Because I like dragons and I can’t help myself. Don’t worry. This one won’t be nearly as long as my usual posts about dragons. If anything, it’s more to show the thought process that goes into my thought experiments.
Let’s dispense with the notion of dragonfire hotter than the surface of the sun, and with biologically-produced antimatter. Let’s pretend that dragons are made of fairly ordinary flesh. They breathe fire from their mouths (naturally), so they’re going to have to be careful not to burn their tongues off. Let’s assume they have funny saliva glands that mist their mucous membranes to stop them getting scalded off by direct contact with hot air and fire. There’s still thermal radiation to deal with.
According to NOAA (who usually talk about weather, but have, in this case, started talking about fire), exposure to thermal radiation at an intensity of 10 kilowatts per square meter will cause severe pain after 5 seconds and second-degree burns (nasty blisters) after 14 seconds. With that in mind, I want to find out how hot dragonfire can be before its thermal radiation is too much for a dragon’s mouth to handle.
Well, let’s assume a dragon’s mouth is a cylinder 1 meter long and 30 centimeters in diameter. Multiply the circumference of that cylinder by its length to get its surface area (minus the ends), and then multiply the area by 10 kilowatts per square meter to get the maximum radiant power that can reach the mucous membranes. The result: 9.425 kilowatts. Now, let’s model the jet of fire as a cylinder (again, without ends) 1 centimeter in diameter and 1 meter long. That cylinder can’t emit more than 9.425 kilowatts as radiant heat. Divide 9.425 kilowatts by the cylinder’s surface area. To stay below 9.425 kilowatts, the jet of flame can’t emit at an intensity higher than 300 kilowatts per square meter. Apply the Stefan-Boltzmann law in reverse to get an estimate of what temperature gas radiates at 300 kilowatts per square meter. That comes out to a disappointing 1,517 Kelvin, which is cooler than the average wood fire.
I’m not satisfied with that, so I’m going to cheat. Sort of. I’m going to assume that the dragon has a bone in its fire-spewing orifice that acts like a supersonic rocket nozzle, which allows it to emit a very narrow, fast-moving stream of burning gas. The upshot of this is that the jet becomes narrower than that of a pressure washer: 1 mm in diameter throughout its transit through the mouth. That’s a bit more encouraging: 2,697 Kelvin, about the temperature of a hydrogen-air flame (which means we can just use hydrogen as the fuel). It’s still nowhere as hot as I want it to be, but I don’t think Sir Knight is going to be walking away from this one.
We could, of course, push the temperature up by taking into account the fact that the dragon’s mouth isn’t a perfect blackbody, and reflects some of the radiation, but like I said, this isn’t a full post. I just wanted to show you guys how I flesh out an idea.
Stay safe out there. And don’t try to breathe fire.
As you might have noticed, I have a minor obsession with dragons. I blame Sean Connery. And, because I can never leave anything alone, I got to wondering about the practical details of a dragon’s life. I’ve already talked about breathing fire. I’m not so sure about flight, but hell, airplanes fly, so it might be possible.
But I’ll worry about dragon flight later. Right now, I’m worried about metabolism. Just how many Calories would a dragon need to stay alive? And is there any reasonable way it could get that many?
Well, there’s more than one type of dragon. There are dragons small enough to perch on your shoulder (way cooler than a parrot), and there are dragons the size of horses, and there are dragons the size of cathedrals (Smaug again), and there are, apparently, dragons in Tolkein’s universe that stand taller than the tallest mountains. Here’s a really well-done size reference, from the blog of writer N.R. Eccles-Smith:
The only downside is that there’s no numerical scale. There is, however, a human. And, if you know my thought experiments, you know that, no matter what age, sex, or race, human beings are always exactly 2 meters tall. Therefore, the dragons I’ll be considering range in size from 0.001 meters (a hypothetical milli-dragon), 1 meter (Spyro, number 3, purple in the image) to 40 meters (Smaug, number 11), and then beyond that to 1,000 meters, and then beyond to the absolutely ludicrous.
In a recent post, I decided that plasma-temperature dragonfire might be feasible, from a physics standpoint. There’s one catch: my solution required antimatter (and quite a bit of it). Antimatter does occur naturally in the human body, though. An average human being contains about 140 milligrams of potassium, which we need to run important stuff like nerves and heart muscle. The most common isotope of potassium is the stable potassium-39, with a few percent potassium-41 (also stable), and a trace of potassium-40, which is radioactive. (It’s the reason you always hear people talking about radioactive bananas. It also means that oranges, potatoes, and soybeans are radioactive. And cream of tartar is the most radioactive thing in your kitchen, unless you’ve got a smoke detector in there.)
Potassium-40 almost always decays by emitting a beta particle (transforming itself into calcium-40) or by cannibalizing one of its own electrons (producing argon-40). But about one time in 100,000, one of its protons will transform into a neutron, releasing a positron (the antimatter counterpart to the electron) and an electron neutrino. The positron probably won’t make it more than a few atoms before it attracts a stray electron and annihilates, producing a gamma ray. But that doesn’t matter, for our purposes. What matters is that there are natural sources of antimatter.
Unfortunately, potassium-40 is about the worst antimatter source there is. For one thing, its half-life is over a billion years, meaning it doesn’t produce much radiation. And, like I said, of that radiation, only 0.001% is in the form of usable positrons.
Luckily, modern medicine gives us another option. Nuclear medicine, specifically (which, by the way, is just about the coolest name for a profession). As you may have noticed by the fact that you don’t vomit profusely every time you go outside, human beings are opaque. We can shoot radiation or sound waves through them to see what their insides look like, but that usually only gives us still pictures, and it doesn’t tell us, for instance, which organs are consuming a lot of blood, and therefore might contain tumors. For that, we use positron-emission tomography (PET) scanners. In PET, an ordinary molecule (like glucose) is treated so that it contains a positron-emitting atom (most often fluorine-18, in the case of glucose). The positron annihilates with an electron, and very fancy cameras pick up the two resulting gamma rays. By measuring the angles of these gamma rays and their timing, the machine can decide if they’re just stray gamma rays or if they, in fact, emerged from the annihilation of a positron. Science is cool, innit?
