Kepler's Third Law
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Kepler's Third Law
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The ratio of the squares of the
periods of revolution
for two planets is equal to the ratio of the cubes of their
semimajor axes.
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In this equation P represents the period of revolution for a
planet and R represents the length of its semimajor axis. The subscripts "1"
and "2" distinguish quantities for planet 1 and 2 respectively.
The periods
for the two planets are assumed to be in the same time units and the lengths
of the semimajor axes for the two planets are assumed to be in the same distance
units in this form of Kepler's third law.
Astronomical Units
Kepler's third law takes a particularly simple form if we agree to measure all periods
in Earth years and all distances in astronomical units. (An astronomical unit
is defined to be the average separation of the Earth and the Sun and is abbreviated AU.
It is approximately equal to 150 million kilometers.)
Kepler's third law for a planet can be stated as
P2 (years) = R3 (AU)
where the notation tells us explicitly that in this form of Kepler's third law
the period P for the planet must be measured in years and the
length of its semimajor axis
R in astronomical units. For example, Venus has an average separation from the
Sun of 0.7233 AU. By Kepler's third law, its period should be
P = R3/2 = (0.7233 AU)1.5 = 0.6151 years = 224.7 days
which is indeed the observed orbital period of Venus.
Testing Kepler's Third Law
The above right image shows a plot of the distance versus period for the planets,
with both quantities plotted on logarithmic scales. The
almost straight line represents the prediction of Kepler's third law
and the symbols represent the actual
data for the planets. One sees from this graph
that the planets fall almost exactly on the line and thus
obey Kepler's third law nearly perfectly.
Animations: Kepler's Laws
This
animation illustrates the difference in motion for elliptical and circular orbits of
the same average radius.
Here is an animation showing Kepler's laws in action:
the actual motion of the inner planets over a two-year period from 1994-1996.
Finally, this Java
applet will allow you
to investigate Kepler's laws quantitatively.