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The Nightmare Tongue, Part 1

This series is going to be a little different. A sort of ongoing project. Don’t worry, I won’t let it derail my other bizarre ramblings.

Anyway, here’s the project: I’m going to construct a language. There’s a whole community dedicated to that, but it wasn’t for nothing that my grade-school teachers kept writing “Doesn’t play well with others” and “Is not very good at taking turns” and “Shithead” on my papers. I’m going to go at this more or less alone. (Unless any of my readers are compelled to hop on this loony rollercoaster with me.)

The premises and requirements of the Nightmare Tongue are simple. Not like lojban, a constructed language based on freakin’ mathematical logic which is so sprawling and complex that the language itself has its own Creative Commons License (I think). I want the premises to be simple, but there’s a reason I’m calling the language The Nightmare Tongue. I want it to be the kind of language demons or evil aliens or sentient hyenas would speak. I want a language that sounds scary. I want a language in which you can use a phrase to express weird thoughts James Joyce couldn’t express in English. (Re-reading that last sentence makes me realize I really need to get more sleep…) Why? Fun, mostly. Because I’ve dabbled in creating languages in the past, but I want to take a serious shot at it. This is something I’ve wanted to do ever since I learned just how much effort and love J.R.R. Tolkein put into Elvish. Tolkein is a famed and respected writer, and Elvish is a beautiful and nuanced language. I remember watching The Fellowship of the Ring on freakin’ VHS when it first came out, and how the actress playing Arwen said she loved speaking Elvish.

Tolkein is a famed and respected writer and scholar (if I remember correctly, he did his own translation of Beowulf). I’m a madman on the Internet with too much time on his hands. The Nightmare Tongue isn’t going to be nearly as pretty as Elvish. But here’s a list of the things I do want it to be:

  • Pronounceable. I don’t want to turn this into some jackass art project where I deliberately try to be as dense as possible. Despite my sentient-hyena example from earlier, I want the Nightmare Tongue to be pronounceable by the human vocal tract. I do intend to stuff as many weird clicks and other bizarre consonants in there as I can, but I want it to be the kind of thing that a person can, with practice, speak fluently and with a nice rhythm.
  • Weird-sounding. Icelandic is an infamously complicated language. Years ago, everybody panicked because an Icelandic volcano erupted and pretty much blocked the flyways through Europe for a week. The name of that volcano is, of course, Eyjafjallajökull. (It probably says something about me that I spelled that right on the first try, but that I still get the I and the E the wrong way around in “receive”…) Eyjafjallajökull is roughly pronounced (forgive me, Icelanders–even in text I’m going to mess it up) “EY-aff-yaht-lah-YO-kut-th.” Those double-Ls are a weird-sounding phoneme we don’t have in English: a voiceless alveolar lateral fricative. It’s (roughly) the kind of sound you make when you try to say the English letters “K” and “L” at the same time. You’ve probably seen this consonant before without realizing it. The name of the feathered serpent, the badass Aztec god Quetazalcoatl has one at the end, so if you want to pronounce it authentically, unless you speak Nahuatl (there it is again), you’re going to end up spitting on the person in front of you. Random fact: I used to work with a guy from Mexico who spoke Nahuatl fluently. It sounded awesome.
  • Complex.  Once again, I don’t want to descend too far into navel-gazing (for one thing, navels are kinda gross). By which I mean I don’t want an impenetrable mess of a language that’s purposely too difficult for anybody to learn. It wouldn’t be hard to make a language like that. After all, as Lewis Caroll once pointed out (I’m paraphrasing), you and I are imperfect speakers of English and imperfect doers of arithmetic because it takes us a lot of effort to decipher the perfectly grammatical sentence “What is the sum of one plus one plus one plus one plus one plus one plus one plus one and the largest prime factor of one plus one plus one plus one all multiplied by one plus one plus one plus one.” I’m sorry you had to see that. My point is, I want the grammar to be bizarre, complex, and alien, but I don’t want some abstract-art nonsense that’s impossible and pretentious.
  • Writeable. The Nightmare Tongue will have a written alphabet. When I first got interested in created languages (thanks to Tolkein), the invented writing was one of my favorite parts. Plus, one of my cousins gave me some sweet calligraphy pens for my birthday, so I’ll be able to write that alphabet in BLOOD RED. (I really need to get more sleep…)
  • Complete. Or as close as I can get. This site is all about thought experiments, but it’s also about fleshing things out. I don’t want this to be an unfinished concept-art project like all the other languages I’ve tried to create. My goal, by the end of this, is to have a weird-sounding, twisted, evil language that you could write a competent dictionary for, and maybe a grammar reader for children. (I’ve seen The Exorcist. I know children can learn demon tongues.) Perhaps someday I’ll find a way to crowbar it into a novel or something.

