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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 planets’ 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:

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 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

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 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.

10.2Newton’s Law of Gravity

The sun’s force on the planets obeys an inverse square law.

Kepler’s laws were a beautifully simple explanation of what the planets

did, but they didn’t 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 Kepler’s laws. It

was clear now that an inward force would be needed to bend the planets’

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 10.2Newton’s Law of Gravity