1. Space Mission Design: Interplanetary Super Highway Hyerim Kim Jan. 12 th 2013 1 st SPACE Retreat

2. Contents Introduction Restricted Three-Body Problem – Lagrange’s Three-Body Solution (General Conic Solutions) – Circular Restricted Three-Body Problem (CR3BP) – Interplanetary Super Highway Applications to Space Mission Design

3. Introduction Classical Solution of Two-Body Problem Orbital elements, Kepler’s law, Geometry of conic solution Classical methods are inadequate to support the complex space mission planned for the near future!

4. Lagrange’s Three-Body Solution General Conic Solutions  Assumptions “It is possible to find solutions to the three-body problem where the shape of the three-body formation does not change in time. In general case, the size or orientation of the fixed three-body formation is free to vary with time.“  Analytical Solutions for the Restricted 3BP 1)Equilateral Triangle Solution 2)Collinear Formation Solution

5. Lagrange’s Three-Body Solution General Conic Solutions By Assumption, we have two conditions

6. Lagrange’s Three-Body Solution General Conic Solutions Express the position vector r i in rotating reference frames and substitute to the angular momentum vector H. Calculate the resultant force vector F i on each mass and comopute with previous three equations of motion. Only Two geometric Solutions 1)Equilateral Triangle 2)Collinear formation

7. Lagrange Solution: Equilateral Triangle Equilateral Triangle Solution The triangle is rotating at some variable angular velocity and the size of the equilateral triangle may be time varying.

8. Lagrange Solution: Equilateral Triangle Example 1. Each side of the equilateral triangle has an initial length of 10 9 m. The initial velocity vector of each mass forms a 40 deg angle with the respective radial position vector. Initial position vectors:

9. Lagrange Solution: Equilateral Triangle

11. Lagrange Solution: Collinear Collinear Formation Solution Assuming the three bodies are aligend along a rotating stragith line, then the rotating vecotrs will be collinear. To simplify solving for the relative distances, the scalar quantity is intorduced as:

12. Lagrange Solution: Collinear Example 2. Scalar parmeter χ is 0.451027 and the scalar distance x 12 is chosen to be 10 9 m. Each velocity vector initially forms a 40 deg angle with the respective radial position vector from the system center of mass. Initial position vectors:

13. Lagrange Solution: Collinear

14. CR3BP Circular Restricted Three-Body Problem Assumed that both m 1 and m 2 are very massive objects compared to the theirs mass m 3. The small mass m 3 could be the Apollo s/c or asteroids and comets moving under the influence of the sun and some planet. Five Lagrange’s Points = libration points (L1, L2, L3, L4, L5)

15. CR3BP These Lagrange points are generate interplanetary super highway!! Note that Earth-Moon Lagrange Points: LL1, LL2, …, LL5 Sun-Earth Lagrange Points: EL1, EL2,…, EL5

16. Interplanetary Super Highway Solar system is interconnected by a vast system of tunnels winding around the Sun. Invariant manifolds are generated by the Lagrange Points of all the planets and their moons and are found to play an important role in the formulation of solar system. It could be exploited to obtain low-energy interplanetary spacecraft trajectories.

17. Interplanetary Super Highway EARTH EARTH L 2 HALO ORBIT MOO N LUNAR L 1 HALO ORBIT LUNAR L 2 HALO ORBIT LUNAR L 1 GATEW AY Close look of invariant manifolds tube Green: Stable Manifolds Red: Unstable Manifolds

18. Interplanetary Super Highway Homoclinic, Heteroclinic Chain Systems

19. Interplanetary Super Highway Take a longer time than usual But significantly small delta V is need!! (almost no fuel needs during entire trip) New Paradigm of Trajectory Design

20. Interplanetary Super Highway

21. Genesis Discovery Mission (NASA, 2001) Lunar Orbit L1L1 L2L2 Halo Orbit Portal Earth

22. References Papers –Dynamical Systems, the Three-Body Problem and Space Mission Design, W. S. Koon, M. Lo et al., 2006 –Low-Thrust Transfers in the Earth-Moon Systems, Including Applications to Libration Point Orbits, M. T. Ozimek and K. C. Howell, Journal of Guidance, Control and Dynamics, 2010 –Control Strategies for Formation Flight in the Vicinity of the Libration Points, K. C. Howell and B. G. Marchand, AAS, 2003 Books –Analytical Mechanics of Space Systems, H. Schaub and J. Junkins, 2003 –Spacecraft trajectory optimization, Bruce A Conway, 2010 Webpage –Martin Lo’s webpage: http://www.gg.caltech.edu/~mwl