Either way, that said, work will begin, with updates as developments warrant. If you’re not interested in this kind of thing, you won’t hurt my feelings by skipping these posts. Don’t worry, I’ll be getting back to my bread and butter–ludicrous thought experiments–as soon as my brain gets unstuck.

Be safe out there.

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Sundiving, Part 2

In the previous post, I figured out how to get a spacecraft to an altitude of one solar radius (meaning one solar radius above the Sun’s surface, and two solar radii from its center). That’s nice and all, but unless we figure out how to get it the rest of the way down intact, then we’ve essentially done the same thing as a flight engineer who sends an astronaut into orbit in a fully-functional space capsule, but forgets to put a parachute on it. (Not that I know anything about that. Cough cough Kerbal Space Program cough…)

The sun is vicious. Anyone who’s ever had a good peeling sunburn knows this, and cringes at the thought. And anyone who, in spite of their parents’ warnings, has looked directly at the sun, also knows this. But I’ve got a better demonstration. I have a big 8.5 x 11-inch Fresnel lens made to magnify small print. I also have a lovely blowtorch that burns MAP-Pro, a gas that’s mostly propylene, which is as close as a clumsy idiot like me should ever come to acetylene. Propylene burns hot. About 2,200 Kelvin. I turned it on a piece of gravel and a piece of terra cotta. It got them both orange-hot, but that was the best it could do. The Fresnel lens, a cheap-ass plastic thing I bought at a drugstore, melted both in seconds (albeit in very small patches), using nothing more than half a square foot of sunlight.

Actually, the area of that magnifier is handy to have around. It’s 0.0603 square meters. On the surface of the Earth, we get (very roughly) 1,300 watts per square meter of sunlight (that’s called the solar constant). To melt terra cotta, I have to get the spot down to about a centimeter across. The lens intercepts about 80 Watts. If those 80 Watts are focused on a circle a centimeter across, then the target is getting irradiated with 770 solar constants, which, if it was a perfect absorber, would raise its temperature to 2,000 Kelvin. If I can get the spot is half a centimeter across, then we’re talking 3,070 solar constants and temperatures approaching 3,000 Kelvin.

And while I was playing around with my giant magnifier, I made a stupid mistake. Holding the lens with one hand, I reached down to re-position my next target. The light spot, about the size of a credit card, fell on the back of my hand. I said words I usually reserve for when I’ve hit my finger with a hammer. This is why you should always be careful with magnifying lenses. Even small ones can burn you and start fires.

The area of a standard credit card is about 13 times smaller than the area of my lens, so my hand was getting 13 solar constants. And even a measly 13 solar constants was more than enough to sting my skin like I was being attacked by a thousand wasps. Even at the limit of my crappy Fresnel lens, somewhere between 770 and 3,070 solar constants, we’re already in stone-melting territory.

At an altitude of 1 solar radius, our Sundiver will be getting to 11,537 solar constants. Enough to raise a perfect absorber to 4,000 Kelvin, which can melt every material we can make in bulk. Our poor Sundiver hasn’t even reached the surface and already it’s a ball of white-hot slag.

Except that I’ve conveniently neglected one thing: reflectivity. If the Sundiver was blacker than asphalt, sure, it would reach 4,000 Kelvin and melt. But why on Earth would we paint an object black if we’re planning to send it to the place where all that heat-producing sunlight comes from? That’s even sillier than those guys you see wearing black hoodies in high summer.

My first choice for a reflective coating would be silver. But there’s a massive problem with silver. Here are two graphs to explain that problem:

(Source.)

Sun Blackbody

(Source is obvious.)

