Kepler worked for another decade looking for patterns in the motions of
the planets. Finally, in 1619, he published his third law - the harmonic
law:
The squares of the sidereal periods of the planets are proportional
to the cubes of the semi-major axes of their orbits.
We have seen that the semi-major axis of
a planet is equal to the mean distance of the planet, so an alternative
form of Kepler's third law is that the cube of the mean distance of a
planet is proportional to the square of its sidereal period.
figure 33:
Kepler's third law. Note that this figure is just a schematic
and so should not be used to infer either the sidereal period or
the semi-major axis of any of the planets.
With his third law, Kepler had derived a relationship between the sizes
of planetary orbits and their orbital periods. This relationship is shown
in Figure 33 - a log-log plot of semi-major
axis versus sidereal period for the planets of the solar system falls
very close to a straight line of slope 2/3. Hence if
a_{1} and T_{1} refer to the semi-major
axis and sidereal period of a planet P_{1} moving
about the Sun,
a_{1}^{3} / T_{1}^{2} =
constant,
the constant being the same for any of the planetary orbits. If
a_{2}, a_{3}, etc., and
T_{2}, T_{3}, etc., refer to the
semi-major axes and sidereal periods of the other planets
P_{2}, P_{3}, etc., moving
about the Sun, then
The most convenient form of the constant is obtained by taking the
planet to be the Earth and expressing the distance in astronomical units
and the time in years. Hence, for the Earth, a = 1 and
T = 1, so the constant becomes unity. For any other planet,
consequently,
a^{3} = T^{2},
showing that if we measure the sidereal period, T, of a
planet in Earth-years, we can obtain its mean distance from the Sun,
a, in AU, as shown in the example
problems.
By 1621, Kepler had shown that the four moons of Jupiter discovered
by Galileo also obeyed his third law, but with a different value of
the constant, confirming its wide applicability. As we shall see
when we discuss Newton's
derivation of Kepler's laws, however,
the so-called constant in Kepler's third law is not really a constant
but is dependent on the masses of the two celestial bodies in orbit about
each other. Nevertheless, when one of the bodies is significantly more
massive that the other, such as the Sun compared to the planets, or
Jupiter compared to its satellites, Kepler's third law is a very close
approximation to the truth. Only when
the outer-most retrograde satellites in the solar system are considered, or
close satellites of a non-spherical planet, do Kepler's laws fail to
describe in their usual highly accurate manner the behaviour of such bodies.
Even then, however, they may be used as a first approximation.