Well, hey, to avoid fighting the hypothetical…
You definitely need to get through the atmosphere, so muons are by far your best choice of particle. They only live for 2 microseconds, though, so they need to be at least 10 GeV in energy to have enough time dilation to get to the surface with any efficiency. They’ll lose a few GeV due to ionization of the atmosphere on the way through, but that’s not too bad. Let’s pick 20 GeV as a working initial energy.
A rough calculation for the RMS scattering angle of a 20 GeV muon penetrating the atmosphere yields 6 mrad, or a spread of about 60 meters around the target. I’ve approximated the atmosphere as 10 km thick; it’s actually thicker than 10 km, but the denser, lower portions will induce most of the scattering.
60 meters of collateral damage isn’t awful as military operations go, but it does introduce substantial inefficiency. A human target presents about 1 m[sup]2[/sup] of area when lying down, so the muon “collection” efficiency is about 0.3%. Thus, we’re clearly not talking about making bullet-like holes, and presumably we aren’t interested in damage that causes symptoms only many hours later. About 20 gray of absorbed dose will incapacitate someone within minutes. (They won’t die for many days, but they will wish otherwise.) That’s about 70 joules of energy absorbed over the whole body. Assuming a 20-cm thick human made of water, each muon will leave about 50 MeV (8x10[sup]-12[/sup] joules) behind during its passage through the human, meaning we need 9x10[sup]12[/sup] muons passing through the human to achieve 20 gray. Given the collection efficiency above, we need 3x10[sup]15[/sup] muons in total.
With that established, we now need a muon source.
Muon production, capture, “cooling” (the process of taking the random momenta muons produced and getting them all marching how you want), and storage is an active area of accelerator research right now, and things have a long way to go. You could do away with the complexities of storage if you could generate the requisite number of muons in one shot.
To produce muons, you start with protons. The J-PARC facility in Japan has the most protons-per-pulse of any fixed-target beam of relevant energy (50-150 GeV), so let’s look there. In the linked picture, notice the purple-ish path and the physical dimensions indicated. That 1-km-across sequence of linac, RCS, and “main ring” synchrotron is needed to get the protons from a stand-still up to 50 GeV. You can’t make the early stages any smaller because you can’t maintain high enough electric fields in the accelerating cavities (“RF” cavities), and you can’t make the main ring any smaller because you’d need magnets bigger than we know how to make to keep the protons circulating.
Okay, so you need to get that whole complex into orbit. You’ll also need to power it, but as long as we’re bringing a whole accelerator complex up into orbit, I figure we can bring a nuclear reactor, too. The weapon doesn’t operate continuously, so we can just power it up when needed.
Anyway… the protons, once up to speed, are directed into a block of material to produce pions, which in turn decay to produce muons, which in turn are collected and directed. Since we’re talking about 20 GeV muons, we’ll actually want somewhat higher energy protons to start with – perhaps 150 GeV. Glossing over the vagaries of proton–>pion–>muon production, muon capture, and muon focusing, you’d never do better than one useful, targeted muon per dozen protons, and that’s probably very generous. So, you need a pulse of 10[sup]17[/sup] to 10[sup]18[/sup] protons. However, J-PARC only achieves 10[sup]14[/sup] protons per pulse, so we’re off by a factor of about 1000, which puts the damage down at the level of a full-body CT scan.
Since the orbiting machine won’t need to operate continuously, some of the normal limitations to proton intensity can be ignored, like accumulated radiation damage and thermal damage to the pion production target. Maybe we could scrape back a factor of 50 or so this way. The last 200x, well, you’re on your own. (Muon storage only gets you so far, since the muons will be decaying as you try to build up a big enough batch. You could always just fire 200 (or, heck, 1000) shots in a row, once every second or so. You’d have to track the target for 15 minutes, but maybe they’re asleep or something, and as long as they don’t stray much more than a few tens of meters from where you want them to be, they’d still be in the damage zone.)
As an aside: aiming this device is much easier in space since you don’t need to worry about swinging a large vacuum pipe around – the vacuum is everywhere, and it is free. You just need a couple of deflecting magnets to kick the focused beam in the direction you want. The whole facility needn’t be rotated.
Right, so, having said all that: yeah, this is patently ridiculous.
