Solar System Dynamics: Orbits and Kepler’s Laws

The planets orbit the Sun in a counterclockwise
direction as viewed from above the Sun’s northpole, 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 werenot 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 difficultiescommon during that era, Kepler was banished
from Graz on August 2nd, 1600. Fortunately, an opportunity to work as an
assistant for the famous astronomer TychoBrahe presented itself and the young Kepler
moved his family from Graz 300 miles acrossthe Danube River to Brahe’s home in Prague.
Tycho Brahe is credited with the most accurateastronomical observations of his time and
was impressed with the studies of Kepler duringan earlier meeting. However, Brahe mistrusted
Kepler, fearing that his bright young internmight eclipse him as the premier astronomer
of his day. He therefore led Kepler see onlypart of his voluminous planetary data. He set Kepler, the task of understanding the
orbit of the planet Mars, the movement ofwhich fit problematically into the universe
as described by Aristotle and Ptolemy. Itis believed that part of the motivation for
giving the Mars problem to Kepler was Brahe’shope that its difficulty would occupy Kepler
while Brahe worked to perfect his own theoryof the solar system, which was based on a
geocentric model, where the earth is the centerof 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 problematicwas 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 ofthe planets are not circles, but were instead
the elongated or flattened circles that geometerscall ellipses, and the particular difficulties
Brahe hand with the movement of Mars weredue to the fact that its orbit was the most
elliptical of the planets for which Brahehad extensive data. Thus, in a twist of irony,
Brahe unwittingly gave Kepler the very partof his data that would enable Kepler to formulate
the correct theory of the solar system, banishingBrahe’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 ellipseis always a constant. The second property
of an ellipse: the amount of flattening ofthe ellipse is called the eccentricity. The
flatter the ellipse, the more eccentric itis. Each ellipse has an eccentricity with
a value between zero, a circle, and one, essentiallya 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 semimajor axis. Knowing then that the orbits of
the planets are elliptical, johannes Keplerformulated three laws of planetary motion,
which accurately described the motion of cometsas well. Kepler’s First Law: each planet’s
orbit about the Sun is an ellipse. The Sun’scenter 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 isconstantly changing as the planet goes around
its orbit. Kepler’s Second Law: the imaginary line joining
a planet and the Sun’s sweeps equal areasof space during equal time intervals as the
planet orbits. Basically, that planets donot move with constant speed along their orbits.
Rather, their speed varies so that the linejoining the centers of the Sun and the planet
sweeps out equal parts of an area in equaltimes. The point of nearest approach of the
planet to the Sun is termed perihelion. Thepoint of greatest separation is aphelion,
hence by Kepler’s Second Law, a planet ismoving 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 proportionalto the cubes of the semi major axes of their
orbits. Kepler’s Third Law implies that theperiod for a planet to orbit the Sun increases
rapidly with the radius of its orbit. Thuswe find that Mercury, the innermost planet,
takes only 88 days to orbit the Sun. The earthtakes 365 days, while Saturn requires 10,759
days to do the same. Though Kepler hadn’tknown about gravitation when he came up with
his three laws, they were instrumental inIsaac Newton deriving his theory of universal
gravitation, which explains the unknown forcebehind Kepler’s Third Law. Kepler and his theories were crucial in the
better understanding of our solar system dynamicsand as a springboard to newer theories that
more accurately approximate our planetaryorbits.

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