CHAPTER 4 | THE ORIGIN OF MODERN ASTRONOMY 65
Kepler’s fi rst law says that the orbits of the
planets around the sun are ellipses with the sun
at one focus. Th anks to the precision of Tycho’s
observations and the sophistication of Kepler’s
mathematics, Kepler was able to recognize the
elliptical shape of the orbits even though they
are nearly circular. Mercury has the most ellipti-
cal orbit, but even it deviates only slightly from
a circle (■ Figure 4-15).
Kepler’s second law says that an imaginary
line drawn from the planet to the sun always
sweeps over equal areas in equal intervals of
time. Th is means that when the planet is closer
to the sun and the line connecting it to the sun
is shorter, the planet moves more rapidly, and
the line sweeps over the same area that is swept
over when the planet is farther from the sun.
You can see how the planet in Figure 4-15
would move from point A to point B in one
month, sweeping over the area shown. But
when the planet is farther from the sun, one
month’s motion would be shorter, from A’ to B’,
and the area swept out would be the same.
Kepler’s third law relates a planet’s orbital period to its aver-
age distance from the sun. Th e orbital period, P, is the time a
planet takes to travel around the sun once. Its average distance
from the sun turns out to equal the semimajor axis of its orbit,
a. Kepler’s third law says that a planet’s orbital period squared is
proportional to the semimajor axis of its orbit cubed. Measuring
P in years and a in astronomical units, you can summarize the
third law as
P^2 y a^3 AU
For example, Jupiter’s average distance from the sun is roughly
5.2 AU. Th e semimajor axis cubed is about 140, so the period
must be the square root of 140, which equals just under
12 years.
Notice that Kepler’s three laws are empirical. Th at is, they
describe a phenomenon without explaining why it occurs. Kepler
derived the laws from Tycho’s extensive observations, not from
any fi rst principle, fundamental assumption, or theory. In fact,
Kepler never knew what held the planets in their orbits or why
they continued to move around the sun.The Rudolphine Tables
Kepler continued his mathematical work on Th e Rudolphine
Tables, and at last, in 1627, it was ready. He fi nanced the print-
ing himself, dedicating the book to the memory of Tycho Brahe.
In fact, Tycho’s name appears in larger type on the title page
than Kepler’s own. Th is is surprising because the tables were not
based on the Tyconic system but on the heliocentric model of
Copernicus and the elliptical orbits of Kepler. Th e reason forFocus FocusStringaKeep the string taut,
and the pencil point
will follow an ellipse.The sun is at one
focus, but the other
focus is empty.■ Figure 4-14
The geometry of elliptical orbits: Drawing an ellipse with two tacks and a
loop of string is easy. The semimajor axis, a, is half of the longest diameter.
The sun lies at one of the foci of the elliptical orbit of a planet.
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Cop■ Table 4-1 ❙ Kepler’s Laws of Planetary
MotionI. The orbits of the planets are ellipses with the sun at one
focus.
II. A line from a planet to the sun sweeps over equal areas in
equal intervals of time.
III. A planet’s orbital period squared is proportional to its average
distance from the sun cubed:
P^2 yr = a^3 AUTh e eccentricity of an ellipse tells you its shape; if e is nearly equal
to one, the ellipse is very elongated. If e is closer to zero, the
ellipse is more circular. To draw a circle with the string and tacks
shown Figure 4-14, you would have to move the two thumbtacks
together because a circle is really just an ellipse with eccentricity
equal to zero. Try fi ddling with real thumbtacks and string, and
you’ll be surprised how easy it is to draw graceful, smooth ellipses
with various eccentricities.
Ellipses are a prominent part of Kepler’s three fundamental
rules of planetary motion. Th ose rules have been tested and con-
fi rmed so many times that astronomers now refer to them as
natural laws (How Do We Know? 4-2). Th ey are com-
monly called Kepler’s laws of planetary motion (■ Table 4-1).