地月空间的远距离逆行轨道族及其分岔研究

陈冠华; 杨驰航; 张晨; 张皓

doi:10.13700/j.bh.1001-5965.2020.0608

地月空间的远距离逆行轨道族及其分岔研究

doi: 10.13700/j.bh.1001-5965.2020.0608

陈冠华^{1, 2},
杨驰航^{1, 2},
张晨¹,
张皓^1, ,

1.
中国科学院空间应用工程与技术中心太空应用重点实验室, 北京 100094
2.
中国科学院大学, 北京 100049

基金项目:

国家重点研发计划 2018YFB1900605

中国科学院重点部署项目 ZDRW-KT-2019-1-0102

详细信息

通讯作者:
张皓, E-mail: hao.zhang.zhr@csu.ac.cn

中图分类号: P173.1;P173.3;P132⁺.2;V412
计量
- 文章访问数: 690
- HTML全文浏览量: 156
- PDF下载量: 112
- 被引次数: 0
出版历程
- 收稿日期: 2020-10-30
- 录用日期: 2021-01-22
- 网络出版日期: 2021-02-24
- 整期出版日期: 2022-12-20

Distant retrograde orbits and its bifurcations in Earth-Moon system

CHEN Guanhua^{1, 2},
YANG Chihang^{1, 2},
ZHANG Chen¹,
ZHANG Hao^{1
, ,}

1.
Key Laboratory of Space Utilization, Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences, Beijing 100094, China
2.
University of Chinese Academy of Sciences, Beijing 100049, China

Funds:

National Key R & D Program of China 2018YFB1900605

Key Research Program of the Chinese Academy of Sciences(CAS) ZDRW-KT-2019-1-0102

More Information

Corresponding author: ZHANG Hao, E-mail: hao.zhang.zhr@csu.ac.cn

摘要

摘要:
地月系统中存在着一类绕月逆行、高度稳定的轨道族，称为远距离逆行轨道族(DRO)。以圆型限制性三体问题(CR3BP)为动力学模型研究了DRO轨道族周边的动力系统结构。利用Broucke稳定性图寻找分叉点，判断分叉类型，基于数值延拓计算分岔后产生的一系列新轨道分支。分叉类型主要有切分叉与多倍周期分叉(从3倍周期开始)，轨道维度包含平面轨道族与三维轨道族。计算新轨道族的特征，包括形状、周期、能量、稳定性、双曲流形结构等。探讨周期轨道的轨道周期与能量的关系，以几何化的方式展现分叉结构、多周期轨道的双曲流形结构等。该动力结构将为基于DRO轨道族的地月空间任务提供重要的理论支持。
- 远距离逆行轨道 /
- 分岔 /
- 多周期轨道 /
- 数值延拓 /
- 轨道稳定性
Abstract:
There exists a type of stable retrograde orbit around the Moon called distant retrograde orbit (DRO) in the Earth-Moon system. The circular restricted three-body problem (CR3BP) is taken as the dynamical model to study the dynamical structure around DRO. It is possible to determine the bifurcation point and type using Broucke's stability diagram. The numerical continuation method is used to calculate several new orbital branches. Tangent and multi-period bifurcations are the two primary forms of DRO bifurcations (starting from period tripling). New orbit families include planar orbits and 3D orbits. The characteristics of the new orbit family are discussed, including shape, period, energy, stability, hyperbolic manifold structure. The relationship between orbital period and energy is discussed. The bifurcation structure and the hyperbolic manifold structure of multi-periodic orbits are presented geometrically. The dynamic structure will provide theoretical support for the mission based on DRO families.
- distant retrograde orbit /
- bifurcation /
- multi-period orbit /
- numerical continuation /
- orbital stability

HTML全文

图 1 圆型限制性三体问题

Figure 1. Circular restricted three-body problem

下载: 全尺寸图片幻灯片

图 2 稳定性图^[26]

Figure 2. Stability diagram^[26]