One of the other nucleides used in PET scanning is carbon-11. Carbon-11 is just about perfect, as far as biological sources of antimatter go. It’s carbon, which the body is used to dealing with. It decays almost exclusively by positron emission. It decays into boron, which isn’t a problem for the body. And its half-life is only 20 minutes, which means it’ll produce antimatter quickly.
There’s one major catch, though. Whereas potassium-40 occurs in nature, carbon-11 is artificial, produced by bombarding boron atoms with 5-MeV protons from a particle accelerator. I may, however, have found a way around this. To explain, here’s a picture of a dragon:
No, those aren’t labels for weird cuts of meat. They’re to explain the pictures that follow.
Living things contain a lot of free protons. They’re the major driver of the awesome mechanical protein ATP synthase, which looks like this:
Sorry. I just really like the way PDB renders its proteins.
Either way, we know organisms can produce concentrations of protons. But in order to accelerate a proton, you need a powerful electric field. The first particle accelerators were built around van de Graaff generators, which can reach millions of volts. Somehow, I doubt a living creature can generate a megavolt.
Actually, you might be surprised. The electric eel (and the other electric fish I’m annoyed my teachers never told me about) produces is prey-stunning shock using cells called electroctyes. These are disk-shaped cells that act a little bit like capacitors. They charge up individually by accumulating concentrations of positive ions, and then they discharge simultaneously. The ions only move a little bit, but there are a lot of ions moving at the same time, which produces a fairly powerful electric current that generates a field that stuns prey. The fact that organisms can produce potential differences large enough to do this makes me hopeful that maybe, just maybe, a dragon could do the same on a nanometer scale, producing small regions of megavolt or gigavolt potential that could accelerate protons to the energies needed to turn boron-bearing molecules into carbon-11-bearing molecules. Here’s how that might work:
There’s going to have to be a specialized system for containing the carbon-11 molecules, transporting them rapidly, and shielding the rest of the body from the positrons that inevitably get loose during transport, but if nature can invent things like electric eels and bacteria with built-in magnetic nano-compasses, I don’t think that’s too big a stretch.
The production of carbon-11 is going to have to happen as-needed, because it’s too radioactive to just keep around. I imagine it’d be part of the dragon’s fight-or-flight reflex. Here’s how I imagine the carbon-11 molecules will be stored:
Note the immediate proximity to a transport duct: when you’ve got a living creature full of radioactive carbon, you want to be able to get that carbon out as soon as you can. Also note the radiation shielding around the nucleus. That would, I imagine, consist of iron nanoparticles. There might also be iron nanoparticles throughout the cytoplasm, to prevent the gamma rays from lost positrons from doing too much tissue damage.
Those positrons are going to have to be stored in bulk once they’re produced, though. This problem is the hardest to solve, and frankly, I feel like my own solution is pretty handwave-y. Nonetheless, here’s what I came up with: a biological Penning trap:
These cells are going to require a lot of brand-new biological machinery: some sort of bio-electromagnet, for one (in order to produce the magnetic component of the Penning trap). For another, cells that can sustain a high electric field indefinitely (for the electric component). Cells that can present positron-producing carbon-11 atoms while simultaneously maintaining a leak-proof capsule and a high vacuum in which to store the positrons. And cells that can concentrate high-mass atoms like lead, because there’s no way to keep all the positrons contained. That’s probably wishful thinking, but hey, nature invented the bombardier beetle and the cordyceps zombie-ant fungus, so maybe it’s not too out there.
The process of actually producing the dragonfire is very simple, by comparison. The dragon vomits water rich in iron or calcium salts (or maybe just vomits blood). The little storage capsules open at the same time, making gaps in their fields that let the positrons stream out. The positrons annihilate with electrons in the fluid (hopefully not too close to the dragon’s own cells; this is another stretch in credibility). The gamma rays produced by the annihilation are scattered and absorbed by the water and the heavy elements in it, and by the time they exit the mouth, they’re on their way to plasma temperatures.
This is not, of course, the kind of thing nature tends to do. Evolution is a lazy process. It doesn’t find the best solution overall (because if you wanna talk about dominant strategies, having a built-in particle accelerator is up there with built-in lasers). It just finds the solution that’s better enough than the competitor’s solution to give the critter in question an advantage. So, although nature has jumped the hurdles to create bacteria that can survive radiation thousands of times the dose that kills a human on the spot, and weird things like bombardier beetles, insect-mind-controlling hairworms, and parasites that make snails’ eyestalks look like caterpillars so birds will eat them and spread the parasites, the leap to antimatter storage is probably asking a bit too much, unless we’re talking about some extremely specific evolutionary pressures.
Which is not to say that nature couldn’t produce something almost as awesome as plasma-temperature dragonfire. Let’s return once again to the bombardier beetle. The bombardier beetle has glands that produce a soup of hydrogen peroxide and quinones. Hydrogen peroxide likes to decompose into water and oxygen, which releases a fair bit of heat (which is why it was used as a monopropellant in early spacecraft thrusters). But at the beetle’s body temperature, the decomposition is too slow to matter. When threatened, however, the beetle pumps the dangerous soup into a chamber lined with peroxide-decomposing catalysts, which makes the reaction happen explosively, spraying the predator with a foul mix of steam, hot water, and irritating quinone derivatives. Here’s what that looks like:
So if nature can evolve something like that, is it too much of a stretch to imagine a dragon producing hydrogen-peroxide-laden fluid, mixing it with hydrogen gas, and vomiting it through a special orifice along with some catalyst that ignites the mixture into a superheated steam blowtorch like the end of a rocket nozzle? Well, look at that beetle! Maybe it’s not as far-fetched as it seems…
Last year, I wrote a post about the physics of the plasma-temperature dragonfire from Dwarf Fortress. Today, because my frontal lobes are screwed on backwards, I wanna work out whether or not biology could produce a plume of 20,000-Kelvin plasma without stretching credibility too far. I have a hunch that the answer will be disappointing, but my hunches are usually wrong. Must be those faulty frontal lobes.
The first thing we need to work out is how much power we’re going to need to heat all that air. Let’s say dragonfire comes out of the dragon’s mouth at 50 meters per second (111 mph, about as fast as a sneeze or a weak tornado). As a rough approximation, let’s assume that a dragon’s mouth has a cross-sectional area of about 0.0600 square meters (about the area of a piece of ordinary printer paper). This is one of those nice situations where we can just multiply our two numbers together and get what we’re looking for: a flow rate of 3.120 cubic meters per second.
