Goals: The student will learn|
- 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.
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 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.)
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.)
Brahe was a Danish astronomer. He lost part of his nose in a duel and wore a false nose of gold-plated brass.
--What occurred in 1572 that started Brahe's interest in astronomy?
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.
--What was Brahe's main contribution to astronomy?
Brahe made very precise measurements of the positions of stars.
--What sort of telescope did Tycho use?
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.
--Did Tycho believe the teachings of Copernicus--that the Sun was at the center of the solar system?
No, he did not agree with Copernicus. He recognized that the planets rotated around the Sun, but claimed the Sun rotated around Earth.
--Who was Johannes Kepler?
Kepler was an astronomer hired by Tycho to help analyze his observations.
--Did Kepler believe the claims of Copernicus?
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.
--What did Copernicus assume about the shapes of the orbits of planets, and the motion of the planets along them?
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.
--How did Kepler test the theory of Copernicus?
He used the theory to calculate the positions in the sky where the planets were expected to be found.
--Did the theory of Copernicus predict the positions of the planets correctly?
Approximately yes, accurately, no. However, Ptolemy's theory did not predict them accurately, either.
--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?
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.
--We say the ellipse is "one of the conic sections." What does this mean?
If we cut a cone with a flat plane, an ellipse is one of the classes of cross-sections we may generate.
--Is the circle a conic section?
Yes, the circle is a special kind of ellipse. We get a circle if the cone is cut perpendicular to its axis.
--What kinds of conic sections do you know?
Ellipses, parabolas, and hyperbolas.
[Demonstrate with a flashlight]
--How do you cut a cone to produce an elliptic cross section?
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.]
--How do you cut a cone to produce a parabolic cross section?
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.]
--How to you cut a cone to produce a hyperbolic cross section?
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.]
--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?
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.
--What did Kepler's 3rd law say?
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.
--If two planets have average distances (a1, a2), and orbital periods (T1, T2), can you use the 3rd law to give a formula connecting (a1, a2, T1 , T2)?
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.
--If instead of "planets" we say "artificial satellites of the Earth" is the same statement still true?
Yes, Kepler's laws apply to artificial satellites as well as planets.
--What if instead of "planets" we say "an artificial satellite and the Moon"?
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.
--What came first: Kepler discovering his laws, or the "pilgrims" landing at Plymouth Rock in Massachusetts?
The pilgrims landed one year after Kepler announced his 3rd law, 11 years after his first laws.