Dyson Sphere Scales

Allow me to indulge myself here in this post. It will be mathematically dense, and by “dense” I mean it could very well be thick-in-the-skull outta-left-field dead wrong.

Anyway, ahem.

Writing my Orbitsville review yesterday, I got to thinking about the Dyson Sphere, a sphere enclosing a sun at a distance of about two AUs out. (An AU is an Astronical Unit, or the average Earth-Sun distance, about 93 million miles.) The book mentions the surface area of the inner side of the sphere compared to the surface area of the Earth. It’s obviously vastly more spacious on the sphere. I think the book quoted something like either 400 million or 400 billion times more.

I actually laid awake last night in bed, unsuccessfully trying to figure out which was the more appropriate figure.

I have to put this matter to rest, and I’m hijacking you along for the ride.

First, the Earth.

Assume the radius of the Earth to be 4,000 miles. It’s surface area would then be

4 * pi * r ^ 2

4 * 3.14159 * (4,000 * 4,000)

= 201,061,760 square miles

Now, let’s assume that 70 percent of the Earth is ocean-covered. That leaves approximately 60 million square miles of land for mankind to spread out and conquer.

Very well.

Now, a Dyson Sphere.

The equation above is the same (the thickness of the sphere is negligible). Except instead of 4,000 miles, we’re going to plug in twice times 93 million miles, to represent that distance of 2 AUs from the ensphered star.

4 * pi * r ^ 2

4 * 3.14159 * (1.86 x 10 ^ 8) ^ 2

= 434,745,790,560,000,000 square miles

or about 435 quadrillion square miles.

Orbitsville does have oceans and seas, but nothing on the magnitude, proportionally, to waterworld Earth. So let’s call it an even 432 quadrillion square miles of habitable land on the inside of that Dyson Sphere.

How does that compare to 60 million square miles?

Divide the bigger by the smaller.

Drum roll ...

The area of the inside of a Dyson Sphere two AUs from its star would be 7.2 billion times the land area of the planet Earth. One Dyson Sphere is equal to 7.2 billion Earth-like planets.

Mind boggling, eh?

N.B. Okay, I had to go back and see how I fared against Shaw’s calculations. This first required going through the novel to find Shaw’s exact quote. Surprisingly, since I remembered generally where I read it, it took only a few minutes. On page 73 of my Baen Paperback edition of Orbitsville, our hero expounds, “... This sphere has a surface area equivalent of 625,000,000 times the total surface of Earth. If we allow for the fact that only a quarter of the Earth’s surface is land and perhaps only half that usable, it means the sphere is equivalent to five billion Earths.”

Whew! I’m satisfied. The figure I quoted in my review was flat-out wrong. 7.2 billion differs from Shaw’s 5 billion in the fact that he’s using 25 percent of Earth’s surface as land and only half that usable – 12.5 percent to my 30 percent figure.

Since 12.5 is about 42 percent of 30, you’d expect my figure to be a bit larger than 7.2 or Shaw’s figure to be a bit smaller than 5. Regardless, I’m satisfied that we’re at least in the same ball park.