So here’s what we know so far: we’ve got a dragon breathing 3.120 cubic meters of air every second. That air has to be heated from 300 Kelvin (roughly room temperature) to 20,000 Kelvin. The specific heat capacity of air is close to 1,020 Joules per kilogram Kelvin over a pretty wide range of temperatures, so we’ll assume that holds even when the air turns to plasma. That means that every second, our dragon has to put out 79.96 million Joules, or 22.2 kilowatt-hours. But we’re not talking about hours here. We’re talking per second. That’s 79.96 megawatts, which is almost twice the power produced by the GE CF6-5 jet engines that power many airliners. That’s a lot of power.
But, much to my surprise, there are some fuels that can deliver that kind of power. Compressed hydrogen burning in pure oxygen could do it. Except I’m basing that assumption entirely on the power required. There’s a lot more physics involved than that. The highest temperature that a combustion reaction can reach, assuming no heat loss, is called the adiabatic flame temperature, and although this is an impressive 3,500 Kelvin for a well-mixed oxy-hydrogen flame, that’s nowhere near the 20,000 Kelvin we need. The only fuels with higher energy densities than hydrogen are things like plutonium and antimatter, and for once, I’m going to be restrained and try not to resort to antimatter if I don’t have to. Let’s see if there’s another way to do it.
In my previous post on dragonfire, I described Dwar Fortress’s dragon’s-breath as a medieval welding arc. So to hell with it–why not use an actual welding arc to heat the air? Well, it turns out that something like this already exists. It’s called an arcjet. Like VASIMR, it’s one of those electric-thruster technologies that has yet to get its day in the spotlight. But arcjets have found another purpose in life: allowing space agencies to test their reentry heat shields on the ground. Here’s a strangely satisfying video of one such arcjet heater being tested on an ordinary metal bolt:
That certainly looks like how my brain tells me dragonfire should look, but from a little research, it seems that the Johnson Space Center’s arcjet only puts out something like 2 megawatts, thirty-five times less than the 79 we need. According to these people, the arc in an arcjet thruster can reach the 20,000 Kelvin we need, but it seems pretty likely that the actual plume temperature is going to be a lot lower.
And besides, our dragon’s powerplant has to be (relatively) biology-friendly, since it has to be inside a living creature. The voltages and currents needed to run an arcjet would probably make our dragon drop dead or explode or both.
So, as much as I hate to do it (I’m kidding; I love to do this) I’ve gotta turn to antimatter.
Antimatter is the ultimate in fuel efficiency. Because almost all of the universe is made of matter (and nobody really knows why), if you release antimatter into the world, it’ll very quickly find its matching non-anti-particle and annihilate, producing gamma rays, neutrinos, and weird particles like kaons. The simplest case is when an electron meets a positron (its antiparticle). The result is (almost) always two gamma rays with an energy of 511 keV, meaning a wavelength of 2.4 picometers, which is right on the border between really high-energy X-rays and really low-energy gamma rays.
This presents yet another problem: hard x-rays and soft gamma rays are penetrating radiation. They pass through air about as well as bullets pass through water (which isn’t an amazing distance, I’ll admit, but I’m still not about to sit in a pool and let someone shoot at me). At 511 keV and ordinary atmospheric density, the mass attenuation coefficient (which tells you what fraction of the radiation in question gets absorbed after traveling a certain distance) is in the neighborhood of 0.013 per meter, which means a beam of 511 keV photons will get 1.3% weaker for every meter it travels.
Working out just what fraction of these photons need to be absorbed is a bit beyond me. If the radiation has to be 1,000 times weaker, it’ll have to pass through 1.6 meters of air. That sounds to me like it’d be enough to burn our dragon’s tongue right off. And indeed, if we run the equation a different way, we see that, after traveling through 30 centimeters (about a foot) of air, the gamma rays will still have 25% of their original strength. I’m trying very hard not to imagine what burning dragon teeth would smell like.
But there’s no reason our dragon has to be making its death-dealing plasma out of air. Water is the most common molecule in biology, so why not use that instead? A 511 keV photon can still travel over 10 centimeters in water, but that’s a heck of a lot better than the 150 centimeters we were looking at before.
Of course, we can add a dash of metal atoms to the mix to absorb more of the x-rays and protect our poor dragon from its own flame. The heaviest metal found in organisms in large quantities is iron, usually in the form of hemoglobin. So let’s just throw some hemoglobin in that water, handwave away how the dragon is producing so many positrons, and call this experiment a success.
Well, it’s not a total success, since what I just described is essentially a dragon vomiting a jet of blood and then turning that into scalding-hot plasma. No wonder everybody’s scared of dragons…
So, I’ve been playing a lot of Dwarf Fortress lately (which goes a long way to explain the lack of new posts). If you don’t know, Dwarf Fortress is a bizarre and ridiculously detailed fantasy game where you send a squad of dwarves into the wilderness to dig for gems and ore and try to stay alive as long as possible. That’s harder than you might think, since all dwarves are born alcoholics who must have booze to function properly, they’re surrounded by horrible creatures that want them dead, the environment is harsh, and they’re…well, they’re a little dim.
I love Dwarf Fortress. I love it because the creators have put such an insane level of love and detail into it. For example, how many other fantasy games do you know where they actually use the specific heat of copper when calculating whether or not your armor is melting?
But one detail in particular caught my eye: Dwarf Fortress’s temperature system. Temperatures in Dwarf Fortress are, to quote the Wiki, “stored as sixteen-bit unsigned integers,” which means temperatures between 0 and 65,535. The cool thing is that Dwarf Fortress doesn’t use some wimpy unspecified temperature scale. There is a direct correspondence between Dwarf Fortress temperatures (measured in degrees Urist. Don’t ask.) and real temperatures. To convert from Dwarf Fortress temperatures to Kelvins, for instance, just do a little simple math: [Temperature in Kelvins] = ([Temperature in degrees Urist] – 9508.33) * (5/9) . As it turns out, the Urist scale is just the Fahrenheit scale shifted downward by 9968 degrees (which, incidentally, means you can go several thousand degrees below aboslute zero, but that’s an issue for another time).
