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All kinds of explosions.

As you’ve probably worked out by now, I’m a big fan of explosions. Since this ridiculous blog began, I’ve blown up rabbits (don’t worry–hypothetical rabbits), stellar-mass balls of gold, grains of neutronium, and I’ve nearly blown up the earth with an ultrarelativistic BB gun. As you’ve probably also figured out, I’m a big fan of bizarre thought experiments, thought experiments based on ideas boiled and distilled down to their absolute essentials. With that in mind, let’s blow some more shit up!

But before we can get started, we have to decide what an explosion is. For the purpose of this thought experiment, we’re going to use a one-meter-diameter sphere of space anchored to the surface of the Earth, just touching the ground. The explosions will consist of energy (in the form of photons of an appropriate wavelength) magically teleported into this sphere. (I still haven’t decided what symbol to use for the magic teleporter in my Feynman diagrams.) With that cleared up, let’s get blastin’.

The smallest possible explosion.

It’s pretty difficult to decide on the lower limit for the size of a one-meter-diameter explosion. First of all, that one-meter sphere is already full of all kinds of energy: solar photons, the kinetic energy of air molecules moving at high speeds, the rest-mass energy of those air molecules, et cetera. But even if the sphere were completely evacuated, quantum mechanics tells us that the lowest amount of energy that a volume of space can possess is greater than zero: its zero-point energy or vacuum energy.

To simplify things (and to keep me from having to learn the entirety of gauge field theory while writing a blog post), we’re going to say that the lowest-energy explosion we can create has the energy of the longest-wavelength (and therefore lowest-energy) photon we can fit in the sphere: 1 meter. That’s towards the high-frequency (short-wavelength) end of the radio spectrum. It’s not quite a microwave (those have wavelengths on the order of a centimeter), but it’s shorter than the photons used to transmit FM radio signals. Needless to say, it would impart a pretty much un-measurable quantity of energy to our sphere of air. A 1-meter photon carries 1.986×10e-25 joules of energy, the same energy as an oxygen molecule puttering along at a grandmotherly 6 miles per hour (10 kph).

The most efficient possible explosion.

But while we’re using our magic vacuum-fluctuation laser (lots of magic in this article…), we might as well see how big an explosion we can make with a single photon. We know the minimum is 1.986×10e-25 joules. But what’s the highest-energy photon we can stick in there? (Sounds like a plot to a horrible science porno…) I’m not a physicist, but I would guess that it’s a photon with a wavelength of 1 Planck length. Here’s my logic: the Planck length is the smallest length that makes sense according to our current laws of physics. To carry energy, an electromagnetic wave must change over time (or space, which work out to be part of the same thing). In order to take two different values, the wave’s crest and trough must be at least 1 Planck length apart. Of course, weird things happen on the Planck scale, so who knows if an electromagnetic field varying by a large quantity over 1 planck length would even make sense, or even behave like a photon, but you can say this with some certainty: a photon with a wavelength shorter than that doesn’t make a lot of sense.

A photon with a wavelength this short would be off-scale, as far as the electromagnetic spectrum goes. By definition, it would be a hard gamma ray, but that’s only because we humans don’t have access to the high-energy regions of the spectrum, so we say “Anything with a wavelength between A and B is an X-ray. Anything with a wavelength between B and zero is a gamma ray. Stupid gamma rays. Who needs ’em?”

This is a gross oversimplification, but photons tend to prefer to interact with objects that are roughly their same size. Visible photons interact with the electron clouds of  large molecules (like chlorophyll, which is good if you like oxygen). Infrared photons interact with large molecules directly, making them rotate or move. Radio-frequency photons interact only with big crowds of mobile electrons, like you find in plasma or radio antennas. In the other direction, ultraviolet photons interact with atoms and their bonds, either knocking outer electrons loose or breaking the bonds. X-ray photons interact with the tightly-bound electrons in lower energy states (I’m afraid to say “closer to the nucleus,” because that’s not quite right, and people smarter than me will make fun of me; but we’ll pretend that’s how it works to speed things along). Gamma-ray photons interact with the nucleus directly. They can push the nucleus into strange high-energy states or, if they’re the right wavelength, pelt protons and neutrons loose.

