187
convinced everyone of the sun-centered system 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 planetsí motion using the sun-
centered assumption. After 900 pages of calculations and many false starts
and dead-end ideas, Kepler finally synthesized the data into the following
three laws:
Keplerís elliptical orbit law: The planets orbit the sun in elliptical orbits
with the sun at one focus.
Keplerís equal-area law: The line connecting a planet to the sun sweeps
out equal areas in equal amounts of time.
Keplerís 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 planetsí orbits are ellipses rather than circles, most are very
close to being circular. The earthís orbit, for instance, is only flattened 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
Keplerís 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 r
3/2
,
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 wouldonly be proportional to r to the first
power. The more drastic dependence on r
3/2
means that the outer planets
must be moving more slowly than the inner planets.
10.2Newtonís Law of Gravity
The sunís force on the planets obeys an inverse square law.
Keplerís laws were a beautifully simple explanation of what the planets
did, but they didnít address why they moved as they did. Did the sun exert a
force that pulled a planet toward the center of its orbit, or, as suggested by
Descartes, were the planets circulating in a whirlpool of some unknown
liquid. Kepler, working in the Aristotelian tradition, hypothesized not just
an inward force exerted by the sun on the planet, but also a second force in
the direction of motion to keep the planet from slowing down. Some
speculated that the sun attracted the planets magnetically.
Once Newton had formulated his laws of motion and taught them to
some of his friends, they began trying to connect them to Keplerís laws. It
was clear now that an inward force would be needed to bend the planetsí
paths. This force was presumably an attraction between the sun and each
An ellipse is a circle that has been dis-
torted by shrinking and stretching
along perpendicular axes.
An ellipse can be constructed by tying
a string to two pins and drawing like
this with the pencil stretching the string
taut. Each pin constitutes one focus
of the ellipse.
If the time interval taken by the planet to move from P to Q is equal to the time
interval from R to S, then according to Kepler's equal-area law, the two shaded
areas are equal. The planet is moving faster during interval RS than it did during
PQ, which Newton later determined was due to the sun's gravitational force accel-
erating it. The equal-area law predicts exactly how much it will speed up.
sunP
Q
R
S
Section 10.2Newtonís Law of Gravity
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