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Astronomical Scale |1
Astronomical
Scale
Astronomy is the study of the universe, and when studying the universe, we often deal
with unbelievable sizes and unfathomable distances. To help us get a better understanding of
these sizes and distances, we can put them to scale. Scale is the ratio between the actual object
and a model of that object. Some common examples of scaled objects are maps, toy model kits,
and statues. Maps and toy model kits are usually much smaller than the object it represents,
whereas statues are normally larger than its analog.
Today, we will create a scaled model of our solar system. To do this, we must find a
ratio. We start by selecting an object [1] we would like the solar system to be scaled to. For
our convenience, we will use a yellow fitness ball as today’s representation for our sun. Next,
look for a common parameter. Let’s use the diameters of the fitness ball and the Sun [2]. The
diameter of our fitness ball is 24” (inches). The diameter of the Sun is 1,400,000𝑘𝑚 (kilometers).
Before we go on, we must determine the system of units [3] that we want to use. Doing so
greatly reduces the chance for miscalculations. In astronomy, like in all sciences, we use the
metric system, also known as the International System of Units (SI for short).
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Astronomical Scale |2
The metric system is a base-10 system, which essentially means it is very easy to change and use
units within this system.
Our fitness ball is in the imperial system. To convert it, let’s use a unit in the metric
system that is similar in size to inches. Here, we’ll change inches to centimeters (𝑐𝑚). One inch
is exactly 2.54𝑐𝑚, and the diameter of our fitness ball is 24”, so multiply 2.54𝒄𝒎 by 24 [4]. We
conclude that the diameter of the fitness ball is about 61𝑐𝑚.
Now that the measurements are in the same system, we need to check if the units are the
same between the fitness ball and the Sun. Our fitness ball is in centimeters and the Sun is in
kilometers, so clearly, they are not the same. Let’s change that. The metric system is based off
the meter (hence the name of the system), so let’s convert the centimeters and kilometers into
meters. “centi-“ means one hundredth (0.01) and “kilo-“ means one thousand (1,000), so 1𝑐𝑚 is
one hundredth of a meter and 1𝑘𝑚 is one thousand meters. We can now change units of the
fitness ball and the Sun with this knowledge. For the fitness ball, multiply 61𝒄𝒎 by 0.01. For
the Sun, multiply 1,400,000𝒌𝒎 by 1,000 [5]. We conclude that the diameter of the ball is
0.61𝑚 and the diameter of the Sun is 1,400,000,000𝑚.
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We can finally determine our ratio! To do this, always divide the size of the object you
want to scale to (the numerator) by the size of the actual object (the denominator). In our case,
divide the diameter of the fitness ball by the diameter of the Sun [6]. We conclude that this
number is about 0.0000000004.4. Note that the units canceled each other out.
This can be written in scientific notation as 4.4 X 10-10 or 4.4E-10, both of which are
clearer to read. We will use 4.4E-10 because it is easier to type this number style into our
calculators.
Now that we have figured out our model’s ratio, we can calculate the scaled sizes of the
planets and their relative distances. As an example, we’ll take Mercury, the closest planet to the
Sun. The diameter of Mercury is roughly 4,900𝑘𝑚. Note that this number’s units are in
kilometers. Our ratio was determined by using meters, however, so we must change the
diameter of the object into meters [7]. Mercury’s actual diameter in meters is 4,900,000𝑚.
With this number, multiply the actual diameter of the object by our ratio [8]. We conclude
that Mercury’s diameter in our model is about 0.0021𝑚 or 0.21𝑐𝑚, about the width of the tip of
a crayon.
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The distance that Mercury is from the Sun varies because Mercury revolves around the
Sun in an elliptical pattern, but we will use the semi-major axis, the farthest distance the
planet is from the Sun. This is nearly 58,000,000𝑘𝑚 for Mercury. As we did previously, we
need to change kilometers into meters. So using 58,000,000,000𝑚 instead, multiply the semimajor axis by our ratio [9]. We conclude that the closest planet is surprisingly far away from
our fitness ball – a distant 25 meters away!
Now that I’ve shown you how to scale down our solar system to the fitness ball standard,
determine with a partner the sizes and distances for the other seven planets. Once you are done,
come up to the desk and select the object that best represents the size for each of your planets. At
the end of the lab, we will go outside and try to visualize just how far our models are from each
other.
