(10)  Kepler and his Laws  

"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web
          http://www.phy6.org/stargaze/Sintro.htm

• About Tycho Brahe, his work and his connection with Kepler.

• About Kepler and his laws--and introduction to the subject.

• About conic sections, qualitatively.

• The mathematical formulation of the third law, and its explicit form for artificial Earth satellites.

• The student will confirm Kepler's 3rd law by comparing orbital periods and mean distances for all major planets.

Terms: Conic sections, ellipse, parabola, hyperbola, astronomical unit (AU).

Stories and extras: Story of Tycho and his supernova, some details about Kepler's life.

   Today we continue the story of the discovery of the solar system. Copernicus, as was seen last time, gave the first logical explanation of the motion of planets in the sky--not just formulas describing those strange motions, but an idea of what the solar system looked like.

   Old habits are hard to break. Copernicus had assumed all planets moved at constant velocities along circles, centered on the Sun, because after all, wasn't the circle the perfect curve, and the Sun the center of it all?

   Kepler tried to test this, and luckily, he could use some very precise observations, made by Tycho Brahe--the most precise astronomical observations before the invention of the telescope. Assuming that the planets moved in circles as Copernicus had proposed, Kepler calculated their expected positions. They did not agree, and the expected positions differed from the observed ones. Kepler had to conclude that the world was not as perfect as Copernicus had suggested. The planets speeded up and slowed down. Their orbits were not exact circles, and the Sun was not at the center of their orbits. Searching for a more accurate representation, he deduced what we now call Kepler's laws.

   About 70 years later Newton showed that all 3 of these laws were a consequence of the laws of motion and of gravitation, which Newton himself was the first to formulate.

That is how science makes progress:

   One, you don't guess what nature should be ("because it is perfect"), but observe what actually occurs.

   And two, you calculate. Kepler had a thorough command of the math needed to calculate planetary motion. Without that he could not have succeeded.

   The story of Kepler begins however with Tycho Brahe, an arrogant Danish nobleman who was also a talented astronomer. (Continue with the material given on the web.)

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--Who was Tycho Brahe? and: What do you know about his nose? (Follow the link from the "Stargazers" section about him.)

Brahe was a Danish astronomer. He lost part of his nose in a duel and wore a false nose of gold-plated brass.

A bright "new star" appeared in the sky, never seen before. It was a nova, an erupting star. It was so bright it could be seen in the daytime.

Brahe made very precise measurements of the positions of stars.

Brahe used no telescope, none was known as yet. He used sharp eyes and a variety of graduated circles and of open sights, like the ones found on guns.

No, he did not agree with Copernicus. He recognized that the planets rotated around the Sun, but claimed the Sun rotated around Earth.

Kepler was an astronomer hired by Tycho to help analyze his observations.

Yes--Kepler believed the Sun was at the center of the solar system, that the planets revolved around the Sun and that the Earth, too, was such a planet.

About the shapes: Copernicus believed all planetary orbits were circles centered by the Sun.

About the motions:He believed that each planet moved along its orbit with constant speed. The greater the distance from the Sun, the slower was the motion.

He used the theory to calculate the positions in the sky where the planets were expected to be found.

Approximately yes, accurately, no. However, Ptolemy's theory did not predict them accurately, either.

Shape: that the orbits were ellipses--elongated circles [for the planets, very slightly elongated], and the Sun was at one focus--off center, shifted towards one end. That is Kepler's first law.

Motion: That the speed of a planet in its orbit depended on its distance from the Sun--the greater the distance from the Sun, the slower the motion. This relation could be expressed mathematically, and that expression was Kepler's second law.

If we cut a cone with a flat plane, an ellipse is one of the classes of cross-sections we may generate.

Yes, the circle is a special kind of ellipse. We get a circle if the cone is cut perpendicular to its axis.

Ellipses, parabolas, and hyperbolas.
[Demonstrate with a flashlight]

Cut the cone with a plane inclined less steeply than the straight lines which form the cone.

[Those lines, also called "generators", are like the poles which hold up an Native American lodge or "teepee. The "lodgepole pine" was particularly favored by the Indians for this use; it is a type of pine growing in the western US with straight thin trunks.]

Cut the cone with a plane parallel to the straight lines that form the cone.

[Orbits of non-periodic comets, the ones that appear unexpectedly, are often very close to parabolas; their sides become closer and closer to parallel as the distance gets larger. They come from the very edges of the solar system. Their orbits may really be very long ellipses, too close to parabolas to be told apart.

Periodic comets like Halley's, which moves in an elongated ellipse and returns every 75 years, presumably started that way, too, but were diverted by the pull of some planet, most likely by Jupiter, into elliptical orbits.]

Cut the cone with a plane inclined more steeply than the straight lines which form the cone.

[The sides of a hyperbola diverge at an angle: the graph y=12/x for instance is a hyperbola whose sides diverge at 90 degrees.

An object approaching the Sun in a hyperbolic orbit is probably coming from outside the solar system, and will never come back.]

He found a mathematical law, which connected the mean distance from the Sun and the time needed to complete one orbit. This was Kepler's third law.

The third law states that the square of the orbital period is proportional to the third power ("cube") of the average distance from the Sun.

Kepler's 3rd law states T² is proportional to a³. Call the factor of proportionality K. Then for two different planets, distinguished by subscripts 1 and 2, T₁² = K a₁³ and T₂² = Ka ₂³, with the same K. It follows that K= T₁²/a₁³ and also K = T₂²/a₂³. Setting the two expressions for K equal to each other, T₁²/a₁³ = T₂²/a₂³.

Yes, Kepler's laws apply to artificial satellites as well as planets.

It is still true. As Newton showed nearly 70 years later, any objects held in orbit by the gravity of a large central object follow Kepler's laws.

The pilgrims landed one year after Kepler announced his 3rd law, 11 years after his first laws.