Built on Facts

Jupiter, Bullets, Antimatter

There’s been some discussion around the web of Jupiter apparently getting walloped by something, probably a comet. As with the more famous Shoemaker-Levy 9 impact, the result is a small dark blot in the clouds of the Jovian upper atmosphere. Small is a relative term – small spots on the surface can easily be as big as the Earth or bigger. Impacts are some of the most violent events in the solar system, and they release their energy entirely via conversion from kinetic energy.

Here on earth people sometimes discuss the history of high energy weapons in terms of energy per weapon mass. Conventional bombs using chemical reactions start the list, fission and fusion with nuclear reactions are more powerful by several orders of magnitude, and the as-yet purely hypothetical antimatter weapons would beat even nuclear weapons by a factor of maybe 100 or so. There the list usually stops. You can’t squeeze more energy out of a lump of mass than with total conversion to energy.

But there’s no reason you have to extract only the internal energy. An ordinary bullet is still the most common implement of warfare despite not releasing any internal energy at all. The damage is done by kinetic means. Comets smashing into planets do precisely the same thing.

The equation for kinetic energy is:

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Now since nothing can go faster than the speed of light, you might expect that letting v = c gives a maximum kinetic energy for a given quantity of mass. But that formula is a purely classical one – if we’re going to be accurate and consider the speed of light limitations then we need to use the exact relativistic kinetic energy formula:

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The lower case Greek gamma is a quantity that varies with the velocity in the following way:

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It increases without bound as v get closer and closer to the speed of light. From the relativistic kinetic energy equation, we can see that the kinetic energy is going to be equal to the rest energy at gamma = 2. This is going to happen at just over 86% of the speed of light. By the time you get up to 99% of the speed of light or so, the kinetic energy is so large, the energy you’d get via total annihilation of antimatter is starting to look pretty small in comparison.

This is mostly theoretical though. To get that much energy out you have to put that much energy in in the first place. That’s not possible at the moment for anything bigger than subatomic particles. But it is interesting to think that in some ways both bullets and comets are very scaled down instances of a kind of energy that can pound-for-pound outmatch antimatter to whatever degree you want.

Comments

  1. #1 Pierce R. Butler
    July 21, 2009

    How much will DARPA pay me for borrowing the above equations and concocting a study on high-velocity antimatter missiles?

  2. #2 ...
    July 21, 2009

    Ultrarelativistic kinetic penetrators are the weapons of the future.

  3. #3 Andrew
    July 21, 2009

    This is of course why, when a decent science-fiction story contains a threat of the form “nice planet you got there, be a shame if anything were to happen to it”, something like a relativistic rock is high on the list of possible ‘accidents’.

  4. #4 Nathan Murphy
    July 21, 2009

    In the relativistic equation explanation, for it to equal the classical equation, don’t you mean gamma = 1.5? (3/2 – 1 = 1/2)

  5. #5 Andrew
    July 21, 2009

    @4: No, when gamma = 2 then the kinetic energy is equal to the energy that you’d get from completely annihilating a stationary version of the same mass.

  6. #6 Uncle Al
    July 21, 2009

    Matter-antimatter annhihalation is oversold. About 50% of hadron-antihadron annihalation energy will imperceptably exit as neutrinos. Hadrons are 99.97% of rest mass. A fusion warhead initially converts zero matter to energy. The numbers of electrons, protons, and neutrons immediately before and immediately after detonation are unchanged. Initial output is merely a reshuffling of binding energies.

    Shaped charge extrusions are overall limited by the speed of propagation of the high explosive shell, topping around 29,800 fps (HMX) to 32,800 fps (DDF). One can imagine being more clever with soft x-ray illumination from a detonating fission pit. Going relativistic… needs more studies.

  7. #7 EastwoodDC
    July 27, 2009

    “Give a person a relativistic rock, and they will shatter a planet today.
    Teach them to do the math themselves, and they will shatter planets for the rest of their lives.”

    You can that on a T-shirt from Ad Astra Games. :-)

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