While Kepler was working as an astronomer and
mathematician in Austria, he
was required to compose an astrological calendar. On the calendar he included
his predictions for a cold winter, and an invasion by the Turks. (At the time
of Kepler’s life, the studies of astronomy and mathematics were very
interchangeable, and it was not at all unusual for a mathematician to create an
astrological calendar.) When Keplers’ calendar was published, and his
pre-dictions came true, the public went wild for him. He was revered by the
people as a prophet, and for the rest of his life, he would turn to creating
astrological charts for extra income. (Kepler did not come from a wealthy
family, and money was always a problem for him. His astrological charts allowed
him to continue his mathematical research and still collect a small amount of
money to survive on.) Kepler did not think particularly highly of the
astrological pursuit, and, in fact, referred to it as “the foolish little
daughter of astronomy.” (Hawking 629) He even felt that the public was equally
foolish for holding it in such high regard. Kepler was quoted as saying “if ever
astrologers are correct, it ought to be credited to luck.” (Hawking 629)
However, Kepler had no problem using the “foolish” pursuit to his own
advantage, whenever money was tight. (In fact, he died while on a journey to
collect money from a chart he created.)
One day while Kepler was teaching, he drew on the
blackboard a circle with an equilateral triangle inside of it. He then drew
another circle in the center of 
the triangle.
It is said that the drawing was a revelation for him, and that it changed the
course of his life’s journey. Somehow, from the drawing, he had the idea that
“the ratio of the circles was indicative of the ratio of the orbits of Saturn
and Jupiter. Inspired by this revelation, he assumed that all six planets known
at the time were arranged around the sun in such a way that the geometric
figures would fit perfectly between them.” (Hawking 629) Kepler initially
tested his theory using two-dimensional figures (square, pentagon, triangle),
but was unsuccessful. He then tested his theory using Pythagorean solids (cube,
tetrahedron, dodecahedron, icosahedron, and octahedron) which are the only five
solids that can be constructed from regular geometric figures. He reasoned that
was why there were only six planets, because they had five spaces between them,
and that the geometric figures explained the difference in the sizes of the
spaces. From this theory, he wrote a book called Mysterium Cosmographicum (Mystery
of the Cosmos), which was published in 1596. As mentioned previously,
Kepler agreed with Copernicus’ heliocentric theory of the planets revolving 
around
the sun. However, Copernicus’ theory had the sun somewhere near the middle of
the orbits, but Kepler’s theory put it directly in the center. (Kepler was
deeply religious and he was determined to understand how God designed the
universe. Based on his beliefs, he placed the sun at the center of the orbit,
and supported his theory by reasoning that God would have put the sun in the
middle because the revolution of the planets was central to God’s design. It
turns out that Kepler’s theory was incorrect, but his conclusions were
accurate, and played a large part in shaping the course of modern science.)
After Kepler’s book was published, he sent copies of it to
other mathematicians and astronomers, such as Galileo and Tycho Brahe (a
wealthy Danish astronomer who held the position of Imperial Mathematician.)
Galileo rejected the work because he felt it was speculative, but Tycho Brahe
found Kepler’s ideas to be exciting. Brahe wrote to Kepler and invited him to
come to his castle in Prague,
where he could continue to work on his research. It is said that Kepler was wary
of Brahe, and was quoted as saying “My opinion of Tycho is this: he is
superlatively rich but he knows not how to make proper use of it, as is the
case with most rich people. Therefore, one must try to wrest his riches from
him.” (Hawking 631) Forever in pursuit of money, Kepler went to Prague to
work with Brahe.
The relationship between the two men was tentative at
best. Kepler was allowed to assist Brahe with work that was portioned out to
him, but he was not allowed access to Brahe’s extensive observational data.
(This highly frustrated Kepler, who wanted to be treated as an equal, and
allowed to do independent research.) Brahe had been studying the planet Mars
and kept very detailed records of its movements. Brahe did not find popularity
in his theories, but his careful research allowed Kepler to make further
mathematical distinctions regarding orbits and motion. Kepler continued to
search for an explanation of the harmony and movements of the planets. Although
still deeply religious, Kepler did not allow religious dictations to force his
mathematical computations into unnatural order (unlike Brahe.)
Kepler was eventually assigned to study the orbit of Mars,
because it had an orbit that was not entirely circular, and that had been a
mystery to Brahe. Only a year and half into the working relationship, Brahe
became ill and died suddenly. Being promoted to Imperial Mathematician, and
being quite the opportunist, Kepler quickly stole all of Brahe’s data. Kepler
himself wrote “I confess that when Tycho died, I quickly took advantage of the
absence, or lack of circumspection, of the heirs, by taking the observations
under my care, or perhaps usurping them.” (Hawking 631) Kepler stole thirty
years worth of Brahe’s planetary observations, and used the information to
compile his Rudolphine Tables (published
in 1627.) Rather than continuing the work using the theories of Tycho Brahe,
Kepler chose to use the data to support his own studies in predicting planetary
orbits. After eight years of continued research, Kepler was eventually able to
prove that the work of Copernicus was incorrect in the theory that planetary
orbits were circular. Through his study of the orbit of Mars, he was able to
discover that the orbit was not just an irregular circle, but more importantly,
that it was a precise ellipse. Kepler realized that all of the planetary orbits
were, in fact, elliptical. This discovery led Kepler to write the first two of
his three laws of planetary motion, and in 1605, he announced them publicly.
(In 1609, the first two laws were published in his book Astronomia Nova.)
