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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 depends on the distance between the two points, f (compared with the length of the piece of string)
The eccentricity of an ellipse gives an indication of the difference between its major and minor axes
© David Hoult 2009
eccentricity = distance between foci / major axis
© David Hoult 2009
The eccentricity of the orbits of the planets is low; their orbits are very nearly circular orbits.
eccentricity = distance between foci / major axis
© 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 0.206 Venus 0.0068
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472
© David Hoult 2009
Mercury 0.206 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0485 Saturn 0.0556 Uranus 0.0472 Neptune 0.0086
© David Hoult 2009
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
© David Hoult 2009
...it can be shown that...
© David Hoult 2009
minor axismajor axis
= 1 - e2
where e is the eccentricity of the ellipse
© David Hoult 2009
minor axismajor 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 axismajor 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
T2
r3= a constant
© David Hoult 2009
F = Gr2
Mm
© David Hoult 2009
F = m r 2
F = Gr2
Mm
© David Hoult 2009
F = Gr2
Mm F = m r 2
© David Hoult 2009
F = Gr2
Mm F = m r 2
r2G M m m r 2
=
© David Hoult 2009
F = Gr2
Mm F = m r 2
= T
2
r2G M m m r 2
=
© David Hoult 2009
T2
r3= 42
GM
© David Hoult 2009
T2
r3= 42
GM
in which we see Kepler’s third law
© David Hoult 2009