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Example Sheet
Determine the Ratio
1. Select the fitness ball as the object that the Sun will be scaled to.
2. The diameter of the fitness ball is 24”, and the diameter of the Sun is 1,400,000𝑘𝑚.
3. Use the metric system.
4. 2.54𝑐𝑚 = 1”, so:
2.54𝑐𝑚
∗ 24 ≈ 61𝑐𝑚
1
5. The fitness ball needs to be in meters, so:
61𝑐𝑚
1𝑚
∗
= 0.61𝑚
1
100𝑐𝑚
And the Sun needs to be in meters as well, so:
1,400,000𝑘𝑚 1,000𝑚
∗
= 1,400,000,000𝑚
1
1𝑘𝑚
6. To get the ratio, we divide the diameter of the fitness ball by the Sun:
0.61𝑚
1,400,000,000𝑚
≈ 4.4E-10
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Determine the Scaled Diameter
7. Our object is Mercury, so:
4,900𝑘𝑚 1,000𝑚
∗
= 4,900,000𝑚
1
1𝑘𝑚
8. Mercury’s actual diameter multiplied by our ratio is:
4,900,000𝑚 ∗ 4.4E-10 ≈ 0.0021𝑚
0.0021𝑚 100𝑐𝑚
∗
= 0.21𝑐𝑚
1
1𝑚
Determine the Scaled Distance
9. The semi-major axis of Mercury is 58,000,000𝑘𝑚, so:
58,000,000𝑘𝑚 1,000𝑚
∗
= 58,000,000,000𝑚
1
1𝑘𝑚
58,000,000,000𝑚 ∗ 4.4E-10 ≈ 26𝑚
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Work Sheet
1. Actual Diameter of Mercury  Scaled Diameter of Mercury
4,900𝑘𝑚
1
∗
1,000𝑚
1𝑘𝑚
= 4,900,000𝑚 : 4,900,000𝑚 ∗ 4.4E-10 = 0.0021𝑚
0.0021𝑚 100𝑐𝑚
∗
= 0.21𝑐𝑚
1
1𝑚
2. Actual Diameter of Venus  Scaled Diameter of Venus
3. Actual Diameter of Earth  Scaled Diameter of Earth
4. Actual Diameter of Mars  Scaled Diameter of Mars
5. Actual Diameter of Jupiter  Scaled Diameter of Jupiter
6. Actual Diameter of Saturn  Scaled Diameter of Saturn
7. Actual Diameter of Uranus  Scaled Diameter of Uranus
8. Actual Diameter of Neptune  Scaled Diameter of Neptune
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1. Actual Distance of Mercury  Scaled Distance of Mercury
5,800,000𝑘𝑚
1
∗
1,000𝑚
1𝑘𝑚
= 5,800,000,000𝑚; 5,800,000,000𝑚 ∗ 4.4E-10 ≈ 26𝑚
2. Actual Distance of Venus  Scaled Distance of Venus
3. Actual Distance of Earth  Scaled Distance of Earth
4. Actual Distance of Mars  Scaled Distance of Mars
5. Actual Distance of Jupiter  Scaled Distance of Jupiter
6. Actual Distance of Saturn  Scaled Distance of Saturn
7. Actual Distance of Uranus  Scaled Distance of Uranus
8. Actual Distance of Neptune  Scaled Distance of Neptune
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Data Sheet
Solar System
Object
Actual Diameter
(𝑘𝑚)
Actual Diameter
(𝑚)
Scaled Diameter
(𝑐𝑚)
Portrayal of
Scaled Object
Sun
1,400,000
1,400,000,000
61
Fitness Ball
Mercury
4,900
4,900,000
0.21
Venus
12,000
Earth
13,000
Mars
6,800
Jupiter
140,000
Saturn
120,000
Uranus
51,000
Neptune
49,000
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A s t r o n o m i c a l S c a l e | 10
Data Sheet
Solar System
Object
Actual Distance
(𝑘𝑚)
Actual Distance
(𝑚)
Scaled Distance
(𝑚)
Sun
0
0
0
Mercury
58,000,000
58,000,000,000
26
Venus
110,000,000
Earth
150,000,000
Mars
230,000,000
Jupiter
780,000,000
Saturn
1,400,000,000
Uranus
2,900,000,000
Neptune
4,500,000,000
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Homework
Scale the Sun to the Sunsphere, and use this new ratio to determine the semi-major axial
distances of the eight planets. The diameter of the Sunsphere is 74’ (feet). Follow the Example
Sheet if you need a reference. Remember that 1 foot equals 12 inches.
Solar System
Object
Actual Distance
(𝑘𝑚)
Actual Distance
(𝑚)
Scaled Distance
(𝑚)
Sun
0
0
0
Mercury
58,000,000
Venus
110,000,000
Earth
150,000,000
Mars
230,000,000
Jupiter
780,000,000
Saturn
1,400,000,000
Uranus
2,900,000,000
Neptune
4,500,000,000
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Homework Work Sheet
1. Determine the Ratio
2. Actual Distance of Mercury  Scaled Distance of Mercury
3. Actual Distance of Venus  Scaled Distance of Venus
4. Actual Distance of Earth  Scaled Distance of Earth
5. Actual Distance of Mars  Scaled Distance of Mars
6. Actual Distance of Jupiter  Scaled Distance of Jupiter
7. Actual Distance of Saturn  Scaled Distance of Saturn
8. Actual Distance of Uranus  Scaled Distance of Uranus
9. Actual Distance of Neptune  Scaled Distance of Neptune
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