April 15, 2013 | 12
I was recently asked a great question about the meteor that hit Chelyabinsk and the asteroid that nearly missed the Earth that same week: “They look small. At what speed would their energy density equal that of TNT? ” Since I like to do physics about (a) space, (b) things moving very fast, and (c) explosions, this seemed like a great excuse to think about just how much energy various things have.
When something falls from the sky and crashes into the ground, it releases its energy in a number of ways. Some of that will go into heating the air, or forming a sonic boom if it’s going faster than sound. Some of it will shake the ground, or burrow through it. At high enough speeds… well, we’ll get to that later. But thanks to the Law of Conservation of Energy, no matter where the energy goes, the total amount of energy released is the same, which means that we can compare levels of destruction fairly reasonably by just looking at how much energy everything carries.
(Note for the nitpickers: The details of the damage will vary tremendously, as explosions have very complicated dynamics involving pressure waves, thermal waves, shocks, etc. That said, if you are struck in the head by, say, a penny moving at orbital speeds, the basic conclusion that you are having a bad day and probably no longer have a head will remain the same.)
Let’s think about how much kinetic energy things have. The basic unit of energy is the Joule (J), which is about the energy required to lift a one-kilogram mass 10 centimeters off the ground. (Or equivalently, the energy of dropping a one-kilogram mass onto your foot from a height of 10 centimeters. For those who live in the US, drop a one-pound weight nine inches onto your foot, instead.) The total amount of kinetic energy (energy of motion) that objects have depends on both their speed and their size, in much the same way that the amount of energy in an explosive depends on both its composition and its size, but the energy density — the amount of energy per kilo of object — depends only on the speed, composition, and so on, and not on the size. That makes it a useful way to compare objects. For things travelling well under the speed of light, the kinetic energy density is given by the formula E / m = ½ v2 (E is the energy, m is the mass, and v is the velocity).
A car travelling at highway speed has an energy density of about 360 Joules per kilogram (J/kg). A fully-loaded Toyota Corolla weighs about 1,700 kilograms, so it has a total energy of about 612,000 Joules, which is the energy it would deliver to you if it hit you and came to a complete stop in the process. That’s enough to cause a lot of damage, but that’s not enough to really be exciting. (Physicists have an interesting notion of excitement.)
Cruising speed for a 747 is about 570 miles per hour (mph). At that speed, it has about 32,000J/kg of energy density. (Notice how fast this is rising! Energy goes up as the square of speed) Energy density depends only on the speed, not on the size of the 747; a penny (3.1 grams) travelling at the speed of a 747 would have an energy of about 100 Joules, which is about the same as the energy of a baseball going at 83mph. (Not quite as much oomph as Nolan Ryan’s record 108mph, but still not something you’d want to hit you in the head.)
That’s not very fast. I want to go faster!
Of course, planes can go faster. An SR-71 Blackbird tops out somewhere around Mach 3.2, which gives it an energy density of about 600,000J/kg, so a one-kilo slug going at this speed would hit you as hard as that same Corolla barrelling down the highway. But that still isn’t enough to match the original request to have something with as much energy as TNT.
TNT has an energy density of 4.6 million J/kg. This means that setting off one kilo of TNT releases 4.6 million Joules of energy, which is enough to raise the temperature of a liter of water by about 1,100 degrees Celsius — or equivalently, to shoot that same liter of water about 470 kilometers straight into the air. (Gasoline actually has nearly ten times the energy density of TNT, but it usually releases that energy in the form of burning rather than exploding, so it’s not quite as exciting.) To have the same energy density as TNT, something would have to be travelling at about 3,000 meters per second, around Mach 8.9. That’s just under the speed record for an airplane: the unmanned NASA X-43 hypersonic plane, on its third and final test flight, made it to Mach 9.68 before disintegrating.
