Exponents Project: Solar System

The following image is of our solar system, produced by NASA.

source: https://www.nasa.gov/ames/press-release/nasa-selects-new-research-teams-to-further-solar-system-exploration-research

It’s a familiar representation of the solar system, and as is expected with a picture of the solar system neither the planets nor their distances from each other are to scale (why?).

What does our solar system actually look like to scale? This project invites you to recreate the solar system, to scale, in a suitable outside location.

For this project, you will need

  1. a large open space
  2. a way to measure up to 50m (trundle wheel, pacing, 10m measuring tape?)
  3. a variety of balls (marbles to exercise balls or bigger if you can find one) or chalk with string.

First, some preliminary mathematics.

The SI – how we agree to measurements

We all know how long a meter is. Did you know that there are seven basic units that are agreed on internationally? We have one unit for length (meter); one for weight (gram); one for time (second); and four others. Other measurements are derived from these – for example, one centimeter is \frac{1}{100} of a meter.

You will notice that the basic units are all quite simple for us to think about – we are probably somewhere between 1 m and 2 m tall. One second is about one heartbeat when you are reasonably healthy and reasonably relaxed!

Indeed, in the early days, according to legend, the unit ‘1 yard’ is said to have been the distance between King Henry VIII’s nose and thumb when he extended his arm. Iron rods were made of this measurement and sent around the country to become the standard for measuring lengths.

In more recent days (that is, 1870), the French determined one meter to be a portion of the Paris meridian (arc drawn from the equator to North Pole that goes through Paris). More specifically, 1 meter = \frac{1}{10\,000\,000}=\frac{1}{1\times 10^{7}} of the Paris Meridian.

You can read more about the history of length here.

Scientific Notation

Now, since our units of measurements are intuitive relative to our size and our daily experience, how to we measure things of different magnitude? The diameter of a human egg? The diameter of the sun? The answer is that we use a certain number of digits (referred to as significant figures) along with a measurement of magnitude (power of 10).

Suppose one year you have a small birthday party – 3 people over for pizza and a movie. Then the next year you have a big birthday party – 20 people over to celebrate. These numbers are ‘small’ and ‘big’ relative to each other. What about a rich and famous person who throws a city wide street party for their birthday, where 12 342 people show up? We’d refer to this as a difference in magnitude. The number of people might be recorded as more than ’12 300′. Really, when referring to this number of people, the plus or minus of 42 is not so important. The most important part of the number is the first digit – 10 000 not 50 000. The next digit is next important, 12 000 not 18 000. The next digit is next important … and so on. A common standard for large numbers is to take the first three digits – the first three significant figures. Rounding rules apply (see question 2 of the exercise below).

Then, when the magnitude is large, we often will write it in short hand – that is, with powers of 10 (scientific notation).

For example, 12\,300 = 1.23 \times 10^4.

The rule for scientific notation is that only the most significant figure stays to the left of the decimal point. It can be tricky to get the right power of ten. What method do you have for getting the right power of ten? Try out your method with the exercise below.

Short Comprehension Exercise:

Write the following correct to three significant figures, then correct to three significant figures in scientific notation:

  1. 23\,447
  2. 3\,459\,321
  3. 4\,622


We are now ready to begin the project.

The Project

The Planets

Do some research to find out the diameter of the planets in our solar system. Here is a helpful sheet Planet Record Sheet for recording the information you find.

Record:

  • the diameter as given to you on your source (e.g. Venus, diameter = 12104 km);
  • the diameter correct to three significant figures (e.g. Venus, diameter = 12100 km (3 s.f.));
  • the diameter, correct to 3 s.f. in scientific notation (eg, Venus, diameter = 1.21 \times 10^4 km (3 s.f.)).

Choose the biggest ball that you have. Hopefully, something like an excercise ball or bigger. Take the diameter of that ball. Now if that ball represents Jupiter, what is the scale of your model? E.g. If you ball measures 50 cm, diameter, and you want it to represent 140000km, then

    \[50 \text{ cm} : 140000\text{ km}\]

that is,

    \[1\text{ cm}: 2800\text{ km}\]

that is,

    \[1\text{ cm}: 2\,800\,000\text{ m}\]

that is,

    \[1\text{ cm}: 280\,000\,000\text{ cm}\]

Now that we have both numbers in the same unit, we have a scale:

    \[1:280000000 = 1:2.8 \times 10^8\]

If your are using chalk to draw planets on the ground, you can decide a scale. I would suggest 1 cm to represent 1000 km, that is, 1:1 \times 10 ^8.

Now use your scale to calculate the diameters of the other planets in your model.  Do your answers make sense? You can do a check by pairing up similar sized planets (Venus with Earth, Uranus with Neptune etc). Now select the best sized ball from your collection for each planet.

Figure out a good way to label your planets, and lay them out in order. Is there some way you can include the sun with this scale?

According to your information, what is the total volume of planets in the solar system? How does rounding to 3 s.f. affect your answer? (Divide and conquer these calculations – do them as a class, or do them with a spreadsheet!). What portion of the volume of the sun is the total volume of the 8 planets (include Pluto if your class votes for it!). Can you write the portion with scientific notation?

The layout

Now unless you have a space with a 16 km radius and are feeling fast today, then we’ll need to use a different scale for the layout.

Let’s use 1 cm to represent 1,000,000 km. What is that written in a scale without units?

Decide on a location for the sun, and lay out the planets.

Did your planets all line up? Do they line up? Where are the planets today, relative to the sun? Are they even all on the same plane? Here is a nice animation of the solar system.

Describe some of the problems you might encounter if you used the same scale for laying out the planets as for the size of the planets.

 

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