The physics of extremely short-wavelength gamma rays is much more complicated. That’s partly because, once a gamma ray passes an energy greater than 1022 keV (the kinds of energies you get when you heat something to several billion Kelvin) have enough mass-energy that they can spontaneously turn into matter: a 1023 keV gamma ray can briefly become an electron and a positron, which rapidly annihilate to re-form the gamma ray. The higher your photon’s energy gets, the more important these bizarre transformations become. Basically, once you get above 1022 keV, your photons start behaving less and less like light and more and more like matter.

Our Planck-length photon would (probably) be small enough to pass through all ordinary matter. It might pop a quark loose of a proton or a neutron, but I don’t really know. This photon would carry an energy far greater than 1022 keV. Indeed, its energy wouldn’t be measured in the peculiar particle-physics unit of the electronvolt, but rather in freakin’ megawatt-hours. This photon would add as much energy to our sphere as the explosion of three tons of TNT, which would form a cube large enough to contain our sphere with room to spare. From one…single…photon. Physics is scary.

Explosions, both conventional and nuclear (ain’t I posh?).

If you Google the phrase “largest conventional explosion,” you’ll probably dig up an article on Operation Sailor Hat, in which the U.S. Navy built a 500-ton Minecraft-style sphere of TNT:

It made a real mess of the decommissioned test ships anchored just offshore, and left a crater in Hawai’i that still exists to this day. It remains one of the largest intentional non-nuclear explosions humans have ever created. This isn’t relevant to our thought experiment, but I’m easily distracted by 500-ton piles of TNT.

In everyday life, a “conventional explosion” is one created by a chemical reaction of some kind. This ranges from the rapid burning of a lot of wheat dust in a grain silo to the bizarre high-tech cube-shaped molecule octanitrocubane. An explosion that gets most of its power from the fission of radioactive elements (or the fusion of light elements) is a nuclear explosion. Simple.

But you can think of it another way: the difference between conventional chemical explosions and nuclear explosions is a matter of temperature. Explosives like TNT produce gases with temperatures of thousands of degrees Kelvin. Nuclear explosions produce temperatures in the million to hundred million Kelvin range.

Since we’ve got a magic sphere of air we can pump energy into (how convenient!), let’s make one of each kind of explosion!

For the conventional explosion, we’ll heat the air to a temperature of 5,000 Kelvin (about as hot as the surface of the sun). Our sphere has a volume of 0.52 cubic meters, and air has a volumetric heat capacity of about 0.001 joules per cubic centimeter per Kelvin. Therefore, heating the air to 5,000 kelvin will require 2.6 million joules. We’re talking about a decent-sized bang here, equivalent to just over half a kilogram of TNT. Not exactly spectacular, but certainly not the kind of explosion you want to hold in your hand. (Actually, thinking about it, why would you want to hold any explosion in your hand? That was silly of me.)

For the nuclear explosion, we’ll heat our sphere to 10 million Kelvin, which is a compromise, since nuclear explosions are complicated beasties and their temperature varies very rapidly in the span of microseconds. I should point out that I’m still using a volumetric heat capacity of 0.001 joules per cubic centimeter per Kelvin, which is absurd. At these temperatures, we’re not just making molecules move faster–we’re breaking them apart. And blasting electrons off the individual atoms. Both of these things take energy, so heating the plasma to 10 million Kelvin will actually require more energy input than my idealized equation suggests. But I digress. We’re looking at an explosive energy of about 5.2 billion Joules, which is about one and a quarter tons of TNT. I’m disappointed. Even the smallest nuclear weapon ever made had a yield larger than that. (Incidentally, the smallest nuclear weapon ever made (by the U.S., at least) was the W54 warhead. This is what they used in the “Davy Crockett,” which was a nuclear warhead launched from a fucking bazooka. Think about that.)

