JOHANNES KEPLER
Johannes Kepler (1571-1630)
Johannes Kepler was born on
December 27, 1571 in Weil der Stadt in Swabia, a wine region in south
west Germany not far from France. His paternal grandfather, Sebald
Kepler, was a respected craftsman who served as mayor of the city;
his maternal grandfather Melchior Guldenmann, was an innkeeper and
mayor of the nearby village of Eltingen. His father, Heinrich Kepler,
was "an immoral, rough and quarrelsome soldier," according
to Kepler, and he described his mother in similar unflattering terms.
As a seven-month child, Kepler was
sickly from birth, he contracted smallpox. His vision was severely
defective, and he had various other illnesses throughout childhood,
some of which may have been hypochondria.
From 1547 to 1576 Johannes lived
with his grandparents; in 1576 his parents moved to nearby Leonberg,
where Johannes entered the Latin school. In 1584 he entered the
Protestant seminary at Adelberg, and in 1589 he began his university
education and the Protestant university of Tubingen. At Tubingen he
studied mainly theology and philosophy, but also mathematics and
astronomy. At the university, Kepler's exceptional intellectual
abilities became apparent. Kepler's teacher in mathematical subjects
was Michael Maestlin, who Kepler admired greatly. Maestlin was one of
the earliest astronomers to believe in Copernicus's heliocentric
theory, the theory that the sun not the earth was the center of the
universe and that all planets including the earth orbited it,
although he did not teach it to his students because Martin Luther
denounced it and he would lose his job if he did. He was forced to
teach the Ptolemaic system or the geocentric theory which said that
the earth was the center of the universe and that everything orbited
the earth, this was named after Ptolemy, the man who first arrived at
this conclusion. Maestlin, however, was able to influence some of his
students to subscribe to the Copernican system, among them was Kepler.
After graduation Kepler was
offered a professorship of astronomy in faraway Graz (in the Austrian
province in Styria), where he went in 1594. One of his duties of this
professorship was to make astrological predictions. Despite earlier
failures at predictions, he predicted a cold winter, and an invasion
by the Turks. Both predictions turned out to be correct, he was
treated with new respect, and his salary was raised.
While lecturing to his math class
in Graz, contemplating some geometric figure involving concentric,
having a common center, circles and triangles on the blackboard,
Kepler suddenly realized that the figures of the type shown (Illus.
3-1) determine a definite fixed ratio between the sizes of the two
circles, provided the triangle has all sides equal, and a different
ratio of sizes will occur for a square between the two circles,
another for a regular, having all sides equal, pentagon, and so on.
He thought this might be the key
to the solar system. He truly believed in the Copernican system, so
he saw the planetary orbits as six concentric circles (only six
planets had been discovered then), meaning the planets revolve around
the sun and have the sun as there common center but have different
radii or distances to the sun. Disappointingly, he found it just
didn't work out---the ratios were wrong. Then he had a real
inspiration. The universe was really three-dimensional, and instead
of thinking of circles, he should be thinking about spheres, with the
planetary orbits being along the equators. Thinking in three
dimensions, the analogue of the above diagram (Illus. 3-1) would be
two concentric spheres with a tetrahedron, or a pyramid shaped four
planed figure, between them, so that the outer sphere passes through
the vertices, or points, of the tetrahedron, and the inner sphere
touches all its sides, but is completely contained in the
tetrahedron. There were just six planets, so five spaces between
spheres, and there are just five regular solids. Thus, if the
distances came out right, the theory provided a complete explanation,
in terms of a geometric model, of why there were just six planets,
and why they are spaced as we find them. Actually, the distances
still didn't come out right, especially for Jupiter, but Kepler was
so sure of the rightness of his work, that he blamed the
discrepancies on errors in Copernicus' tables. He titled his work
Mysterium Cosmographicum--the mystery of the universe (explained).
The crucial illustration from his book (also his model) is shown
(Illus. 4-1), the outer sphere being the orbit of Saturn.