The top spectrum shows the reflectivities of aluminum (Al), silver (Ag, because Latin), and gold (Au) at wavelengths between 200 nanometers (ultraviolet light; UV-C, to be specific: the kind produced by germicidal lamps) and 5,000 nanometers (mid-infrared, the wavelength heat-seeking missiles use).

The bottom spectrum is the blackbody spectrum for an object at a temperature of 5,778 Kelvin, which is a very good approximation for the solar spectrum. See silver’s massive dip in reflectivity around 350 nanometers? See how it happens, rather inconveniently, right around the peak of the solar emission spectrum? Sure, a silver shield would be good at reflecting most of the infra-red light, but what the hell good is that if it’s still soaking up all that violet and UV?

Gold does a little better (and you can see from that spectrum why they use gold in infrared mirrors), but it still bottoms out right where we don’t want it to. (Interesting note: see how gold is fairly reflective between 500 nanometers and 1,000 nanometers, but not nearly as reflective between 350 nanometers and 500 nanometers? And see how silver stays above 80% reflectivity between 350 and 1,000? That’s the reason gold is gold-colored and silver is silver-colored. Gold absorbs more green, blue, indigo, and violet than it does red, orange, and yellow. Silver is almost-but-not quite constant across this range, which covers the visible spectrum, so it reflects all visible light pretty much equally. Spectra are awesome.)

Much to my surprise, our best bet for a one-material reflector is aluminum. My personal experiences with aluminum are almost all foil-related. My blowtorch will melt aluminum, so it might seem like a bad choice, but in space, there’s so little gas that almost all heat transfer is by radiation, so it might still work. And besides, if you electropolish it, aluminum is ridiculously shiny.

(Image from the Finish Line Materials & Processes, Ltd. website.)

That’s shiny. And it’s not just smooth to the human eye–it’s smooth on scales so small you’d need an electron microscope to see them. They electro-polish things like medical implants, to get rid of the microscopic jagged bits that would otherwise really annoy the immune system. So get those images of crinkly foil out of your head. We’re talking a mirror better than you’ve ever seen.

Still, aluminum’s not perfect. Notice how its reflectivity spectrum has an annoying dip at about 800 nanometers. The sun’s pretty bright at that wavelength. Still, it manages 90% or better across almost all of the spectrum we’re concerned about. (Take note, though: in the far ultraviolet, somewhere around 150 nanometers, even aluminum bottoms out, and the sun is still pretty bright even at these short wavelengths. We’ll have to deal with that some other way.)

So our aluminum Sun-shield is reflecting 90% of the 15.7 million watts falling on every square meter. That means it’s absorbing the other 10%, or 1.57 million watts per square meter.

Bad news: even at an altitude of 1 solar radius, and even with a 90% reflective electropolished aluminum shield, the bastard’s still going to melt. It’s going to reach over 2,000 Kelvin, and aluminum melts at 933.

We might be able to improve the situation by using a dielectric mirror. Metal mirrors reflect incoming photons because metal atoms’ outer electrons wander freely from one atom to another, forming a conductive “sea”. Those electrons are easy to set oscillating, and that oscillation releases a photon of similar wavelength, releasing almost all the energy the first photon deposited. Dielectric mirrors, on the other hand, consist of a stack of very thin (tens of nanometers) layers with different refractive indices. For reference, water has a refractive index of 1.333. Those cool, shiny bulletproof Lexan windows that protect bank tellers have a refractive index of about 1.5. High-grade crystal glassware is about the same. Diamonds are so pretty and shiny and sparkly because their refractive index is 2.42, which makes for a lot of refraction and internal reflection.

These kind of reflections are what make dielectric mirrors work. The refractive index measures how fast light travels through a particular medium. It travels at 299,792 km/s through vacuum. It travels at about 225,000 km/s through water and about 124,000 km/s through diamond. This means, effectively, that light has farther to go through the high-index stuff, and if you arrange the layers right, you can set it up so that a photon that makes it through, say, two layers of the stack will have effectively traveled exactly three times the distance, which means the waves will add up rather than canceling out, which means they’re leaving and taking their energy with them, rather than canceling and leaving their energy in your mirror.