Better yet, Dwarf Fortress has DRAGONS! I love dragons, far more than any twenty-six-year-old adult male probably should. I turn into a hyperactive eight-year-old boy when I think about dragons. And Dwarf Fortress combines two of my great loves: dragons, and being unnecessarily specific about things. Here’s a typical encounter between a human swordsman in bronze armor (the @ symbol; the graphics take some getting used to) and an angry dragon (the D symbol).
Round 1. FIGHT!
The dragon breathes fire. The human’s chainmail pants are now filled with poo.
The human is engulfed in dragonfire and begins burning almost immediately.
To nobody’s surprise, the dragon wins. I’d also like to note that this dragon is a real jerk: while his poor prey was burning to death, it swooped in and knocked the human’s teeth out…
Dragon fights in Dwarf Fortress end very quickly. That’s because, as the wiki tells us, dragonfire has a temperature of 50,000 degrees Urist. Which translates to a horrifying 22,495.372 Kelvins (22,222.222 ºC, 40,032 ºF). That’s higher than the boiling point of lead. It’s higher than the boiling point of iron. It’s higher than the boiling point of tungsten, for crying out loud. In fact, it’s sixteen thousand degrees hotter than tungsten’s boiling point. Dwarf Fortress dragons don’t breathe fire like those wimpy Hollywood dragons. They breathe jets of freakin’ plasma. Plasma hotter than the surface of the sun. Plasma almost as hot as a lightning bolt.
With this in mind, we can take a scientific (and somewhat gruesome) look at what happened to our unfortunate human swordsman just now.
From the images above, let’s say the dragon’s plasma jet reached a maximum length of 10 meters before the dragon stopped spitting. Just before it struck our adventurer, it was spread out in a rough cone 10 meters long and 5 meters wide at the base. It was broiling away at a temperature of 22,500 Kelvin. When you’re working with absurd temperatures like this, the radiated heat and light do as much or more damage than the plasma itself. This kind of thing (unfortunately) also happens in more mundane circumstances: when high-voltage, high-current equipment shorts out, it can produce an arc flash, an electric discharge that produces a dangerous explosion, a deadly flash, a flare of plasma, and a shower of molten metal.
Arc flashes are horrifying. They’re a serious source of danger to electrical engineers. They’re also not terribly funny. But they give us an idea of the effects of dragonfire.
At a temperature of 22,500 Kelvin, the front surface of the fireball would radiate about 0.285 terawatts of energy. The formula for a blackbody spectrum tells us that the fireball will be brightest at a wavelength of 128.79 nanometers, which is in the far ultraviolet. That’s more energetic than the ultraviolet light from germicidal lamps, which is already more than enough to cause burns and damage the eyes. So our unlucky swordsman would be looking at instant UV flash-burns.
Lucky for him, he probably won’t have long to worry about those burns. The fireball is radiating at 1.453e10 watts per square meter. If we assume the swordsman knew he was about to fight a dragon and therefore put on some sort of bizarre medieval bronze spacesuit and polished it to a mirror finish. He’s still dead meat: copper, one of the main components of bronze (the other is most often tin) is a terrible reflector at the wavelengths in question here, bottoming out at around 30%. That means our foolish knight is still going to be absorbing 70% of the radiant heat, which will (given a long enough exposure) raise its temperature to around 20,500 Kelvin, more than hot enough to flash-vaporize the outer layers.
But if we’re nice and pretend the knight was smart enough to have his bronze armor coated with something decently reflective at all wavelengths (like ye olde dwarven electropolished electroplated aluminum), he would only absorb about 5% of the incident radiation. Well, bad news, sir knight: your armor’s still heating up to 7,600 Kelvin, which is much hotter than the surface of the sun.
Of course, producing a plume of 22,000-degree plasma takes a lot of energy (I’ll resist the urge to nitpick the biology of that), and even if we put that aside, according to the game’s own internal logic, dragonfire doesn’t hang around very long. Each in-game tick (in adventure mode or arena mode) lasts one second, and our bronze swordsman was only exposed to these ridiculous temperatures and irradiances for around 10 ticks, or 10 seconds. If we consider the fact that the plume of dragonfire is going to lose a lot of energy to radiation and thermal expansion, our knight probably wouldn’t evaporate right away. But he will probably wish to his randomly-generated deity that he did.
Metals are good conductors of heat, and copper is one of the most conductive metals, heat-wise. Therefore, although our knight only got exposed to that horrifying draconic welding arc for a few seconds, his armor’s going to soak up a lethal amount of heat from that exposure. Arc flashes, lightning, and nuclear explosions can cause second- and third-degree burns from just a few seconds’ exposure, so our night is going to be blind and scorched, and then he’s going to poach like an egg inside his armor.
Don’t worry, though–he probably won’t feel it. Unless he has superhuman willpower (and is therefore able to hold his breath while the rest of his body is bursting into flames), he’s going to take a panicked gasp, and that’ll put an end to his battle very, very quickly.
The inhalation of superheated gas kills very rapidly. The inhalation of gas at thousands of degrees (meaning: the dragon’s plume and every cubic centimeter of air in contact with it) kills instantly. So our knight would probably lose consciousness either instantly, or within 15 seconds, which is how long it takes you to pass out when your heart and/or lungs quit working. And what would be left? A knight cooked Pittsburgh rare, wrapped in a blanket of broken bronze welding slag.
So, if you think you’ve outgrown being scared of dragons, imagine this: a scaly reptilian horror older than a sequoia, fixing you with its piercing gaze and then spewing a jet of gas as hot and bright as a welding arc. That’s good–I didn’t need to sleep tonight, anyway…
When I was little, there was always that one kid on the playground who thought he was clever. We’d be drawing horrifying killer monsters (we were a weird bunch). I would say “My monster is a thousand feet high!” Then Chad would say “My monster is a mile high!” Then I would say “Nuh-uh, my monster is a thousand miles high!” Then Taylor would break in, filling us with dread, because we knew what he was going to say: “My monster is infinity miles high!” There would then follow the inevitable numeric arms race. “My monster is infinity plus one miles high!” “My monster is infinity plus infinity miles high!” “My monster is infinity times infinity miles high!” Our shortsighted teachers hadn’t taught us about Georg Cantor, or else we would have known that, once you hit infinity, pretty much all the math you do just gives you infinity right back.
