1. A Dynamical Model—Co-orbit Restricted Problem,and its Application in Astronomy and Astronautics
Zhaohua Yi 1,2, Guangyu Li 1
Gerhard Heinzel 3, Oliver Jennrich 4
1. Purple Mountain Observatory ,CAS, Nanjing
2. Nanjing University
3. Max Planck Institute for Gravitational Physics
4. European Space Research and Technology Center

2. Astronomical Background
Asteroid Family; pointed out by Japanese astronomer Hirayama in 1940s.It is composed by some asteroids almost located in same orbit. In 1980s,Liesk and Williams(JPL) pointed out that there were more than 70 asteroid families in main band of minor planet.
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

3. Astronomical Background
Trojians and Greeks group in Jupiter’s orbit.
Co-orbit phenomena in KBO (Kuiper Band Objects).
Co-orbit satellites around major planets.
Arms in Galaxy.
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

4. Astronautical Background
Co-orbit satellite constellations of Earth.
LISA ( Laser Interferometer Space Antenna), 3 spacecraft formed as an equilateral triangle whose center of mass locates on earth’s orbit and moves on same orbit with earth.
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

5. Restricted Problem Classical restricted problem
1 Restricted 3-body problem Two large bodies moves on an orbit of 2-body problem (circle, ellipse, parabola, hyperbola), to study a massless body’s motion under the gravitation of 2 large bodies.
2 Problem of two fixed centers To study a body’s motion under the gravitation of two fixed bodies.
3 Hill’s problem.
4 Fatou’s problem.
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

6. Modern restricted problem
In 1983,USA astronomer V.Szybehely pointed out restricted N+K problem. There are N large bodies and K small bodies;To study any large body’s motion only consider the gravitation of all other large bodies,and to study any small body’s motion must consider the gravitation of all large bodies ;the gravitation of other small bodies may be considered according to the real situation.
To study the orbit of LISA may be looking as a 2+3 co-orbit restricted problem.
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

7. Co-orbit circular restricted 3-body problem
Let the earth (plus moon) E moves around the sun S on a circular orbit; and the orbital plane denotes xy-plane with x-axis located from origin S to E. This is a rotating coordinate system. At the origin of time, C, the center of mass of 3 spacecraft locates on earth’s orbit, and co-moves with earth. The motion of C is a planar problem. The equations of motion are:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

8. Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
Jacobi integral
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

9. Transforming to polar coordinate,
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

10. The undisturbing solution is co-orbital
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

11. The approximating disturbed solution:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

12. The approximating disturbed solution:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

13. The comparative result with précised numerical integration
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

14. The comparative result with précised numerical integration
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

15. The configuration of 3 spacecraft
At the origin of time, let Sc1,Sc2,Sc3 denote 3 spacecraft, and the original positions of them are shown in figure
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

16. Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
The orbiter elements of them can be calculated as:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

17. Orbital elements of 3 spacecrafta w W M0
SC1 1
270°
180°
SC2
267°.38503
31°.17427
61°.14603
SC3
272°.59502
146°.90162
298°.85374
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

18. Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
At first, the variation of armlengths be discussed in 2-body problem. The positon vectors of them are:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

19. The square of distance of Sc1 and Sc2:
The approximated development to 2 degrees of e and sin i are:
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

20. The optimal inclination angle can be calculated as
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

21. Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
The next work
To discuss more precise solution;
To study it on elliptic restricted 3-body problem;
To study stability of co-orbit solution;
To explore the application to astronomical and other astronautical problems
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing

22. Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
THANK YOU！
Zhaohua Yi： A Dynamical Model—Co-orbit Restricted Problem, ….
July 14-16, Beijing