Wednesday, June 29, 2016

This Island Earth

I finally finished Neal Stevenson's Seveneves, a monumental end-of-the-world novel. I tend to "read" this kind of book as an audiobook while driving, which means it takes a while to get through one of this length.
Seveneves begins with the moon blowing up, although as many reviewers have pointed out, this isn't a spoiler since it is the first sentence in the book. On the other hand, there will be some spoilers here, so I urge you to read the book first.

As I started to listen to the story, once it became apparent that the moon was going to continue to break up and ultimately produce a "hard rain" of meteors that deposit enough energy in the atmosphere (and surface) that it would glow red hot--and it does glow red hot in the story-- I realized I had seen it before. One of the more visually spectacular (if scientifically dodgy) SF movies of the 50s, This Island Earth, has the planet Metaluna being attacked by forces which use meteors as weapons, and in the movie, you see the attack succeed, the rain of meteors overwhelm the defenders' forcefields, and ... yep, the planet glowing red hot as the aliens finish it off.

(Oh, and as long as we're on 50s SF, the parallels to When Worlds Collide where there is a frantic rush to build rockets to escape the doomed Earth with a remnant of humanity is too obvious to mention.  So I won't.)

This is not a review of Seveneves -- that would take a year to write properly -- but just an investigation of one point. Namely, does that really work? Could a meteor bombardment make a planet glow red hot, and if so for how long? In the story, it is some appreciable fraction of 5000 years.

We are used to vast energies in our weather system dwarfing what mere puny humans can do with our our little machines. But all of that energy is just side effects of the incoming sunlight on its way to being re-emitted as IR. For the planet as a whole, that energy flux works out to about 300 watts per square meter (and in equals out to within roughly a tenth of a percent).

For the entire planet to glow red hot, it would have to be about 1000 degrees Kelvin, and it would radiate a LOT more energy into space. To do the calculation right, it gets complicated, in lots of ways, so what I'm about to do is an extreme simplification. But for a back of the envelope, the average temperature of the earth now is about 288 K. Planck's Law says black-body radiated power scales as the cube of the temperature, so the red-hot Earth would be radiating about 40 times as much as now, i.e. 12 kilowatts per square meter. Earth is about 5e14 m^2, so it would be putting out 6e18 watts.

One kilogram at the height of the moon's orbit has a potential energy relative to the surface a little over 60 megajoules (the kinetic energy of its 1 km/sec orbital velocity is just half a megajoule). Thus you need to drop 1e11 kg of moon per second on the earth to keep it red hot. The moon masses 7e22 kg, so you get 7e11 seconds of bombardment if you want to keep the Earth red hot.

7e11 seconds is 22,000 years; so yes, it works. You need to use less than a quarter of the moon.

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