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 © 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 © David Hoult 2009
Mercury Venus © David Hoult 2009
Mercury Venus Earth © David Hoult 2009
Mercury Venus Earth Mars © David Hoult 2009
Mercury Venus Earth Mars Jupiter © David Hoult 2009
Mercury Venus Earth Mars Jupiter Saturn © David Hoult 2009
Mercury Venus Earth Mars Jupiter Saturn Uranus © David Hoult 2009
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune © David Hoult 2009
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 0.25 © David Hoult 2009
...it can be shown that... © David Hoult 2009
minor axis major axis = 1 - e 2 where e is the eccentricity of the ellipse © David Hoult 2009
minor axis major axis = 1 - e 2 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 - e 2 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
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
T2T2 r3r3 = a constant © David Hoult 2009
F = G r2r2 Mm © David Hoult 2009
F = m r 2m r 2 F = G r2r2 Mm © David Hoult 2009
F = G r2r2 Mm F = m r 2m r 2 © David Hoult 2009
F = G r2r2 Mm F = m r 2m r 2 r2r2 G M mG M m m r 2m r 2 = © David Hoult 2009
F = G r2r2 Mm F = m r 2m r 2 = T 2 2 r2r2 G M mG M m m r 2m r 2 = © David Hoult 2009
T2T2 r3r3 = 4242 GM © David Hoult 2009
T2T2 r3r3 = 4242 GM in which we see Kepler’s third law © David Hoult 2009