## Welcome home, Atlantis.

For the last time, that is.

If it’s any consolation that the shuttle program is ending, we got a neat picture out of it (courtesy NASA):

There goes the shuttle and – is that thin layer the atmosphere?

Wolfram Alpha tells me that earth’s radius is 63.7 times larger than the height of the atmosphere above earth’s surface. But I’m not content just being told that. Let’s see if we can determine the ratio from this image. It’s pretty simple by astrophysics standards of math – just obvious ratios, no crazy equations.

First, we need to extrapolate a circle from the curved earth and atmosphere shown in the picture. I’m using an image editing program to overlay two translucent circles on the image.

I’m assuming that this image doesn’t distort the the earth into an ellipse, which is likely a bad assumption, but it’s the best we can do. (Astrophysicists make this assumption all the time, and are happy with answers precise only to a power of ten.) Another confounding factor is that the atmosphere doesn’t just stop at a certain altitude but makes a smooth gradient into the vacuum. Anyway, zooming out:

Wow. That really is small difference.

At full size, the blue circle (which we’ll call a for atmosphere) is 18234.8 pixels in diameter. The green one, e for earth, is 17985.9 pixels. The height of the atmosphere in pixels, h, is obtained by subtracting the radius of a from the radius of e.

$h=r_a-r_e=124.5~pixels$

Now we can divide by the radius of e (8993 pixels) and we get what we came for: the ratio of the height of earth’s atmosphere to earth’s radius.

$\cfrac{124.5~pixels}{8993~pixels}\approx\cfrac{1}{72}$

Hey, science works! I figure that to be an 11.5% error – not bad at all.

Also, while we’re saying goodbye to the shuttle, take a listen to Neil deGrasse Tyson on what space exploration really buys.

UPDATE: Those with red/cyan glasses can see the image in 3D.

### 5 responses to this post.

1. Spooky — I got 1/63.35 on my first try, using GeoGebra. Great little project!

2. Wow! That’s a 0.549% error – probably more certain than the Kármán line, slightly arbitrarily set at 100km.

3. Probably more a freak accident of how I set things up. Slight reasonable changes give answers anywhere from 1/85 to 1/60:

http://bit.ly/neKMZu