Starlight Starship: An Interstellar Voyage of 3-D Shapes and the Inverse Square Law

Your Interstellar Spaceship is Nearly Complete!

Only one technical challenge remains: finding a power source.

You’ve tried fusion, fission, rocket fuel, and even briefly accepted the generous offer of a thousand badgers to power your ship by running on their tiny treadmills. But none of it generated enough power for the voyage you want to undertake.

There’s only one way left to try.

You need the power of an entire star.

You’ve located such a star and have fixed it in your tractor beams. You’ve built an assortment of perfectly square solar panels, too. Your plan is to build a cube of solar panels around the star and use them to capture all the star’s light.

As with all scientific endeavors, it may take you a few tries to get things right.

First, you decide to make a cube that consists of a single solar panel on each face.

Assemble “Cube A”!

Head to the Technical Manual for instructions on assembling your first cube of solar panels!

Click the image to download the technical manual

Time to Test out Cube A!

You cleverly assemble your first cube of solar panels and place it around a star. (If you have a small LED light or flashlight, you could put the light inside your newly constructed cube to simulate the project, just make sure the light isn’t warm so you don’t catch the paper on fire!) Immediately, you notice a few things. The first is that the panels are, unfortunately, on fire.

It turns out that one sixth* of an entire star’s sunlight is a bit much for a single solar panel.

Ah, well, these things happen. That’s why you have so many backups!

You do notice one other thing. Something mathematical.

Each solar panel is exactly half a panel’s width from the star. Interesting.

No time for thinking about things like that, though! It’s time to try again!

*There are six faces, so each panel gets exactly one sixth of the total light of the star, as long as the star is exactly in the middle and the cube is symmetrical.

Assemble Cube B!

Head back to your technical manual and follow the instructions to assemble Cube B. Notice that this time each face is made up of exactly four solar panels.

Why four solar panels?

Well, in order to have a cube, each face must be a square. Try cutting out some small squares and play around with arranging them to make more squares. What do you notice?

You should notice that your squares can be 1 x 1, 2 x 2, 3 x 3, etc.

This means that the total number of solar panels is 1, 4, 9, 16, 25 etc. They will all be perfect squares!

You carefully set Cube B up around the star. This time the panels are farther away, and the light of the star is spread out over 24 panels instead of 6.

Each panel is now only absorbing 1/24 of the total light, instead of 1/6. This seems much more reasonable.

This also means that each panel is absorbing one fourth of the sunlight that the original panels did. This makes sense because each face has 4 panels instead of 1 now!

This time, the panels work for several seconds before lighting on fire! This seems very promising!

The 1,000 badgers who decided to come along on your travels agree that this seems like great progress.

Assemble Cube C!

Head back to the technical manual and follow the instructions to assemble Cube C.

How many panels will be on each face of the cube this time? How does this affect the amount of light each panel gets compared to one of the panels on Cube A? (Warning: spoilers below!)

Cube C is a cube where each face has 9 panels on (it’s a 3 x 3 square of panels on each side.) This means that each panel gets 1/54 of the total light of the star, which is 1/9 of the light that the original panel got.

Success!

Cube C works perfectly. Each panel is able to absorb 1/54 of the light of the star without any problems. Your spaceship hums to life, and you and the 1,000 badgers go adventuring through the cosmos.

However, soon other aspiring interstellar explorers start writing to you, asking you how to set up their own star-powered spaceships. You decide to expand your technical manual to make it useful to others who come after you. Because, after all, they might have different types of solar panels, or they might wish to use stars of different sizes…

Create a Reference Table for Other Interstellar Explorers

See if you can fill out the table below for other possible cube sizes (this table is also included in the Technical Manual). Make as many of your own cubes or drawings as you need in order to figure it out! (This section involves some tough math. Feel free to dive in, or to skip it and go onto the next section, which will still make sense even if you skip this section.)

One Final Observation

One thing about math: it’s all about noticing patterns. There’s a pattern hidden in that table above. A secret relationship between the distance a panel is from the star and the amount of energy it absorbs. It’s pretty tricky to notice, though. Feel free to go back and look if you’d like! I’m going to make a slightly different version of the table now, which might make it clearer.

It’s still hard to see, but do you notice that when we triple the distance (go from 0.5 to 1.5 panels away from the star) the energy divides by 9? And when the distance is multiplied by 4, the energy divides by 16?

This is something called an “Inverse Square Relationship” and we see it in a lot of really interesting places. Light energy works this way, electric fields work this way, and gravity works this way (if you double your distance from Earth, the gravity you experience will be ¼, if you quadruple your distance from Earth, the gravity you experience will be 1/16th.)

For all of these phenomena, you can always go back and imagine these cubes of solar panels. When you make them farther away, you have to make each face of the cube bigger. When we make the edge of each face bigger, it changes the area. If we double the edge, we multiply the area by 4. If we triple the edge, we multiply the area by 9, etc. It’s always the square of the side length.

Because as we get farther away each side has more and more solar panels, each panel gets less and less light. This situation where one thing increasing causes another thing to decrease is called an “inverse” relationship. The distance from the star is increasing, and the energy each panel gets is decreasing.

Not only that, but because each face is going from 1 to 4 to 9 to 16 to 25 panels each time, the energy goes from 1 to 1/4 to 1/9, to 1/16, to 1/25. It’s getting smaller and smaller with these fractions of perfect squares.

That’s why we call it an inverse square relationship!