Built on Facts

Shoot the Moon

There’s a comic strip called Non Sequitur. Not one of my favorites, but occasionally there’s a pretty good one. Here’s an example. We have a kid speculating about firing a gun on the moon. He asks:

If you were on the moon, which is a vacuum, and tried to shoot a gun, would it fire? After all combustion needs an atmosphere with oxygen… or does the casing of the bullet create its own atmosphere? If so, the bullet would travel farther in the reduced gravity, but would it travel faster than it does on the earth?

i-02aed4facee28fde7a6ecf199f8ee5dc-3.jpg

Let’s take ‘em one at a time.

First, the bullet would fire. Explosives including gunpowder don’t need atmospheric oxygen to fire. They react far too quickly, and atmospheric oxygen can’t be pulled in from the surrounding air fast enough. Instead, explosives contain their own oxidizers and will work in a vacuum just fine.

Willthe bullet travel farther? Sure it will. A bullet stops moving once it hits the ground, and if it’s falling more slowly in the low gravity it will go farther, all other things being equal.

Will the bullet travel faster? No, not appreciably. It won’t lose any speed until it hits the ground and stops, but the lack of air won’t make it come out of the barrel any faster. I suppose there won’t be any air resistance in the barrel itself, but that shouldn’t do anything significant.

Now the comic itself shows the bullet being fired, coming all the way around the moon, and hitting a rock near the shooter. Is that doable? Well, an object in circular motion has a relationship between acceleration, velocity, and the radius of the circle given by:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

On the moon, the acceleration due to gravity is about a = 1.622 m/s2.

So we can use that fact and find the velocity necessary for a bullet to be in orbit just above the moon’s surface:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

That’s about 5,500 feet per second. A standard 9mm bullet travels at about 1000 feet per second. Not enough. A .30-06 rifle bullet reaches around 2,900 feet per second. Still not there. The fastest commercially available cartridge I know of is the .17 Remington, which screams along at 4,400 or so feet per second. That’s close but not enough either, and it’s already an uncommon specialty round that’s about at the limit of the abuse a barrel can realistically take. So I’d say that part of the comic isn’t possible.

For the best, I think. The last thing the moon needs is a lot of bullets orbiting a few feet above the surface. That would probably be a bit hazardous for future exploration.

Comments

  1. #1 CRM-114
    September 12, 2008

    Now repeat the analysis with a CO2-powered pellet gun and a large asteroid.

  2. #2 NoAstronomer
    September 12, 2008

    Diagree partly on the “explosives contain their own oxidizers” statement. A large portion of the chemicals in most explosives (eg TNT) is not oxidized at all.

    A very nice article regardless.

  3. #3 Dunc
    September 12, 2008

    I suppose there won’t be any air resistance in the barrel itself, but that shouldn’t do anything significant.

    Never assume friction or air resistance are negligible. They may be, but it’s rarely a good assumption – especially when dealing with supersonic projectiles.

  4. #4 Matt Springer
    September 12, 2008

    That’s a good point. My reasoning is that if the bullet doesn’t lose much speed in the first few feet after the barrel, it shouldn’t be resisted much in the barrel either.

    On the other hand, a barrel is enclosed and so the air can’t get out of the way as easily. Especially at those very supersonic speeds of the faster bullets that might be an important effect.

    If someone has a good-sized vacuum chamber, a chronograph, a gun, and a sandbag to catch the bullet, I’d be very interested to see what actually happens.

  5. #5 Uncle Al
    September 12, 2008

    Why did the gun go “BLAM” and the bullet go “PING!”?

  6. #6 Mathless Joe
    September 12, 2008

    So, if the .17 Remington goes at 4/5 the orbital velocity, how far around the moon would it travel??

  7. #7 Winter Toad
    September 12, 2008

    I read a short story once, I believe it was by Ben Bova, but I could be wrong about that, in which a shooting war between cold war enemies on the moon led to a situation in which orbiting bullets fired months earlier periodically riddled the moon bases. They had stopped asking for ammunition, and were asking for more computing power so they could better predict when they would be subjected to another peppering.

