University of Michigan - Department of Astronomy

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Version: Long out

The Size of the Solar System

According to the map, we've only gone about four inches

-- -Harry  ("Dumb and Dumber," New Line Cinema, 1994)

Overview

Introduction

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.


Constructing the Model

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)
Object
Radius (km)
semi-major axis (km)
Sun
6.96 x 105
--
Mercury
2.44 x 103
5.83 x 107
Venus
6.05 x 103
1.08 x 108
Earth
6.38 x 103
1.50 x 108
Moon 1.74 x 103 3.84 x 105
Mars
3.40 x 103
2.27 x 108
Jupiter
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
Saturn
6.03 x 104
1.43 x 109
Titan 2.58 x 103 1.22 x 106
Uranus
2.56 x 104
2.87 x 109
Neptune
2.43 x 104
4.50 x 109
Pluto
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:
the model must be a reasonable distance to walk;
the planets must be visible when drawn to scale;
the Sun must be a reasonable size to draw.
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.

Table 2: Scaled distances
Object Radius Distance from Sun Distance from Previous
Sun    0.0 0.0
Mercury      
Venus      
Earth      
Moon      
Mars      
Jupiter      
Io      
Ganymede      
Saturn      
Titan      
Uranus      
Neptune      
Pluto      
Quaoar      
Sedna      

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.

Table 3: Sun Size
Standing at Apparent Size Compared to Earth
Mercury    
Earth    1.0
Saturn    
Pluto    

Observations from the Model

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.

  1. Look at the pictures of the planets, and at table 2. Are all the pictures the right size? Can you tell the difference between Jupiter and Neptune from the pictures? How about Neptune and Uranus? Can you tell the difference between Earth and Mars? Explain your answers.











  2. How does the spacing of the terrestrial planets compare to the spacing of the Jovians?











  3. Look at the Earth and Moon. Is the Moon relatively close to or far from Earth?











  4. Look at the other planets with moons. Which one has the farthest moon from the planet? Which one is the closest? Is there a big difference?











  5. Are there any problems with this model? How would you solve those problems? What would be the problems with the new model?












  6. Your GSI probably did not have you go all the way out to Quaoar and Sedna. Why not? How far would you have to walk to get to them? Once you are back inside, use a map to estimate where they would be.









  7. How does the spacing of the KBOs compare to the spacing of the Jovians and Terrestrials?












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.


Questions

Show your work and explain your answers for full credit

  1. The largest asteroid is Ceres, which is an average of 4.50x102 in radius and 4.14x108 km from the Sun on average. Calculate its size in our model and draw a sketch of it below. Calculate its position in our model and mark its position on the map.












  2. Human space vehicles like the space shuttle or International Space Station generally orbit around 250 km above the Earth's surface. How far would that be from The sketch of the Earth? How does that distance compare to the distance to the Moon? How does that distance compare to the distance to Mars?











  3. Solar power is a great way to power a spacecraft in orbit around Earth. Would it also be useful around Mercury and Saturn? Explain.











  4. The nearest star is alpha centauri, 4.3 light-years away. Where would you have to go to post its picture?











  5. Alpha Centauri is actually a multiple star system, but one of the stars is nearly identical to the Sun. If you included a picture of a planet the size of Earth with your picture of alpha Centauri, would you be able to see your picture of the other planet while standing at the picture of Earth (assuming you had a clear line of sight: no trees, buildings, or Earth in the way)? Explain.











  6. Summarize the observations you can make about the solar system from this model.












Last modified: 6/16/05

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