But the real temperature of a young nuclear fireball (good name for a band) is probably higher than this. After the fission of the nuclear material is complete, there’s a brief period where the nuclear explosion is mostly made up of high-energy nuclear radiation (hard x-rays, gamma rays, and particles). Then, it gets absorbed by the evaporating bomb casing and the surrounding air and (mostly) turns into heat. It’s not unreasonable to assume that the temperature in a baby nuclear fireball (bad idea for a child’s doll) is closer to 100 million Kelvin, which works out to almost twelve and a half tons of TNT. Still not nuclear in the traditional sense, but certainly more than enough to bust up a whole city block.

Turning it up to 10.

I’m tired of messing around. I want a real boom. We’re sticking a full megaton into that sphere, and damn the consequences!

Well, with energy densities this high, you can’t really even pretend that the volumetric heat capacity equation applies. Even at 100 million Kelvin, we were getting into the region where atoms are stripped of most or all of their electrons, which is where matter stops behaving even remotely the way it does at ambient temperatures. Much hotter, and we’re going to start splitting the atoms themselves.

Instead, to describe the conditions in our magic sphere when 1 megaton is pumped into it, I’m going to make two assumptions: the energy is deposited over the course of 1 nanosecond, and the superheated sphere radiates like a thermodynamic blackbody, which is to say the same way red-hot steel does and stars (mostly) do.

An energy release of 1 megaton over 1 nanosecond gives us 4.184 trillion trillion watts of radiated power. The sphere will very briefly shine as bright as a small star, and will have a surface temperature of 69 million Kelvin. Then, ba-boom, say goodbye to your city. Next time you build a city, don’t go letting people like me build magic energy-spheres in the middle. I hope you’ve learned your lesson.

This thing goes to 11.

Sorry. Couldn’t help myself.

Let’s stick in 50 megatons now. That’s about the energy released by the Tsar Bomba, a Soviet hydrogen bomb that produced a mushroom cloud that rose above the top of the stratosphere. For comparison, really impressive explosions and big thunderstorms and volcanic eruptions and such are lucky if they can make it into the stratosphere at all, let alone breaking all the way through the damn thing.

I can do all the math here just like before: 210 trillion trillion watts, a temperature of 185 million Kelvin, yadda yadda yadda. The thing is, as I wrote in “A Piece of a Neutron Star“, up to a point, an explosion is an explosion. The energy of an explosion gets spread over a large volume fairly quickly, and ordinary fluid dynamics and thermodynamics take over. Ever notice how a conventional explosive produces a mushroom cloud that looks an awful lot like the mushroom from a nuclear explosion? That’s because both are powered by the same thing: a rising plume of hot air. While it’s true that in the nuclear explosion there’s a lot more air and it’s a hell of a lot hotter, it’s easy to see that the two are related. They’re part of the same spectrum. A nuclear explosion is like a conventional explosion, only it’s larger, and it produces a lot more radiant heat.

This thing goes to 111.

This scaling law (which real physicists have investigated in great detail) seems to hold for larger and larger energies. Small asteroid impacts create explosions that are very much like nuclear explosions (with some extra effects from high-velocity ejecta and from the entry trail and so on). Even impacts like Chicxulub (which may or may not have helped kill off the dinosaurs, but certainly ruined everybody’s day for several thousand years) produce a fairly ordinary shockwave, and then a fireball that reaches an enormous, but reasonable, size before cooling to ordinary temperatures. The excellent Impact Effects Calculator tells me that a Chicxulub-like impact would produce a visible fireball with a radius of about 66 kilometers. This would reach up through the stratosphere, and about halfway to the edge of space, so it would probably be flattened, with a blurry, undefined edge at the top, but it would still be very much like what it is: a 100 million megaton explosion.

But like all scaling laws, the explosion spectrum eventually gives out. Once you start imagining blasts with the same kinetic energy as, say, a 100-kilometer-wide stone asteroid, you’ve passed over a threshold. Beyond this threshold, the assumptions that let us imagine our earlier explosions break down. Assumptions like “Shockwaves and heat plumes travel through the atmosphere, but the atmosphere doesn’t go flying off or anything” and “The crust is firmly attached to the rest of the Earth.” (It’s nice that we live in a time when we can make assumptions like that. Not that we could live in a time when the crust wasn’t attached, but it’s nice to know we (probably) have that kind of stability). We move out of the realms of meteorology and geology and into the realms of astrophysics. When you talk about 100-kilometer asteroid impacts, you’re no longer talking about “an asteroid hitting the Earth.” You’re talking about “two celestial bodies colliding.” Things like organic life and atmospheres crumble (and burn and evaporate) before energies like this.

