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KEPLERS LAWS

People have observed the movements of the planets, stars, and other celestial bodies

for thousands of years. In early history, scientists regarded the Earth as the centerofthe Universe. This so-called geocentric model was elaborated and formalized by the

Greek astronomer Claudius Ptolemy (c. 100c. 170) in the second century A.D. and

was accepted for the next 1 400 years. In 1543 the Polish astronomer Nicolaus

Copernicus (14731543) suggested that the Earth and the other planets revolved in

circular orbits around the Sun (the heliocentric model).

The Danish astronomer Tycho Brahe (1546

1601) wanted to determine how the heavens were

constructed, and thus he developed a program to

determine thepositions of both stars and planets. It

is interesting to note that those observationsof the

planets and 777 stars visible to the naked eye were

carried out with only alarge sextant and a compass.

(The telescope had not yet been invented.)

The German astronomer Johannes Kepler

was Brahes assistant for a short while before

Brahes death, whereupon he acquired his mentors

astronomical data and spent 16 years trying to

deduce a mathematical model for the motion ofthe

planets. Such data are difficult to sort out because

the Earth is also in motion around the Sun. After

many laborious calculations, Kepler found that

Brahes dataon the revolution of Mars around the

Sun provided the answer.

Keplers analysis first showed that the concept of circular orbits around the Sun

had to be abandoned. He eventually discovered that the orbit of Mars could be

accurately described by an ellipse. Figure 14.5 shows the geometric description of an

ellipse. The longest dimension is called the major axis and is of length 2a, where a is

the semimajor axis. The shortest dimension is the minor axis, of length 2b, where bisthe semiminor axis. On either side of the center is a focal point, a distance cfrom the

center, where a2= b

2+ c

2. The Sun is located at one of the focal points of Marss

orbit. Kepler generalized his analysis to include the motions of all planets.

14. 4

Johannes Kepler Germanastronomer(15711630)The German

astronomer Johannes Kepler is bestknown for developing the laws ofplanetary motion based on the careful

observations of Tycho Brahe. (ArtResource).

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For more information about JohannesKepler, visit our Web site at

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The complete analysis is summarized in three statements known asKeplers laws:

1. All planets move in elliptical orbits with the Sun at one focal point.2.

The radius vector drawn from the Sun to a planet sweeps out equal areasin equal time intervals.

3. The square of the orbital period of any planet is proportional to the cubeof the semimajor axis of the elliptical orbit.

Most of the planetary orbits are close to

circular in shape; for example, the semimajor

and semiminor axes of the orbit of Mars differ

by only 0.4%. Mercury and Pluto have the most

elliptical orbits of the nine planets. In addition to

the planets, there are many asteroids and comets

orbiting the Sun that obey Keplers laws. Comet

Halley is such an object; it becomes visible

when it is close to the Sun every 76 years. Its

orbit is very elliptical, with a semiminor axis

76% smaller than its semimajor axis.

Although we do not prove it here, Keplers first law is a direct consequence of thefact that the gravitational force varies as 1/r

2. That is, under an inverse-square

gravitational-force law, the orbit of a planet can be shown mathematically to be an

ellipse with the Sun at one focal point. Indeed, half a century after Kepler developed

his laws, Newton demonstrated that these laws are a consequence of the gravitational

force that exists between any two masses. Newtons law of universal gravitation,

together with his development of the laws of motion, provides the basis for a full

mathematical solution to the motion of planets and satellites.

Figure 14.5 Plot of an ellipse.The

semimajor axis has a length a,and thesemiminor axis has alength b. Thefocal points are locatedat a distance cfrom the center,where a2 = b2 + c2.