(10) Kepler and his Laws 
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "Kepler and his Laws," section #10: on disk Skeplaws.htm, on the web http://www.phy6.org/stargaze/Skeplaws.htm
"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web 
Goals: The student will learn
Terms: Conic sections, ellipse, parabola, hyperbola, astronomical unit (AU). Stories and extras: Story of Tycho and his supernova, some details about Kepler's life.
Start of the lesson: 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 skynot 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 Brahethe 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.) " Guiding questions and additional tidbits (Suggested answers included) Who was Tycho Brahe? and: What do you know about his nose? (Follow the link from the "Stargazers" section about him.)
What occurred in 1572 that started Brahe's interest in astronomy?
What was Brahe's main contribution to astronomy?
What sort of telescope did Tycho use?
Did Tycho believe the teachings of Copernicusthat the Sun was at the center of the solar system?
Who was Johannes Kepler?
Did Kepler believe the claims of Copernicus?
What did Copernicus assume about the shapes of the orbits of planets, and the motion of the planets along them?
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.
How did Kepler test the theory of Copernicus?
Did the theory of Copernicus predict the positions of the planets correctly?
To explain the motion of the planets, what did Kepler assume in his first two laws about the shape of the orbits of planets, and the motion of the planets along them?
Motion: That the speed of a planet in its orbit depended on its distance from the Sunthe greater the distance from the Sun, the slower the motion. This relation could be expressed mathematically, and that expression was Kepler's second law. We say the ellipse is "one of the conic sections." What does this mean?
Is the circle a conic section?
What kinds of conic sections do you know?
[Demonstrate with a flashlight]
How do you cut a cone to produce an elliptic cross section?
[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.]
How do you cut a cone to produce a parabolic cross section?
[Orbits of nonperiodic 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.]
How to you cut a cone to produce a hyperbolic cross section?
[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.]
In fitting the observed motion of the planets to the theory of Copernicus, as modified by his own two laws, Kepler also had to estimate the relative size of the orbits. Copernicus already knew that the further away from the Sun a planet was, the slower it moved. How did Kepler improve on this?
What did Kepler's 3rd law say?
If two planets have average distances (a_{1}, a_{2}), and orbital periods (T_{1}, T_{2}), can you use the 3rd law to give a formula connecting (a_{1}, a_{2}, T_{1} , T_{2})?
If instead of "planets" we say "artificial satellites of the Earth" is the same statement still true?
What if instead of "planets" we say "an artificial satellite and the Moon"?
What came first: Kepler discovering his laws, or the "pilgrims" landing at Plymouth Rock in Massachusetts?

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Author and Curator: Dr. David P. Stern
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Last updated: 12.17.2001