Kepler’s Laws of Planetary
Motion
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
The eccentricity of an ellipse gives an indication of
the difference between its major and minor axes
© David Hoult 2009
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)
© David Hoult 2009
eccentricity = distance between foci / major axis
© David Hoult 2009
eccentricity = distance between foci / major axis
The eccentricity of the orbits of the planets is low;
their orbits are very nearly circular orbits.
© David Hoult 2009
Law 1
Each planet orbits the sun in an elliptical
path with the sun at one focus of the ellipse.
© David Hoult 2009
Mercury
0.206
© David Hoult 2009
Mercury
Venus
0.206
0.0068
© David Hoult 2009
Mercury
Venus
Earth
0.206
0.0068
0.0167
© David Hoult 2009
Mercury
Venus
Earth
Mars
0.206
0.0068
0.0167
0.0934
© David Hoult 2009
Mercury
Venus
Earth
Mars
Jupiter
0.206
0.0068
0.0167
0.0934
0.0485
© David Hoult 2009
Mercury
Venus
Earth
Mars
Jupiter
Saturn
0.206
0.0068
0.0167
0.0934
0.0485
0.0556
© David Hoult 2009
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
0.206
0.0068
0.0167
0.0934
0.0485
0.0556
0.0472
© David Hoult 2009
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
0.206
0.0068
0.0167
0.0934
0.0485
0.0556
0.0472
0.0086
© David Hoult 2009
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
0.206
0.0068
0.0167
0.0934
0.0485
0.0556
0.0472
0.0086
0.25
© David Hoult 2009
...it can be shown that...
© David Hoult 2009
minor axis
major axis
= 1 - e2
where e is the eccentricity of the ellipse
© David Hoult 2009
minor axis
major axis
= 1 - e2
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:
© David Hoult 2009
minor axis
major axis
= 1 - e2
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
© David Hoult 2009
In calculations we will consider the orbits
to be circular
© David Hoult 2009
Eccentricity of ellipse much exaggerated
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
© David Hoult 2009
Law 2
A line from the sun to a planet sweeps out
equal areas in equal times.
© David Hoult 2009
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.
© David Hoult 2009
r3
= a constant
T2
© David Hoult 2009
Mm
F = G
r2
© David Hoult 2009
Mm
F = G
r2
F = m r w2
© David Hoult 2009
Mm
F = G
r2
F = m r w2
© David Hoult 2009
Mm
F = G
r2
GMm
=
2
r
F = m r w2
m r w2
© David Hoult 2009
Mm
F = G
r2
F = m r w2
GMm
=
2
r
m r w2
2p
w =
T
© David Hoult 2009
r3
T2
=
GM
4p2
© David Hoult 2009
r3
T2
=
GM
4p2
in which we see Kepler’s third law
© David Hoult 2009