下载: 全尺寸图片幻灯片

图 3 地月空间DRO

Figure 3. DRO in Earth-Moon system

下载: 全尺寸图片幻灯片

图 4 DRO稳定性指数

Figure 4. Stability indices of DRO

下载: 全尺寸图片幻灯片

图 5 DRO轨道族的Broucke稳定性图

Figure 5. Broucke stability diagram of DRO family

下载: 全尺寸图片幻灯片

图 6 三维DRO

Figure 6. 3D DRO

下载: 全尺寸图片幻灯片

图 7 三维DRO的稳定性指数

Figure 7. Stability indices of 3D DRO

下载: 全尺寸图片幻灯片

图 8 P3DRO

Figure 8. P3DRO

下载: 全尺寸图片幻灯片

图 9 P3DRO的几何特性

Figure 9. Geometric properties of P3DRO

下载: 全尺寸图片幻灯片

图 10 P3DRO的稳定性指数

Figure 10. Stability indices of P3DRO

下载: 全尺寸图片幻灯片

图 11 P4DRO分支1的延拓及其稳定性指数

Figure 11. Stability indices of P4DRO-1

下载: 全尺寸图片幻灯片

图 12 P4DRO分支2的延拓及其稳定性指数

Figure 12. Stability indices of P4DRO-2

下载: 全尺寸图片幻灯片

图 13 P5DRO平面分支1的延拓及其稳定性指数

Figure 13. Stability indices of 2D-P5DRO-1

下载: 全尺寸图片幻灯片

图 14 P5DRO平面分支2的延拓及其稳定性指数

Figure 14. Stability indices of 2D-P5DRO-2

下载: 全尺寸图片幻灯片

图 15 P5DRO三维分支1的延拓及其稳定性指数

Figure 15. Stability indices of P5DRO-3D-1

下载: 全尺寸图片幻灯片

图 16 P5DRO三维分支2的延拓及其稳定性指数

Figure 16. Stability indices of P5DRO-3D-2

下载: 全尺寸图片幻灯片

图 17 同一能量对应多个P5DRO

Figure 17. Different P5DRO with same Jacobi energy

下载: 全尺寸图片幻灯片

图 18 不同稳定性指数下的稳定流形

Figure 18. Manifold in different stability indices

下载: 全尺寸图片幻灯片

图 19 不同稳定性指数下的稳定流形(J=3.003 5, v_max=6.634 4)

Figure 19. Manifold in different stability indices (J=3.003 5, v_max=6.634 4)

下载: 全尺寸图片幻灯片

图 20 y=0截面处二维P3DRO的稳定流形(J=2.58, v_max=55.495 2)

Figure 20. Manifold section of 2D P3DRO in y=0 section (J=2.58, v_max=55.495 2)

下载: 全尺寸图片幻灯片

图 21 x=1-μ截面处三维P5DRO的稳定流形(J=3.003 5, v_max=6.634 4)

Figure 21. Manifold section of 3D P5DRO in x=1-μ section(J=3.003 5, v_max=6.634 4)

下载: 全尺寸图片幻灯片

图 22 分岔图

Figure 22. Bifurcation chart

下载: 全尺寸图片幻灯片

图 23 能量-周期关系

Figure 23. Relationship between energy and period

下载: 全尺寸图片幻灯片

图 24 P5DRO的能量-周期关系放大图

Figure 24. Relationship between energy and period of P5DRO and its local magnification

下载: 全尺寸图片幻灯片

表 1 分岔类型^[25-28]

Table 1. Bifurcation type^[25-28]

分岔类型	方程	稳定性
切分岔	β=-2α-2	变化
2倍周期分岔	β=2α-2	变化
3倍周期分岔	β=α+1	不变
4倍周期分岔	β=2	不变
5倍周期分岔	β=α/(2cos(4π/5))-(cos(8π/5)+1)/cos(4π/5)β=α/(2cos(8π/5))-(cos(16π/5)+1)/cos(8π/5)	不变
二次Hopf分岔	β=α²/4+2，-4 < α < 4	变化
修正二次Hopf分岔	β=α²/4+2，α∈[-∞, -4]∪[4, ∞]	不变

