Newton wouldn’t have been able to gure out why the planets
b / Tycho Brahe made his name
as an astronomer by showing that
the bright new star, today called
a supernova, that appeared in
the skies in 1572 was far beyond
the Earth’s atmosphere. This,
along with Galileo’s discovery of
sunspots, showed that contrary
to Aristotle, the heavens were
not perfect and unchanging.
Brahe’s fame as an astronomer
brought him patronage from King
Frederick II, allowing him to carry
out his historic high-precision
measurements of the planets’
motions. A contradictory charac-
ter, Brahe enjoyed lecturing other
nobles about the evils of dueling,
but had lost his own nose in a
youthful duel and had it replaced
with a prosthesis made of an
alloy of gold and silver. Willing to
endure scandal in order to marry
a peasant, he nevertheless used
the feudal powers given to him by
the king to impose harsh forced
labor on the inhabitants of his
parishes. The result of their work,
an Italian-style palace with an
observatory on top, surely ranks
as one of the most luxurious
science labs ever built. He died
of a ruptured bladder after falling
from a wagon on the way home
from a party — in those days, it
was considered rude to leave the
dinner table to relieve oneself.
move the way they do if it hadn’t been for the astronomer Tycho
Brahe (1546-1601) and his protege Johannes Kepler (1571-1630),
who together came up with the rst simple and accurate description
of how the planets actually do move. The di culty of their task is
suggested by gure c, which shows how the relatively simple orbital
motions of the earth and Mars combine so that as seen from earth
Mars appears to be staggering in loops like a drunken sailor.
tensive data on the motions of the planets over a period of many
years, taking the giant step from the previous observations’ accuracy
of about 10 minutes of arc (10/60 of a degree) to an unprecedented
1 minute. The quality of his work is all the more remarkable consid-
ering that his observatory consisted of four giant brass protractors
mounted upright in his castle in Denmark. Four di erent observers
would simultaneously measure the position of a planet in order to
check for mistakes and reduce random errors.
With Brahe’s death, it fell to his former assistant Kepler to try
to make some sense out of the volumes of data. Kepler, in con-
tradiction to his late boss, had formed a prejudice, a correct one
as it turned out, in favor of the theory that the earth and planets
revolved around the sun, rather than the earth staying xed and
everything rotating about it. Although motion is relative, it is not
just a matter of opinion what circles what. The earth’s rotation
and revolution about the sun make it a noninertial reference frame,
which causes detectable violations of Newton’s laws when one at-
tempts to describe su ciently precise experiments in the earth- xed
frame. Although such direct experiments were not carried out until
the 19th century, what convinced everyone of the sun-centered sys-
tem in the 17th century was that Kepler was able to come up with
a surprisingly simple set of mathematical and geometrical rules for
describing the planets’ motion using the sun-centered assumption.
After 900 pages of calculations and many false starts and dead-end
ideas, Kepler nally synthesized the data into the following three
laws:
Kepler’s elliptical orbit law
The planets orbit the sun in elliptical orbits with the sun at
one focus.
Kepler’s equal-area law
The line connecting a planet to the sun sweeps out equal areas
in equal amounts of time.
Kepler’s law of periods
The time required for a planet to orbit the sun, called its
period, is proportional to the long axis of the ellipse raised to
the 3/2 power. The constant of proportionality is the same
for all the planets.
Although the planets’ orbits are ellipses rather than circles, most
are very close to being circular. The earth’s orbit, for instance, is
only attened by 1.7% relative to a circle. In the special case of a
planet in a circular orbit, the two foci (plural of “focus”) coincide
at the center of the circle, and Kepler’s elliptical orbit law thus says
that the circle is centered on the sun. The equal-area law implies
that a planet in a circular orbit moves around the sun with constant
speed. For a circular orbit, the law of periods then amounts to a
statement that the time for one orbit is proportional to r3/2, where
r is the radius. If all the planets were moving in their orbits at the
same speed, then the time for one orbit would simply depend on
the circumference of the circle, so it would only be proportional to
r to the rst power. The more drastic dependence on r3/2 means
that the outer planets must be moving more slowly than the inner
planets.
d / An ellipse is a circle that