# 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.