下载: 导出CSV

参考文献(30)

[1]	STRANGE N, LANDAU D, MCELRATH T, et al. Overview of mission design for NASA asteroid redirect robotic mission concept[C] // 3rd International Electric Propulsion Conference, 2013: 1-12.
[2]	HAMBLETON K. Deep space gateway to open opportunities for distant destinations[EB/OL]. (2018-04-25)[2020-10-27]. https://www.nasa.gov/feature/deep-space-gateway-to-open-opportunities-for-distant-destinations/.
[3]	GATES M, MUIRHEAD B, NAASZ B, et al. NASA's asteroid redirect mission concept development summary[C] // 2015 IEEE Aerospace Conference. Piscataway: IEEE Press, 2015: 1-13.
[4]	BEZROUK C J, PARKER J. Long duration stability of distant retrograde orbits[C] // AIAA/AAS Astrodynamics Specialist Conference. Reston: AIAA, 2014: 4424.
[5]	BEZROUK C J, PARKER J. Long term evolution of distant retrograde orbits in the Earth-Moon system[J]. Astrophysics and Space Science, 2017, 362(9): 176. doi: 10.1007/s10509-017-3158-0
[6]	彭超, 温昶煊, 高扬. 地月空间DRO与HEO (3: 1 / 2: 1) 共振轨道延拓求解及其稳定性分析[J]. 载人航天, 2018, 24(6): 703-718. doi: 10.3969/j.issn.1674-5825.2018.06.001 PENG C, WEN C X, GAO Y. Solution and stability analysis of resonance orbits of DRO and HEO (3: 1 / 2: 1) in Earth Moon space[J]. Manned Spaceflight, 2018, 24(6): 703-718(in Chinese). doi: 10.3969/j.issn.1674-5825.2018.06.001
[7]	吴小婧, 曾凌川, 巩应奎. DRO计算及其在地月系中的摄动力研究[J]. 北京航空航天大学学报, 2020, 46(5): 48-57. doi: 10.13700/j.bh.1001-5965.2019.0353 WU X J, ZENG L C, GONG Y K. DRO calculation and its perturbation in the Earth-Moon system[J]. Journal of Beijing University of Aeronautics and Astronautics, 2020, 46(5): 48-57(in Chinese). doi: 10.13700/j.bh.1001-5965.2019.0353
[8]	CHEN H R, CANALIAS E, HESTROFFER D, et al. Effective stability of quasi-satellite orbits in the spatial problem for Phobos exploration[J]. Journal of Guidance, Control, and Dynamics, 2020, 43(12): 1-12. doi: 10.2514/1.G002893.c1
[9]	CAPDEVILA L R, GUZZETTI D, HOWELL K C. Various transfer options from Earth into distant retrograde orbits in the vicinity of the Moon[J]. Advances in the Astronautical Sciences, 2014, 152: 3659-3678.
[10]	CAPDEVILA L R, HOWELL K C. A transfer network linking Earth, Moon, and the triangular libration point regions in the Earth-Moon system[J]. Advances in Space Research, 2018, 62(7): 1826-1852. doi: 10.1016/j.asr.2018.06.045
[11]	ZHANG R K, WANG Y, ZHANG H, et al. Transfers from distant retrograde orbits to low lunar orbits[J]. Celestial Mechanics and Dynamical Astronomy, 2020, 132(8): 1-30.
[12]	曾豪, 李朝玉, 彭坤, 等. 地月空间NRHO与DRO在月球探测中的应用研究[J]. 宇航学报, 2020, 41(7): 910-919. https://www.cnki.com.cn/Article/CJFDTOTAL-YHXB202007010.htm ZENG H, LI C Y, PENG K, et al. Application of Earth Moon space NRHO and DRO in lunar exploration[J]. Journal of Astronautics, 2020, 41(7): 910-919(in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-YHXB202007010.htm
[13]	CAVALLARI I, PETITDEMANGE R, LIZY-DESTREZ S. Transfer from a lunar distant retrograde orbit to Mars through Lyapunov orbits[C]//27th International Symposium on Space Flight Dynamics (ISSFD). Melbourne: Engineers Australia Press, Royal Aeronautical Society, 2019: 1393-1398.