Except for Mercury, Kepler's
construction produced remarkably accurate results. Because of his
talent as a mathematician, displayed in his work and his book, Kepler
was invited by the great Tycho Brahe to Prague to become his
assistant and calculate new orbits from Tycho's observations. Kepler
moved to Prague in 1600.
Kepler and Brahe did not get along
well. Brahe apparently mistrusted Kepler, fearing that his bright
young assistant might eclipse him as the prominent astronomer of his
day. He therefore only let Kepler see part of his numerous data.
He set Kepler to the task of
understanding the orbit of the planet Mars, which was particularly
troublesome. It is believed that part of the reason 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. Ironically, 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. When Brahe died in 1601, Kepler stole the
data Brahe had been keeping from him, and began to work with it.
Once Kepler had secured Tycho's
data, he set himself to the task of once and for all determining the
exact orbit of Mars. A preliminary analysis showed the orbit to be
very close to a circle, of radius about 142 million miles, but the
sun was not at the center of the circle---it was at a point 13
million miles away from the center. Also, it was clear that Mars
varied in speed as it went around this orbit, moving fastest when it
was closest to the sun (at perihelion) and slowest when it was
furthest from the sun (aphelion). Everybody (including Kepler)
believed that the motion of planets must be a simple steady motion,
or at least made up of simple steady motions, if only it were looked
at in the right way. They wondered how the motion of Mars described
above could be seen as some kind of steady motion.
Actually, a possible solution to
this problem had been given long before, by Ptolemy. The method was
to introduce another point, called the equant, on the line through
the sun and the center of the circular orbit, the equant being on the
opposite side of the center from the sun.
The idea is to try to position
this point so that the planet moves around the equant at a steady
angular speed. This steady motion about the equant is somewhat
believable, because the planet is observed to be moving slowest when
it is furthest from the sun, which is when it is closest to the
equant, and vice versa, so if you imagine a spoke going out from the
equant point to the planet and sweeping around with the planet, maybe
this spoke could be turning at a steady rate.
Ptolemy had shown that
observations of the movement of Mars in its orbit were in fact well
accounted for by a model of this sort, with the equant point the same
distance from the center as the sun, but on the other side (as in
Illus. 5-1). Kepler, however, had far more accurate records of the
movement of Mars, and he was interested in seeing if the model still
held up under this closer scrutinizing. He found it didn't. Even
adjusting independently the radius of the orbit, the distance of the
sun from the center, and the distance of the equant from the center,
he found the best possible orbit of this type was still in error by
eight minutes of arc (8/60 of a degree) in accounting for the
observations. Such an error could not have been detected before
Tycho's work. Kepler knew Tycho's work was accurate to about one
minute, and so the model had to be thrown out.
Having thrown out the equant
model, though, it was difficult to see what to do next. The natural
thought for an astronomer at that time would have been to add an
epicycle, that is, to imagine Mars to be going in a small circular
orbit about a point which itself goes along the orbit shown above.
But Kepler didn't like that approach. the whole business with cycles
and epicycles was purely descriptive---trying to account for the
observed planetary motions with a suitable combination of circular
motions. Kepler, in contrast, was trying to think dynamically, that
is, to understand the planetary motions somehow in terms of a force
stemming from the sun sweeping them around in their orbits. Thinking
in those terms adding an epicycle looks unattractive---what force
could be pushing the planet around the small circle, which has
nothing at its center?
Kepler realized that to get the
kind of precision he needed in analyzing the orbit of Mars, he first
needed to have very accurate picture of the earth's orbit, since all
measurements of Mars' position were conducted from the earth, to pin
down Mars' position relative to the sun it was necessary to know the
earth's position relative to the sun to the required precision. He
wondered how he could pin down the earth's position in space
accurately. This is like being in a boat some distance from shore. If
you can see only one landmark, such as a lighthouse, and you have
both a compass and a map, that is not enough to really fix your
position, because you cannot tell very accurately just how far away
the lighthouse is. On the other hand, if you can see two landmarks,
in different directions, and measure with your compass the exact
directions they lie from your boat, that is enough to fix your
position exactly without guessing about distances. You just take out
your map, draw lines through the two landmarks on the map in the
direction your boat lies from each of them in turn, and the point
where the two lines intersect on the map is your location.
