Copernican Revolution

 

 Reading: Chapter 6 - particularly pages 141-148

Review: Chapter 4, especially pages 105-110

 

The Heliocentric Model of the Solar System

While the Greek astronomer Aristarchus proposed that because the Sun is bigger than the Earth it might make sense for the Sun, rather than the Earth, to be at the center of the universe, the geocentric view held fast until Copernicus. Copernicus lived in the time of the Renaissance - when new ideas in science, philososphy and the arts were blossoming after the intellectual stagnation of the Dark Ages. Time was ripe for the Copernican Revolution!
 
 
(1) What was the apparent motion of Mars that early astronomers were struggling to explain with their models of the solar system?

(2) What was the actual motions in the models of (a) Ptolemy and (b) Copernicus that explained these apparent motions?

(3) Why was the distance of the stars an important issue in resolving the geocentric vs. heliocentric issue?

(4) Why did the early astronomers want to make the planets move in circles (rather than any other shapes)?

 
Ptolemy vs. Copernicus Computer Experiment: This Ptolemy vs. Copernicus site has some interactive material which beautifully illustrates how more accurate data (of the retrograde motion of Mars, obtained by Tycho Brahe) was able to distinguish between Ptolemy's epicycle model of the solar system and Kepler's ellipses. Unfortunately, you need to be running Netscape 4.0+ on a PC or Internet Explorer 4.5+on a Mac for the Java applets to work - and even so, they can be temperamental and may have problems.

 

Kepler's Laws of Planetary Motion

In 1619, Kepler published what he described as his major achievement, not a book on mathematics of astronomy but on the music of the heavens: Harmonice Mundi (Harmony of the Worlds). The divine law he had been seeking was uncovered at last: the velocities of the planets and their orbits are related to specific musical scales! Earlier astrologer-astronomers had assigned single musical notes to individual planets; but Kepler claimed to have realized a specific series of tones for each planet: Saturn, Jupiter, Mars, Earth, Venus and Mercury--as well as the Moon, "hic locum habet etiam" ("The Moon always occupies this position").
 
 
(5) Why did he not have musical tones for Uranus, Neptune or Pluto?

 

Getting back to our roots! For this session we will need squares, cubes and their roots. Here is a reminder of how they work and some limbering-up exercises - do these first if your math is a little rusty.

Kepler's 1st Law of Planetary Motion: Each planet moves in an ellipse with the Sun at one focus.
(6) If the Sun is at one focus, what is at the other? Anything?
Taking 2 thumb tacks and a piece of string, draw an ellipse as follows (there is another, rather better, diagram in Figure 6.8 on page 145): 

Draw another, varying the distance between the foci. (Yes, one focus, 2 foci--how do you pronounce this word?!)
(7) Measure the Major Axis (A) and the Minor Axis (B) of each of your ellipses:

(8) What is the value of each of the Semi-Major Axes? 

The eccentricity of an ellipse tells you how "squished" it is: ECCENTRICITY= e = (A-B) / A
(9) Sketch ellipses with eccentricities of e = 0 and e=10.

For comparison, the orbit of the Earth has an eccentricity of e=0.017 while that of Pluto is e=0.246.

Kepler's 2nd Law of Planetary Motion: The line between the Sun and the planet sweeps over equal areas in equal time intervals.
(10) Halley's Comet has a very eccentric orbit (e = 0.983). It orbits between 35 and 0.6 A.U, being visible for only about a year, when it is inside about 2 A.U. Hoes Kepler's 2nd law of planetary motion explain that the comet spends most of its 76 year orbit outside 5 A.U. and only a couple of years inside 5 A.U.?

Kepler's 3rd Law of Planetary Motion: The ratio of the cube of the semi-major axis (a3) to the square of the orbital period (P2) is the same for each planet.