But that’s not what I’m getting at here. As we got older and started (unfortunately) to care about money, the concept of “infinite money” inevitably started coming up. As I got older still and descended fully into madness, I realized that having an infinite amount of printed money was a really bad idea, since an infinite amount of mass would cause the entire universe to collapse into a singularity, which would limit the number of places I could spend all that money. Eventually, my thoughts of infinite wealth matured, and I realized that what you really want is a machine that can generate however much money you want in an instant. With nanomachines, you could conceivably assemble dollar bills (or coins) with relative ease. As long as you didn’t create so much money that you got caught or crashed the economy, you could live really well for the rest of your life.
But that’s not what I got hung up on. I got hung up on the part where I collapsed the universe into a Planck-scale singularity. And that got me thinking about one of my favorite subjects: weird objects in space. I’ve been mildly obsessed with creating larger and larger piles of objects ever since. Yes, I do know that I’m weird. Thanks for pointing that out.
Anyway, I thought it might be nice to combine these two things, and try to figure out the largest pile of money I could reasonably accumulate. My initial thought was to make the pile from American Gold Eagle coins, but I like to think of myself as a man of the world, and besides, those Gold Eagles are annoyingly alloyed with shit like copper and silver, and I like it when things are pure. So, instead, I’m going to invent my own currency: the Hobo Sullivan Dragon’s Egg Gold Piece. It’s a sphere of 24-karat gold with a diameter of 50 millimeters, a mass of 1,260 grams, and a value (as of June 29, 2014) of $53,280. The Dragon’s Egg bears no markings or portraits, because when your smallest unit of currency is worth $53,280, you can do whatever the fuck you want. And you know what? I’m going to act like a dragon and pile my gold up in a gigantic hoard. But I don’t want any arm-removing Anglo Saxon kings or tricksy hobbitses or anything coming and taking any of it, so instead of putting it in a cave under a famous mountain, I’m going to send it into space.
Now, a single Dragon’s Egg is already valuable enough for a family to live comfortably on for a year, or for a single person to live really comfortably. But I’m apparently some kind of ridiculous royalty now, so I want to live better than comfortably. As Dr. Evil once said, I want one billion dollars. That means assembling 18,769 Dragon’s Eggs in my outer-space hoard. Actually, now that I think about it, I’m less royalty and more some kind of psychotic space-dragon, which I think you’ll agree is infinitely cooler. 18,769 Dragon’s Eggs would weigh in at 23,649 kilograms. It would form a sphere with a diameter of about 1.46 meters, which is about the size of a person. Keen-eyed (or obsessive) readers will notice that this sphere’s density is significantly less than that of gold. That’s because, so far, the spheres are still spheres, and the closest possible packing, courtesy of Carl Friedrich Gauss, is only 74% sphere and 26% empty space.
You know what? Since I’m being a psychotic space-dragon anyway, I think I want a whole golden planet. Something I can walk around while I cackle. A nice place to take stolen damsels and awe them with shining gold landscapes.
Well, a billion dollars’ worth of Dragon’s Eggs isn’t going to cut it. The sphere’s surface gravity is a pathetic 2.96 microns per second squared. I’m fascinated by gravity, and so I often find myself working out the surface gravity of objects like asteroids of different compositions. Asteroids have low masses and densities and therefore have very weak gravity. The asteroid 433 Eros, one of only a few asteroids to be visited and mapped in detail by a space probe (NEAR-Shoemaker), has a surface gravity of about 6 millimeters per second squared (This varies wildly because Eros is far from symmetrical. Like so many asteroids, it’s stubbornly and inconveniently peanut-shaped. There are places where the surface is very close to the center of mass, and other places where it sticks way up away from it.) The usual analogies don’t really help you get a grasp of how feeble Erotian gravity is. The blue whale, the heaviest organism (living or extinct, as far as we know) masses around 100 metric tons. On Earth, it weighs 981,000 Newtons. You can also say that it weighs 100 metric tons, because there’s a direct and simple equivalence between mass and weight on Earth’s surface. Just be careful: physics dorks like me might try to make a fool out of you. Anyway, on Eros, a blue whale would weigh 600 Newtons, which, on Earth, would be equivalent to a mass of 61 kilograms, which is about the mass of a slender adult human.
But that’s really not all that intuitive. It’s been quite a while since I tried to lift 61 kilograms of anything. When I’m trying to get a feel for low gravities, I prefer to use the 10-second fall distance. That (not surprisingly) is the distance a dropped object would fall in 10 seconds under the object’s surface gravity. You can calculate this easily: (0.5) * (surface gravity) * (10 seconds)^2. I want you to participate in this thought experiment with me. Take a moment and either stare at a clock or count “One one thousand two one thousand three one thousand…” until you’ve counted off ten seconds. Do it. I’ll see you in the next paragraph.
In those ten seconds, a dropped object on Eros would fall 30 centimeters, or about a foot. For comparison, on Earth, that dropped object would have fallen 490 meters. If you neglect air resistance (let’s say you’re dropping an especially streamlined dart), it would have hit the ground after 10 seconds if you were standing at the top of the Eiffel Tower. You’d have to drop it from a very tall skyscraper (at least as tall as the Shanghai World Financial Center) for it to still be in the air after ten seconds.
But my shiny golden sphere pales in comparison even to Eros. Its 10-second fall distance is 148 microns. That’s the diameter of a human hair (not that I’d allow feeble humans on my golden dragon-planet). That’s ridiculous. Clearly, we need more gold.
Well, like Dr. Evil, we can increase our demand: 1 trillion US dollars. That comes out to 18,768,769 of my golden spheres. That’s 23,649,000 kilograms of gold. My hoard would have a diameter of 14.6 meters and a surface gravity of 29.6 microns per second squared (a 10-second fall distance of 1.345 millimeters, which would just barely be visible, if you were paying close attention.) I am not impressed. And you know what happens when a dragon is not impressed? He goes out and steals shit. So I’m going to go out and steal the entire world’s economy and convert it into gold. I’m pretty sure that will cause Superman and/or Captain Planet to declare me their nemesis, but what psychotic villain is complete without a nemesis?