[14]	CONTE D, SPENCER D B. Mission analysis for Earth to Mars-Phobos distant retrograde orbits[J]. Acta Astronautica, 2018, 151: 761-771. doi: 10.1016/j.actaastro.2018.06.049
[15]	SCOTT C J, SPENCER D B. Calculating transfer families to periodic distant retrograde orbits using differential correction[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(5): 1592-1605. doi: 10.2514/1.47791
[16]	OCAMPO C A, ROSBOROUGH G W. Transfer trajectories for distant retrograde orbits of the Earth[R]. [S. l. ]: NASA STI/Recon Technical Report A, 1993, 95: 1323-1337.
[17]	徐明, 徐世杰. 绕月飞行的大幅值逆行轨道研究[J]. 宇航学报, 2009, 30(5): 1785-1791. https://www.cnki.com.cn/Article/CJFDTOTAL-YHXB200905006.htm XU M, XU S J. Exploration of distant retrograde orbits around Moon[J]. Journal of Astronautics, 2009, 30(5): 1785-1791(in Chinese). https://www.cnki.com.cn/Article/CJFDTOTAL-YHXB200905006.htm
[18]	WANG W B, SHU L Z, LIU J K, et al. Joint navigation performance of distant retrograde orbits and cislunar orbits via LiAISON considering dynamic and clock model errors[J]. Navigation, 2019, 66(4): 781-802.
[19]	LAM T, WHIFFEN G J. Exploration of distant retrograde orbits around Europa[C] //15th AAS/AIAA Space Flight Mechanics Meeting. Reston: AIAA, 2005.
[20]	WALLACE M, PARKER J, STRANGE N, et al. Orbital operations for Phobos and Deimos exploration[C] // AIAA/AAS Astrodynamics Specialist Conference. Reston: AIAA, 2012: 5067.
[21]	GIL P J S, SCHWARTZ J. Simulations of quasi-satellite orbits around Phobos[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(3): 901-914.
[22]	赵育善, 师鹏, 张晨. 深空飞行动力学[M]. 北京: 中国宇航出版社, 2016: 131-157. ZHAO Y S, SHI P, ZHANG C. Dynamics of deep space flight[M]. Beijing: China Aerospace Publishing House, 2016: 131-157(in Chinese).
[23]	KOON W S, LO W Y, MARSDEN J E, et al. Dynamical systems, the three-body problem and space mission design[C] //International Conference on Differential Equations, 2000: 1167-1181.
[24]	MOON F C. Nonlinear dynamics[M]//MEYERS R A. Encyclopedia of physical science and technology. 3rd ed. Amsterdam: Elsevier, 2003: 523-535.
[25]	HOWELL K C, CAMPBELL E T. Three-dimensional periodic solutions that bifurcate from halo families in the circular restricted three-body problem[J]. Advances in the Astronatical Sciences, 1999, 102: 891-910.
[26]	ZIMOVAN-SPREEN E M, HOWELL K C, DAVIS D C. Near rectilinear halo orbits and nearby higher-period dynamical structures: Orbital stability and resonance properties[J]. Celestial Mechanics and Dynamical Astronomy, 2020, 132(5): 1-25.
[27]	DOUSKOS C, KALANTONIS V, MARKELLOS P. Effects of resonances on the stability of retrograde satellites[J]. Astrophysics and Space Science, 2007, 310(3-4): 245-249.
[28]	WINTER O C. The stability evolution of a family of simply periodic lunar orbits[J]. Planetary and Space Science, 2000, 48(1): 23-28.
[29]	CAMPBELL E T. Bifurcations from families of periodic solutions in the circular restricted problem with application to trajectory design[D]. West Lafayette: Purdue University, 1999.
[30]	BOSANAC N. Leveraging natural dynamical structures to explore multi-body systems[D]. West Lafayette: Purdue University, 2016.