One of the great things about “Fermi Problems” is that there are multiple ways of attacking them. So, for example, when considering the death ray plot yesterday, I used medical devices as an example system to assess the plausibility of the plot, while Physics Buzz talked total energy. But those aren’t the only ways to approach this, and turning it over last night, I thought of another approach.

As I mentioned in yesterday’s post, I normally think of light in terms of photon flux or energy flux, and one way to go at this would be to ask how many photons of light you’d be dealing with. The conversion to radiation dose is a messy business, though, which makes it a little tricky. The radiation units reference I used yesterday, though, offers a path to something like a reasonable connection, noting that one Roentgen of radiation works out to around 3×10^{10} photons per square centimeter at an x-ray energy of 60keV. One Roentgen then corresponds, under some set of simple assumptions, to a dose of about 0.03 Gray; 30 Gray is the number Wikipedia suggests for fatal radiation poisoning. A tiny bit of math, then, suggests that a flux of one photon per square centimeter corresponds to 10^{-12} Gray, which gets things more into my language.

So how many photons are we talking? Well, the news story I read specified that they were hoping to use a 2000 Watt battery to power it, which means that it could produce a maximum of 2000J of energy every second. If we made the ridiculously optimistic assumption that every bit of that two kilowatts came out as x-rays at 60keV, that would correspond to around 2×10^{17} photons/second.

This needs an area to convert it into the right units for the medical conversion, though– I need photons/cm^{2}. But we can make the simplifying assumption that these aren’t focused at all, and spray out equally in all directions. In which case, at a distance of 1m, they would be spread over a surface of 4π square meters, or around 120,000 cm^{2}. That’s a flux of about 1.7×10^{12} photons per square centimeter per second, or a dose of 1.7 Gray per second at that distance. In other words, 20 seconds to kill a person.

That sounds disturbingly plausible, but then we’ve made a wildly optimistic assumption, namely that 100% of the energy from the battery turned into x-rays. The real number would be much, much lower– 1% is the figure Physics Buzz used, and I suspect that’s pretty reasonable. In which case, the flux drops by a factor of 100, which means the time to kill somebody goes up by the same amount. 2000 seconds is half an hour, making it a much slower death ray.

And, of course, that distance estimate is wildly optimistic as well– you’re not really going to secretly kill anybody with a death ray that has to be 1m away. 10m is probably a more reasonable estimate, and the flux goes as the square of the distance, so that’s another factor of 100. 200,000 seconds is about 55 hours, or better than two full days. And that’s not taking into account any shielding effects due to, say, hiding the device inside a van.

So, again, we come around to the same basic conclusion: this wasn’t a terribly realistic scheme. Which isn’t particularly surprising, but it’s nice to have general consistency between methods…

(And at this point, I’ve probably spent more time working on the math of this than the alleged plotters did…)