Laws of Planetary Motion
In the interplay between quantitative observation and theoretical construction that characterizes the development of modern science, we have seen that Brahe was the master of the first but was deficient in the second. The next great development in the history of astronomy was the theoretical intuition of Johannes Kepler (1571-1630), a German who went to Prague to become Brahe's assistant.
Brahe's Data and Kepler
Kepler and Brahe did not get along well. Brahe apparently mistrusted Kepler,
fearing that his bright young assistant might eclipse him as the premiere
astonomer of his day. He therefore let Kepler see only part of his voluminous
data.
He set Kepler the task of understanding the orbit of the planet Mars, which was particularly troublesome. It is believed that part of the motivation for giving the Mars problem to Kepler was that it was difficult, and Brahe hoped it would occupy Kepler while Brahe worked on his theory of the Solar System. In a supreme irony, it was precisely the Martian data that allowed Kepler to formulate the correct laws of planetary motion, thus eventually achieving a place in the development of astronomy far surpassing that of Brahe.
Kepler and the Elliptical Orbits
Unlike Brahe, Kepler believed firmly in the Copernican system. In retrospect, the reason that the orbit of Mars was particularly difficult was that Copernicus had correctly placed the Sun at the center of the Solar System, but had erred in assuming the orbits of the planets to be circles. Thus, in the Copernican theory epicycles were still required to explain the details of planetary motion.
It fell to
Kepler to provide the final piece of the puzzle: after a long struggle, in
which he tried mightily to avoid his eventual conclusion, Kepler was
forced finally to the realization that
the orbits of the planets were not the circles demanded by Aristotle and
assumed implicitly by Copernicus, but were instead
the "flattened circles" that geometers call
ellipses
(See adjacent figure; the planetary orbits are only slightly
elliptical and are not as flattened as in this example.)
The irony noted above lies in the realization that the difficulties with the Martian orbit derive precisely from the fact that the orbit of Mars was the most elliptical of the planets for which Brahe had extensive data. Thus Brahe had unwittingly given Kepler the very part of his data that would allow Kepler to eventually formulate the correct theory of the Solar System and thereby to banish Brahe's own theory!
Some Properties of Ellipses
Since the orbits of the planets are ellipses, let us review a few basic properties of ellipses.
1. For an ellipse there are
two points called foci (singular: focus) such that the
sum of the distances to the foci from any point on the ellipse is a constant.
In terms of the diagram shown to the left,
with "x" marking the location of the foci, we
have the equation
a + b = constant
that defines the ellipse in terms of the distances a and b.2. The amount of "flattening" of the ellipse is termed the eccentricity. Thus, in the following figure the ellipses become more eccentric from left to right. A circle may be viewed as a special case of an ellipse with zero eccentricity, while as the ellipse becomes more flattened the eccentricity approaches one. Thus, all ellipses have eccentricities lying between zero and one.
3. The long axis of the ellipse is called the major axis, while the
short axis is called the minor axis (adjacent figure). Half of the
major axis is termed a semimajor axis. The
length of a semimajor axis is often termed the size of the ellipse. It can
be shown that the average separation of a planet from the Sun as it goes around
its elliptical orbit is equal to the length of the semimajor axis. Thus,
by the "radius" of a planet's orbit one usually means the length
of the semimajor axis. For a more detailed investigation of the properties of
ellipses, see this
Java applet.
The Laws of Planetary Motion
Kepler obtained Brahe's data after his death despite the attempts by Brahe's family to keep the data from him in the hope of monetary gain. There is some evidence that Kepler obtained the data by less than legal means; it is fortunate for the development of modern astronomy that he was successful. Utilizing the voluminous and precise data of Brahe, Kepler was eventually able to build on the realization that the orbits of the planets were ellipses to formulate his Three Laws of Planetary Motion.
Kepler's First Law:
| I. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. |
 |
Kepler's First Law is illustrated in the image shown above.
The Sun is not at the center of the ellipse, but is instead at one focus
(generally there is nothing at the other focus of the ellipse). The planet
then follows the ellipse in its orbit, which means that the Earth-Sun distance
is constantly changing as the planet goes around its orbit. For purpose of
illustration we have shown the orbit as rather eccentric; remember that the
actual orbits are much less eccentric than
this.
Kepler's Second Law:
| II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. |
 |
Kepler's second law is illustrated in the preceding figure.
The line joining the Sun and planet sweeps out equal areas in
equal times, so the planet moves faster when it is nearer the Sun. Thus,
a planet executes elliptical motion
with constantly changing angular speed as it moves about
its orbit.
The point of nearest approach of the planet to the Sun is termed
perihelion; the point of greatest separation
is termed aphelion. Hence, by Kepler's second law,
the planet moves fastest when it is
near perihelion and slowest when it is near aphelion.
Kepler's Third Law:
| III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes: |
 |
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.
Kepler's Third Law implies that the period for a planet to orbit the Sun increases
rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost
planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto)
requires 248 years to do the same.
Calculations Using Kepler's Third Law
A convenient unit of measurement for periods is in Earth years, and a convenient unit of measurement for distances is the average separation of the Earth from the Sun, which is termed an astronomical unit and is abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators in the preceding equation are numerically equal to unity and it may be written in the simple form
or for the length of the semimajor axis, given the period of the planet,
Some References
