The planets orbit the Sun in a counterclockwise
direction as viewed from above the Sun's north
pole, and the planets' orbits all are aligned
to what astronomers call the ecliptic plane.
The story of our greater understanding of
planetary motion could not be told if it were
not for the work of a German mathematician
named Johannes Kepler. Kepler lived in Graz,
Austria during the tumultuous early 17th century.
Due to religious and political difficulties
common during that era, Kepler was banished
from Graz on August 2nd, 1600.
Fortunately, an opportunity to work as an
assistant for the famous astronomer Tycho
Brahe presented itself and the young Kepler
moved his family from Graz 300 miles across
the Danube River to Brahe's home in Prague.
Tycho Brahe is credited with the most accurate
astronomical observations of his time and
was impressed with the studies of Kepler during
an earlier meeting. However, Brahe mistrusted
Kepler, fearing that his bright young intern
might eclipse him as the premier astronomer
of his day. He therefore led Kepler see only
part of his voluminous planetary data.
He set Kepler, the task of understanding the
orbit of the planet Mars, the movement of
which fit problematically into the universe
as described by Aristotle and Ptolemy. It
is believed that part of the motivation for
giving the Mars problem to Kepler was Brahe's
hope that its difficulty would occupy Kepler
while Brahe worked to perfect his own theory
of the solar system, which was based on a
geocentric model, where the earth is the center
of the solar system. Based on this model,
the planets Mercury, Venus, Mars, Jupiter,
and Saturn all orbit the Sun, which in turn
orbits the earth. As it turned out, Kepler,
unlike Brahe, believed firmly in the Copernican
model of the solar system known as heliocentric,
which correctly placed the Sun at its center.
But the reason Mars' orbit was problematic
was because the Copernican system incorrectly
assumed the orbits of the planets to be circular.
After much struggling, Kepler was forced to
an eventual realization that the orbits of
the planets are not circles, but were instead
the elongated or flattened circles that geometers
call ellipses, and the particular difficulties
Brahe hand with the movement of Mars were
due to the fact that its orbit was the most
elliptical of the planets for which Brahe
had extensive data. Thus, in a twist of irony,
Brahe unwittingly gave Kepler the very part
of his data that would enable Kepler to formulate
the correct theory of the solar system, banishing
Brahe's own theory.
Since the orbits of the planets are ellipses,
let us review three basic properties of ellipses.
The first property of an ellipse: an ellipse
is defined by two points, each called a focus,
and together called foci. The sum of the distances
to the foci from any point on the ellipse
is always a constant. The second property
of an ellipse: the amount of flattening of
the ellipse is called the eccentricity. The
flatter the ellipse, the more eccentric it
is. Each ellipse has an eccentricity with
a value between zero, a circle, and one, essentially
a flat line, technically called a parabola.
The third property of an ellipse: the longest
axis of the ellipse is called the major axis,
while the shortest axis is called the minor
axis. Half of the major axis is termed a semi
major axis. Knowing then that the orbits of
the planets are elliptical, johannes Kepler
formulated three laws of planetary motion,
which accurately described the motion of comets
as well. Kepler's First Law: each planet's
orbit about the Sun is an ellipse. The Sun's
center is always located at one focus of the
orbital ellipse. The Sun is at one focus.
The planet follows the ellipse in its orbit,
meaning that the planet to Sun distance is
constantly changing as the planet goes around
its orbit.
Kepler's Second Law: the imaginary line joining
a planet and the Sun's sweeps equal areas
of space during equal time intervals as the
planet orbits. Basically, that planets do
not move with constant speed along their orbits.
Rather, their speed varies so that the line
joining the centers of the Sun and the planet
sweeps out equal parts of an area in equal
times. The point of nearest approach of the
planet to the Sun is termed perihelion. The
point of greatest separation is aphelion,
hence by Kepler's Second Law, a planet is
moving fastest when it is at perihelion and
slowest at aphelion.
Kepler's Third Law: the squares of the orbital
periods of the planets are directly proportional
to the cubes of the semi major axes of their
orbits. 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. The earth
takes 365 days, while Saturn requires 10,759
days to do the same. Though Kepler hadn't
known about gravitation when he came up with
his three laws, they were instrumental in
Isaac Newton deriving his theory of universal
gravitation, which explains the unknown force
behind Kepler's Third Law.
Kepler and his theories were crucial in the
better understanding of our solar system dynamics
and as a springboard to newer theories that
more accurately approximate our planetary
orbits.
