Nuclear Winter

So, in Battlestar Galactica, the Cylons have just launched a nuclear attack on the Twelve Colonies, including the Earth-like world of Caprica. This is, of course, terribly sad, but we will assuage our grief with science.

Each bomb the Cylons use kicks up a lot of dust into the planet's atmosphere. And increasing the amount of optically thick dust in the atmosphere reduces the sunlight that reaches the planet's surface, which results in lower surface temperatures. Can we figure out the relationship between the amount of dust in the atmosphere and the cooling of the surface?

OK, so suppose the Cylons drop N bombs, and each bomb moves a mass M of dust into the atmosphere. Each dust particle has a a mass density rho and a radius r (assume a spherical dust particle).

First, we should find the total number of particles moved to the atmosphere. Note that



where mp is the mass per particle and Vp is the volume per particle. So we can write



where the units of mp are kg/particle. NM is the total mass of particles. So if we take



we have units of particles, so Np is the total number of particles.

Next we need to find the fraction of flux that gets through the dust and to the planet's surface. We'll assume that



is the surface area of a particle of dust, and



is the surface area of the planet, with R being comparable to the Earth's radius. So then we have



for the ratio of the flux that reaches the planet's surface to the flux that reaches the planet. Since temperature is related to flux by



we can find the factor relating the original temperature to the new with



The only problem with this formulation is with the flux ratio. When the second term is greater than one, which occurs for large numbers of bombs or masses of dust, the result is a negative ratio, which doesn't make sense.

Worked out with David and Mee.

Also, on a semi-related note, if you are a fan of nuclear winter and/or post-apocalyptic fiction, I recommend The Road by Cormac McCarthy.