It’s pretty much impossible to be certain how much money is in the world economy, but estimates seem to be on the order of US$50 trillion (in 2014 dollars). That works out to 938,438,439 gold balls (you don’t know how hard I had to fight to resist calling my currency the Hobo Sullivan Golden Testicle). That’s a total mass of 1.182e9 kilograms (1.182 billion kilograms) and a diameter of 54 meters (the balls still aren’t being crushed out of shape, so the packing efficiency is still stuck at 74%). 54 meters is pretty big in human terms. A 54-meter gold ball would make a pretty impressive decoration outside some sultan’s palace. If it hit the Earth as an asteroid, it would deposit more energy than the Chelyabinsk meteor, which, even though it exploded at an altitude of 30 kilometers, still managed to break windows and make scary sounds like this:
This golden asteroid would have a surface gravity of 0.108 millimeters per second squared, and a 10-second fall distance of 0.54 centimeters. Visible to the eye if you like sitting very still to watch small objects fall in weak gravitational fields (and they say I have weird hobbies), but still fairly close to the kind of micro-gravity you get on space stations. I can walk across the room, get my coffee cup, walk back, and sit down in 10 seconds (I timed it), and my falling object would still be almost exactly where I left it.
Obviously, we need to go bigger. Most small asteorids do not even approach hydrostatic equilibrium: they don’t have enough mass for their gravity to crush their constituent materials into spheres. For the majority of asteroids, the strength of their materials is greater than gravitational forces. But the largest asteroids do start to approach hydrostatic equilibrium. Here’s a picture of 4 Vesta, one of the other asteroids that’s been visited by a spacecraft (the awesome ion-engine-powered Dawn, in this case.)
You’re probably saying “Hobo, that’s not very fucking spherical.” Well first of all, that’s a pretty damn rude way to discuss asteroids. Second of all, you’re right. That’s partly because of its gravity (still weak), partly because its fast rotation (once every 5 hours) deforms it into an oblate spheroid, and partly because of the massive Rheasilvia crater on one of its poles (which also hosts the solar system’s tallest known mountain, rising 22 kilometers above the surrounding terrain). But it’s pretty damn spherical when you compare it to ordinary asteroids, like 951 Gaspra, which is the shape of a chicken’s beak. It’s also large enough that its interior is probably more similar to a planet’s interior than an asteroid’s. Small asteroids are pretty much homogenous rock. Large asteroids contain enough rock, and therefore enough radioactive minerals and enough leftover heat from accretion, to heat their interiors to the melting point, at least briefly. Their gravity is also strong enough to cause the denser elements like iron and nickel to sink to the center and form something approximating a core, with the aluminosilicate minerals (the stuff Earth rocks are mostly made of) forming a mantle. Therefore, we’ll say that once my golden asteroid reaches the same mass as 4 Vesta, the gold in the center will finally be crushed sufficiently to squeeze out the empty space.
It would be convenient for my calculations if the whole asteroid melted so that there were no empty spaces anywhere. Would that happen, though? That’s actually not so hard to calculate. What we need is the golden asteroid’s gravitational binding energy, which is the amount of energy you’d need to peel the asteroid apart layer by layer and carry the layers away to infinity. This is the same amount of energy you’d deposit in the asteroid by assembling it one piece at a time by dropping golden balls on it. A solid gold (I’m cheating there) asteroid with Vesta’s mass (2.59e20 kg) would have a radius of 147 kilometers and a gravitational binding energy of about 1.827e25 Joules, or about the energy of 37 dinosaur-killing Chicxulub impacts. That’s enough energy to heat the gold up to 546 Kelvin, which is less than halfway to gold’s melting point.
But, you know what? Since I don’t have access to a supercomputer to model the compressional deformation of a hundred million trillion kilograms of close-packed gold spheres, I’m going to streamline things by melting the whole asteroid with a giant draconic space-laser. I’ll dispense with the gold spheres, too, and just pour molten gold directly on the surface.
You know where this is going: I want a whole planet made of gold. But if I’m going to build a planet, it’s going to need a name. Let’s call it Dragon’s Hoard. Sounds like a name Robert Forward would give a planet in a sci-fi novel, so I’m pleased. Let’s pump Dragon’s Hoard up to the mass of the Earth.
Dragon’s Hoard is a weird planet. It has the same mass as Earth, but its radius is only 66% of Earth’s. Its surface gravity is 22.64 meters per second squared, or 2.3 earth gees. Let’s turn off the spigot of high-temperature liquid gold (of course I have one of those) for a while and see what we get.
According to me, we get something like this:
The heat content of a uniform-temperature sphere of liquid gold depends on its volume, but since it’s floating in space, its rate of heat loss depends on surface area (by the Stefan-Boltzmann law). The heat can move around inside it, but ultimately, it can only leave by radiating off the surface. Therefore, not only will the sphere take a long time to cool, but its upper layers will cool much faster than its lower layers. Gold has a high coefficient of thermal expansion: it expands more than iron when you heat it up. Therefore, as the liquid gold at the surface cools, it will contract, lose density, and sink beneath the hotter gold on the surface. It will sink and heat up to its original temperature, and will eventually be displaced by the descent of cooler gold and will rise back to the surface. When the surface cools enough, it will solidify into a solid-gold crust, which is awesome. Apparently, my fantasies are written by Terry Pratchett, which is the best thing ever. I’ve got Counterweight Continents all over the place!
Gold is ductile: it’s a soft metal, easy to bend out of shape. Therefore, the crust would deform pretty easily, and there wouldn’t be too many earthquakes. There might, however, be volcanoes, where upwellings of liquid gold strike the middle of a plate and erupt as long chains of liquid-gold fountains. It would behave a bit like the lava lake at Kilauea volcano, in Hawai’i. See below:
What a landscape this would be! Imagine standing (in a spacesuit) on a rumpled plain of warm gold. To your right, a range of gold mountains glitter in the sun, broken here and there by gurgling volcanoes of shiny red-hot liquid. Flat, frozen puddles of gold fill the low spots, concave from the contraction they experienced as they cooled. To your right , the land undulates along until it reaches another mountain range. In a valley at the foot of this range is an incandescent river of molten gold, fed by the huge shield volcano just beyond the mountains. Then a psychotic space-dragon swoops down, flying through the vacuum (and also in the face of physics), picks you up in his talons, carries you over the landscape, and drops you into one of those volcanoes.
