E. Earth and Space Science: Space Exploration · Space Exploration: The Sun, the Solar System and the UniverseBuilding Scale Models of the Solar System and Beyond
E2.5 A1.3 A1.4About 70 minutes
Learning goals- I can use a scale (ratio) to calculate model sizes and model distances.
- I can explain why most pictures of the solar system are not to scale.
- I can build and evaluate a scale model of distances in the solar system.
- I can describe how empty space is, even inside the solar system.
Words to know
scale
The ratio between a measurement on a model and the real measurement, such as 1 cm : 100 000 km.
scale factor
The number you divide (or multiply) by to change a real measurement into a model measurement.
scale model
A model in which every measurement is shrunk (or enlarged) by the same scale factor, so the proportions stay true.
proportion
An equation stating that two ratios are equal, such as 1 cm/100 000 km = x cm/1 392 000 km.
relative size
How big something is compared with something else, for example 'Jupiter is about 11 times as wide as Earth'.
A question to start
If you drew Earth the size of a peppercorn, how big would the Sun be? And how far away would you have to put it? Most textbook pictures show the planets lined up close to a slightly bigger Sun. Those pictures are useful, but they are not to scale. Today you will find out what the solar system really looks like when everything is shrunk by the same amount.
How scale works
A scale is a ratio. A scale of 1 cm : 100 000 km means that 1 cm on the model stands for 100 000 km in real life. To find a model measurement, divide the real measurement by the scale factor:
model size = real size ÷ scale factor
So the Sun (diameter 1 392 000 km) becomes 1 392 000 ÷ 100 000 = 13.9 cm: about the size of a grapefruit.
You can also set up a proportion: 1 cm / 100 000 km = x / 1 392 000 km, so x = 13.9 cm. This is the same proportional reasoning you used in Grade 8 math.
| Object | Real diameter (km) | Model diameter | Real distance from Sun (km) | Model distance from Sun |
|---|
| Sun | 1 392 000 | 13.9 cm (grapefruit) | | |
| Mercury | 4 879 | 0.5 mm | 5.8 × 10⁷ | 5.8 m |
| Venus | 12 104 | 1.2 mm | 1.08 × 10⁸ | 10.8 m |
| Earth | 12 756 | 1.3 mm (pinhead) | 1.50 × 10⁸ | 15 m |
| Mars | 6 792 | 0.7 mm | 2.28 × 10⁸ | 22.8 m |
| Jupiter | 142 984 | 1.4 cm (marble) | 7.78 × 10⁸ | 78 m |
| Saturn | 120 536 | 1.2 cm | 1.43 × 10⁹ | 143 m |
| Uranus | 51 118 | 5.1 mm | 2.87 × 10⁹ | 287 m |
| Neptune | 49 528 | 5.0 mm | 4.50 × 10⁹ | 450 m |
What the model shows
- The Sun is a grapefruit. Earth is a pinhead 15 m away: about the length of a classroom and a half.
- Jupiter is a marble 78 m away, most of the length of a football field.
- Neptune is a 5 mm bead 450 m away, several city blocks.
- The Moon is a tiny speck (0.35 mm) only 3.8 cm from Earth.
- The nearest star after the Sun, Proxima Centauri, would be about 4000 km away: more than the distance from Toronto to Calgary.
The big lesson: space is mostly empty. The planets are tiny compared with the distances between them.
Why most pictures are not to scale
If a picture showed sizes and distances at the same scale, the planets would be too small to see, or the page would need to be hundreds of metres long. So most diagrams use one scale for sizes and a completely different one for distances (or no scale at all). That's fine, as long as you know it. Always look for 'not to scale' in a caption.
Worked exampleExample 1: Find a model size
Using the scale 1 cm : 100 000 km, how big would Saturn (diameter 120 536 km) be?
- model size = real size ÷ scale factor
- = 120 536 km ÷ 100 000 km per cm
- ≈ 1.2 cm
About 1.2 cm, a bit smaller than the Jupiter marble (without its rings).
