A review of periodic orbits in the circular restricted three-body problem

doi:10.23919/JSEE.2022.000059

Abstract

Abstract:

This review article aims to give a comprehensive review of periodic orbits in the circular restricted three-body problem (CRTBP), which is a standard ideal model for the Earth-Moon system and is closest to the practical mechanical model. It focuses the attention on periodic orbits in the Earth-Moon system. This work is primarily motivated by a series of missions and plans that take advantages of the three-body periodic orbits near the libration points or around two gravitational celestial bodies. Firstly, simple periodic orbits and their classi?cation that is usually considered to be early work before 1970 are summarized, and periodic orbits around Lagrange points, either planar or three-dimensional, are intensively studied during past decades. Subsequently, stability index of a periodic orbit and bifurcation analysis are presented, which demonstrate a guideline to ?nd more periodic orbits inspired by bifurcation signals. Then, the practical techniques for computing a wide range of periodic orbits and associated quasi-periodic orbits, as well as constructing database of periodic orbits by numerical searching techniques are also presented. For those unstable periodic orbits, the station keeping maneuvers are reviewed. Finally, the applications of periodic orbits are presented, including those in practical missions, under consideration, and still in conceptual design stage. This review article has the function of bridging between engineers and researchers, so as to make it more convenient and faster for engineers to understand the complex restricted three-body problem (RTBP). At the same time, it can also provide some technical thinking for general researchers.

Key words: periodic orbit, restricted three-body problem, classification, orbit family, bifurcation

Renyong ZHANG. A review of periodic orbits in the circular restricted three-body problem[J]. Journal of Systems Engineering and Electronics, 2022, 33(3): 612-646.

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Applications of periodic orbit missions"

Mission	Operator	Orbit/Type	Scienti?c objective	Year	Launch/Dry mass/kg	References
ISEE-3	NASA, ESA	L₁/L₂-Halo¹ 1st mission	Solar wind, Earth’s magnetic ?eld	1978	479/390	[206–212]
WIND	NASA	L₁-Quasi-halo¹	Solar wind/Earth’s magnetosphere monitor	1994	1250/950	[13–17]
SOHO	NASA, ESA	L₁-Halo¹	Solar observatory	1996	1850/610	[213–217]
ACE	NASA	L₁-Lissajous¹	Energetic particles solar wind	1997	757/562	[218–220]
WMAP	NASA	L₂-Lissajous¹	Cosmic microwaves, Background radiation	2001	835/763	[221, 222]
Genesis	NASA	L₁-Quasi-Halo¹	Solar wind particles samples and particles	2001	636/494	[20, 175, 223]
Herschel	ESA	L₂-Halo¹	Far-infrared telescope, Formation of galaxies	2009	3400/2800	[224–233]
Chang’e?2	China	L₂-Halo¹	Extend mission, Visited asteroid	2010	2480/1180	[234–237]
GAIA	ESA	L₂-Halo¹	Galactic structure, Astrometry	2013	2029/1392	[238–241]
LISA	ESA	L₁-Quasi-Halo¹	Gravitational wave	2015	1910/810	[242, 243]
DSCOVR	NASA, ESA	L₁-Lissajous	Space weather/climate Earth observation	2015	570/307	[244–246]
Artemis	NASA	L₁-Quasi-Halo¹ 1st mission	Extend mission, Lunar magnetosphere	2007	128/77	[1, 160, 244–248]
Que-qiao	China	L₂-Halo²	Communication relay	2018	425/325	[249–251]
DRO	China	DRO	Earth-Moon space exploration	2022	#	#
TESS	NASA	2:1Resonant²	Search exoplanets	2018	362/317	[9–12]
LOP-G	NASA, Russia	NRHO²	Lunar station, Space gateway	#	#	[4–8]
JWST	NASA, ESA	L₂-Halo¹	Space telescope, Universe observatory	2021	6500/#	[3, 252, 253]
IXO	NASA, ESA, JAXA	L₂-Lissajous¹	International X-ray observatory	#	#	[254, 255]
Stellar imager	NASA	L₂-Halo¹	Interferometry of stellar surface	#	#	[177, 256–259]
Eddington	ESA	L₂-Halo¹	Earth-like planets, Stellar observations	2003*	#	[260–262]
Darwin	ESA	L₂-Halo¹	Search for life	2007*	#	[263]
TPF	NASA	L₂-Halo¹	Detecting planets	2007*	#	[264, 265]

Table 3

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Family	Around	Direction in RS	Direction in FS	Symmetric	Interpretation
a	L₃	Retrograde	Inexistence	x axis	Lyapunov, Halo
b	L₁	Retrograde	Inexistence	x axis	Lyapunov, Halo
c	L₂	Retrograde	Inexistence	x axis	Lyapunov, Halo
d	L₄	—	—	—	Inexistence for μ = 0.5
e	L₅	—	—	—	Inexistence for μ = 0.5
f	m₁	Retrograde	Direct	x axis	Earth orbit
g	m₁	Direct	Direct	x axis	Earth orbit
h	m₂	Retrograde	Direct	x axis	DRO
i	m₂	Direct	Direct	x axis	DPO
k	m₁,m₂	Direct	Direct	x axis	See [33]
l	m₁,m₂	Retrograde	Direct	x axis	See [33]
m	m₁,m₂	Retrograde	Retrograde	x axis	See [33]
n	—	Retrograde	—	Asymmetric	See [33]
o	—	Retrograde	—	Asymmetric	Asymptotic
r	—	Retrograde	—	y axis	Asymptotic

Reference title	Principal contributor	Year	General description	μ
n-body problem	Newton	1687	Solution to the n-body problem	Optional
Circular restricted	Euler	1772	Formulation	Optional
Three-body problem	Lagrange	1772	Equilibrium solutions	Optional
	Jacobi	1836	Jacobi constant	Optional
	Hill	1877–1878	Motion of the Moon	0
	Poincare	1892–1899	Existence of periodic orbit	Optional
Copenhagen category	Darwin	1897–1910	Families of periodic orbits	1/11
	Moulton	1900–1917		1/5,1/2
	Stromgren	1913–1939		1/2
Periodic lunar orbit	Egorov	1957	Families of periodic orbits	~0.012
	Newton	1958
	Broucke	1962
	Huang	1962
	Arenstorf	1963
Motion around the triangular	Rabe	1961	Families of periodic orbits	~0.000 95
Motion around the triangular	Rabe	1962	Families of periodic orbits	~0.012
Lagrangian points	Deprit	1965	—	—
Lunar trajectories	Egorov	1957	Families of special non-periodic orbits	~0.012
	Thüring	1959
	Buchheim	1959
	Ehriche	1962
	Szebehely	1964
	Pierce	1965
	Standish	1965
Application to binary systems	Kuiper	1941	Families of non-periodic orbits	0.1–0.5
	Kopal	1956
	Abhyankar	1959
	Gould	1959
Additional periodic orbits	Message	1959	2:1 commensurability	~0.00095
	Deprit	1965	asymptotic-periodic	several
	Szebehely	1965	1:3 commensurability	0.2–0.24
	Knowles	1959	—	—
Application to Earth- Moon systems	Farquhar	1968–2017	Families of	0–0.5
	Howell	1984–2017	halo (NRHO) orbits
	Restrepo	2017	and all planar group

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