Kepler’s Laws

1. Kepler’s Laws • The planets move in ellipses with the sun at on focus. • Areas swept out by the radius vector from the sun to a planet in equal times are equal. • The square of the period of revolution is proportional to the cube of the semi-major axis.

2. Kepler’s Second Law 2. Areas swept out by the radius vector from the sun to a planet in equal times are equal. Angular momentum is conserved Gravity is a central force rdq r dq

3. Conservative Force Pseudo Potential Gravity as a Central Force Kepler’s 0th Law: Planets orbits are bound.

4. K>0 1 turning point K<0 & E>0 1 turning point K<0 & Emin<E<0 2 turning points “Potential” for 1/r2 Force

5. Kepler’s Laws • The planets move in ellipses with the sun at on focus. • Areas swept out by the radius vector from the sun to a planet in equal times are equal. • The square of the period of revolution is proportional to the cube of the semi-major axis. Is there another way to find path (r(f))?

6. Solution for Orbits Formal Solution to Gravity Problems To find orbit, need to combine r(t) and f(t). Can’t we find r(f) directly???

7. r(f) Trick Instead

8. Change Force Equation to u(f)

9. Equation for a conic section Solution

10. Conic Sections Standard Form + branch Hyperbola - branch Parabola Ellipse (Circle)

11. rminturning point if rmaxturning point if Solution (continued)

12. 0 at turning point Energy at Turning Points

13. Ellipse

14. Ellipse

15. Second Law Kepler’s Laws • The planets move in ellipses with the sun at on focus. • Areas swept out by the radius vector from the sun to a planet in equal times are equal. • The square of the period of revolution is proportional to the cube of the semi-major axis.

16. If then Kepler’s Third Law holds. Kepler’s Third Law

17. Force is attractive. Force is 1/r2. Force is radial; angular momentum is conserved. Force is proportional to mass. Kepler’s Laws • The planets orbits are bound. • The planets move in ellipses with the sun at on focus. • Areas swept out by the radius vector from the sun to a planet in equal times are equal. • Thesquare of the period of revolution is proportional to the cube of the semi-major axis.