) $\varphi $ of the 1st eigenvector. (

**e**) $\varphi $ of the 2nd eigenvector. (

**f**) $\varphi $ of the 3rd eigenvector. (

**g**) $\theta $ of the 1st eigenvector. (

**h**) $\theta $ of the 2nd eigenvector. (

**i**) $\theta $ of the 3rd eigenvector.

**Figure 12.**The distribution of the eigenvalues of $M\left(t\right)$ over ${10}^{6}$ seconds, where the initial angular velocity of the secondary is $1.4{\omega}_{0}$ (

**a**) and $1.5{\omega}_{0}$ (

**b**).

**Figure 13.**The maximum $\varphi $ value under different perturbation terms and initial angular velocities within 1 year.

**Figure 14.**The maximum $\varphi $ value under different $a/b$, $b/c$, and initial angular velocity within 1 year.

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## Share and Cite

**MDPI and ACS Style**

Gao, Y.; Cheng, B.; Yu, Y.; Lv, J.; Baoyin, H.
Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid. *Mathematics* **2022**, *10*, 3757.
https://doi.org/10.3390/math10203757

**AMA Style**

Gao Y, Cheng B, Yu Y, Lv J, Baoyin H.
Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid. *Mathematics*. 2022; 10(20):3757.
https://doi.org/10.3390/math10203757

**Chicago/Turabian Style**

Gao, Yunfeng, Bin Cheng, Yang Yu, Jing Lv, and Hexi Baoyin.
2022. "Stability Analysis on the Moon’s Rotation in a Perturbed Binary Asteroid" *Mathematics* 10, no. 20: 3757.
https://doi.org/10.3390/math10203757