This, of course, only works for a wavelength that matches up with the thickness of your layers. Still, close to the target frequency, a dielectric mirror can do better than 99.9% reflectivity. And if you use some scary algorithms to optimize the thicknesses of the different layers, you can set it up so that it reflects over a much broader spectrum, by making the upper layers very thin to reflect short-wavelength light (UV, et cetera) and the deeper layers reflect red and infra-red. The result is a “chirped mirror,” which is yet another scientific name that pleases me in ways I don’t understand. Here’s the reflection spectrum of a good-quality chirped mirror:

(Source.)

Was I inserting that spectrum just an excuse to say “chirped mirror” again? Possibly. Chirped mirror.

Point is, the chirped mirror does better than aluminum for light between 300 and 900 nanometers (which covers most or all of the visible spectrum). But it drops below 90% for long enough that it’s probably going to overheat and melt. And there’s another problem: even at an altitude of 1 solar radius, the Sundiver’s going to be going upwards of 400 kilometers per second. If the Sundiver crosses paths with the smallest of asteroids (thumbnail-sized or smaller), or even a particularly bulky dust grain, there’s going to be trouble. To explain why, here’s a video of a peanut-sized aluminum cylinder hitting a metal gas canister at 7 kilometers per second, 57 times slower than the Sundiver will be moving:

We have a really, really hard time accelerating objects anywhere near this speed. We can’t do too much better than 10 to 20 km/s on the ground, and in space, we can at best double or triple that, and only if we use gravity assists and clever trajectories. On the ground, there are hypersonic dust accelerators, which can accelerate bacterium-sized particles to around 100 km/s, which is a little better.

But no matter the velocity, the news is not good. A 5-micron solid particle will penetrate at least 5 microns into the sunshade (according to Newton’s impact depth approximation). Not only will that rip straight through dozens of layers of our carefully-constructed chirped mirror, but it’s also going to deposit almost all of its kinetic energy inside the shield. A particle that size only masses 21.5 picograms, so its kinetic energy (according to Wolfram Alpha) is about the same required to depress a computer key. Not much, but when you consider that this is a bacterium-sized mote pressing a computer key, that’s a lot of power. It’s also over 17,000 times as much kinetic energy as you’d get from 21.5 picograms of TNT.

As for a rock visible to the naked eye (100 microns in diameter, as thick as a hair), the news just gets worse. A particle that size delivers 110.3 Joules, twenty times as much as a regular camera’s flash, and one-tenth as much as one of those blinding studio flashbulbs. All concentrated on a volume too small to squeeze a dust mite into.

And if the Sundiver should collide with a decent-sized rock (1 centimeter diameter, about the size of a thumbnail), well, you might as well just go ahead and press the self-destruct button yourself, because that pebble would deliver as much energy as 26 kilos (over 50 pounds) of TNT. We’re talking a bomb bigger than a softball. You know that delicately-layered dielectric mirror we built, with its precisely-tuned structure chemically deposited to sub-nanometer precision? Yeah. So much for that. It’s now a trillion interestingly-structured fragments falling to their death in the Sun.

My point is that a dielectric mirror, although it’s much more reflective than a metal one, won’t cut it. Not where we’re going. We have to figure out another way to get rid of that extra heat. And here’s how we’re going to do it: heat pipes.

The temperature of the shield will only reach 2,000 Kelvin if its only pathway for getting rid of absorbed heat is re-radiating it. And it just so happens that our ideal shield material, aluminum, is a wimp and can’t even handle 1,000 Kelvin. But aluminum is a good conductor of heat, so we can just thread the sunshade with copper pipes, sweep the heat away with a coolant, and transfer it to a radiator.

But how much heat are we going to have to move? And has anybody invented a way to move it without me having to do a ridiculous handwave? To find that out, we’re going to need to know the area of our sunshade. Here’s a diagram of that sunshade.

Sunshade 2

I wanted to make a puerile joke about that, but the more I look at it, the less I think “sex toy” and the more I think “lava lamp.” In this diagram, the sunshade is the long cone. The weird eggplant-shaped dotted line is the hermetically-sealed module containing the payload. That payload will more than likely be scientific instruments, and not a nuclear bomb with the mass of Manhattan island, because that was probably the most ridiculous thing about Sunshine. Although (spoiler alert), Captain Pinbacker was pretty out there, too.

