1. 1 Class #22 Celestial engineering Central Forces DVD The power of Equivalent 1-D problem and Pseudopotential  Kepler’s 3 rd law Orbits and Energy  The earth-moon flywheel Energy and Eccentricity :02

2. 2 Two particles with central forces

3. 3 The power of :20

4. 4 The power of - part 2 :25

5. 5 Equivalent 1-D problem :30 Relative Lagrangian Radial equation Total Radial Force

6. 6 Pseudopotential and Energy :35

7. 7 Pseudopotential and Energy :37

8. 8 Earth and Moon 1. What is the reduced mass for the earth-moon system (in kg)? 2. How many percent different is it than the lunar mass 3. What is theta-dot? 4. What is the radius of a circular orbit? 5. How would this change if the earth were fixed in space by the hand of God or a Borg tractor beam?

9. 9 Orbital Energy :47 1. Last class we derived values for omega, mu and r for the earth- moon system 2. Total energy consumption on Earth is 1000 Terajoules/day 3. If we could power human activites by stealing the angular kinetic energy from the earth-moon system, how much should omega change to give us 100 years of power?

10. 10 Converting. :30

11. 11 Properties of ellipses :30

12. 12 E, L and Eccentricity :30 The physics is in E and L. Epsilon is purely a geometrical factor. Epsilon equation applies to ALL conic sections (hyperbolae, ellipses, parabolas).

13. 13 Energy and Eccentricity :30 EccentricityEnergyOrbit E<0 Circle E<0 Ellipse E=0 Parabola E>0 hyperbola

14. 14 Kepler’s 1 st, 2 nd and 3 rd laws (1610) :37 1 st Law – Planets move in ellipses with sun at one focus Third law demonstrated previously relates period to semi-minor radius 2 nd law is direct consequence of momentum conservation “Equal areas are swept out in equal times” True for ALL central forces

15. 15 E, L and Eccentricity :30 The physics is in E and L, and it transfers to quantum mechanics.

16. 16 Predicting collisions :37 Case All objects hit the earth, regardless of initial distance. At turning point

17. 17 Predicting collisions :37 Case Need

18. 18 Predicting collisions :37

19. 19 Predicting collisions :37 For v-infinity comparable to v-escape, R-impact is approx same as R-earth. For v-infinity < v-escape, R-impact is many times R- earth

20. 20 Planetary Scattering Angle :37