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.

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5 responses to this post.

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

    Reply

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

    Reply

  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

    Reply

  4. Ah crap. Sorry to rain on the parade, but check out my latest blog post. (And if you can refute it, please please do! I liked our previous nice neat results better.)

    Reply

  5. Okay, I dug around and did some further research. As the NASA page you linked says, that thin orange layer in the picture is the airglow. According to several sources, the airglow actually is just about 100km above the Earth’s surface! See the following video at 1:05, for example:

    So, it turns out that the perspective projection geometry I was playing around with at my blog did not take into account the fact that the camera was pointed almost at the horizon, which, as it happens, makes it okay to use the simple calculations you and I did in the first place.

    Good to know that I can use this little project in class some time (without mentioning the non-issue of perspective.)

    Reply

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