Which is a (very) roundabout way of saying that a 90-billion-megaton explosion isn’t really an explosion anymore. It’d blow the atmosphere off a pretty big portion of the planet, peel back the crust, and pave some or all of the Earth in magma. Larger explosions could put sizeable dents in the Earth, or blow off enough material to push it into a different orbit. Not that we care: we’ll all be dead. Again. Sorry. Planet-killing is a bad addiction to have.

Hey! Where’d my explosion go?

This transition from fluid dynamics to astrodynamics has an ultimate limit. You can only squeeze so much energy into a 1-meter-diameter sphere and have it come back out. For this, you have Albert Einstein and Karl Schwarzschild to thank.

Energy and mass are equivalent. Not only can one be converted into the other, but they also produce and react to gravity in the same way. Because of this, if you try to cram more than 3.029e43 Joules (7239 trillion trillion megatons) of energy into a sphere 1 meter across, you won’t get an explosion at all. Your energy, no matter what form it’s in, will vanish behind an event horizon. This is bad news for the Earth. Partly because this energy will weigh 56 times as much as the Earth (over a fifth the mass of Jupiter), and will therefore screw up its orbit and either freeze us or boil us. Of course, the slow and painful destruction of the entire human race, all of our infrastructure and achievements, everything you and I and everybody else cares about, and also all life on earth, is the least of our problems. Because now we’ve got a 1-meter black hole sitting in the middle of a public park somewhere (I moved it away from downtown, to be nice. Sorry.) The Earth will swirl into the hole like a sinkful of water down a drain. That’s not just me failing to be poetic, that’s about how it’ll be. The whole Earth wont’ fall into the hole in an instant, nor will it be instantly spaghettified. Remember that, until you get close to it, a black hole behaves like any other object of the same mass. So the bulk of the Earth will fall towards the hole (thereby putting the hole at the middle of the planet) and everything else will collapse around it. We will, briefly, have a planet cracked and hollowed like a broken Kinder egg. Then we will have a ball of incandescent gas around a tiny black hole. Then we will have an accretion disk whose molecules do not resemble the farmers, sailors, priests, road signs, eggs benedict, and fire ants they were once part of. This is the point at which the explosion stops being an explosion. This is the limit.

Hey, this thing goes to 10.999.

That’s no fun. Well, I tell a lie–black holes are great fun to play with. (From a large distance.) But we want explosions, not black holes! We can talk about black holes some other time (hint hint). What if we put a little less than 3.029e43 Joules into our magic sphere? If that happens, boy oh boy do we get some fireworks!

The first thing that’ll happen is that the spacetime around the sphere will go from the gentle curvature (gravity) produced by the Earth, Sun, and other celestial bodies to a violent light-bending time-warping curvature. This does not end well for us. (why would you think it would? You’re silly.) A single powerful pulse of gravity waves races out at the speed of light, turning the Earth to gravel in a few milliseconds. Right on its heels comes a wave of radiation with almost the power of a supernova (almost, as in, 0.3 times as energetic as a supernova. Serious shit.) The Earth does not have time to fall into the region of ridiculous energy density and turn it into a black hole. That’s because the Earth is busy turning into a pancake of purple-white plasma and racing outwards at high speed. If the pancake hits anything, the flash will destroy the solar system. (Which sounds like it should be a line from Spaceballs). If it doesn’t hit anything, the supernova will destroy the solar system.

The moral of today’s story is: Don’t trust a man who says he has a magic sphere. He might kill everything.

The other moral of today’s story is: If you study a little physics, you can take any thought experiment to its absolute logical limit. Which, as I’m discovering, is pretty damn fun.

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A piece of a neutron star.

In the previous article, I talked about neutron stars, and like pretty much everybody else who’s ever tried to describe a neutron star’s absurd density, I explained that a piece of a neutron star the size of a 500-micron grain of sand would weigh as much as a small cargo ship.

