## The Size of the Solar System

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 (*Kuiper Belt Object 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

(avg. distance from Earth)

Mars 3.40 x 103 2.27 x 108

Ceres 4.73 X 102 4.14 X 108

Jupiter 7.14 x 104 7.78 x 108

Io 1.82 x 103 4.22 x 105

(avg. distance from Jupiter)

Ganymede 2.63 x 103 1.07 x 106

(avg. distance from Jupiter)

Saturn 6.03 x 104 1.43 x 109

Titan 2.58 x 103 1.22 x 106 (average distance from Saturn)

Uranus 2.56 x 104 2.87 x 109

Neptune 2.43 x 104 4.50 x 109

Pluto 1.19 x 103 5.91 x 109

Charon (moon of Pluto) 6.35 x102 1.96 x104

(avg. distance from Pluto)

Quaoar* 5.84 X 102 6.49 X 109

Eris* 1.16 X 103 1.02 X 1010

Sedna* 7.45 x 102 7.51 x 1010

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. Suppose the length of a hallway in the campus in meters is 10 meters. We can choose a scale factor, so that we can fit all

the planets from the Sun to Uranus in this hallway. Then, the scale value can be obtained through the following procedure:

If 10 meters are assigned to 2.87 x 109 Km

• Then the scale factor for distances from the Sun is: 1 meter / (2.87 x 108 Km)

For the size of the planets, we can choose in our scaled model the radius of the Sun to be 10 centimeters. Then, the scale value can be obtained through the following procedure:

If 10 centimeters are assigned to 6.96 x 105 Km

• Then the scale factor for radii is: 1 centimeter / (6.96 x 104 Km)

Use the scale factors to calculate the size of your object and the distance of the object from the Sun (round two decimal digits). Fill in these values 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 in Table 2.

As an example, below you will find the calculations for the first three rows:

• Scaled radius of the Sun: 6.96 x 105 Km * [1 cm / (6.96 x 104 Km)] = 10 cm

• Scaled radius of Mercury: 2.44 x 103 Km * [1 cm / (6.96 x 104 Km)] = 0.04 cm

• Scaled Distance Mercury to Sun: 5.83 x 107 Km * [1 meter / (2.87 x 108 Km)] = 0.20 m

• Scaled radius of Venus: 6.05 x 103 Km * [1 cm / (6.96 x 104 Km)] = 0.09 cm

• Scaled Distance Venus to Sun: 1.08 x 108 Km * [1 meter / (2.87 x 108 Km)] = 0.38 m

• Scaled Distance Venus to Mercury:

[(distance Venus to Sun) – (distance Mercury to Sun)] * Distance Scale Factor =

[(1.08 x 108 Km) – (5.83 x 107 Km)] * [1 meter / (2.87 x 108 Km)] = 0.17 m

OR

• From Scaled Distances table directly:

[Venus Distance from Sun – Mercury Distance from Sun = Distance of Venus from Previous Planet] = [0.38 m – 0.20 m = 0.17 m]

Table 2: Scaled Distances

Object Radius (cm) Distance from Sun (m) Distance from Previous Planet (m) Distance of Moon from Planet (m)

Sun 10.00 0.0 0.0 N/A

Mercury 0.04 0.20 0.20 N/A

Venus 0.38 0.17 N/A

Earth N/A

Moon N/A N/A

Mars N/A

Ceres N/A

Jupiter N/A

Io N/A N/A

Ganymede N/A N/A

Saturn N/A

Titan N/A N/A

Uranus N/A

Neptune N/A

Pluto N/A

Charon N/A N/A

Quaoar* N/A

Eris* N/A

Sedna* N/A

Now that we have our scaled values, we can make a scale model of the Solar System.

Using the information from Table 2, draw a scale picture of your objects 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.

Decide at which end of the hallway to start. Tape the picture of the Sun to the wall. Then measure from the wall and place Mercury on the floor at the appropriate distance. Then measure from

Mercury to Venus and tape Venus to the floor. Continue until all 8 planet pictures are taped to the floor.

Now with Neptune, Pluto, Eris, Quaoar, and Sedna figure out how many times longer the hallway would have to be to fit those objects 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. Answer the following questions and give explanations for your answers.

(a) Are all the pictures the right size? Can you tell the difference between Jupiter and Neptune from the pictures?

(b) How about Neptune and Uranus?

(c) Can you tell the difference between Earth and Mars?

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

3. How does the spacing between the Kuiper Belt Objects compare to the spacing between 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.

(a) Which one has the farthest moon from the planet?

(b) 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?