The shield is cone-shaped for many reasons. One is that, for any given cross-sectional radius, you’re going to be absorbing the same amount of heat no matter the shield’s area, but the amount you can radiate depends on total, not cross-sectional, area. Let’s say the cone is 5 meters long and 2 meters in diameter at the base. If it’s made of 90% reflective electropolished aluminum, it’s going to absorb 4.93 megawatts of solar radiation at an altitude of 1 solar radius. Its cross-section is 3.142 square meters, but its total surface area is 16.02 square meters. That means that, to lose all its heat by radiation alone, the shield would have to reach a blackbody temperature of 1,500 Kelvin. Still almost twice aluminum’s melting point, but already a lot more bearable. If we weren’t going to get any closer than an altitude of 1 solar radius, we could swap the aluminum mirror out for aluminum-coated graphite and we could just let the shield cool itself. I imagine this is why the original solar probe designs used conical or angled bowl-shaped shields: small cross-sectional area, but a large area to radiate heat. But where we’re going, I suspect passive cooling is going to be insufficient sooner or later, so we might as well install our active cooling system now.

Heat pipes are awesome things. You can find them in most laptops. They’re the bewildering little copper tubes that don’t seem to serve any purpose. But they do serve a purpose. They’re hollow. Inside them is a working fluid (which, at laptop temperatures, is usually water or ammonia). The tube is evacuated to a fairly low pressure, so that, even near its freezing point, water will start to boil. The inner walls of the heat pipe are covered with either a metallic sponge or with a series of thin inward-pointing fins. These let the coolant wick to the hot end, where it evaporates. Evaporation is excellent for removing heat. It deposits that heat at the cold end, where something (a passive or active radiator or, in the case of a laptop, a fan and heat sink) disposes of the heat.

Many spacecraft use heat pipes for two reasons. 1) The absence of an atmosphere means the only way to get rid of heat is to radiate it, either from the spacecraft itself, or, more often, by moving the heat to a radiator and letting it radiate from there; heat pipes do this kind of job beautifully; 2) most heat pipes contain no moving parts whatsoever, and will happily go on doing their jobs forever as long as there’s a temperature difference between the ends, and as long as they don’t spring a leak or get clogged.

On top of this, some heat pipes can conduct heat even better than solid copper. Copper’s thermal conductivity is 400 Watts per meter per Kelvin difference, which is surpassed only by diamond (and graphene, which we can’t yet produce in bulk). But heat pipes can do better than one-piece bulk materials: Wikipedia says 100,000 Watts per meter per Kelvin difference, which my research leads me to believe is entirely reasonable. (Fun fact: high-temperature heat pipes have been used to transport heat from experimental nuclear reactor cores to machinery that can turn that heat into electricity. These heat pipes use molten frickin’ metal and metal vapor as their working fluids.)

The temperature difference is going to be the difference between the temperature of the shield (in this case, around 1,500 Kelvin at the beginning) and outer space (which is full of cosmic background radiation at an effective temperature of 2.3 Kelvin, but let’s say 50 Kelvin to account for things like reflected light off zodiacal dust, light from the solar corona, and because it’s always better to over-build a spacecraft than to under-build it).

When you do the math, at an altitude of 1 solar radius, we need to transport 4.93 megawatts of heat over a distance of 5 meters across a temperature differential of 1,450 Kelvin. That comes out to 680 Watts per meter per Kelvin difference. Solid copper can’t quite manage it, but a suitable heat pipe could do it with no trouble.

But we still have to get rid of the heat. For reasons that will become clear when Sundiver gets closer to the Sun, the back of the spacecraft has to be very close to a flat disk. So we’ve got 3.142 square meters in which to fit our radiator. Let’s say 3 square meters, since we’re probably going to want to mount things like thruster ports and antennae on the protected back side. Since we’re dumping 4.93 megawatts through a radiator with an area of 3 square meters, that radiator’s going to have to be able to handle a temperature of at least 2,320 Kelvin. Luckily, that’s more than manageable. Tungsten would work, but graphite is probably our best choice, because it’s fairly tough, it’s unreactive, and it’s a hell of a lot lighter than tungsten, which is so dense they use it in eco-friendly bullets as a replacement for lead (yes, there’s such a thing as eco-friendly bullets). Let’s go with graphite for now, and see if it’s still a good choice closer to the Sun. (After graphite, our second-best choice would be niobium, which is only about as dense as iron, with a melting point of 2,750 Kelvin. I’m sticking with graphite, because things are going to get hot pretty fast, and the niobium probably won’t cut it. (Plus, “graphite radiator” has a nicer ring to it than “niboium radiator.”)