Essentially, this is just Thales' method used in geometry-- the two
landmarks form the base of a triangle, and we know the direction of
the boat from the two ends of the base, so we can construct a
triangle on this base with the boat at the other vertex. Just knowing
the angle between the two lines isn't enough, we have to know their
individual directions relative to the base, which is what we can find
with the map and the compass.
The idea was to use this same
technique repeatedly to find the location of the earth, and thereby
map out its orbit. The problem was, we need two fixed lighthouses to
form the base, and we only had one, the sun. The fixed stars wouldn't
do, they were infinitely far away, and just play the role of the
compass, giving a fixed direction. Kepler solved the problem of the
second lighthouse by a very clever trick. He used Mars. Of course,
Mars is moving all the time, and the orbit of Mars is what he was
trying to find, so this didn't seem to be a promising approach. But
one thing Kepler did know is that if Mars was in a certain location
at a certain time, it would be in exactly that same place 687.1 days
later. Kepler was able to use Brahe's volumes of data to find the
exact direction of Mars from the earth at a whole series of times at
687.1 day intervals. By finding the direction of Mars and that of the
sun at those times, he had a steady Mars-sun base to use in
constructing the earth's orbit.
In contrast to the orbit of Mars,
Kepler found the earth's orbit to be essentially a perfect circle.
(It is actually off by about one part in 10,000.) However, the center
of the circle is about 1.5 million miles away from the sun, and the
speed of the earth in its orbit varies, being greatest at the closest
approach to the sun. At the furthest point, the earth is 94.5 million
miles from the sun, and it is moving around its orbit at a speed of
18.2 miles per second. At the point of closest approach to the sun,
the earth is 91.4 million miles from the sun, and moving around at a
speed of 18.8 miles per second. Kepler noticed that there was an
interesting relationship among these numbers. The ratio of speeds,
18.8/18.2= 1.03, is the inverse of the ratio of the corresponding
distances, 91.4/94.5= 1/1.03. Kepler's interpretation of this was
that the force he believed to be emanating from the sun, pushing the
planets around, was weaker at the greater distance, and that was why
the earth was being pushed more slowly.
This relationship between speeds
and distances at the extreme points of the orbit enabled Kepler to
develop his second law of planetary motion (described later). Along
with his knowledge about the orbit of the earth, he was able to give
a sufficiently precise account of the earth's position in space as a
function of time to be able to go on to the main business, the
plotting of the more interesting Martian orbit.
Kepler knew the orbit of Mars was
not a circle, in fact he had plotted it and found it to be and oval
shape that could be fitted inside a circle as shown here, and
deviated from the circle by at most 0.00429 of the oval's
half-breadth (half-width) MC, about one-half of one percent (see
Illus. 10-1).
This means the ratio of
AC/MC=1.00429. Kepler's figure was constructed directly from Tycho's
data. He also measured the angle CMS. Mars subtended on the baseline
consisting of the sun and the center of the orbit when the planet was
in the position shown. The value of this angle was 5 18' (5 degrees
18 minutes). He stumbled entirely by chance on the fact that the
ratio of lengths SM/Cm, was 1.00429.
Kepler felt that this could not
just be a coincidence--there must be a similar relationship between
angle SMC and the distance from the sun at all points on the orbit.
He found from data that this was so, but was still unable to figure
out what the curve must be. Still, he had stared at his plot long
enough to believe that the curve was and ellipse, with the sun at one
focus, and he then constructed an ellipse by a different approach. At
this point it dawned on him that his original analysis also led to an ellipse.
In fact, with hindsight, it is not
difficult to show how the numerical coincidence Kepler encountered
follows for an ellipse. Kepler found AC/MC=1.00429=MS/MC. This meant AC=MS.
The analysis of the Martian orbit,
with many wrong turnings and dead ends, took Kepler six years and
thousands of pages of calculations. It led to two simple laws.