    In real life, the moon is too inhomogeneous to support this. The quadrupole and higher order moments of the gravitational field perturb the orbits of bodies traveling in non-synchronous orbits around the moon (and there are no synchronous orbits because of the long rotational period and the influence of the earth and sun). These perturbations lengthen the major axis of the ellipse, resulting in inevitable intersection with the surface. There are no stable low-altitude lunar orbits. That’s one reason lunar-orbiting probes are often deliberately crashed into the moon, it’s going to happen pretty quickly anyway, might as well get some use out of it by choosing a deliberate crash point that might allow you to do some science.

  8. #8 Carl Brannen
    September 12, 2008

    The muzzle velocity of the M1A1 is 1575 meters per second, considerably faster than a rifle round, and close to orbital velocity for the moon.

    Let’s see if I can find something faster… Ah, velocities up to 3000 m/sec have at least been considered. Hope that’s not classified.

    By the way, I put myself through school by working part time at an armaments testing facility, I think the only one in the US attached to a college.

  9. #9 intransigentia
    September 12, 2008

    I read that story too, but when I was learning how to do the math, I wasn’t thinking about the story. Thanks for sorting that out for me.

  10. #10 I spend too much time daydreaming about moon wars
    September 12, 2008

    Let’s upgrade the kid’s handgun to a tank. A quick Google shows anti-tank uranium projectiles to have speeds of >1,700 m/s [1], fast enough to orbit the moon. To do this, they use an extremely high ratio of propellant to bullet mass, so high that the barrel diameter is much wider than the projectile (a “sabot” [2]).

    There’s really no air resistance on the moon; its atmosphere is a ultrahigh vacuum on the order of 10^-9 Pa [3][4]. This is too thin to think in terms of pressure waves and sound; instead you have individual gas molecules which don’t interact at all (perfect ideal gas), typically going thousands of miles [5] without interacting (mean free path). So the interaction between a bullet and the atmosphere is really a ballistic problem – a series of two-body collisions (gas molecule + bullet). The bullet collides with every atom in the volume it traverses, and pushes it forwards, losing momentum.

    Let’s get a rough figure on air resistance. Let’s look at the first high-speed bullet from [1]: has a cross-sectional area of 3.8cm^2 and a mass of 4.8kg (uranium is really dense). The lunar atmosphere’s temperature has huge day-night variations, and it’s chemical composition is not really known; let’s assume it’s all Ar. Then at typical temps, the speeds of Ar atoms [6] (Maxwell-Boltzmann thermodynamic distribution) are small [7] compared to 1,700 m/s; we simplify and assume they’re at rest. Then each collision accelerates an Ar atom by (at most [8]) a ΔV of 1,700 m/s, for a ΔKE of 10^(-19) J. Adding them; at 5*10^4 atoms/cm^3 [4], the power dissipation of the projectile moving at speed v is

    (ΔKE/atom) * (atoms/second)

    = (ΔKE/atom) * (atoms/volume) * (volume traversed/second)

    = [40 / (6.022 * 10^26) kg) * 1/2 v^2] * (5*10^10 m^-3) * 4*10^-4 m^2 * v

    = 3*10^-9 W

    To slow down to sub-orbital speeds (1,700m/s -> 1,670 m/s) is a ΔV^2 of 101,000 m^2/s^2, or a ΔKE of about 500 kJ. So everything else being equal, the orbit lifetime due to air resistance would be 500 kJ / (3 nW) = 2*10^14s, or 5.3 million years.

    There are much more serious problems: for instance, the moon is not flat, and the bullet will almost certainly collide into a crater wall or mountain unless fired from a tall mountain. Even then, the orbit will typically be at least slightly elliptic [9], spending most of its time at high altitude rather than at the surface (unless the launch speed were *precisely* controlled). Also, the bullet will not return to its starting place, because its orbit will appear to precess because to the (slow) lunar rotation [10]. And the earth’s tides would shift it in yet different directions.

    [1] http://www.atk.com/customer_solutions_missionsystems/cs_ms_w_tgs_120ammo.asp

    [2] http://en.wikipedia.org/wiki/Sabot

    [3] http://www.tsgc.utexas.edu/tadp/1995/spects/environment.html

    [4] http://en.wikipedia.org/wiki/Vacuum#Examples

    [5] http://en.wikipedia.org/wiki/Mean_free_path#Mean_free_path_in_kinetic_theory

    [6] http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution

    [7] If they were not, the air resistance would be much higher, because of large (ΔV)^2 from atoms moving in the opposite direction of the bullet

    [8] If the bullet is streamlined, collisions would happen at angles, not flat-on

    [9] http://en.wikipedia.org/wiki/Eccentricity_(orbit)

    [10] http://www.thetech.org/exhibits/online/satellite/4/4b/4b.1.html

  11. #11 Greg Laden
    September 12, 2008

    Nice equations!!!