Our radiator’s going to be glowing orange-hot. We’ll need a lot of insulation to minimize thermal contact between the shield-and-radiator structure and the payload, but we can do that with more mirrors, more heat pipes, and insulating cladding made from stuff like like calcium silicate or thermal tiles filled with silica aerogel.

Of course, all the computations so far have been done for an altitude of 1 solar radius. And I didn’t ask for a ship that could survive a trip to 1 solar radius. I want to reach the freakin surface! Life is already hard for our space probe, and it’s going to get worse very rapidly. So let’s re-set our clock, with T=0 seconds being the moment the Sundiver passes an altitude of 1 solar radius.

Altitude: 0.5 solar radii

T+50 minutes, 46 seconds

Speed: 504 km/s

Solar irradiance: 28 megawatts per square meter (20,600 solar constants)

Temperature of a perfect absorber: 4,700 Kelvin (hot enough to boil titanium and melt niobium)

Total heat flux: 8.79 megawatts

Temperature of a 90% reflective flat shield: 2,700 Kelvin (almost hot enough to boil aluminum)

Temperature of Sundiver’s conical shield (radiation only): 1,764 Kelvin (still too hot for aluminum)

Radiator temperature: 2,600 Kelvin (manageable)

Required heat conductivity: 1,000 Watts per meter per Kelvin difference (manageable)

Altitude: 0.25 solar radii

T+1 hour, 0 minutes, 40 seconds

Speed: 553 km/s

Solar irradiance: 40.3 megawatts per square meter (29,600)

Temperature of a perfect absorber: 5,200 Kelvin (hot enough to boil almost all metals. Not tungsten, though. Niobium boils.)

Total heat flux: 12.66 megawatts

Temperature of a 90% reflective flat shield: 2,900 Kelvin (more than hot enough to boil aluminum)

Temperature of Sundiver’s conical shield (radiation only): 1,900 Kelvin (way too hot for aluminum)

Radiator temperature: 2,900 Kelvin (more than manageable, but the radiant heat would probably hurt your eyes)

Required heat conductivity: 1,400 Watts per meter per kelvin difference (manageable)

Altitude: 0.1 solar radii

T+1 hour, 5 minutes, 25 seconds

Speed: 589 km/s

Solar irradiance: 52 megawatts per square meter

Temperature of a perfect absorber: 5,500 Kelvin (tungsten melts, but still doesn’t boil; tungsten’s tough stuff; niobium is boiling)

Total heat flux: 16.34 megawatts

Temperature of a flat shield: 3,000 Kelvin (tungsten doesn’t melt, but it’s probably uncomfortable)

Temperature of our conical shield: 2,000 Kelvin (getting uncomfortably close to aluminum’s boiling point)

Radiator temperature: 3,100 Kelvin (tungsten and carbon are both giving each other worried looks; the shield can cause fatal radiant burns from several meters)

Required heat conductivity: 1,600 watts per meter per kelvin difference (still manageable, much to my surprise)

Altitude: 0.01 solar radii (1% of a solar radius)

T+1 hour, 8 minutes, 11 seconds

Speed: 615 km/s

Irradiance: 61 megawatts per square meter

Temperature of a perfect absorber: 5,700 Kelvin (graphite evaporates, but tungsten is just barely hanging on)

Total heat flux: 19.38 megawatts

Temperature of a flat shield: 3,200 Kelvin (most materials have melted; tungsten and graphite are still holding on)

Temperature of our conical shield: 2,100 Kelvin (titanium melts)

Radiator temperature: 3,200 Kelvin (tungsten and graphite are still stable, but at this point, the radiator itself is almost as much of a hazard as the Sun)

Required heat conductivity: 1,900 Watts per meter per Kelvin difference (we’re still okay, although we’re running into trouble)

The Sundiver finally strikes the Sun’s surface traveling at 618 kilometers per second. Except “strike” is a little melodramatic. The Sundiver’s no more striking the Sun than I strike the air when I jump off a diving board. The Sun’s surface is (somewhat) arbitrarily defined as the depth the sun’s plasma gets thin enough to transmit over half the light that hits it. At an altitude of 0 solar radii, the Sun’s density is a tenth of a microgram per cubic centimeter. For comparison, the Earth’s atmosphere doesn’t get that thin until you get 60 kilometers (about 30 miles) up, which is higher than even the best high-altitude balloons can go. Even a good laboratory vacuum is denser than this.

