The Purple Mountain Observatory and the Department of Astronomy, Nanjing University, in Nanjing China-Research, cooperating with the Albert Einstein institute, in Hannover, Germany, put forward a model of the planar co-orbital circular restricted three-body problem, gave the equations of motion, a set of approximation formulas, and an approximate semi-analytical solution. The study is reported in Issue 53, No. 1 of *SCIENCE CHINA Physics*, Mechanics & Astronomy because of its significant research value.

As a branch of the famous three-body problem in the field of celestial mechanics, the co-orbital motion of natural bodies were paid attention to by astronomers a long time ago. In the year 1918 the Japanese astronomer T. Hirayama found the orbital elements of some asteroids close to each other. He proposed that asteroids with three orbital elements semi-major axis a, eccentricity e, and orbital inclination i close to each other, be called co-orbital. The most well-known cases of co-orbits in natural objects are Trojan groups of asteroids, of which the number of the registered asteroids already found has exceeded 2900. There also exist co-orbital phenomena both in the motion of stars in the same spiral arm of the Milky Way galaxy, and in the motion of natural and man-made satellites.

The planned gravitational wave observatory LISA is regarded approximately as an example of co-orbital circular restricted three-body problem. A rough method, proposed by the authors, has been used to design and optimize the LISA orbit with a 10 years designed life.

After further analysis and study, the authors greatly improved their previous method. And the new method has extended the applicable region from 10 years to over the timescale of 1000 years for most of the range of the initial values. As a typical example, they found that the barycenter of the LISA constellation moves by a horse-shoe-shaped orbit around the Sun near the Earth’s orbit during a period of 468 years: first, along the neighboring outer circumference of the horse-shoe-shaped orbit during 233.7 years, then in the neighborhood along the inner circumference of the horse-shoe-shape during 234.4 years.

The authors think their work is a new development in the research field of restricted three-body problem, it would be extended to the study of the stability of this kind of orbit and the effect of orbital eccentricity of planets, the exploration of a more accurate analytical solution, the application to the study of the orbital evolution of the Trojan asteroids, the application to the study of the orbits of the near-earth asteroids, the application to the study of the orbit design of spacecraft being approximately co-orbital, and the investigation of the motion of co-orbital stellar families in the Milky Way galaxy.

Purple Mountain Observatory is one of the significant research institutes in the field of celestial mechanics. The group, headed by Prof. LI GuangYu, cooperating with Prof. YI ZhaoHua in Naming University and Researcher Gerhard HEINZEL in AEI, Germany, has obtained great achievements in the field of NEO exploration, planetary and lunar ephemeris, and orbit design and optimization of the planetary spacecraft.

Funding from the National Natural Science Foundation of China (Grant No. 10503013, 10933004), and the Foundation of Minor Planets of Purple Mountain Observatory supported this research.

Yi Z H, Li G Y, Heinzel G, et al. The co-orbital restricted three-body problem and its application. *Sci China Phys* Mech Astron, 2010, 53: 171-178

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