According to the map, we've only gone about four inches
-- -Harry ("Dumb and Dumber," New Line Cinema, 1994)
It is easy to flip to the index of an astronomy textbook to discover that, say, the Sun lies 150 million kilometers away from Earth. It is far more difficult (if not impossible), however, to picture this distance in the human mind. In this exercise, we will learn to access the often unpalatable distances encountered in astronomy by simply scaling the huge distances to more recognizable, pedestrian numbers. So long as every distance within the system of interest is scaled by the same factor, we retain the meaningful information about relative distances between objects. This is exactly the same principle employed by map makers so that they can fit Texas onto a turnable page.
Table 1 gives current measurements for the actual sizes and orbital distances of the nine planets.Table 1: Measured Astronomical Distances in Solar System (*KBO radii are not well known)
||semi-major axis (km)
||6.96 x 105
||2.44 x 103
||5.83 x 107
||6.05 x 103
||1.08 x 108
||6.38 x 103
||1.50 x 108
|Moon||1.74 x 103||3.84 x 105|
||3.40 x 103
||2.27 x 108
||7.14 x 104
||7.78 x 108
|Io||1.82 x 103||4.22 x 105|
|Ganymede||2.63 x 103||1.07 x 106|
||6.03 x 104
||1.43 x 109
|Titan||2.58 x 103||1.22 x 106|
||2.56 x 104
||2.87 x 109
||2.43 x 104
||4.50 x 109
||1.16 x 103
||5.91 x 109
|Charon||6.35 x102||1.96 x104|
|Quaoar*||5.84 x 102||6.49 x1012|
|Sedna*||7.45 x 102||7.51 x 1013|
As you can see, even when expressed in the one of the largest units (km) used to describe Earth-bound distances, the sizes of and distances to the planets require numbers raised to large powers of ten. In order to fully appreciate the relative sizes and distances within the solar system, it is necessary to scale these numbers down to values small enough so that we can "see" them in terms of more familiar distances. We can accomplish this by dividing every number in Table 1 by some constant scale value.
Each group should begin by determining a reasonable scale value. Then as a class you will pick the best scale value for everyone to work with. Some things you need to consider:
Determine what the scale value is in terms of how many km of real distance = how many m of scale distance. Describe your scale and how you arrived at it:
The class will pick the best scale for everyone to work with. Record it here: _____________________km (in space) = ________________m (in the model)
Your GSI will assign an object to you. Place a star next to your object's name in table 2.
Use the scale factor to calculate the size of your object and the distance of the object from the Sun. Fill in these values on the board and in table 2. To make it easier to make the model, find the distance from the previous object to the current object. Again, record the distance on the board and in table 2. Copy the information for the other objects from the board.
|Object||Radius||Distance from Sun||Distance from Previous|
Using the information from table 2, draw a scale picture of your object on plain white paper. If you have the Sun, you may need to tape some paper together, and make sure it is dark enough to see the sketch from a long ways away (e.g. if you want to color it in with yellow highlighter, make sure it's outlined in black marker!) If your object is a moon, you should include your sketch on the same paper as the planet it orbits. Label each of the sketches in the picture.
When everyone has their picture, you will gather outside in front of the Union. Each person without a picture should have a ruler and the lab. The person with the Sun will stand in front of the Union in a convenient location so you can see down South U. The person with Mercury should measure from the Sun and stand with their picture of Mercury at the appropriate distance. Do not stand in the road!
Everyone else should note the location of the Sun and Mercury on their map. If a planet ends up in the road, mark where it should be on the map, but have the person stand on the closest curb. The people with rulers should hold a ruler at arm's length and measure the angular size of the Sun: close one eye and align the picture of the Sun with the marks on the ruler. Record this value in table 3.
The person with Venus can then measure from Mercury to Venus and stand there with the picture of Venus while everyone else marks their map. Continue until either all the objects are arranged on South U or your GSI tells you to stop. Measure the angular size of the Sun again from the positions of Earth, Saturn, and Pluto.
|Standing at||Apparent Size||Compared to Earth|
The people who were recording the positions of the planets on the map should trade off with people holding the pictures so everyone gets a chance to look at the model.
Once you have looked at the model, filled out your map and completed the middle column in table 3 you are ready to go back inside. Make sure all the people who were holding pictures get a copy of the map and table 3. In the third column of table 3, calculate the ratio of the size of the Sun viewed from another planet compared to the size viewed from Earth, then answer the questions.
Last modified: 6/16/05Copyright Regents of the University of Michigan.