Topics in Astronomy: Lesson 1

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   :   Orbital Mechanics: Kepler and his laws
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Orbital Mechanics: Kepler and his laws

If we had a sensational press in the Seventeenth Century, the hottest headline for 1601 could easily have read “Major Astronomical Observatory Closed, assistant chief astronomer absconds with priceless data.” Such was the case with the observatory of Tycho Brahe. Brahe had just died, and the Danish government had decided to close his astronomy observatory which had been producing observations since 1576. Johannes Kepler, his assistant claimed the data, which oddly enough, no one else wanted.

Kepler was one of the most brilliant mathematical minds of his century, and probably there was no better repository for the data than him. By rigorous mathematics, Kepler came to the conclusion that not only was the sun the center of the solar system, something that Copernicus had believed, but that the orbits around the sun were not circles, as Aristotle had believed, but were in fact ellipses.

Kepler then posited three laws dealing with orbits. The first simply stated that all orbits are ellipses. The second law is that equal areas are measured out in equal time lengths, and finally for us the most important law that the square of the period (the amount of time that it takes to complete a single orbit) is equal to the cube of the semi-major axis (roughly the average distance from the sun). This A figure for our solar system is measured in Astronomical Units, the distance from the Earth to the Sun. The formula is expressed as A^2=P^3.

Mars, for instance has a period of 1.88 years. What is its average distance from the Sun? The answer is 1.52 AU, or about 1.5 times the distance of the Earth. For those of you who have calculators that can compute cubes, it is an interesting exercise to complete the solar system. Mercury’s period is .24 year, Venus is .61, Jupiter is 11.86, Saturn is 29.46, Uranus is 84.01, Neptune is 164.79, and lonely Pluto is 248.54. If you have a calculator up to the task, go ahead and calculate the distance. Just multiply the number by itself and that product by itself again if you don’t have a cube function. I will provide the answers at the end of the lesson so that you can check yourselves.

The reason that Kepler’s laws were so useful is that with them combined with the idea that the Sun was the center, for the first time it was possible to determine with scientific accuracy the order of the planets, that is which is closest, and which is farthest from the sun.

A second effect of Kepler was that with his laws it was possible to explain retrograde motion of the outer planets. Planets outside of the orbit of the earth have a rather strange aspect to their orbits. For a while they cease moving against the background stars, and appear to move backward for a while, then again stopping and starting in the original direction.

Explore the Planetary Orbits site for illustrations of these laws and then return to this page.

At the same time that Kepler was busy discovering his laws a gentleman by the name of Galileo Galilei was also busy. Galileo had acquired a top secret military weapon called a telescope. The telescope was a military instrument in that its heretofore sole use was for one general to use to look and see what another general was about to do to him. Galileo’s genius was to turn the telescope up and look at the heavens. One bright little light that particularly interested him was Jupiter. Galileo discovered four little dots of light that seemed to revolve around the larger disk of Jupiter. He named these Medician Moons in honor of his sponsors the Medici family. Now classical Aristotelian theory stated that all objects orbited around the Earth, yet clearly here were four objects that orbited around Jupiter. If Aristotle was wrong about these four, could he have been in error about others? This attack combined with the mathematical attack of Kepler was the beginning of the end for Aristotle’s theories about the cosmos.

For the practical portion of this lesson, you will need to use the shareware program Skyglobe or Starry Night that you downloaded during Orientation. If you haven’t downloaded the software and become familiar with it, please do that now. Please remember, shareware is not freeware, if you like the program, be honest and pay the fee and register it. After you are running, enter your location or the location closest to where you are. This will give you a map view from an area close to your home. Play with the program for a while and get the feel of it. After you have become familiar with the program, make sure you can see both the constellations and the planets. What planets are visible in the sky at this time from your home? Jump the time by a month at a time and observe Venus, Jupiter, and Saturn.

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Questions

Please email answers to these questions to Janet Hallmark

1. Keeping in mind Kepler’s laws, explain how we know that Jupiter is farther out from the Sun than Venus.

2. From your observations using Skyglobe or Starry Night, describe the theory by which retrograde motion can be explained. Hint: Only planets outside the orbit of the Earth exhibit retrograde motion. Is it possible that the placement of the planets has something to do with retrograde motion?

3. How could you explain retrograde motion if the Earth was the center of the solar system? Could it be explained at all?

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Math Answers

Answers to distance problems for self-checking:

: Mercury .39AU
Venus .72AU
Mars 1.88AU
Jupiter 5.2AU
Saturn 9.54AU
Uranus 19.19AU
Neptune 30.06AU
Pluto 39.53AU

 Supernova
Remnants --
photo by
NASA

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Reminder of Optional Activity: You may use the starchart you printed from the downloaded software or information from other sources to locate and observe objects in the night sky.

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About this Site

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webspinner: Janet Hallmark
© 1998 -- 2006 jhallmark@hughes.net
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Netword: Janet Hallmark

K.B. & Janet Hallmark's photos taken at Loblolly Observatory. The Loblolly Observatory logo was created by Pogebait. NASA images used with permission for educational purposes. Some clipart images by Absolute Web Graphics Archive. If I have forgotten to credit any clipartist for his/her work, I apologize and will gladly make corrections as soon as notified. Special thanks to K.B. Hallmark III, Subject Matter Expert for this course. K.B. holds a Masters' Degree in Astronomy at the University of Western Sydney in Nepean, Australia. Janet has received a Masters' Degree in Education at the University of Houston -- Clear Lake, Texas and a Educational Specialist Degree at Nova Southeastern University, Fort Lauderdale, Florida.

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