Kepler’s Laws of Planetary Motion © David Hoult 2009

The eccentricity of an ellipse gives an indication of the difference between its major and minor axes

The eccentricity of an ellipse gives an indication of the difference between its major and minor axes The eccentricity depends on the distance between the two points, f (compared with the length of the piece of string)

eccentricity = distance between foci / major axis

eccentricity = distance between foci / major axis The eccentricity of the orbits of the planets is low; their orbits are very nearly circular orbits.

Law 1 Each planet orbits the sun in an elliptical path with the sun at one focus of the ellipse.

Mercury 0.206

Mercury 0.206 Venus 0.0068

Mercury 0.206 Venus 0.0068 Earth 0.0167

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 Neptune 0.0086

Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 Neptune 0.0086 Pluto 0.25

...it can be shown that...

minor axis = 1 - e2 major axis where e is the eccentricity of the ellipse

minor axis = 1 - e2 major axis where e is the eccentricity of the ellipse which means that even for the planet (?) with the most eccentric orbit, the ratio of minor to major axis is only about:

0.97 minor axis = 1 - e2 major axis where e is the eccentricity of the ellipse which means that even for the planet (?) with the most eccentric orbit, the ratio of minor to major axes is only about: 0.97

In calculations we will consider the orbits to be circular

Eccentricity of ellipse much exaggerated

Law 2 A line from the sun to a planet sweeps out equal areas in equal times.

Law 3 The square of the time period of a planet’s orbit is directly proportional to the cube of its mean distance from the sun.

r3 = a constant T2

Mm F = G r2

Mm F = G F = m r w2 r2

Mm F = G F = m r w2 r2

Mm F = G F = m r w2 r2 G M m m r w2 = r2

Mm F = G F = m r w2 r2 G M m m r w2 = r2 2 p w = T

r3 GM = T2 4p2

r3 GM = T2 4p2 in which we see Kepler’s third law