That’s the kind of scientific example I like: it uses to comprehensible objects to illustrate something that would otherwise be pretty much impossible to visualize. I know the mass of a cargo ship isn’t exactly intuitive, but it’s more intuitive than saying 2e17 kilograms per cubic meter.

But one thing such examples gloss over is just how hard it is to pack this much mass so close together. In order to reach the pressures and temperatures necessary for fusion, we need the mass of the entire Sun, and that still only compresses the Sun’s core to 1.5e5 kilograms per cubic meter. It takes a truly massive star that has no way to maintain its internal temperature to compress matter the rest of the way and keep it there.

Neutron stars are supported almost entirely by neutron-degeneracy pressure, which, to seriously oversimplify matters, is a product of the fact that neutrons don’t like to occupy the same quantum state, and therefore don’t like to be brought too close together. It produces a lot of pressure. Enough to support 1.38 solar masses or more against a surface gravity of 100 billion gees. It also means that if we got really literal and took an actual piece out of a neutron star, it would not end well.

Let’s say we teleported a cube of neutron fluid, at a density of 2e17 kilograms per cubic meter and 3,000,000 Kelvin, into the air a meter over an empty field on Earth. The pressure exerted by all those neutrons packed too close together is complicated to calculate, but would probably be in the neighborhood of 5e33 pascals, or about 5e18 atmospheres. That’s a million trillion times higher than the pressure during the detonation of a hydrogen bomb.

That’s a lot of energy in one place, but as we’ve learned while trying to kill humanity with a BB gun and contemplating killer asteroids, even when you deposit a ridiculous amount of energy into a small volume, if there’s enough matter around it, eventually, it’ll be converted into a more ordinary form. This is another way of saying that, up to a certain limit, all explosions are going to behave a lot like scaled-up nuclear explosions.

But a whole lot of interesting shit is going to happen very rapidly before we get to that point. First, our grain of neutronium (which, admit it, sounds way cooler than “neutron superfluid,” cool as that one is) will expand rapidly. This will cause its pressure to decrease, and so it’ll be a lot like ascending through the layers of a neutron star, moving from outer core to crust. When the pressure drops low enough, many of the neutrons will decay into protons, emitting electrons and neutrinos. Neutrinos are infamous for carrying off energy, and also for refusing to interact with things. They might heat the ground below them by a few fractions of a degree, but considering that they pass right through the Earth without difficulty, they’re probably not important, except in the fact that they’ll cool the nuclear matter down.

Now, our grain of neutronium is a slightly larger grain of protons and neutrons all mashed together. Without the surrounding pressure to force them unnaturally close, the protons will naturally repel each other. They’ll still be attracted to each other and to the neutrons via the strong force, but once again, without the ridiculous pressures provided by the bulk of a neutron star, their clustering will be limited by the short range of the strong force. That is to say, they’ll stop being a soup of nucleons and go back to being atomic nuclei.

These nuclei will start out quite heavy, but the falling pressure will cause them to rapidly fission and give off a lot of radiation. There’ll be a lot of gamma rays, a lot of stray protons and neutrons, a lot of alpha particles, and probably a lot of beta decays producing protons and electrons from neutrons. It’d take a particle physicist to tell you exactly what elements to expect in the fallout, but I’d wager it’d mostly be lead, iron, hydrogen, and helium, with a smattering of lighter and heavier elements.

By now, we’re dealing with energies too low for massive neutrino emission, so the only way this expanding sphere of plasma can lose energy is by emitting traditional electromagnetic radiation and by expanding. It is now, for all intents and purposes, an extremely hot and extremely small version of a nuclear fireball.

