Kepler’s Laws of Planetary Motion

A bit of History

• Tycho Brahe – Danish Astronomer 1547• He was given an island in the North Sea to set up an observatory. Tycho observed the stars for 20 years and collected data.

• The man with the gold replacement nose!

Johannes Kepler

Polish Astronomer born 1571

Interpreted Brahe’s data.

Kepler’s Laws

• 1st Law - All objects in the solar system travel in elliptical paths.

• An ellipse has two focal points.

• The sun is at one focal point.• The eccentricity of an ellipse is a ratio

of the two major axes.

Kepler’s Laws

• 2nd Law – All objects sweep out equal areas in equal times.

• S = Sun is at one of the focal points.

Kepler’s Laws

• 2nd Law – • Area of triangle

ASB(green) equals

area of triangle CSD(blue)

This is true because of the conservation of angular momentum.

Kepler’s Laws

• Point B is summer for New York

• Point D is winter for New York

• Is the speed of the Earth constant?

• When is the Earth moving fastest in its orbit around the sun?

Kepler’s Laws

• 3rd Law - R2/T3 is a constant for every object in the solar system moving around the sun.

• This is true because of Newton’s 2nd Law of motion

• F = G M1M2/r2 = M2v2

Sir Isaac Newton

• Describes why the planets behave the way they do – Gravity – It’s everywhere! Not just on the Earth!

Gravitation

• What variables affect the gravitational attraction between two objects.

1) Mass of object #1

Earth

This is a direct and linear relationship.

F

M earth

Gravitation

2) Mass of object #2

You

This is a direct and linear relationship.

F

M you

Gravitation

• 3) Distance between the centers of the two

objects - r.

This is an inverse square relationship.

F

r

Newton’s Law of Gravitational Attraction

• F = G M1M2/r2

• G = Universal Gravitational Constant =

6.7 x 10-11 N-m2/kg2

Newton’s Law of Universal Gravitation

• It is only an attractive force. • It acts at a distance.• It is an inverse square law.• It describes the gravitational

force that exists between all bodies.

Force of Attraction

• What is the force of attraction between a 50 Kg girl and a 70 Kg boy standing 1 meter apart?

Force of Attraction

• F = G M1M2/r2

• M = 50 Kg

• M = 70 Kg

• r = 1 m

• G = 6.7 x 10-11

• F = ?

The Cavendish Experiment -1798

• Proves that even small objects have gravitational attraction for each other.

Newton’s 2nd Law

• F = m2a = G MeM2/r2

So a = G Me/r2

This is the acceleration due to gravity on a planet with mass - Me and a radius of r.

Distance from the Earth’s Center

Local g

m/s/s

Local Weight w = mg

½ Re 39.2 2800 N

1 Re 9.8 700 N

2 Re 9.8/4 = 2.45 172 N

3 Re 9.8/9 = 1.1 77 N

4 Re 9.8/16 = 0.61 43 N

Suddenly, through forces not yet fully understood, Darren Belsky’s apartment became the center of a

new black hole.