University of Michigan - Department of Astronomy

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Version: Long In

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.  

To determine the scale value you'll need to know how much space you have. Measure the length of the hallway in meters: ____________.
Choose a scale factor so that you can fit all the planets from the Sun to Uranus in the hallway. Record your scale value and describe how you arrived at it here:

The entire class will need to use the same scale factor. 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. If your object is a moon, you should include your sketch on the same paper as the planet it orbits. Label the picture.

The person/group with the Sun will decide which end of the hallway to start at. Tape the picture of the Sun to the wall. The person/group with Mercury should measure from the wall and place Mercury on the floor at the appropriate distance. The person/group with Venus can then measure from Mercury to Venus and tape Venus to the floor. Continue until all 8 planet pictures are taped to the floor.

The people/groups with Neptune, Pluto, Quaoar and Sedna should figure out how many times longer the hallway would have to be to fit their object in using this scale (i.e. it would have to be 1.5 times longer, twice as long, 10 times longer...) Note that distance on the picture and tape the pictures to the wall opposite the Sun.


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











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











    5. 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?











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













Stand at the position of Earth and hold a ruler at arms length. Close one eye and measure the size the Sun appears to be. Enter this into table 3. Stand at the position of Mercury and Saturn and measure the size again. Enter these values into table 3. Calculate the ratio of the size of the Sun at that planet to the size of the Sun at Earth.

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

Questions

Show your work and explain your answers for full credit1.

  1. The largest asteroid is Ceres, which is an average of 4.50x102 in radius and 4.14x108 km from the Sun on average. How big would its picture be (in sensible units)? Where would it be placed in the model (both from the Sun and from the "previous object".) Why was the radius specified as an average?











  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? It takes a few hours to get into orbit, and about 3.5 days to get to the moon. What does this tell you about the planned program to send humans 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)?











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












Last modified: 6/16/05 by SAM, original by MW

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