Tanning in perspective

September 2, 2009 in Math

Information is Beautiful tipped me off to this poster from GOOD:

Skeptic that I am, I immediately questioned the headline as propaganda. The Sun very well may produce that much energy, but how much of it reaches the Earth - in other words, how much of it can actually be harnessed? This makes for a good, if straightforward, Fermi problem.

Consider the sun is a point which radiates energy in all directions. The Earth revolves about that point with an orbital radius of 150mm km. Let's think of the Earth as a small 2-D patch on the surface of a sphere with the Sun at the center. The 500,000 years of energy are distributed along that sphere, and the Earth will only receive a fraction of that energy in proportion to the amount of the sphere's surface that it covers.

The radius of the Earth is 6,400 km; therefore its projection on the sphere has an area of roughly 129mm km (that's pi-r-squared at work!). The surface area of the sphere is 2.83 x 1017 km. Thus, the Earth intercepts just 4.55 x 10-8 percent - a nothingth! - of the total radiated energy. If the total output per second is equal to 500,000 years of power, then the Earth really receives enough energy each second to power the planet for just 200 hours. But "just 200 hours" is nothing to sneeze at: in every hour the Earth receives enough radiation to power civilization for more than a year. I think that's an astounding statistic.

If you didn't want to go through all this work, there's always this Wikipedia article. It shows that the Earth absorbs 3,850,000 exojoules (EJ) from the sun each year, and humans use just 487 EJ per year. Happily, this works out to receiving enough energy every hour to power the Earth for just over a year, validating my approximation.

It should be noted that these numbers are probably still overstated - they account for ever single photon that hits our atmosphere; many of those are reflected back into space or are impractical to collect. Nonetheless, while 500,000 years/second may be a sensational number, it turns out that it does capture the spirit of the message. The Earth receives an absolutely massive amount of solar energy. Think about that next time you're at the beach.

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