Overview
- Become familiar with the scale of the planets vs. their distances.
- Get an overview of the solar system.
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 (*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:
OR
- From Scaled Distances table directly:
| 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
- Look at the pictures of the planets and at Table 2. Answer the following questions and give explanations for your answers.
- 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?
- How does the spacing between the terrestrial planets compare to the spacing between the Jovians?
- How does the spacing between the Kuiper Belt Objects compare to the spacing between the Jovians and Terrestrials?
- Look at the Earth and Moon. Is the Moon relatively close to or far from Earth?
- 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?
- Are there any problems with this model? How would you solve those problems? What would be the problems with the new model?
Compare Relative Sizes of Sun
Stand at the position of Earth and hold a ruler about 15 cm (6 inches) from your face. Close one eye and measure the size your model Sun appears to be. Enter this into Table 3. Stand at the positions of Mercury, Venus, 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. Make sure you enter the values in metric; convert from inches if necessary! [The first row is completed for you.] [As an example, if the Apparent Size of the Sun from Ceres in our model were 1.5 cm, then the ratio could be expressed as 1.5 / 2.5 OR 0.6 OR 1.5 : 2.5]
| Table 3: Sun Size | ||
| Standing at | Apparent Size | Compared to Size from Earth |
| Earth | ~2.5 cm | 1 |
| Mercury | ||
| Venus | ||
| Saturn | ||
Questions
Show all of your mathematical calculations. Any verbal explanation is not needed and redundant.
- 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 it took about 3.5 days to get to the moon during the Apollo program.
- Using this information, about how long would it take (with that technology) to get to Mars?
- Solar power is a great way to power a spacecraft in orbit around Earth. Would it also be useful around Mercury and Jupiter? Explain and use examples.
- The nearest star is Alpha Centauri, 4.3 light-years away. In our scale model, how far away would this star be placed? (Show calculations. Express in meters [m]. No verbal explanation).
- TRAPPIST-1, a small star with seven, possibly terrestrial, planets lies about 40 light-years away.
(a) What is the distance to this star using our scaled model? (express in meters [m]
No verbal explanation)
- Is our scaled distance to TRAPPIST-1 on the surface of the Earth, inside the actual orbit of the Moon, or outside the actual orbit of the Moon? (Show numbers to be compared).
- In #3 above, you found the scaled distance to Alpha Centauri. 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, and given our response to #3, would you be able to see your picture of the other planet while standing at the picture of Earth in your model (assuming you had a clear line of sight: no trees, buildings, or terrain in the way)?
- Summarize the observations you can make about the solar system from this model. Are the sizes of the solar system’s objects big as compared to the system’s size and distances between the objects? Is the solar system big as compared to distances to the nearest stars?
