Kepler’s Laws of Planetary Motion

1. Kepler’s Laws of Planetary Motion • The orbits of the planets are ellipses with the sun at one focus. c Eccentricity e = c/a

2. Elliptical orbits Parameters: perihelion Rp, aphelion Ra, semimajor axis a = (Rp+Ra)/2, eccentricity e Ra - Rp e = Ra + Rp

3. LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes: For the Earth P2 = 1 yr, a2 = 1 AU Note units!!

4. 0 A New Era of Science Mathematical foundation for physics

7. 0 Newton’s Laws of Motion (1) • A body continues at rest or in uniform motion in a straight line unless acted upon by some net force. An astronaut floating in space will continue to float forever in a straight line unless some external force is accelerating him/her.

8. Velocity and Acceleration Acceleration (a) is the change of a body’s velocity (v) with time (t): a a = Dv/Dt Velocity and acceleration are directed quantities (vectors)! v Different cases of acceleration: • Acceleration in the conventional sense (i.e. increasing speed) • Deceleration (i.e. decreasing speed) • Change of the direction of motion (e.g., in circular motion)

9. Newton’s Laws of Motion (2) • The accelerationa of a body is inversely proportional to its mass m, directly proportional to the net forceF, and in the same direction as the net force. a = F/m F = m a

11. Curved path  there must be some force! p. 63

12. Gravity: by far the most important force in the Universe m1 m2

13. 0 Newton’s Laws of Motion (3) • To every action, there is an equal and opposite reaction. The same force that is accelerating the boy forward, is accelerating the skateboard backward.

14. Understanding orbital motion No other force but gravity! Why Moon does not crash into Earth? Why Earth does not crash into the Sun?

16. Understanding Orbital Motion The universal law of gravity allows us to understand orbital motion of planets and moons: Example: • Earth and moon attract each other through gravitation. Dv • Since Earth is much more massive than the moon, the moon’s effect on Earth is small. v v’ • Earth’s gravitational force constantly accelerates the moon towards Earth. Moon F • This acceleration is constantly changing the moon’s direction of motion, holding it on its almost circular orbit. Earth

17. Center of Mass (SLIDESHOW MODE ONLY)

18. m1 m2 If m1 << m2, then d2 << d1

19. m r M Uniform circular motion On the Earth’s surface r = R = 6400 km; M = 6x1024 kg;

20. m r M Uniform circular motion - continued III Kepler’s law:

21. Orbital Motion (2) In order to stay on a closed orbit, an object has to be within a certain range of velocities: Too slow => Object falls back down to Earth Too fast => Object escapes Earth’s gravity

22. Newton’s Cannon (SLIDESHOW MODE ONLY)

23. Orbital Motion (3) Geosynchronous Orbits

24. Geosynchronous Orbit (SLIDESHOW MODE ONLY)

25. Escape condition: Kinetic Energy K  Gravitational Potential Energy U At threshold: Note: total energy E = K + U; E < 0 for bound orbits E  0 for unbound trajectories

26. Object Mass Escape velocity Ceres (largest asteroid) 1021 kg 0.64 km/s The Moon 7x1022 kg 2.38 km/s The Earth 6x1024 kg 11.2 km/s Jupiter 2x1027 kg 60 km/s The Sun 2x1030 kg 618 km/s

27. Celestial mechanics - summary It all started with Galileo!

28. Clockwork universe