Worked exampleExample 2: Choose a scale to fit a space
You want to make a model of distances from the Sun to Neptune (4.5 × 10⁹ km) that fits in a 45 m school hallway. What scale should you use, and where does Earth go?
- Model distance for Neptune = 45 m = 4500 cm.
- Scale factor = real ÷ model = 4.5 × 10⁹ km ÷ 4500 cm = 1 000 000 km per cm.
- So the scale is 1 cm : 1 000 000 km.
- Earth: 1.5 × 10⁸ km ÷ 1 000 000 km per cm = 150 cm = 1.5 m from the Sun.
Scale 1 cm : 1 000 000 km; Earth goes 1.5 m from the Sun. At this scale Earth itself would be only about 0.13 mm wide, far too small to see.
Code it: a scale calculator (A1.4)
This Python program converts real distances (km) into model distances (m) for any scale. Change the scale to design your own model.
scale = 100000 # km per cm
planets = {"Earth": 1.5e8, "Jupiter": 7.8e8, "Neptune": 4.5e9}
for name, km in planets.items():
print(name, km / scale / 100, "m")
# Output:
# Earth 15.0 m
# Jupiter 78.0 m
# Neptune 450.0 m
Dividing by the scale gives centimetres; dividing by 100 more turns centimetres into metres. 1.5e8 is Python's way of writing 1.5 × 10⁸.
Hands-on activityBuild it: a toilet-paper solar system
You need: a roll of toilet paper (or a long strip of paper or adding-machine tape), a marker, a hallway or a driveway/sidewalk, and small labels or sticky notes.
Scale: 1 sheet = 1 AU (about 150 million km).
- Mark the Sun at the start of the roll.
- Unroll and mark each planet at its distance in sheets: Mercury 0.4, Venus 0.7, Earth 1, Mars 1.5, Jupiter 5.2, Saturn 9.5, Uranus 19.2, Neptune 30.1 (estimate the part-sheets).
- Mark the asteroid belt (about 2.2 to 3.2 sheets) and the start of the Kuiper belt (about 30 sheets).
- Walk along the model and look back at the Sun.
- Evaluate your model: what does it show well? What does it not show? (Think about planet sizes, the fact that planets are not lined up, and the Oort cloud.)
What to look for: the four inner planets are crowded into the first two sheets, while the outer planets are spread over 30 sheets. A good evaluation notes that the model shows relative distances accurately but not sizes (a planet at this scale would be microscopic), that real planets are spread around their orbits rather than in a line, and that the Oort cloud would be thousands of sheets away. If you work outside, stay on the sidewalk, away from traffic.
Just for funIf the Sun were a grapefruit in Toronto at our 1 cm : 100 000 km scale, the next star would be another little ball about 4000 km away. Space has a lot of personal space.
Your turn
1Using 1 cm : 100 000 km, how big would Earth's Moon be (diameter 3475 km)?
3475 ÷ 100 000 = 0.035 cm = 0.35 mm: a tiny speck.
2At the same scale, how far from the Sun would Mars be (2.28 × 10⁸ km)?
2.28 × 10⁸ ÷ 10⁵ = 2280 cm = 22.8 m.
3Why are most diagrams of the solar system not to scale?
If sizes and distances used the same scale, the planets would be too small to see or the diagram would have to be enormous. So diagrams use different scales (or none) for sizes and distances.
4Jupiter's diameter is about 11 times Earth's. If Earth is a 1 cm bead, how wide is Jupiter?
5In the toilet-paper model (1 sheet = 1 AU), how many sheets from the Sun is Saturn?
6What does the scale model tell you about how much of the solar system is empty space?
Almost all of it: the planets are tiny specks separated by huge distances.
7The Python scale program prints 'Earth 15.0 m'. What would it print for Mars at 2.28e8 km?
