What is the Origin of our Moon? • Giant impactor theory – Mars-sized object collides with Earth – Our Moon is formed • Possible impactor formation: – Method: Planetesimal accretion – Location: 1 AU from Earth • Belbruno and Gott – Impactor formed at Sun-Earth L4 or L5 • Sun-Earth L4 or L5 planetessimal samples to confirm Moon origin Source: http://sdo.gsfc.nasa.gov/gallery/main.php?v=item&id=65 Source: http://www.nasa.gov/mission_pages/LRO/multimedia/moonimg_06.html L4 L4 Circular Restricted Three Body Problem (CR3BP) x • Two massive bodies, e.g., Sun and Earth, orbit about system center of Earth ˆ mass in circular orbits 60° x • Third body (significantly less massive), e.g., spacecraft, motion dictated 60° by gravity of other two bodies X x Earth • Example - Sun-Earth CR3BP: Sun Sun • Inertial View: Earth moves around Sun (dashed blue), Sun retains Xˆ orientation shown in Fig. 1. L5 • Rotating View (Common View): Sun and Earth retain orientation shown in Fig. 2. L5 Where and What are L4 and L5? Figure 1: Schematic of Sun-Earth system, Figure 2: Schematic of Sun- inertial view. Earth system, rotating view • Sun, Earth and L4 form triangle above Sun-Earth line (dashed green) in Figs. 1, 2, 3. Sun, Earth and L5 form triangle below the Sun-Earth line (see Figs. 1, 2, 3) • L4 travels 60 degrees ahead of Earth in orbit around Sun, L5 travels 60 DROs degrees behind (see Fig. 1) L4/L5 Short xˆ • L4 and L5 are linearly stable equilibrium points, i.e., if third mass is Period placed at either point (no velocity), it will remain there Lyapunovs 8 Periodic Orbits 10 x • Trajectory type in CR3BP: Third mass motion repeats after one period. • Periodic Orbit Family: Group of “similar” periodic orbits • Sample Families: Distant Retrograde Orbits (DROs) about Earth, L4/L5 Xˆ Short-Period Lyapunovs (see Fig. 3) Research Agenda • Design Newton-Raphson type differential corrector (numerical algorithm) to find orbit-to-orbit trajectory x 108 • Find transfer trajectories between Distant Retrograde Orbits (DROs) and L4/L5 Short-Period Lyapunovs Figure 3: Periodic orbits, rotating view. Distant Retrograde Orbits (DROs) • Determine transfer trajectory cost and time of flight (blue), L4/L5 Short-Period Lyapunov (green). Note: Sun and Earth are scaled for visual purposes (Sun x20, Earth x500). Background Image: NASA’s Solar Sentinels mission spacecraft.