    I think the air matters.

    But we would hope that the second amendment does not apply to the moon, so none of this matters.

  12. #12 Greg Laden
    September 12, 2008

    Ah, in fact, let us examine the last frame of the cartoon!

    Girl: “Only a boy would think of bringing a gun to the moon!”

    Boy: “See? This is what happens every time I try to have a serious discussion with a girl!”

  13. #13 eddie
    September 12, 2008

    NASA done this already;

    http://science.nasa.gov/headlines/y2007/14mar_marbles.htm

    They even have a “vertical gun range”. I remember Horizon, on the BBC doing a prog on Bill Cooke’s work on meteor strikes.

  14. #14 eddie
    September 12, 2008

    As for the ‘ping’ and ‘blam’, that cliche is as old as star trek itself.

  15. #15 Nomen Nescio
    September 12, 2008

    air resistance in the gun barrel should be negligible compared to frictional resistance against the lands of the rifling.

    there’s atmosphere in a gun barrel when it’s fired (and not in space), but it’s not as if that atmosphere is stationary; blow-by of combustion gasses exit the muzzle measurably before the bullet does, so any cold atmosphere would have been pushed out of the way by gasses that would have been lost to the job of accelerating the bullet anyway.

  16. #16 Brando
    September 12, 2008

    As a 12 year Army vet with experience shooting extreme range rifles (1600-2200m) and a total physics geek…this is the best blog entry I’ve read in a while!

  17. #17 Tom
    September 13, 2008

    Been there (metaphorically), done that (though it was a quickie)

  18. #18 Matt Springer
    September 13, 2008

    Very cool, I’m glad to see we came to the same conclusions. I didn’t see the NS comic or your post when it first came out, I just had it sent to me.

    One of your commenters mentioned the air in the barrel as a possible limiting factor in velocity that would be removed in a vacuum. This may be true, but you’ll quickly run into another problem – the bullet can’t reach a velocity higher than the average thermal speed of the propellant gas molecules no matter how much powder you have. This is going to be pretty fast, but not nearly infinite.

  19. #19 Carl Brannen
    September 13, 2008

    “the bullet can’t reach a velocity higher than the average thermal speed of the propellant gas molecules no matter how much powder you have”

    In the trade, this is what we call “assigning yourself a homework problem”, it happened to me when I opened my mouth in class.

    Typical explosive gasses are CO2, N2, H2O, maybe some H2S or SO2. Typical energies per atom? I can only guess. Typical chemical energies per molecule is something like 1 eV; maybe a molecule of explosive would make two or three molecules of gas.

    So. Just what is that “average thermal velocity”? A good practice PhD qualifier question.

  20. #20 Winter Toad
    September 13, 2008

    The expansion velocity of propellants is why we have something called “light gas guns”. You get a higher speed per molecule when the molecules are lighter. An orbital gun design I’ve seen somewhere involves firing extremely hot hydrogen into a long barrel from “sidings” (for lack of a better term) along the length of the barrel, each additional injection timed to provide additional impetus to the projectile running down the barrel.

  21. #21 James
    September 14, 2008

    One question not asked. would you be able to fire the weapon a second time or would rounds 2/3/4 etc cook off on their own. Any weapon I’ve ever worked with was air cooled, and counted heavily on that cooling to keep working. However in the near vacuum of space this isn’t a reliable method to use. Additionally would it work differently in the shadows, from how it does in full sunlight?

  22. #22 Freemon Sandlewould
    September 14, 2008

    What if I had a Saddamn super gun? I bet I could fire it around the moon!

  23. #23 P-K-One
    December 2, 2010

    Are you completely sure about “explosives contain their own oxidizers”?

    Let’s take termite (I know, not an explosive) as example. Termite gets it’s oxygen from the iron-oxide reduction. But only the burning aluminum creates enough heat to reverse the iron oxidation. Without a certain amount of oxygen the initial reaction of aluminum and oxygen can not start.

    So although the termite reaction is able to run without oxygen it is not able to start without it unless you use an extremely hot heat source to start it.