 

For roughly circular orbits the semi-major axis is equal to the semi-minor axis--we can use the radius of the orbit for a. Even for eccentric orbits, if we measure a in A.U. and P in years we can say 
  | a  |3      |  P  |2
  |----|   =   |-----| 
  |A.U.|       |years|

(11) (a) A new asteroid (called Redloub) is discovered with a roughly circular orbit of radius 3 A.U. Use Kepler's 3rd law to derive its orbital period (in years). 
(b) A distance planet (called Revned) is found to have an orbital period of 150 years--how far from the Sun is it?
(12) The table below tests Kepler's 3rd law with current measurements of the planet's orbits. How accurate is Kepler's 3rd law? At which significant figure does the law fail? For which planet is it least accurate? Any guesses why?


Planet a (A.U.) P (years) a3 P2
Mercury 0.387 0.241 0.058 0.058
Venus 0.723 0.615 0.378 0.378
Earth 1.000 1.000 1.000 1.000
Mars 1.523 1.881 3.533 3.538
Jupiter 5.203 11.86 140.85 140.66
Saturn 9.539 29.46 867.98 867.89
Uranus 19.18 84.01 7055.79 7057.68
Neptune 30.06 164.8 27162.32 27159.04
Pluto 39.44 248.4 61349.46 61762.56

 
Kepler's Laws Computer Experiment: This is an exercise that leads you through Kepler's 3 Laws of planetary motion. Unfortunately, you need to be running Netscape 4.0 on a PC or Internet Explorer 4.5 on a Mac for the Java applets to work. If you are taking a lab course, you may be doing these in your lab period.
Here are a couple of web sites about these Famous Men of Astronomy:
  • Ptolemy - This shows a picture of Ptolemy - but it is probably just an artist's imagination since no-one probably knows what he really looked like. 
  • C opernic us - This is about the life and achievements of Copernicus. 
  • Tycho Brahe - A picture and some info about Tycho Brahe - but does not show his silver nose, unfortunately. 
  • Kepler - A page about Kepler.

Galileo*: The Birth of Modern Astronomy

 

For centuries people had used lens-shaped pieces of glass to counteract short-sightedness or magnify writing on a page. (Incidentally, "lens" comes from the Latin word for "lentil"). In the early 1600s, it was realized that using 2 lenses you could build a device that would allow you to see far ("tele-scope"). In Italy, like others in different parts of Europe, Galileo built telescopes and pointed them at the sky. His telescope was about 1 1/2 inches in diameter, about a foot long and magnified distant objects by about a factor of 3. He might have been left alone if he had stuck with pointing the telescopes at distant ships. Instead, Galileo looked at various objects in the sky which convinced him--but not the Inquisition--that Copernicus was right and the church doctrine incorrect. Astronomy was a dangerous topic in those days!

*Galileo Galilei - Galileo for short - Tycho Brahe is also known by his first name. Odd lot these astronomers.

Above is a drawing Galileo made of the collection of stars call the Pleiades (or Subaru in Japanese). With his telescope, Galileo was able to detect many more stars than the "seven sisters" that you can see with a naked eye.

Click on the words to see a Drawing of Spots on the Sun

Galileo was not the first to discover sunspots, but Galileo was the first to make a convincing case that the spots were actually on the Sun, not material passing in front of the Sun. This drawing shows the "blemishes" on the Sun's surface, illustrating that the Sun is not a "perfect, flawless sphere" as proposed by Aristotle and maintained by philosophers and theologians of Galileo's time.

Above is a drawing by Galileo of the Moon. With his telescope, Galileo was able to detect mountains and valleys on the Moon. This showed that the Moon was not Aristotle's "perfect celestial sphere" but another world, just like Earth. So, perhaps Earth is not so special.

Click on the words to see the Discove ry of Galilean moons.

This drawing shows Galileo's discovery of the "stars" orbiting Jupiter. He called them the Medecean Stars after his sponsor, the Grand Duke of Tuscany. This showed that at least some objects clearly did not orbit the Earth (perhaps other objects might not also).

For more info on the life of Galileo, check out these web links:
  • Galileo - Summary of the scientist 
  • The Galileo Project, a web site devoted to providing info on the life and work of the man and his science. Contains pictures of pages from his notebooks showing drawings of the observations that he made, as well as info on many of his contemporaries. 
  • Observations - a catalog of his observations and description of his observatory



Phases of Venus

This composite shows photos of Venus in a diagram illustrating how the phases of Venus would appear under the Ptolemaic vs. Copernican systems.

Here is another illustration of what the correct phases of Venus look like at 3 locations along its orbit. Notice both the change in size and illumination.

(13) How did these discoveries provide supporting evidence for the Copernican (heliocentric) universe rather than the Ptolemaic (geocentric) universe?

 

Law of Inertia

Galileo studied many aspects of the physical world--he was a prolific physicist and mathematician. The famous (but probably fabricated, in actuality) image of Galileo dropping various objects from the top of the leaning Tower of Pisa is used to illustrate his search for general laws of motion. While Galileo still thought of natural motion of planets was circular, his experiments led him to postulate the law of inertia for regular, Earthly objects:
Law of Inertia: Objects not acted on by a force, travel in straight lines at constant speeds. Or if they are at rest, they stay at rest.

Let's think about this. Many people today have the idea that you need to apply a force to something to keep it moving. Au contraire. A turtle ice-skating on a frictionless rink would sail on for ever and ever and ever.... The law of inertia says that you need a force only to change an object's state of motion. Such a force is friction, which slows down most Earth-bound objects, including, eventually, real skaters on real ice rinks.

Galileo, though apparently for the wrong reasons, also arrived at the correct relationships amongst distance, velocity and acceleration. Let us remind ourselves of these.
(14) You take a trip from Boulder to Santa Fe, roughly 600 kilometers (km). You have exceeded the speed limit at times, going 140 km/hr, and you have been stopped by the police, during which time your speed was 0 km/hr. But you made the trip in 6 hours overall.
    (a) What was your average speed? 

    (b) You are driving a new Honda VX (it does 55 miles to the gallon!) and, for a small engine, it has good acceleration: You can accelerate from 0 MPH to 60 MPH (100 km/hr) in 8 seconds. Explain how this is a measure of acceleration. 

    (c) If you accelerate from 0 to 100 km/hr, at a constant acceleration for 8 seconds, it is pretty easy to believe that your average speed is 50 km/hr (just (0+100)/2). How far did you travel in those 8 seconds?

(15) What property gives an object inertia? Color? Mass? Smell? Speed? Direction? Price?

(16) Did the Copernican revolution take place overnight? If not, how long would you say it took? Why?

The Size of the Solar System

Below are a couple of diagrams that illustrate where we see Mercury and Venus in the sky. The dashed lines show the orbits of Mercury and Venus - they remain close to the ecliptic plane and do not extend very far from the Sun. Venus and Mercury are at greatest elongation when they are farthest from the Sun.

(17) Woops! There is a mistake in the lower picture - is this sunSET or sunRISE? The sun is just below the horizon and we are looking East.

Above was a diagram showing the view from standing on Earth. Below is the view we would get from above the north pole of the Sun, looking down on the orbits of Earth and Mercury. The diagram illustrates the geometry of Mercury when it is at greatest elongation.

The angles of greatest elongation for these two planets are 46 degrees for Venus and 22 degrees for Mercury. Measurement of the angle of greatest elongation and some simple geometry allows one to determine the distance of the planet from the Sun, relative to the Earth's distance from the Sun - the astronomical unit, 1 A.U.
(18) Note that when Mercury is at greatest elongation, the Earth-Mercury-Sun angle is a right-angle. 
(a) Using 22 degrees for Mercury's angle of greatest elongation, work out the distance of Mercury from the Sun in astronomical units (A.U.). 

Model answers to the comprehension questions.