1 Class #19 of 30 Celestial engineering - II Reduced 2-body problem Kepler 1 st, 2 nd and 3 rd laws Cometary collision prediction A whiff of scattering theory DVD – Gravitational slingshot :02

2 Reduced two-body problem :15

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

4 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

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

6 Predicting collisions :37 Case Need

7 Predicting collisions :37

8 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

9 Properties of ellipses :30

10 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).

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

12 Planetary Scattering Angle :37

13 Strong Force Scattering Angle :37 Quantum mechanics is needed, but… Particle physicists probe strong interactions by scattering particles just as we can probe gravity by scattering meteors

14 Problem L19-1 :30 Two rocks are scattering off the Earth’s gravitational field One has orbit with Epsilon= Other has orbit with Epsilon=3.0 How much are their paths deflected by the Earth? Sketch the two orbits.

15 Problem L19-2 :30 A 1000 kg rock is aimed in the vicinity of Earth Out beyond the orbit of pluto, it has a kinetic energy of 50 billion joules and an angular momentum of 96 trillion kg-m^2/s. Calculate epsilon Will the rock hit earth? If not, what is the minimum radius?

16 Class #19 Windup Epsilon is function of L and E and GMm Office hours today 3-5, tomorrow 4- 5:30 :60 KEPLER 1 st Law – Planets move in ellipses 2 nd law dA/dt=const 3 rd law Period goes as semi-minor radius to 3/2 power