Kepler’s first law of planetary motion is the law of ellipses:
The above diagram
shows that the sun is not at the center of the ellipse, but rather, at one of
the foci. As the months change and the planet continues along its elliptical
orbit, its position changes from being closer to the sun, to being farther away
from the sun. This means that the Sun-Planet distance is constantly changing.
For any ellipse,
there are two points contained within it called foci. The sum of the
distances to the foci from any point
on the ellipse is a constant. The equation that defines the ellipse in
terms of distances a and b is:
a + b = constant
An ellipse is a
circle that has been slightly flattened. The flattening of the circle is
referred to as “eccentricity.” When a circle is full, it has an “eccentricity”
of zero, and when the circle has become completely flattened into an ellipse,
it has an “eccentricity” of 1. Therefore the eccentricity of all ellipses lies
on a scale between Zero and 1. (The elliptical orbit of a planet is only very
slight, and would not look as extreme as the egg shapes in the diagram below.)
Equation for an ellipse:
Kepler’s second law of planetary motion is the law of equal areas:
Kepler’s second law shows that because of the change
in speed and distance affected by the orbit, the planet will cover equal areas
in equal time. (The gravitational pull of the Sun will cause the orbit to
accelerate, allowing it to cover the same distance as it would when it is farthest
from the Sun and moving slower.)
After Kepler published his first two laws, he was forced
to leave Prague when
it came under imminent threat of religious upheaval. Kepler returned to Austria, and
in 1618 published Harmonies of the World,
in which he detailed his third planetary law.
Kepler’s third law of planetary motion is the law of squares:
Equation
for the ratio of squares:
Kepler’s
third law explains that the period of time that it takes for a planet to orbit
the Sun increases rapidly with the expanded radius of the planets’ orbit. For example:
The innermost planet of Mercury will orbit the Sun in 88 days, but in contrast,
the outermost planet of Pluto will take 248 years to orbit
the Sun (see calculations following.)
In the ratio of
squares, the letter P represents the
“period of revolution for a planet” and the letter R
represents the “length of its semimajor axis. The subscripts of 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.” (http://csep10.phys.utk.edu)
“The long axis of the ellipse is called the major axis, while the short
axis is called the minor axis.
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.” (http://csep10.phys.utk.edu)
Following
are calculations using the third law’s ratio of squares equation:
“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
A.U.” (http://csep10.phys.utk.edu) The denominators
from the equation of the ratio of squares are equal, so the equation can be
represented as follows:
This
equation can be solved for “the period P of the planet, given the length of the
semimajor axis”:
or it can be solved for the “length of the
semimajor axis, given the period P of the planet”
(http://csep10.phys.utk.edu):
The following
is a calculation of the radius of the orbit of Mars (radius refers to the
length of the semimajor axis of the orbit.) The time for Mars to
orbit the Sun has been recorded as 1.88 Earth years. If we insert 1.88 into the
equation where years are represented, the length of the semimajor axis for the
orbit of Mars comes out to be 1.52 A.U. This answer (1.52 A.U.) has been
recorded as being the exact measured average distance from the Sun to Mars.
The
planet Pluto has been recorded to have an average separation from the Sun of
39.44 Astronomical Units. If we insert 39.44 into the equation where A.U. is
represented, the number of years turns out to be 248.
After
Kepler published his three laws, he proclaimed that they applied not only to
Mars, but to all planets, including Earth. In truth, Kepler never actually
verified that any of his laws applied to other planets – his proclamation was
just conjecture. However, it turns out that he was right, and that his laws not
only apply to all planets, but also to comets, airplane flight, and satellite
orbits. Keplers’ work helped to prove that the theological explanation for
circular perfection was incorrect, and that the scientific idea of the orbits
being mystical no longer held true. In his time, Kepler had no way of knowing
that the mathematical laws of science were universal, and that one day they
would be readily accepted as truth.
Kepler never figured out why the planets move,
but for the remainder of his life, he continued to search for the cause. At
some point, Kepler vaguely touched on the notion that human bodies have a
magnetic attraction, and extrapolated that the Sun may have an attractive force
on the planets in much the same manner. Kepler did not uncover the laws of
gravity in his lifetime, but he did pave the way for Isaac Newton to formulate
his theory of motion involving gravity as the cause of planetary motion.
Kepler’s contributions to science did not receive much notoriety during the
time of his life, but his work has contributed greatly to the modern scientific
age.
Works Cited
Kepler, Johannes. Biography from Leaders of the
Information Age (2003) Copyright (c) by The H. W.
Wilson Company. All rights reserved.
(credit:
images, as noted.)
Hawking,
Stephen.
“Johannes Kepler: His Life and Work.” On The Shoulders of Giants. Ed.
Stephen Hawking. Pennsylvania: Running Press, 2002.
627-723
Kepler, Johannes. Harmonies
of the World, Book Five. Anapolis: St. John’s Bookstore, 1939.
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html,
03/20/04
(credit:
image of Johannes Kepler on page 1 and other images, as noted.)
http://home.cvc.org/science/kepler.htm,
03/20/04
(credit:
images, as noted.)
http://story.news.yahoo.com/news?tmpl=story&cid=96&ncid=753&e=10&u=/space/20040317/sc_space/distantsednaraisespossibilityofanotherearthsizedplanetinoursolarsystem,
03/27/04
http://mathworld.wolfram.com/EquilateralTriangle.html, 03/27/04
(credit:
image, as noted.)
Leanne Collins 