People have gone faster, of course. The human airspeed record comes from the Space Shuttle’s reentry, where it hits the atmosphere at 17,500mph. At this speed, the shuttle (and the people in it) have an energy density of about 31 million J/kg. A housefly propelled at this speed would have about as much energy as a .45ACP bullet at point-blank range. (In related news, I have the least useful idea for a weapons system ever.) As the Shuttle strikes the air, it forms a shock wave (the bow of its sonic boom) which compresses the air in front of it; that compression alone heats the air to 7,800 Kelvin, (about 13,600°F) causing it to glow blue-hot. For comparison, the surface of the Sun is about 5,500K. (Interestingly, when I look up “7800K light,” the top results discuss the relative merits of 7800K lights versus 6500K lights for growing hydroponic pot plants. Science!)
Needless to say, humans travelling at speeds where the air becomes hotter than the surface of the Sun, a simple housefly would do as much damage as a bullet, and their own impact against something would explode with more energy than their weight in TNT aren’t going to be having a very good day if anything goes wrong. Clearly, in the name of safety, we should stop now.
Safety? We don’t need no steenkin’ safety. Let’s go faster!
We started out talking about rocks in space, so let’s look at those. The typical orbital speed of a rock in our Solar System is around 19 kilometers per second (km/s), or about 42,500mph. (If you want a mental picture of this, 19 kilometers is about the distance from you to the horizon when you’re looking out the window of the tenth story of a building on a clear day at an open field. You notice a faint blur of a rock on the horizon; one second later, the rock hits the building.)
The Earth is travelling at roughly this speed around the Sun, and generally if things get misaligned and something runs into something else within the Solar System, it will be doing that colliding at about this speed. At these speeds, space rocks have an energy density of about 180 million J/kg. A penny travelling at this speed would have the same kinetic energy, and hit you just as hard, as our Corolla travelling down the highway.
However, these meteors tend to average a bit bigger than the average penny: the Chelyabinsk meteor was estimated to weigh about 11,000 tons, giving it a total energy about 28 times that of the atomic bomb which destroyed Hiroshima. Fortunately for us, meteors aren’t famous for their structural integrity, so the meteor disintegrated 23 kilometers in the air. Unfortunately, conservation of energy still applies, so all that energy was still released in the form of heat, a sonic boom, and so on. Most of it dispersed harmlessly, but that which remained was still enough to shatter windows for miles around and injure 1,500 people.
Surprisingly, the sight of a meteor the size of a semi exploding with the force of two dozen nuclear bombs directly over their city appears to have left many of the locals unfazed.
Huh. That didn’t really blow a whole lot up, I guess. I want to go faster!
Once you move above those speeds, things tend to not stay in the Solar System; anything as far out as Jupiter moving at 19km/s will actually escape the Sun’s gravitational pull. (As close in as the Earth, you need to be moving at about 47km/s to do that) By the same token, anything you encounter that’s moving faster than these speeds almost certainly came from outside the Solar System. Typical debris that’s floating free in the Milky Way is moving at about 200km/s, with an energy density of about 20 billion J/kg. A typical 21-gram African swallow (a South African cliff swallow, in particular) travelling at galactic speeds would have an energy of 420 million Joules, which is approximately the impact energy of an adult bull sperm whale (40 tons) plummeting off Canada’s Mount Thor, the tallest vertical cliff in the world (1,250 meters). Carrying a typical five-pound coconut, it would instead have an energy of about 45 billion Joules, approximately equivalent to 20 GBU-27 “smart bombs” of the sort used in Iraq.
In a related note, it’s surprisingly difficult to find comparisons at this energy scale.
I wanna go faster, dammit!
