1. Kepler’s Laws

2. Announcements We have 6 classes left and topics left to cover are:
Simple Harmonic motion
Pendulums
Then we will have revision classes until the end of term

3. Goal of the class To understand Kepler’s laws for solar systems
Question of the day: How do we calculate the eccentricity of an orbit?

4. Johannes Kepler Discovered three laws of planetary motion
Johannes Kepler (German: [ˈkʰɛplɐ]; December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer. A key figure in the 17th century scientific revolution, he is best known for his laws of planetary motion, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.
So Kepler's laws are part of the foundation of modern astronomy and physics.[3]
Johannes Kepler

5. Kepler’s First Law
Planets move in elliptical orbits with the sun at one focus.
X2/a2 + y2/b2 = 1
AF1 + AF2 = 2a
c/a=e eccentricity
If e=0 we get a circle
0<=e<=1

6. Kepler’s Second Law
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
Torque exerted on a planet by Force of gravity
τ = rxF = 0
Therefore L=constant
L=r x mv
dA/dt = L/2m = constant

7. xkcd

8. Kepler’s Third Law
The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
F= G Mm/(RE+h)2 = mv2/r
v=(GME/r)1/2
v=2πr/T
Calculate the height of a geosynchronous (geostationary) orbit. h=3.6x107 m

9. Kepler’s Third Law

10. Mass of the Sun Use Kepler’s laws to find the mass of the sun.
RE = 149.6x109m = 1AU
T=1yr = 31.56x106 s
T2 = 4π2 R3 / GMs
Reaarange and sub in numbers
P391 in book

11. Problem A 1000 kg satellite orbits the earth at an altitude of 200 km.
Find the satellite’s speed and time period
How much energy must be added to the system to move the satellite into a circular orbit with altitude 400 km.