Abstract
A frozen orbit is beneficial for observation owing to its stationary apsidal line. The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J2 and limited high-order terms, which cannot meet the stringent demands of all missions. In this study, the gravitational field is expanded to J15 terms and the Hamiltonian canonical form described by the Delaunay variables is used. The zonal harmonic coefficients of the Earth are chosen as the sample. Short-periodic terms are eliminated based on the Hori–Lie transformation. An algorithm is developed to solve all equilibrium points of the Hamiltonian function. A stable frozen orbit with an argument of perigee that equals neither 90◦ nor 270◦ is first reported in this paper. The local stability and topology of the equilibrium points are obtained from their eigenvalues. The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods. The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case. The analytical results can be applied to other Earth-like planets and asteroids.
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Abbreviations
- U :
-
potential of spacecraft subject to the attraction of an axisymmetric body
- F :
-
Hamiltonian function
- μ :
-
gravitational parameter of the oblate body
- R :
-
disturbing potential
- a e :
-
mean equatorial radius of the oblate body
- r :
-
distance of the spacecraft from the center of the body
- φ :
-
latitude of the spacecraft
- J n :
-
zonal harmonics coefficients of degree n
- C mn, S mn :
-
tesseral harmonics coefficients of degree m, n
- P n(x) :
-
Legendre polynomials
- P mn(x) :
-
associated Legendre polynomials
- a :
-
semi-major axis
- e :
-
eccentricity
- i :
-
inclination
- ω :
-
argument of perigee
- Ω:
-
right ascension of the ascending node (RAAN)
- M :
-
mean anomaly and its derivative in terms of time
- λ:
-
geographical longitude
- f :
-
true anomaly
- L, l, G, a, H, h :
-
Delaunay variables
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Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Nos. 11772024 and 11432001) and Qian Xuesen Youth Innovation Foundation of China Aerospace Science and Technology Corporation.
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Yuechen Ma received his B.S. degree from the College of Mechatronics and Control Engineering in Beihang University in 2015 and his M.S. degree in spacecraft design & engineering from Beihang University in 2018. He is currently studying for his Ph.D. degree in spacecraft design & engineering at Beihang University. His research interests include high-order gravity, spacecraft trajectory design and optimization, and invariant relative orbit.
Yanchao He received his B.S. degree from the College of Mechatronics and Control Engineering in Beihang University in 2015 and his M.S. degree in spacecraft design & engineering from Beihang Universityin 2018. He is currently studying for his Ph.D. degree in spacecraft design & engineering at Beihang University. His research interests include high-order gravity, spacecraft trajectory design and optimization, and invariant relative orbit.
Ming Xu received his B.S. and Ph.D. degrees in aerospace engineering from Beihang University in 2003 and 2008, respectively. He served as an engineer of orbital design and operation in DFH Satellite Co., Ltd., China Academy of Space Technology, Beijing, China, until 2010. Then, he joined Beihang University as an assistant professor and then was promoted as an associate professor in 2012. His current research interests include the applications of dynamical systems theory into astrodynamics and orbital control. Dr. Xu serves as associate editors for Astrodynamics and Advances in Aircraft and Spacecraft Science. Dr. Xu recieved National Top 100 Excellent Doctoral Dissertation Award nomination in 2010 and Third Class Prizes of the National Defense Technology Invention Award in 2016. He has 50 publications in journals, books, and proceedings.
Yaru Zheng received her B.S. degree from the College of Engineering from Ocean University of China in 2017 and her M.S. degree in aerospace engineering from Beihang University in 2020. She is currently working in the 9th Designing of CASIC, Wuhan, China. Her research interests include robust control, spacecraft trajectory design and optimization, and the dynamics of formation flying.
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Ma, Y., He, Y., Xu, M. et al. Global searches of frozen orbits around an oblate Earth-like planet. Astrodyn 6, 249–268 (2022). https://doi.org/10.1007/s42064-021-0104-2
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DOI: https://doi.org/10.1007/s42064-021-0104-2