Some things are moving even beyond typical galactic speeds. The star S-2 appears to be trapped in a tight orbit around the supermassive black hole at the center of the Milky Way galaxy, circling it at a distance of 17 light-hours and orbiting once every 15.5 years. That gives it an orbital speed of about 5,000 km/s, fast enough to fly from New York to London in just over a second. That gives it a kinetic energy density of about 12.5 trillion Joules per kilogram. (And, being more than ten times the size of the Sun, it has quite a few kilograms)
Now, normally I would calculate the impact energy of this against someone’s forehead in the usual way, just multiplying it out. But it turns out that this is a “magic” energy range where more interesting things can happen. Instead of measuring the energy in Joules per kilogram, let’s measure it in electron volts per proton. (The electron volt, or eV, is 1.6·10-19 Joules and is a convenient unit for studying atoms; a typical electron is held into its atoms by an energy of a few eV.) That’s going to tell us how much kinetic energy each and every proton in this star is carrying, and the answer turns out to be about 130,000 eV per proton. This happens to be around the start of the energy range at which atomic nuclei stop simply bouncing off one another and start undergoing nuclear fusion. The exact rate of fusion depends on the precise speed and chemical composition of projectile and target, but each fusion reaction which happens will release about 100 times as much energy as the collision itself. (However, in a real collision, only a small percentage of the atoms would actually fuse, so rather than getting an overall 100 times power boost we should see more like a one to five times power boost)
Pennies are actually full of metals, which don’t fuse well, but our friend the swallow is full of carbon and hydrogen and other things which react interestingly. Even if the speeds come out all wrong and no fusion whatsoever happens, a coconut-laden swallow will have an impact energy of around 30 trillion Joules, about a third of the explosive force of the Fat Man atomic bomb. If the numbers are right and fusion happens, we’re talking about a bird with more kick than all the atomic bombs ever used in warfare.
That should be enough energy, right?
No! I wanna go faster!
OK. To accelerate macroscopic objects to much higher speeds requires a fairly impressive burst of power, and one of the very few things that can do that is a supernova. As a star explodes, it tends to propel about one-third of its mass outwards at a speed of roughly 30,000 kilometers per second.
At these speeds, we can no longer get away with using our simple formula for energy and velocity; this is one-tenth the speed of light, and the special theory of relativity kicks in. The exact formula for kinetic energy is
(For small values of v, this reduces to E/m = ½v2 + ⅛v4/c2 + … You’ll also note that it would require an infinite amount of kinetic energy to reach the speed of light; this is not a coincidence.) However, we no longer have to worry about fusion: at these speeds, the particles will be moving past each other too fast to react.
So, OK. The ejecta of a supernova, the fastest-moving macroscopic objects that you’re likely to encounter, have an energy density of about 450 trillion Joules per kilogram. There’s a handy fuel conversion formula: 13.7 grams of ethanol, the amount in a Standard Alcoholic Drink, moving at this speed contains almost exactly as much energy as is stored in the fuel tanks of a fully loaded Boeing 747-100B.
Macroscopic objects are for wusses. That’s barely even relativistic. I wanna go faster!
Beyond this speed, very few macroscopic objects are around, for the simple reason that anything which has enough energy to accelerate things to this sort of speed is either concentrating a lot of energy into a very small space, or is blowing up something very big, and if it’s something very big then there’s generally a lot of stuff moving at this speed and that would mean more energy than a supernova, which is not often available.
Fortunately, we’re pretty good at manufacturing faster microscopic objects. In a typical fission reaction (say, at a nuclear reactor) part of the energy is released in the form of very energetic electrons, with energies of about 3 million eV per electron. Electrons weigh only about 1/2000 as much as protons, so this translates to a much higher energy density: about 5.3·1017 J/kg. The electrons are travelling at about 99% of the speed of light.
At this point, the electrons have an interesting new way to lose energy. The speed of light through various substances is slower than the speed of light in vacuum, because the light keeps bouncing off the atoms. (This is the reason lenses work: by making the light go slower through the glass, and making different parts of the light go through different amounts of glass because of the way the lens is shaped, you can make some parts of the image “catch up” with others, and thus deform an image.) Nuclear reactors generally sit in tanks of cooling water, and the speed of light in water is only about 75% of its speed in vacuum. That means that these electrons, when they hit the water, are travelling faster than the local speed of light.