Yeah. It would be something like that.
As fun as my golden planet is, I think we could go bigger. Unfortunately, the bigger it gets, the more unpredictable its properties become. As we keep pouring molten gold on it, its convection currents will become more and more vigorous: it will have more trapped heat, a larger volume-to-surface area ratio, and stronger gravity, which will increase the buoyant force on the hot, low-density spots. Eventually, we’ll end up with convection cells, much like you see in a pot of boiling water. They might look like this:
Those are Rayleigh-Bénard cells, which you often get in convective fluids. I used that same picture in my Endless Sky article. But there, I was talking about supercritical oxygen and nitrogen. Here, it’s all gold, baby.
Eventually, the convection’s going to get intense enough and the heat’s going to get high enough that the planet will have a thin atmosphere of gold vapor. If it rotates, the planet will also develop a powerful magnetic field: swirling conductive liquid is believed to be the thing that creates the magnetic fields of Earth, the Sun, and Jupiter (and all the other planets). This planet is going to have some weird electrical properties. Gold is one of the best conductors there is, second only to copper, silver, graphene, and superconductors. Therefore, expect some terrifying lightning on Dragon’s Hoard: charged particles and ultraviolet radiation from the Sun will ionize the surface, the gold atmosphere (if there is one) or both, and Dragon’s Hoard will become the spherical terminal in a gigantic Van de Graaff generator. As it becomes more and more charged, Dragon’s Hoard will start deflecting solar-wind electrons more easily than solar-wind protons (since the protons are more massive), and will soak up protons, acquiring a net positive charge. It’ll keep accumulating charge until the potential difference explosively equalizes. Imagine a massive jet or bolt of lightning blasting up into space, carrying off a cloud of gold vapor, glowing with pink hydrogen plasma. Yikes.
After Dragon’s Hoard surpasses Jupiter’s mass, weird things will begin happening. Gold atoms do not like to fuse. Even the largest stars can’t fuse them. Therefore, the only things keeping Dragon’s Hoard from collapsing altogether are the electrostatic repulsion between its atoms and the thermal pressure from all that heat. Sooner or later, neither of these things will be enough, and we’ll be in big trouble: the core, compressed to a higher density than the outer portions by all that gold, will become degenerate: its electrons will break loose of their nuclei, and the matter will contract until the electrons are squeezed so close together that quantum physics prevents them from getting any closer. This is electron degeneracy pressure, and it’s the reason white dwarf stars can squeeze the mass of a star into a sphere the size of a planet without either imploding or exploding.
The equations involved here are complicated, and were designed for bodies made of hydrogen, carbon, and oxygen instead of gold (Those short-sighted physicists never consider weird thought experiments when they’re unraveling the secrets of the universe. The selfish bastards.) The result is that I’m not entirely sure how large Dragon’s Hoard will be when this happens. It’s a good bet, though, that it’ll be somewhere around Jupiter’s mass. This collapse won’t be explosive: at first, only a fraction of the matter in the core will be degenerate.As we add mass, the degenerate core will grow larger and larger, and more and more of it will become degenerate. It will, however, start to get violent after a while. Electron-degenerate matter is an excellent conductor of heat, and its temperature will equalize pretty quickly. That means that we’ll have a hot ball of degenerate gold (Degenerate Gold. Add that to the list of possible band names.) surrounded by a thin layer of hot liquid gold. Because of the efficient heat transfer within the degenerate core, it will partly be able to overcome the surface-area-versus-volume problem and radiate heat at a tremendous rate. The liquid gold on top, though, will have trouble carrying that heat away fast enough, and will get hotter and hotter, and thanks to its small volume, will eventually get hot enough to boil. Imagine a planet a little larger than Earth, its surface white-hot, crushed under a gravity of 500,000 gees, bubbles exploding and flinging evaporating droplets of gold a few kilometers as gaseous gold and gold plasma jet up from beneath. Yeah. Something like that.
But in a chemical sense, my huge pile of gold is still gold. The nuclei may be uncomfortably close together and stripped of all of their electrons, but the nuclei are still gold nuclei. For now. Because you know I’m going to keep pumping gold into this ball to see what happens (That’s also a line from a really weird porno movie.)
White dwarfs have a peculiar property: the more massive they are, the smaller they get. That’s because, the heavier they get, the more they have to contract before electron degeneracy pressure balances gravity. Sirius B, one of the nearest white dwarfs to Earth, has a mass of about 1 solar mass, but a radius similar to that of Earth. When Dragon’s Hoard reached 1.38 solar masses, it would be even smaller, having a radius of around 3000 kilometers. The stream of liquid gold would fall towards a blinding white sphere, striking the surface at 3% of the speed of light. The surface gravity would be in the neighborhood of 2 million gees. If the gravity were constant (which it most certainly would not be), the 10-second fall distance would be 2.7 times the distance between Earth and moon. Now we’re getting into some serious shit.
Notice that I specified three significant figures when I gave that mass: 1.38 solar masses. That is not by accident. As some of you may know, that’s dangerously close to 1.39 solar masses, the Chandrashekar limit, named after the brilliant and surprisingly handsome Indian physicist Subrahmanyan Chandrasekhar. (Side note: Chandrashekhar unfortunately died in 1995, but his wife lived until 2013. Last year. She was 102 years old. There’s something cool about that, but I don’t know what it is.) The Chandrasekhar limit is the maximum mass a star can have and still be supported by electron degeneracy pressure. When you go above that, you’ve got big trouble.
When ordinary white dwarfs (made mostly of carbon, oxygen, hydrogen, and helium) surpass the Chandrasekhar limit by vacuuming mass from a binary companion, they are unable to resist gravitational contraction. They contract until the carbon and oxygen nuclei in their cores get hot enough and close enough to fuse and make iron. This results in a Type Ia supernova, which shines as bright as 10 billion suns. It’s only recently that our supercomputers have been able to simulate this phenomenon. The simulations are surprisingly beautiful.
I could watch that over and over again and never get tired of it.
Unfortunately, even though it’s made of gold (which, as I said, doesn’t like to fuse), the same sort of thing will happen to Dragon’s Hoard. When it passes the Chandrasekhar limit, it will rapidly contract until the nuclei are touching. This will trigger a bizarre form of runaway fusion. The pressure will force electrons to combine with protons, releasing neutrinos and radiation. Dragon’s Hoard will be heated to ludicrous temperatures, and a supernova will blow off its outer layers. What remains will be a neutron star, which, as I talked about in The Weather in Hell, is mostly neutrons, with a thin crust of iron atoms and an even thinner atmosphere of iron, hydrogen, helium, or maybe carbon. Most or all of the gold nuclei will be destroyed. The only thing that will stop the sphere from turning into a black hole is that, like electrons, neutrons resist being squeezed too close together, at least up to a limit.
But you know what? That tells us exactly how much gold you can hoard in one place: about 1.38 solar masses. So fuck you, Taylor from kindergarten! You can’t have infinity dollars! You can only have 0.116 trillion trillion trillion dollars (US, and according to June 2014 gold prices) before your gold implodes and transmutes itself into other elements! So there!
But while I’m randomly adding mass to massive astronomical objects (that’s what space dragons do instead of breathing fire), let’s see how much farther we can go.
The answer is: Nobody’s exactly certain. The Chandrasekhar limit is based on pretty well-understood physics, but the physics of neutron-degenerate matter at neutron star pressures and temperatures (and in highly curved space-time) is not nearly so well understood. The Tolman-Oppenheimer-Volkoff limit (Yes, the same Oppenheimer you’re thinking of.) is essentially a neutron-degenerate version of the Chandrasekhar limit, but we only have the TOV limit narrowed down to somewhere between 1.5 solar masses and 3 solar masses. We’re even less certain about what happens above that limit. Quarks might start leaking out of neutrons, the way neutrons leak out of nuclei in a neutron star, and we might get an even smaller, denser kind of star (a quark star). At this point, the matter would stop being matter as we know it. It wouldn’t even be made of neutrons anymore. But to be honest, we simply don’t know yet.
Sooner or later, though, Karl Schwarzschild is going to come and kick our asses. He solved the Einstein field equations of general relativity (which are frightening but elegant, like a hyena in a cocktail dress) and discovered that, if an object is made smaller than its a certain radius (the Schwarzschild radius), it will become a black hole. The Schwarzschild radius depends only on the object’s mass, charge, and angular momentum. Dragon’s Hoard, or rather what’s left of it, doesn’t have a significant charge or angular momentum (because I said so), so its Schwarzschild radius depends only on mass. At 3 solar masses, the Schwarzschild radius is 8.859 kilometers, which is only just barely larger than a neutron star. Whether quark stars can actually form or not, you can bet your ass they’re going to be denser than neutron stars. Therefore, I’d expect Dragon’s Hoard to fall within its own Schwarzschild radius somewhere between 3 and 5 solar masses. Let’s say 5, just to be safe. There are suspected black holes with masses near 5.
That’s the end of Dragon’s Hoard. The physics in the center gets unspeakably weird, but the gold-spitting space dragon doesn’t get to see it. He’s outside the event horizon, which means the collapse of his hoard is hidden to him. He just sees a black sphere with a circumference of 92.77 kilometers, warping the images of the stars behind it. It doesn’t matter how much more gold we pour into it now: it’s all going to end up inside the event horizon, and the only noticeable effect will be that the event horizon’s circumference will grow larger and larger. But fuck that. If I wanted to throw money down a black hole, I’d just go to Vegas. (Heyo!) Dragon’s Hoard isn’t getting any more of my draconic space-gold.
But one last thing before I go. Notice how I suddenly went from saying Schwarzschild radius to talking about the event horizon’s circumference. That’s significant. Here’s a terrible picture illustrating why I did that:
Massive objects create curvature in space-time. Imagine standing at the dot on circle B, in the top picture. If you walk to the center-point along line A, you’ll measure a length a. If you then walk around circle B, you’ll get the circle’s circumference. You’ll find that that circumference is 2 * pi * a. The radius is therefore (circumference) / (2 * pi) But that only holds in flat space. When space is positively curved (like it is in the vicinity of massive objects), the radius of a circle will always be larger than (circumference) / (2 * pi). That is to say, radius C in the bottom picture is significantly longer than radius A in the top one, and longer than you would expect from the circumference of circle D.
In other words, the radius of a massive object like a star, a neutron star, or a black hole, differs from what you would expect based on its circumference. The existence of black holes and neutron stars has not actually been directly confirmed (because they’re so small and so far away). It is merely strongly suspected based on our understanding of physics. The existence of spacetime curvature, though, has been confirmed in many experiments.
Imagine you’re standing in a field that looks flat. There’s a weird sort of bluish haze in the center, but apart from that, it looks normal. You walk in a circle around the haze to get a better look at it. It only takes you fifteen minutes to walk all the way around and get back to your starting point. The haze makes you nervous, so you don’t walk straight into it. Instead, you walk on a line crossing the circle so that it passes halfway between the haze and the circle’s edge at its closest approach. Somehow, walking that distance takes you twenty minutes, which is not what you’d expect. When you walk past the haze at a quarter-radius, it takes you an hour. When you walk within one-eighth of a radius, it takes you so long you have to turn back and go get some water. Each time, you’re getting closer and closer to walking along the circle’s radius towards its center, but if you actually tried to walk directly into its center, where the haze is (the haze is because there’s so much air between you and the stuff beyond the haze, which is the same reason distant mountains look blue), you would find that the distance is infinite.
That’s how black holes are. They’re so strongly-curved that there’s way more space inside than there should be. The radius is effectively infinite, which is why it’s better to talk about circumference. As long as the black hole is spherically symmetric, circumferences are still well-behaved.
But the radius isn’t actually infinite. When you consider distance scales close to the Planck length, Einstein’s equations butt heads with quantum mechanics, and physicists don’t really know what the fuck’s going on. We still don’t know what happens near a black hole’s central singularity.
Incidentally, the Planck length compared to the diameter of an atom is about the same as the diameter of an atom compared to the diameter of a galaxy. The Universe is a weird place, isn’t it?
Sublime Curiosity 