Space is Huge

The other day I had an idea for a potential new Illogical product. What if I could find a way to visualize the Earth’s orbit in a way that is both unique and scientifically accurate? I thought maybe I could use some fun math to help me out.

The diameter of the Earth is 7,918 miles. In addition to that, the Earth’s average distance from the sun is about 93,000,000 miles. So my question was this: if two people stood on opposite ends of the earth and aimed a laser at the sun, what angle would their beams meet at? In addition to that, I wanted to get a nice visualization of the math as it pertained to Earth’s orbit (remember, that was the original goal) so I wanted to know what an arc of that measure would look like. By cutting that angle out of a circle, we could get an approximate visualization of how much perceivable curvature the Earth’s orbit has over the distance that the Earth occupies on it. In other words, if you bent a giant wire in a circle around the Sun so that it matched up perfectly with the Earth’s orbit (yes, I know the Earth doesn’t orbit in a circle, but remember we are averaging here), what would the section of that wire that actually intersects our planet look like?

To answer this question, we need to first find the angle of our arc. To do so, let’s draw some wildly disproportionate diagrams.

orbit

Here’s a visualization of our problem with the numbers from before plugged in. Obviously this diagram is completely inaccurate, but it’s only here to help us illustrate what is happening. The Earth’s orbit, represented by a black curve, passes through it at two points. Drawing a line from those points to the center of our Sun gives us two lines that are tangent to the Earth. Now, some quick-witted readers might have noticed that because those two lines are tangent to the same point the distance between their intersection points isn’t actually the diameter. And  you’re right. But when considering the huge scale of the distances involved and the already-present margin of error from the data I acquired, that extra few miles won’t really make much of a difference. And, as you’ll see later, the angle is so slight that those points might as well be perfectly parallel. “extra few miles” is a generous assumption on my part. This is a good time to point out that pretty much every aspect of what I’m doing here is approximate and only really provides accurate results because of the extreme scale. If the Earth and Sun actually looked like the diagram above, performing this calculation this way would give us a larger, more noticeable margin of error. Anyway, back to the diagram.

The dashed lines you see are there to help us plan out our equation. The vertical one is our orbital distance, so I’ve labeled it with the appropriate 93,000,000 miles. The horizontal one is our diameter, so I’ve labeled it 7,918 miles. Because these two lines are perpendicular, we can use them to calculate the angle. We do that by plugging them into a simple inverse tangent equation:

invtan

Notice how we are dividing our diameter by 2 in the equation? That’s because inverse tangents are based on right triangles. And if you look at our diagram, we actually have two right triangles. So we divide our diameter by two, plug it into an inverse tangent equation, and get a result of 0.002439. Now we can multiply that by 2 to get the angle of both triangles together, which is our final angle. That angle is 0.004878 degrees. Doesn’t sound like much, does it?

The resulting angle is so small that it might as well be 0. And 0 degrees isn’t really that interesting to look at, as it turns out. In the end, I wasn’t able to use this data for a product. But I was able to use it to draw out a diagram of the earth’s average orbit to-scale. And that’s kind of cool. Here’s what that looks like:

That’s a screenshot from Illustrator zoomed in 6400%. That horizontal line is actually a circle as large as you can possibly create one in Illustrator. Those lines are our tangents and they are not, in fact, parallel. Fun fact about this diagram: the arc of the circle between those two lines is so slight that when I attempted to separate it to scale it up it’s height was smaller than the smallest digits Illustrator could handle. This resulted in it being treated as a perfectly straight line and remaining the same height no matter how much I expanded it.

So this will not, unfortunately, be coming out as a new product any time soon. Oh well. At least we learned something about our tiny place in this big, big universe. I’ve been doing a lot of random research like this trying to come up with new products, so if you found this interesting let me know and I may do some more similar posts in the future.

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Space is Huge

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