When this happens, a shock wave forms out of light that’s very similar to the one that forms out of vibrations when something tries to travel faster than the speed of sound. This is called Čerenkov radiation, and the electrons proceed to shed a huge fraction of their energy very quickly in the form of a beautiful blue light.
Hey, particle accelerators! I bet those can go pretty fast!
The biggest particle accelerator in the world is the Large Hadron Collider in Geneva, which accelerates protons to an energy of 4 trillion eV (TeV) each; when it reopens in 2015, it should be able to go as high as 6.5 TeV. That’s 6.2·1020 Joules per kilogram, or a speed 99.9999997% of the speed of light. (You can see why we measure the energy in nice, convenient units like TeV per particle…) At this energy, a single penny would have more energy than South Korea used in 2009, nearly ten times as much energy as the largest nuclear bomb ever detonated. A coconut-laden swallow would have as much energy as the total oil reserves of Saudi Arabia, or about 53 times the total world nuclear arsenal.
You might think that this swallow could destroy the Earth. You would be close: even unladen, the swallow would produce an explosion 16 times greater than the eruption of Krakatoa in 1883, and with its coconut in tow, its impact on the Earth would be truly colossal… but still not even 4% of the energy released by the 2004 Indian Ocean earthquake. The Earth is big, you see, and nature full of forces so powerful that they are hard for us to imagine.
And in fact, nature can make things go a lot faster than we can. The fastest particle of any sort ever observed was the “Oh My God Particle,” a cosmic ray believed to be a proton which struck the upper atmosphere over Utah on October 15, 1991, with an energy of about 3·1020eV — about the same energy as a baseball going 60mph. This particle (and the few dozen others of similar energies observed since) are considered an interesting mystery, because their energy exceeds what’s called the GZK limit: the maximum speed that any particle should be able to achieve in interstellar space, simply because it would occasionally bounce off of stray particles in the universe. The mechanism which accelerated this particle to such speeds is a matter of research and debate, with current speculation focusing on quasars.
Earlier we talked about the energy of baseballs, and now we’ve concentrated the energy of a baseball into a single proton, at an energy density of about 3·1028 Joules per kilogram. Unladen and flying at this speed, our swallow would deliver a blow of 6·1026 Joules to the Earth, about 1,200 times the explosive force of the Chicxulub impact which killed the dinosaurs. Armed with its coconut, this avian harbinger of the Apocalypse would theoretically deliver a blow of 6.5·1028 Joules, enough to knock the Moon out of its orbit. (But not enough to destroy the planet: Theia, the Mars-sized object which crashed into the Earth 4.5 billion years ago and formed the Moon, would have had an impact energy of about 1030 Joules)
Fortunately for our continued orbital stability, this demoniac bird would have trouble delivering its full force to the Earth: thanks to relativity, it would see the Earth as a giant pancake, 12,000 kilometers across but only 40 microns thick at its center, and would punch a swallow-shaped hole right through it, depositing less than 0.05% of its energy in the process and leaving with only the vaguest awareness that there was a planet there.
Unfortunately, that small percentage still amounts to 3·1025 Joules, or about 60 dinosaur-killing asteroids. The swallow-shaped hole would only last for a brief fraction of a second, before the vibrations it sent flying through the Earth began to tear apart mountain ranges, unleash volcanos and earthquakes, and so on.
Or possibly not: I haven’t done the calculations, but it’s also possible that the tremendous speed of the swallow would transfer the energy much more uniformly through the bulk of the planet than an asteroid forming a crater possibly could, and that rather than creating an immense tsunami and propelling so much dirt into the upper atmosphere that decades of perpetual winter cause the environment to collapse, there would simply be a very loud “thump.” And sitting around late at night, doing the calculations to answer that sort of question, is exactly the sort of thing that makes people become physicists.
Images: Swallow: a collage by A. V. Flox of photos by Alan Manson, Luis Tamayo, and Alexis Breaux.