How big would the fireball ultimately get? That depends on a lot of things: first, on how much energy was initially contained in our deadly granule. Second, on how much of that energy got carried off by the snobbish non-interacting neutrinos. It’s hard to be certain how much potential energy would have been in the grain to start with, but I’ve read that the neutron degeneracy pressure of neutronium is one third of its mass density. E = m * c^2, so mass density is just energy density. One third of the energy density of our grain of neutronium comes out to about 7.5e23 joules, which is of the same order of magnitude as the Chicxulub impact. So, even though we’re dealing with a very exotic explosion, we know that it’s not the kind of explosion that’s going to blow off the entire atmosphere or boil all the oceans. And actually, since so much energy is likely to be lost to neutrinos (neutrinos carry off 99% of the energy in supernovae, which considering that they still shine as bright as 10 billion suns, is horrifying to contemplate), it could be an almost-ordinary thermonuclear explosion. But, because I don’t know exactly how much energy we’re losing to neutrinos here, I’m going to assume the whole 7.5e23 joules is going to get deposited in the atmosphere.

Using this number, we can estimate the relevant parameters by using the excellent Impact Effects program, written by some very nifty folks. This program is, as far as I’m concerned, justification enough for the existence of the Internet all by itself. By assuming a stony asteroid 12 kilometers across, impacting perpendicular to the ground at 22 kilometers per second, we get an impact energy in the right ballpark.

The fireball would grow to massive proportions. As we learned from nuclear tests, hot plasma is pretty much completely opaque to radiation, since it’s got electrons flying around loose, and since photons like to bounce off of electrons. An initial burst of gamma rays would escape, but much of the radiation from our exploding grain of neutronium would be trapped in the plasma bubble, bouncing around while the bubble expanded at high speed. This bubble would reach a radius of 95 kilometers, reaching vertically to near the edge of space and pushing a massive shockwave out in front of it. Anything that happened to be caught within the fireball wouldn’t be destroyed. It wouldn’t even be vaporized. It would be flash-ionized into hot plasma. But, once the bubble had expanded to 95 kilometers in radius, it would finally have cooled enough to de-ionize and become transparent to ordinary radiation again.

This is very briefly good news for the people in the surrounding area, since it means they’re not going to get smacked in the face with a wall of plasma at 5000 degrees. Then, it becomes very bad news, since there’s a lot of thermal energy in that fireball that can now suddenly escape. The fireball would be visible from 1,100 kilometers away, and possibly farther, if you’re unlucky enough to be in an airliner or on a mountain. And if this fireball is visible to you, that pretty much means you’re dead. We’re looking at flash-fires and third-degree burns for five hundred miles in every direction.

About an hour later, the people at 1,200 kilometers, for whom the fireball was below the horizon, would stop being lucky: the blast wave would arrive, bringing overpressures of almost 2 atmospheres (enough to blow down just about any building) and wind speeds of 610 miles an hour (enough to blow down just about any building).

But the disaster would only just be beginning. Here, the peculiar origin of the explosion would make itself apparent: there would likely be a lot of radioactive fallout, and it would be made of peculiar isotopes generated in a flash when those protons and neutrons were separating into nuclei again. Not only that, with all the ionizing radiation, there would be even more nitric oxide in the plume than usual. Imagine if you will a pancake-shaped incandescent cloud hundreds of kilometers across–the size of a country. This cloud glows from within a larger, dark-red cloud of nitric oxide, ozone, iron, lead, and radioactive dust. Over the course of hours, the upper half of the cloud collapses downward as it cools, while the other half rises buoyantly upward. Within days, there’s a sheet of opaque vapor thousands of kilometers across, trapped in the stratosphere, blowing with the winds, fed from below by a firestorm of a kind not seen for 65 million years. Smoke and dust circle the planet within weeks. Temperatures drop far below freezing. People and animals are poisoned by toxic gases. With the sunlight blotted out, plants die. People starve. There’s a mass extinction. Only the hardiest species survive. After the dust settles out and the climate rebounds, new creatures populate the Earth. The only reminder of the catastrophe is a thin layer of exotic elements, and a crater 160 kilometers across and 2 kilometers deep. Perhaps if Earth ever spawns another species that spawns paleontologists, they will think the crater came from an asteroid impact. But it didn’t. It was created by an object the size of a grain of sand.

So take this as a grim warning: Under no circumstances should you take a useful scientific analogy so literally that you actually remove a piece of an exotic compact star and transport it to a planet. And they say you can’t learn anything from psychotic bloggers!

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