Sunday, February 10, 2008


I vaguely remember reading an anecdote once where Edison gave a newspaper reporter a list of things that every scientist should know. I don't remember the whole list, but one of them was the speed of sound in air. Another reporter showed the list to Einstein, who confessed to having no idea how fast sound propagated through air.

Edison, being partly deaf, was somewhat more interested in sound than Einstein, who was more of a light man, as it were. Still the speed of sound, as a principle, is mighty important; it just varies with a lot of things that were, to be fair, of interest to Einstein as well.

Sound propagates when atoms bump into each other, so it's important how fast the atoms can go, and the nature of the bumping. In solids and liquids, where molecules are sitting right next to each other, as it were, the forces between them, the elastic modulus is the critical factor, as is the nature of the wave that is being transmitted. Molecular movement in solids is also quantized, with the pseudo-particle being the phonon, which represents the quantum levels of forces transmitted from one molecule to another.

The speed of sound (SOS) in gases depends on how fast the individual molecules of the gas are moving, since any individual particle must actually traverse the distance between it and the next particle for momentum to be transferred. So there we get into all sorts of cool things like ideal gas laws, heat capacities, and statistical mechanics, some of which Einstein did have in his thoughts.

In the simplest approximations, the speed of sound for a gas is determined by two factors, the molecular weight of the gas and its temperature. The speed of sound is always limited by the RMS (root mean squared) molecular speed; the two are related via a fairly simple relationship:

RMS/SOS = Sqrt(3/BM)

where BM is the bulk modulus of the gas, it's resistance to pressure. For a diatomic gas, the bulk modulus is 1.4, so the ratio of RMS to SOS is about 3/2.

In rockets, the oomph that any given propellant will give is limited by the velocity of the exhaust gases. So basically you want your exhaust to be very hot, with the lightest molecular weight you can manage. In Rocket Ship Galileo, Heinlein had his protagonists use zinc as the propellant (heated via nuclear reactor), and has one of them muse that he'd have preferred to use mercury. This is, of course, almost exactly backwards, and Heinlein did a better job later, in, for example, Space Cadet, where "monoatomic hydrogen" is supposedly used.

Monoatomic hydrogen would indeed be a good rocket propellant, pretty much the best possible, if you could use it. However, the temperature at which diatomic hydrogen (which is to say, hydrogen gas) dissociates into atomic hydrogen is mighty high, in the thousands of Kelvin, and would probably destroy any rocket nozzle that could ever be built. As I recall, Heinlein had tanks of monoatomic hydrogen on his ships, no doubt made out of unobtainium metal, with a bolonium catalyst to keep the hydrogen atoms from recombining.

Rockets are, as I've said before, a horribly inefficient method of travel, since conservation of momentum means that you're hurling huge masses of material out the back end, and it�s taking most of your energy supply with it. In fact, the more "efficient" your rocket in terms of payload to fuel ratio, the higher the percentage of your energy supply is going into your exhaust stream.

Also, with chemical reactions as your energy source, you can't really use hydrogen as your exhaust gas, because it isn�t the product gas of the energetic reactions you'd like to use, always assuming that you don't actually have tanks of monoatomic hydrogen lying around. MH would produce some pretty hot molecular hydrogen when it recombined, so that would work. Too bad about the world wide unobtainium shortage.

All the speed of sound issues apply to explosively driven projectiles, aka "guns," as well, though such projectiles are much more efficient than rockets, energetically speaking. Mass drivers of all sorts have the advantage of using the Earth as a big momentum sink, and when you use something that large to absorb the recoil, it doesn't get much of the energy in the bargain.

You can't generally use hydrogen and liquid oxygen in a bullet (though there are some cannon designs that do), so typical muzzle velocities are limited by the average mass of the molecules in gases like nitrogen and carbon dioxide. Those have greater masses and hence lower particle velocities than does water vapor, to say nothing of hydrogen.

But then we come to gas guns, where the projectile is driven by compressed gas. Sure, you usually can't get the pressures in a compressed gas cylinder as high as you get from an explosive, but you can then use hydrogen, or helium as the gas. Helium, being honestly monoatomic, has only twice the mass of a hydrogen molecule, so its RMS and speed of sound is still pretty fast, which is why you get a high pitched voice if you inhale helium.

If you use a compressed gas cylinder, you have what is called a "single stage gas gun," which rather demands an answer to what a "two stage gas gun" is, right? Ah, there it gets interesting. In a two stage gas gun, you use an explosively driven piston to ram the gas into the compression chamber. Then, when it reaches a nice, high pressure (and remember, it's also been heated via compression), it ruptures a perforated valve and slams into the projectile, which is then propelled out of the barrel of the gun. Some designs preheat the original gas as well; you can exceed the melt temperatures for parts of the device for brief periods of time, and gun shots are nothing if not brief.

Lawrence Livermore Laboratory has a nice two stage gas gun that can propel a projectile weighing 5 kilograms to 3 kilometers per second. There were plans in the early 1990s, to upgrade the thing and to use lower weight projectiles, which would reach 8 kilometers per second, and LLL wanted to try putting things into orbit with it. Instead, absent the $1 billion upgrade, they had to content themselves with firing the thing into a liquid hydrogen target, experimentally demonstrating the existence of the previously only theoretical metallic phase of hydrogen. And even without quite so lavish funding, they do seem to have managed to get up to the 8 km/sec range, albeit with pretty light projectiles.

Theory doesn�t quite run out of oomph at 8 km/sec, however. As you go to higher and higher temperatures in hydrogen, you begin to get molecular dissociation. Heat your original gas hot enough, and compress it enough, and you can get a gas containing significant amounts of--wait for it--monoatomic hydrogen. I've seen a design document from The Rand Corporation on how to build one of those, and its theoretical top projectile velocity exceeds 10 km/sec. That's flirting with escape velocity and it's well over orbital velocity. It may also be getting close to the velocity necessary to compress inertial fusion materials to the point where a tritium-deuterium burn can occur, but that's a different essay, for another time.

No comments: