A Toyota on Mars (Cars, Part 1)

I’ve said this before: I drive a 2007 Toyota Yaris. It’s a tiny economy car that looks like this:


(Image from RageGarage.net)

The 2007 Yaris has a standard Toyota 4-cylinder engine that can produce about 100 horsepower (74.570 kW) and 100 foot-pounds of torque (135.6 newton-metres). A little leprechaun told me that my particular Yaris can reach 110 mph (177 km/h) for short periods, although the leprechaun was shitting his pants the entire time.

A long time ago, [I computed how fast my Yaris could theoretically go]. But that was before I discovered Motor Trend’s awesome Roadkill YouTube series. Binge-watching that show led to a brief obsession with cars, engines, and drivetrains. There’s something very compelling about watching two men with the skills of veteran mechanics but maturity somewhere around the six-year-old level (they’re half a notch above me). And because of that brief obsession, I learned enough to re-do some of the calculations from my previous post, and say with more authority just how fast my Yaris can go.

Let’s start out with the boring case of an ordinary Yaris with an ordinary Yaris engine driving on an ordinary road in an ordinary Earth atmosphere. As I said, the Yaris can produce 100 HP and 100 ft-lbs of torque. But that’s not what reaches the wheels. What reaches the wheels depends on the drivetrain.

I spent an unholy amount of time trying to figure out just what was in a Yaris drivetrain. I saw some diagrams that made me whimper. But here’s the basics: the Yaris, like most front-wheel drive automatic-transmission cars, transmits power from the engine to the transaxle, which is a weird and complicated hybrid of transmission, differential, and axle. Being a four-speed, my transmission has the following four gear ratios: 1st = 2.847, 2nd = 1.552, 3rd = 1.000, 4th = 0.700. (If you don’t know: a gear ratio is [radius of the gear receiving the power] / [radius of the gear sending the power]. Gear ratio determines how fast the driven gear (that is, gear 2, the one being pushed around) turns relative to the drive gear. It also determines how much torque the driven gear can exert, for a given torque exerted by the drive gear. It sounds more complicated than it is. For simplicity’s sake: If a gear train has a gear ratio greater than 1, its output speed will be lower than its input speed, and its output torque will be higher than its input torque. For a gear ratio of 1, they remain unchanged. For a gear ratio less than one, its output speed will be higher than its input speed, but its output torque will be lower than its input torque.)

But as it turns out, there’s a scarily large number of gears in a modern drivetrain. And there’s other weird shit in there, too. On its way to the wheels, the engine’s power also has to pass through a torque converter. The torque converter transmits power from the engine to the transmission and also allows the transmission to change gears without physically disconnecting from the engine (which is how shifting works in a manual transmission). A torque converter is a bizarre-looking piece of machinery. It’s sort of an oil turbine with a clutch attached, and its operating principles confuse and frighten me. Here’s what it looks like:


(Image from dieselperformance.com)

Because of principles I don’t understand (It has something to do with the design of that impeller in the middle), a torque converter also has what amounts to a gear ratio. In my engine, the ratio is 1.950.

But there’s one last complication: the differential. A differential (for people who don’t know, like my two-months-ago self) takes power from one input shaft and sends it to two output shafts. It’s a beautifully elegant device, and probably one of the coolest mechanical devices ever invented. You see, most cars send power to their wheels via a single driveshaft. Trouble is, there are two wheels. You could just set up a few simple gears to make the driveshaft turn the wheels directly, but there’s a problem with that: cars need to turn once in a while. If they don’t, they rapidly stop being cars and start being scrap metal. But when a car turns, the inside wheel is closer to the center of the turning circle than the outside one. Because of how circular motion works, that means the outside wheel has to spin faster than the inside one to move around the circle. Without a differential, they have to spin at the same speed, meaning turning is going to be hard and you’re going to wear out your tires and your gears in a hurry. A differential allows the inside wheel to slow down and the outside wheel to spin up, all while transmitting the same amount of power. It’s really cool. And it looks cool, too:


(Image from topgear.uk.net)

(Am I the only one who finds metal gears really satisfying to look at?)

Anyway, differentials usually have a gear ratio different than 1.000. In the case of my Yaris, the ratio is 4.237.

So let’s say I’m in first gear. The engine produces 100 ft-lbs of torque. Passing through the torque converter converts that (so that’s why they call it that) into 195 ft-lbs, simultaneously reducing the rotation speed by a factor of 1.950. For reference, 195 ft-lbs of torque is what a bolt would feel if Clancy Brown was sitting on the end of a horizontal wrench 1 foot (30 cm) long. There’s an image for you. Passing through the transmissions first gear multiplies that torque by 2.847, for 555 ft-lbs of torque. (Equivalent to Clancy Brown, Keith David, and a small child all standing on the end of a foot-long wrench.) The differential multiplies the torque by 4.237 (and further reduces the rotation speed), for a final torque at the wheel-hubs of 2,352 ft-lbs (equivalent to hanging two of my car from the end of that one-foot wrench, or sitting Clancy Brown and Peter Dinklage at the end of a 10-foot wrench. This is a weird party…)

By this point, you’d be well within your rights to say “Why the hell are you babbling about gear ratios?” Believe it or not, there’s a reason. I need to know how much torque reaches the wheels to know how much drag force my car can resist when it’s in its highest gear (4th). That tells you, to much higher certainty, how fast my car can go.

In 4th gear, my car produces (100 * 1.950 * 0.700 * 4.237), or 578 ft-lbs of torque. I know from previous research that my car has a drag coefficient of about 0.29 and a cross-sectional area of 1.96 square meters. My wheels have a radius of 14 inches (36 cm), so, from the torque equation (which is beautifully simple), the force they exert on the road in 4th gear is: 495 pounds, or 2,204 Newtons. Now, unfortunately, I have to do some algebra with the drag-force equation:

2,204 Newtons = (1/2) * [density of air] * [speed]^2 * [drag coefficient] * [cross-sectional area]

Which gives my car’s maximum speed (at sea level on Earth) as 174 mph (281 km/h). As I made sure to point out in the previous post, my tires are only rated for 115 mph, so it would be unwise to test this.

I live in Charlotte, North Carolina, United States. Charlotte’s pretty close to sea level. What if I lived in Denver, Colorado, the famous mile-high city? The lower density of air at that altitude would allow me to reach 197 mph (317 km/h). Of course, the thinner air would also mean my engine would produce less power and less torque, but I’m ignoring those extra complications for the moment.

And what about on Mars? The atmosphere there is fifty times less dense than Earth’s (although it varies a lot). On Mars, I could break Mach 1 (well, I could break the speed equivalent to Mach 1 at sea level on Earth; sorry, people will yell at me if I don’t specify that). I could theoretically reach 1,393 mph (2,242 km/h). That’s almost Mach 2. I made sure to specify theoretically, because at that speed, I’m pretty sure my tires would fling themselves apart, the oil in my transmission and differential would flash-boil, and the gears would chew themselves into a very fine metal paste. And I would die.

Now, we’ve already established that a submarine car, while possible, isn’t terribly useful for most applications. But it’s Sublime Curiosity tradition now, so how fast could I drive on the seafloor? Well, if we provide compressed air for my engine, oxygen tanks for me, dive weights to keep the car from floating, reinforcement to keep the car from imploding, and paddle-wheel tires to let the car bite into the silty bottom, I could reach a whole 6.22 mph (10.01 km/h). On land, I can run faster than that, even as out-of-shape as I am. So I guess the submarine car is still dead.

But wait! What if I wasn’t cursed with this low-power (and pleasantly fuel-efficient) economy engine? How fast could I go then? For that, tune in to Part 2. That’s where the fun begins, and where I start slapping crazy shit like V12 Bugatti engines into my hatchback.


The Biology of Dragonfire

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:

Whole 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:

(The Protein DataBank is awesome!)

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:

Storage Zone

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:

Usage Zone

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…


Centrifuging Fruit

In my last post, I detailed some of the very gory things that would happen to a human being in a high-gee centrifuge. Then I remembered that I have access to a high-gee centrifuge. Sort of. You see, I’ve got one of those fancy front-loading washing machines. It saves time on drying by spinning your clothes at a ridiculous speed at the end of the wash cycle. And when I say “ridiculous speed”, I’m talking 1,100 RPM (at least, according to the manufacturer). That’s 18 revolutions per second! I measured the drum’s diameter at 55 centimeters. If you do the math, it tells you that the acceleration on the inner surface of the drum, when the thing’s running full pelt, is 372 gees. Okay, so it’s not ultracentrifuge material, but that’s still a lot of acceleration.

And I thought, you know what? We’ve got some fruit in the refrigerator that would be just as tasty pulped as it would be whole. Let’s see what 372 gees does to it! (Sometimes, I worry about how close I came to growing up to be a serial killer…)

I’ve tried this experiment once before (for another blog, which is why this one might look a bit familiar). Let’s do it again, but this time, with our gory, scary centrifugal thought experiment in mind. Here are our astronauts:

1 - The Astronauts

That’s a plum and a lime. The plum was pretty soft. We had a firmer one, but it wouldn’t fit in the container, and, crazy as I am, I didn’t want to risk splattering the inside of my washing machine with plum pulp. The lime, on the other hand, was so hard it could probably cut glass. Either way, these are our volunteers (it was a pain in the bum getting them to sign the release forms, let me tell you…) Let’s seal them in their space capsule.

2 - The Capsule

I must say, they look pretty brave, as far as produce goes. Note the extra precautions: each fruit sealed in an individual bag, and packing tape to seal up the container. I didn’t want it flipping over during spin-up and seeping stuff everywhere. But enough talk! Let’s get ’em in the centrifuge!

3 - In Position

There they are, their capsule strapped into place. Can you tell how worried I was I’d end up painting the inside of my washing machine with fruit?

4 - Spin

I could’ve sworn I heard a high-pitched shriek when the washer reached maximum spin. Then again, I’ve been hearing a high-pitched shriek ever since the Exploding Kikkoman Bottle Incident, so perhaps it’s just me.

5 - Max G

This is what the picture above would have looked like if I’d remembered to turn the flash on. Believe it or not, the drum is actually spinning here. Sometimes, I’m impressed by what my cheap little point-and-shoot camera can do. And then I remember that it’s got no time-lapse or long-exposure settings, and I stop being impressed. Either way, in this picture, the top of the container is experiencing about 237 gees (2,322 m/s^2). The bottom is experiencing 372 gees (3,649 m/s^2). The difference is because the top of the container is significantly closer to the axis of rotation than the bottom, and the acceleration is the distance from the axis times the square of the angular velocity. I’m surprised how well the space capsule tolerated the gees. I shouldn’t be surprised. After all, lives are at stake here. The capsule was engineered to survive all conditions. Still, considering how many times that capsule has been through the dishwasher, I’m impressed that it didn’t collapse.

Other things, however, did collapse…

6 - There Can Be Only One

This is almost exactly how our juicy astronauts were when I pulled them out of the centrifuge. I moved them around for photographic purposes, but that’s it, and even I’m not clumsy enough to completely obliterate a plum just by touching it. At least not when I’ve had my coffee.

The lime did remarkably well. It was noticeably flattened on the bottom, but it was very much intact. Now, under 1 gee (earth surface gravity), my scale says that a similar lime (someone ate my surviving astronaut; the nerve!) weighed around 100 grams. Under 372 gees, that lime weighed the equivalent of 37 kilograms. That’s 82 pounds. That’s the size of the dumbbells those really gigantic scary guys with the tattoos are always curling at the gym. It’s heavier than a gold bar. But the lime had little trouble. It’s the toughest substance known to man. I feel sorry for whoever ate it.

The plum, as you can see, didn’t fare so well. Here’s a gory close-up:

7 - The Aftermath

(Just an aside: I wonder if there’s anybody who’s genuinely upset by the sight of squashed fruit. Not in a “that’s a waste of food” sense, but in a visceral sense, the way some people can’t stand the sight of blood. If that’s you, I apologize. And you might want to consider some counseling. I’d give you the number of my therapist, but she lives on Jupiter.)

That plum is flattened. It looks like it was squashed under a very heavy weight. Which is exactly what happened. I don’t have a similarly-sized plum for comparison, but I’d say it’s reasonable to assume that, without all that weird white pithy stuff to decrease the density, the plum was at least twice as heavy as the lime, meaning, at maximum acceleration, it weighed almost 80 kilograms (176 pounds). That’s as much as my cousin. (I would invite her over for a comparison test, but even I recognize that “Will you come to my house and stand on a plum for me?” is a pretty weird request.)

But here you have an excellent practical demonstration of what I talked about in the last article. Under high acceleration, the weight of the plum exceeded its structural strength, and it split and oozed horribly all across the bottom of its bag. If the pit had been denser, it might very well have squelched down through the pulp and ended up on the bottom, but even my terrifying washing machine has its limits.

Oh, and before anybody complains that I’m wasting food on silly experiments… First of all, NYEH. Second of all, I didn’t waste it. I ate the plum. Somebody else ate the lime (for some reason). And you know what? That plum was one of the most delicious things I’ve ever eaten. I’m serious. It was all squishy and ripe. I used it because I thought it had gone over the edge already. But no. It was perfect. So not only did I get to centrifuge something, but I got some lovely fruit, too! I might have to try these practical experiments more often…

Or perhaps not. I must remember the Exploding Kikkoman Bottle Incident…



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).

Dragon Fight 1

Round 1. FIGHT!

Dragon Fight 2

The dragon breathes fire. The human’s chainmail pants are now filled with poo.

Dragon Fight 3

The human is engulfed in dragonfire and begins burning almost immediately.

Dragon Fight Aftermath

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…


Return of the submarine-car!

About a month ago, I tried to figure out how fast my 2007 Toyota Yaris would be able to drive underwater. I came up with 15 MPH (23 km/h, 6.52 m/s), which is respectable for a submerged economy car, but not that impressive for an underwater vehicle. Plus, as commenter Azure James correctly pointed out, propelling a car underwater isn’t as simple as having the horsepower to overcome drag: you have to make sure your drivetrain, from engine to gearbox to wheels, can generate enough torque to overcome the drag force.

As usual, I took this as an opportunity to imagine something ridiculous and draw a stupid picture. I present to you, the Hobo Sullivan Cruiser, the world’s most impractical underwater vehicle!

Underwater Car

It would be about the size of a large van or a small delivery truck, and that’s mostly because it has to accommodate a spherical pressure hull. Because if I’m going to make an underwater car, I’m for damn sure going to make it able to drive into the Challenger Deep! A lot of the other stuff is self-explanatory. The outer hull has a half-torpedo shape (Or if you prefer, half-cigar shape. Or, if you’re really picky, half a Sears-Haack body.) That’s to minimize drag. I chose to power it with a fuel cell and electric motors because those allow a greater range of rotation speeds (which, as my commenter correctly pointed out, would limit my speed on Mars, since a normal gearbox can’t get the wheel speeds up high enough to propel an ordinary car at 300+ MPH). Plus, with electric motors, exhaust isn’t as serious a problem.

The power for the electric motors is provided by a methanol-oxygen fuel cell. I chose methanol for one reason: space. Hydrogen, even compressed and cooled to cryogenic temperatures, takes up a lot of room. If you look at the hydrogen-burning stages of the Saturn V, the tanks are mostly hydrogen with a little spheroid at the end for the oxygen. Methanol fuel cells don’t provide as much power, but methanol’s a hell of a lot easier to transport. I’d still have to do something about the exhaust from the fuel cell, but for now, let’s assume I run the fuel cell hot enough that the pressure pushes it out a one-way valve.

You know, the more I look at it, the more “screw wheels” makes it look like I’ve got some weird grudge against wheels. In reality, those funny-looking objects are screw-drive wheels, which are so good at all-terrain they make tanks drive home to their hangars and weep transmission fluid into their berths.

That glorious contraption is the 1929 Fordson snow machine. It’s got huge pontoon-shaped wheels that turn on an axis parallel to the direction of travel, rather than perpendicular, with the force provided by many small segments of the helical screw pushing against the snow as the wheels turn. Because the pontoon wheels have such a large footprint, they don’t sink even in deep snow. And there is, of course, the cool factor, which is almost off the scale here. And now, let me send it careening completely off the scale by showing you the Russian incarnation of the screw-drive all-terrain vehicle:

When they say this is an all-terrain vehicle, they mean all terrain. Including the surface of a fucking lake. It’s hard not to be impressed.

Okay. So screw-wheels are cool. Is that reason enough to complicate the design of my submarine car to throw four of them on there? Well, yes. The seafloor is unpredictable. Without vegetation to anchor it or ordinary algae and fungi to bind it, the seafloor can have a wide variety of textures, and almost all of them would be a nightmare for a wheeled vehicle: silt six feet deep, loose sand, gravel rocks, weird clay, mud, sulfide sludge spewed from seafloor volcanoes… A wheel is very likely to sink into a morass like that. I could make the wheels larger, but a pneumatic wheel will deflate at seafloor pressures. I could make the wheels non-pneumatic, but they’re still big and bulky and taking up a lot of room, and they’re going to increase drag, which I’m already spending all my energy fighting. Tracks might work, but they contain an awful lot of joints and moving parts, and would probably get silted up.

Of course, screws as small as the one in the illustration probably wouldn’t fare too well in soft silt, either, so I’d have to make them larger, but nonetheless, I’d say screw wheels are the optimal solution. And besides that, they’re fucking awesome!

Of course, all this is really moot. There’s a very good physics reason we don’t turn all cars into planes (And plenty of other reasons, if you see how people drive on the ground.): it’s hard to generate lift in air. For a long time, many believed heavier-than-air flying machines were impossible, and they only became really successful a century ago. Air isn’t all that dense. An average 80-kilogram (172-pound) human could comfortably stand in a box 2 meters on an edge. That box would be seriously oversized, but the air in it would still only weigh 10 kilograms.

A box of water 2 meters on an edge, though, would weigh 8,000 kilograms, or about the same amount as an African elephant. Water is dense. Denser fluids resist being pushed around, and therefore more readily generate force. In my earlier article, I figured out that my car would theoretically be able to go 135 miles an hour in air, but only 15 miles an hour in water. But when it comes to flying, that resistive force comes in handy, because you don’t need very big wings to turn your submarine car into a submarine…submarine. Here’s a picture to illustrate how much better wings work in water than they do in air:



That is a boat. Being lifted almost completely out of the water by wings that look to me to be smaller than the wings on a Cessna.

And there’s another thing that makes an underwater vehicle confined to a surface kind of silly: buoyancy. Like I said, air’s not very dense, and for a bag full of gas to lift anything, it has to be less dense than the surrounding medium. There aren’t many gases less dense than air. There are hydrogen and helium, of course, but both are made of such small atoms that they leak out of containers no matter what you do, and hydrogen is flammable. There’s hot air, but that’s a lot denser than hydrogen or helium, and after a certain altitude, it gets hard to heat it up enough to provide useful lift.

Now here are some things that are less dense than water: Myself. Wood. Warmer water. Air. Compressed air. Methanol. Ethanol. Propanol. Butanol. Low-density polyethylene. In other words, around half the matter (by volume) in a submarine is less dense than water. Submarines actually have to take on extra water to sink. I had to stick depleted uranium ballast weights in my ocean car just to make sure it wouldn’t float.

But really, why shouldn’t it float? It’s not like I couldn’t stick landing skids or water-filled pontoons on a regular submarine and then have a submarine that can go to the bottom. And with a little creativity and plenty of power (provided by a wide variety of sources), you can adjust your buoyancy on-the-fly.

So forget the submarine car. I just want a submarine.

Actually, that’s a lie. I want a submarine car with screw-wheels. But I’m going to use it on land. Because, holy shit, look at those motherfuckers in the videos up above!


Endless sky.

I’ve been a nerd pretty much my whole life, so as a kid, I wondered nerdy things. I wondered, for instance, if it was possible to have an endless sky: clouds below you and clouds above. As a slightly older kid, I realized that skies don’t work that way. A sky is just an atmosphere, just a layer of gas wrapped around a solid (or liquid) planet.

Now, though, with a fair bit of obscure physics knowledge under my belt, I can finally decide not only whether or not an all-sky planet is possible, but if it is, what that planet would be like. I knew there had to be some good things about being a grownup.

Because I want my planet to be all sky, I’m going to build it from dry air and water vapor (the water vapor is so we can have clouds). Can you build a planet out of nothing but air? Well-informed intuition suggests you can: after all, Jupiter and the Sun are mostly hydrogen, and hydrogen is less dense and therefore harder to squeeze together than air. But as it turns out, we can get more precise. We can ask how large a cloud of air would have to be to collapse into a planet. This number is called the “Jeans length” or “Jeans radius” (and thanks to XKCD for making me aware of this formula). For air at Earth surface density (1.2 kilograms per cubic meter) and room temperature (68 Fahrenheit, 20 Celsius, or 293.15 Kelvin), the Jeans length is 35,400 kilometers, which is just over half the equatorial radius of Jupiter. You might think that would mean that the silly planet known as Endless Sky would be quite massive. In fact, it’s only got three times the mass of the moon.

Unfortunately, this manageably-small gas planet wouldn’t collapse very much under its own gravity. Its gravity would be weak, and as it collapsed, compression would heat it up and probably evaporate most or all of it. It’s important to note that the Jeans equation was intended to be used on nebulae, not inexplicable pockets of air floating in space for no reason. I guess Sir James Jeans just wasn’t thinking ahead. (Incidentally, Sir James Jeans would be a really pretentious name for a line of denim pants).

But no matter the details, what would Endless Sky actually be like? This is where the fun begins.

As anybody who’s ever looked into hydrodynamics knows, fluids are an enormous pain in the ass to deal with. They’re always swirling around and compressing and carrying pressure waves and expanding and contracting. You can simulate fluid behavior using the Navier-Stokes equations, which are frightening:


(From the Wikipedia article.)

We’ve got partial derivatives and dot products and divergence operators all over the fucking place. And to actually turn these equations into a computer program, you need a whole set of conservation equations, as well as initial conditions and boundary values. These equations are complicated because things like air and water are swirly and their mass moves around all the time. So you’d think the equation for the pressure in an atmosphere would be horrifying. Actually, it’s not. It’s

(pressure at the surface) * exp(-1 * (altitude) / (scale height))

The scale height is the altitude at which the pressure is e times smaller than it is at the surface (where e is about 2.71). This means that pressure corresponds quite closely to altitude. The scale height on Earth is (roughly) 8500 meters, so at an altitude of 8.5 kilometers, the pressure is 1/e atmospheres  (0.37 atmospheres or 37 kilopascals). Because this dependency is pretty stable, you can also say that, if you’re measuring a pressure of 37 kilopascals, you’re at an altitude of 8.5 kilometers.

We can use this nifty formula to figure out what our Endless Sky will be like. Before I get to the amusing pictures, here’s a caveat: I’m assuming constant Earth surface gravity throughout the atmosphere, which is inaccurate. That’s why this is a thought experiment, and why I majored in English instead of physics.

But here’s roughly, what the atmosphere of Endless Sky would look like:

 An Endless Atmosphere

You can learn two things from this picture right away: First, how thin tropospheres are, which tend to be where human beings and similar organisms hang out. Second, don’t buy cheap graph-paper notebooks, because the paper can apparently detect when you’re trying to tear it out gently and violently self-destructs.

Because I decided to make the parameters of my endless sky pretty much the same as Earth’s atmosphere, it all looks pretty familiar from the 1 atmosphere level to the 0.00001 atmosphere level (where we pass the Kármán line and go into space). Because Endless Sky has an oxygen atmosphere and (I’ve just decided) a sun, it’ll have an ozone layer much like Earth’s. I always wondered why thunderstorms grow to a certain height and then stop and spread out. It turns out to be because, below a certain level (called the tropopause), the higher you go, the colder the air gets. But above this level, and in fact throughout the stratosphere, the air actually gets warmer as you go higher. This is partly because of the ozone layer. Thunderstorms happen when hot, moist air rises into the lower-pressure air above, expands, and its water condenses. But this can only happen if it’s warmer than the surrounding air, and that pretty much becomes impossible at the tropopause, since the air above is already warmer.

Basically, if you were flying around Endless Sky in a hot-air balloon, and you sat so that the gondola obscured the horizon, it would be easy to think you were flying around Earth: bright blue skies, fluffy white clouds (no happy trees, though, unfortunately).

But if you looked down, you would see something horrifying. Namely, you would see no ground. Depending on atmospheric conditions below you, you might get different kinds of exotic clouds, but these would be smaller than the ones you see on Earth, because the higher pressure wouldn’t allow them to expand as much.

Looking into the sky below you would probably be quite a lot like looking into a deep, clear ocean: it would grow a deeper and darker blue. The blue is due to Rayleigh scattering, which (combined with the spectrum of light the Sun puts out) is why the sky above is blue. The darker is because less of the sunlight would make it through all that air.

You might think that the pressure would just keep going up and up, and the air would just get denser and denser as you got deeper. On most planets, solid, liquid, and gaseous alike, temperatures tend to go up as you go down. In the case of rocky planets, this is partly because of radioactive decay. But no matter what kind of planet, there’s internal heat left over from its formation. Therefore, it gets pretty hot down there. And when the temperature and pressure rise above the so-called critical point of the gases involved, something magical happens: the gases don’t quite liquefy, but they don’t remain gaseous, either. They sort of forget what they are, and become supercritical fluids. Supercritical fluids are amazing. They can be as dense as water (or denser), but they compress and expand like a gas and fill their containers. Here’s a video from awesome YouTuber Ben Krasnow showing you what supercritical carbon dioxide looks like:

Nitrogen’s critical point is lower than oxygen’s, so at a depth of about 128 kilometers (below the 1-atmosphere level), you would encounter a broiling opalescent sea of semi-liquid nitrogen containing a lot of dissolved oxygen. Looking down, you would see city-sized Bénard cells, which look approximately like this:

Benard Cells

(Image courtesy of NOAA, the US government atmospheric science people, who are pretty neat.)

They probably wouldn’t be quite that orderly, though. But there would be nothing but broiling opalescent clouds rising and falling as far as the eye could see, twisted into peculiar shapes or into alien jet streams by Coriolis forces.

Being denser, the supercritical ocean would attenuate light much faster than the gaseous part of the atmosphere. You would see deep, dark blue down to the opalescent layer, and then nothing. But, farther down still (about 200 kilometers deep), the sky would no longer be endless: the pressure would pass 88,000 atmospheres, which is the pressure at which oxygen solidifies into pretty blue crystals. This means there would be another layer of weather on Endless Sky: down there, the crust of oxygen and nitrogen “ice” would evaporate into the supercritical fluid above, rise, and cool, driving powerful convection currents and stirring the deep, and where it fell and cooled, the supercritical fluid would condense into oxygen and nitrogen snow. Perhaps it would form enormous dune fields like the ones we see on Earth and, amazingly, on Saturn’s moon Titan:


(Earth dunes on top. Titan dunes (probably made of water ice) on the bottom. Image courtesy of NASA, via Wikipedia.)

I hope my eight-year-old self is vindicated. He’s probably dancing around like a lunatic, the hyperactive little freak…


What’s the most interesting place in the universe?

We humans are extremely susceptible to information overload. Yeah, our brains are impressive, but there’s an upper limit on how much they can process before they start to overheat and misbehave. The unfortunate cases of ailments like PTSD and career burnout testify to that. In fact, the brain is remarkably good at filtering out data it deems to be irrelevant. I’ve just looked down at my desk. On my desk is a penny, worth US$0.01. I didn’t put it there recently, so it’s been there for at least a few days. Maybe as long as a week. You know how many times I’ve consciously noticed it since I put it there? Zero. Even though it sits right by my keyboard, where I work every day, my brain just glossed over the existence of a small disc of copper and zinc inexplicably pressed into currency. 

There are all sorts of interesting things all around me, items with thousands or millions of little salient details. I’ve just picked up a ruler from the floor. It’s from Office Depot. It has ’80s-station-wagon-style fake wood down the center. A particular machine in a particular factory in a particular place on Earth made this ruler from a particular load of plastic derived from a particular load of crude oil. If I measured them with high precision, I’m sure I’d find that the tick marks on the ruler have their own unique pattern of variation. Each of the numbers is printed slightly differently, even from identical numerals on the same ruler. I laugh when people say “No two snowflakes are alike,” because if you think about it, no two anythings are alike.

But like I said, it’s hard for us to think about this during our day-to-day lives. Imagine if, rather than just stopping to smell the roses, you stopped to smell them, compare the scents between flowers, check how much pollen each flower had, and count each flower’s petals. You’d still only be covering a tiny fraction of the information you could learn about the roses, but you’d still stall on the sidewalk for a few minutes and get some really strange looks from passerby (trust me on that one…). 

But think about it: each of us only passes through a tiny volume of space in a given day, and we still see all kinds of crazy stuff that we never think about. In the driveway outside, most of the gravel is (I think) some kind of blue granite. Each stone has a different shape. If you studied one of them long enough, you would start to find interesting things about it. Maybe you’d find a cool double-stripe in one of them, or a bumpy spot that looked a little like a face, or a surprisingly sharp edge, or a near-perfect pyramid. And we must remember that these stones, being granite, are little chunks of ancient lava and magma. That is to say, ancient fluid rock that forced its way through other rocks and squeezed out like toothpaste

You might be thinking that this article is kind of loopy and hippy-dippy. Never fear. Because I am borderline obsessive-compulsive, you’d better believe I’m going to quantify the hell out of how awesome and complicated the world is.

Let’s start off with a simple thought experiment. If you took a garden trowel and dug up a cube of dirt 10 centimeters on an edge, you’d end up with 1000 cubic centimeters (one liter) of soil, which, in the area where I live, would contain a few dozen or a few hundred individual stalks of grass, a collection of interesting weeds, organic debris in various stages of decomposition, somewhere between zero and five worms, lots of interesting rocks of different kinds, tough red clay, and, in all likelihood, some extremely unhappy ants. Imagine you wanted to a do a complete analysis of this cube of dirt. The first thing you’d want to do would be to lift it up in one piece and photograph it from all sides. Then you’d want to weigh it and record subjective observations like smell and texture. Then, if you were being really scientific, you could freeze the cube of dirt, embed it in paraffin, and slice it into ten-nanometer slices with a laser microtome. Then you scan each slice with a transmission electron microscope at a resolution of 0.5 nanometers, which would be good enough to show you detail down to the subcellular (almost the molecular) level. This is quite achievable with current technology. Good luck storing all that data, though. At 0.5-nanometer resolution, each individual slice would contain 160,000 terabytes of data (assuming you stored them as uncompressed 32-bit grayscale images, which seems reasonable). That’s at least a whole server farm. Probably several. Possibly several hundred. And that’s only to store the data from a single one of the hundred-million slices you’d end up with. And I’ll remind you that we’re working from a cube of dirt you could easily fit into a breadbox, a grocery bag, a suitcase, or (as a really cruel joke) a cake box. But that ludicrous quantity of data would tell you much of what was going on in the cube of dirt at the time you dug it up. Of course, it wouldn’t tell you things like the temperature distribution or magnetic fields or moisture or anything like that, but you could theoretically work those out, too, or just run the whole cube through a mass spectrometer and get an exact read on its composition. Hope you’ve got another hundred server farms handy.

And, once again, I will remind you that this is for a one-liter cube of dirt. There’s that much interesting stuff happening in 1000 cc’s of soil. And do you know how many cubes this size the Earth alone contains? Upwards of 8.5 trillion trillion. That’s a lot of stuff. Atoms are small, but if you had a diamond containing 8.5 trillion trillion carbon atoms, that diamond would be the size of eight sugar cubes stacked together ( which I can guarantee you I would immediately sell to stop people hunting me down). And don’t forget, that each carbon atom in this diamond represents a cubic hole in your lawn large enough to twist an ankle in, containing more information than you can probably store in an average server farm.

I could go on. I could go on for days. I could try to convey how achingly massive the universe really is, and how intricate many parts of it are, but even my head, accustomed to such bizarre calculations, is already near bursting. 

There’s one more way to begin to grasp how complex the world is. Say you wanted to simulate the whole universe, down to the physics of individual protons, neutrons, and electrons. You could divide the universe into a cubic grid with cells one femtometer (about the radius of a proton) on an edge, and label each cell with the relevant variables (what kinds of particles it contains, their velocities, magnetic fields, et cetera). But in some ways, the universe is like a black-and-white line drawing. Yeah, there are a lot of interesting details, but those details exist only in the shapes of the lines. You can use a nifty mathematical thing called a quadtree to turn that image into pixels without storing each individual pixel of blank white space. The most simpleminded kind of quadtree works like this: take the original image. Divide in half in both directions, so you get four subsquares. If a subsquare contains more than one kind of pixel (meaning it’s not all white or all black), then subdivide it into four smaller squares. Otherwise, store its coordinates and its edge length and write “This square is all white,” which, especially for large squares, saves a lot of memory. Eventually, you’ll subdivide until you’ve got a shitload of squares that are two pixels by two pixels. Some of these you won’t have to subdivide, but many of them will be on the edges of lines, and you’ll have to subdivide them and store the individual pixel values. But a large portion of the image (meaning: all the parts that contain large swaths of white or black) don’t have to be stored at nearly so high a resolution.

You can imagine dividing the universe in the same way. For simplicity’s sake, let’s just consider the physics of individual subatomic particles: protons, neutrons, and electrons. Let’s pick a cubic volume of the universe that fits within our current cosmological horizon, that is, a cube about 90 billion light-years on an edge that contains every object whose light has had time to reach Earth (except for the spherical bits that stick out on the sides; you could deal with those using a different subdivision scheme, but let’s keep things simple for now). Divide that cube in half along each axis, giving you eight separate cubes. If a cube contains nothing, say so and leave it alone. If a cube contains a single subatomic particle, store that particle’s variables in the cube and leave it alone. But if the cube contains more than one particle, subdivide it into eight again. This is called the octree method, and it’s the basis of the absolutely brilliant Barnes-Hut algorithm, which makes gravitational simulations for large numbers of particles run a hell of a lot faster by not doing ultra-high-precision calculations for low-density regions.

We’re going to have to do a lot of subdividing, though. Even in the vacuum of intergalactic space, you’ll probably have to subdivide down to cubes 20 centimeters on an edge to make sure you only have one particle per cube. That’s pretty big in particle-physics terms, but shockingly tiny in universe terms. Even in the vacuums between stars and between planets, you’ll have to subdivide down to one-centimeter cubes, and often smaller. When you reach the density of water (which you do when you approach pretty much any planet or star or grain of zodiacal dust), you have to subdivide down to cubes 0.1 nanometers on an edge. When you’re dealing with something as dense as lead, the cubes are 0.03 nanometers, smaller than most atoms. In the core of the sun, your cubes get down to 0.01 nanometers. 

But actually, even within atoms, you have to subdivide. Atoms are, as our science teachers kept telling us, mostly empty space. The nucleus of an atom has a density of about 2e17 kilograms per cubic meter. So, although for the most part an atom can be described by cubes on the order of maybe a tenth of a picometer on an edge (depending on how many electrons there are), in the nucleus, the cubes are two femtometers on an edge, just large enough to enclose a single proton.

What’s the point of all this? Good question. The point of all this is that, in human terms, or in terms of information, the most interesting places are those where the cubes are very small. That is to say, places where there’s a lot of matter packed close together doing interesting things. These are the places that astronomers study: nebulae and galaxies and stars and planets. But when you subdivide small enough, you begin to see that, really, the most interesting place in the universe is the nucleus of an atom, because you basically have no choice but to describe all the individual particles to get a decent description of the nucleus, which is just another way of saying the cubes involved are the size of the particles involved (protons and neutrons). 

But actually, nuclei aren’t the most interesting place in the universe. They’re dense, yeah, but they’re surrounded by a lot of empty space.

Neutron stars, however, formed when large enough stars go supernova and their cores collapse inwards, contain the smallest amount of empty space possible, according to our current understanding of physics (if they were any denser, you’d end up with a black hole, which, once again, is pretty much all empty space). Large neutron stars have about the same density as atomic nuclei (and possibly higher near their centers), but they’re the size of cities. Which is to say, there’s stuff happening in every cubic femtometer of a neutron star, just like an atomic nucleus, but unlike an atomic nucleus, its size is measured in kilometers, not femtometers.

We’ll be meeting neutron stars again, because they’re such incredibly extreme objects. But for now, I’ll leave you with the echoes of what I’ve been babbling about: The universe is complicated and fascinating and frequently horrifying and headache-inducing. Technically speaking, neutron stars are the most interesting places, but really, it’s all pretty weird. And I like weird. Weird is fun.