2.28e8 ÷ 100 000 ÷ 100 = 22.8, so it prints Mars 22.8 m.
More practice: warm-up
1What does a scale of 1 cm : 100 000 km mean?
1 cm on the model stands for 100 000 km in real life.
2To find a model size, do you multiply or divide the real size by the scale factor?
3At 1 cm : 100 000 km, what household object is about the size of the Sun?
A grapefruit (about 14 cm).
4How many Earth diameters fit between Earth and the Moon?
5True or false: the planets are usually lined up in a row.
False. They are spread around their orbits at different positions.
More practice: on your own
1A model uses 1 cm : 50 000 km. How big is Earth?
12 756 ÷ 50 000 ≈ 0.26 cm = 2.6 mm.
2At 1 cm : 1 000 000 km, how far from the Sun is Jupiter (7.8 × 10⁸ km)?
7.8 × 10⁸ ÷ 10⁶ = 780 cm = 7.8 m.
3A student draws the Sun 10 cm wide and Earth 2 cm wide. Is this to scale? Explain.
No. The Sun's diameter is about 109 times Earth's, so with a 10 cm Sun, Earth should be about 10 ÷ 109 ≈ 0.09 cm (less than 1 mm).
4Choose a scale so that Neptune (4.5 × 10⁹ km) is 9 m from the Sun.
9 m = 900 cm; 4.5 × 10⁹ ÷ 900 = 5 × 10⁶, so 1 cm : 5 000 000 km.
5List one strength and one limitation of the toilet-paper model.
Strength: it shows relative distances accurately and how spread out the outer planets are. Limitation: it does not show sizes (planets would be microscopic), the orbits' shapes, or the planets' real positions.
Challenge
1At 1 cm : 100 000 km, the next star (4.0 × 10¹³ km) would be how far away in km?
4.0 × 10¹³ ÷ 10⁵ = 4.0 × 10⁸ cm. Since 1 km = 10⁵ cm, that is 4.0 × 10⁸ ÷ 10⁵ = 4000 km.
2If you made a model where the Milky Way (about 100 000 ly across) fit across a 100 m soccer field, how far apart would the Sun and Proxima Centauri (4.24 ly) be?
Scale: 100 000 ly → 100 m, so 1 ly → 0.001 m = 1 mm. 4.24 ly → about 4.2 mm. Even the nearest star is a few millimetres away on a galaxy-sized model.
3Explain why it is impossible to make a single poster that shows both the sizes of the planets and their distances to the same scale, with all planets visible.
With distances to scale on a poster about 1 m wide, Neptune would be 1 m from the Sun, so the scale is 1 cm : 4.5 × 10⁷ km, and Earth would be only about 0.0003 cm wide: invisible. To make Earth visible (say 1 mm), Neptune would have to be about 350 m away. The ratio between sizes and distances is too large for one poster.
Quiz
1. At 1 cm : 100 000 km, a planet with diameter 50 000 km would be:
2. Why do diagrams of the solar system usually say 'not to scale'?
3. A scale model shows that the solar system is:
4. To make a model 10 times smaller than another, you:
5. About how many Earth diameters fit between Earth and the Moon?
6. In a model with 1 sheet = 1 AU, Neptune is about:
Big idea: E2.5 asks students to develop an understanding of relative distances and sizes. Scale models are the clearest way to do this, and they connect directly to proportional reasoning from math. Evaluating the model's strengths and limits is part of the engineering design process (A1.3), and the Python calculator supports coding (A1.4).
Common difficulties:
- Multiplying instead of dividing by the scale factor. Ask: 'Should the model be bigger or smaller than the real thing?'
- Unit changes (cm, m, km). Encourage your teen to write units at each step.
- Believing textbook pictures are to scale.
How to help: do the toilet-paper model together in a hallway or on a sidewalk (away from traffic). Standing at 'Neptune' and looking back at the 'Sun' is a memorable moment.