But even this thin plasma is a problem. The problem isn’t necessarily that the Sundiver is crashing into too much matter, it’s that it’s that the matter it is hitting is depositing a lot of kinetic energy. Falling at 618 kilometers per second, it encounters solar wind protons traveling the opposite direction at upwards of 700 kilometers per second, for a total velocity of 1,300 kilometers per second. Even at photosphere densities, when the gas is hitting you at 1,300 kilometers per second, it transfers a lot of energy. We’re talking 17 gigawatts per square centimeter, enough to heat the shield to a quarter of a million Kelvin.

This spells the end for the Sundiver. It might survive a few seconds of this torture, but its heat shield is going to be evaporating very rapidly. It won’t get more than a few thousand kilometers into the photosphere before the whole spacecraft vaporizes.

In fact, even at much lower densities (a million hydrogen atoms per cubic centimeter), the energy flux due to the impacts of protons alone is greater than one solar constant. (XKCD’s What-If, the inspiration for this whole damn blog, pointed this out when talking about dropping tungsten countertops into the sun.) At 1.0011 solar radii, the proton flux is more than enough to heat the shield up hotter than a lightning bolt. As a matter of fact, when the solar wind density exceeds 0.001 picograms per cubic centimeter (1e-15 g/cc), the energy flux from protons alone is going to overheat the shield. It’s hard to work out at what altitude this will happen, since we still don’t know very much about the environment and the solar wind close to the sun (one of the questions Solar Probe+ will hopefully answer when (if) it makes its more pedestrian and sensible trip to 8 solar radii.) But we know for certain the shield will overheat by the time we hit zero altitude. The whole Sundiver will turn into a wisp of purplish-white vapor that’ll twist and whirl away on the Sun’s magnetic field.

But even if heating from the solar wind wasn’t a problem, the probe was never going to get much deeper than zero altitude. Here’s a list of all the problems that would kill it, even if the heat from the solar wind didn’t:

1) This close to the Sun, the sun’s disk fills half the sky, meaning anything that’s not inside the sunshade is going to be in direct sunlight and get burned off. That’s why I said earlier that the back of the Sundiver had to be very close to flat.

2) The radiator will reach its melting point. Besides, we would probably need high-power heat pumps rather than heat pipes to keep heat flowing from the 2,000 Kelvin shield to the 3,000-Kelvin radiator. And even that might not be enough.

3) Even if we ignore the energy added by the proton flux, those protons are going to erode the shield mechanically. According to SRIM, the conical part of the shield (which has a half-angle of 11 degrees) is going to lose one atom of aluminum for every three proton impacts. At this rate, the shield’s going to be losing 18.3 milligrams of aluminum per second to impacts alone. While that’s not enough to wear through the shield, even if it’s only a millimeter thick, my hunch is that all that sputtering is going to play hell with the aluminum’s structure, and probably make it a lot less reflective.

4) Moving at 618 kilometers per second through a magnetic field is a bad idea. Unless the field is perfectly uniform (the Sun’s is the exact opposite of uniform: it looks like what happens if you give a kitten amphetamines and set it loose on a ball of yarn), you’re going to be dealing with some major eddy currents induced by the field, and that means even more heating. And we can’t afford any extra heating.

5) This is related to 1): even if the Sun had a perfectly well-defined surface (it doesn’t), the moment Sundiver passed through that surface, its radiator would be less than useless. In practical terms, the vital temperature differential between the radiator and empty space would vanish, since even in the upper reaches of the photosphere, the temperature exceeds 4,000 Kelvin. There simply wouldn’t be anywhere for the heat to go. So if we handwaved away all the other problems, Sundiver would still burn up.

6) Ram pressure. Ram pressure is what you get when the fluid you’re moving through is too thin for proper fluid dynamics to come into play. The photosphere might be, as astronomers say, a red-hot vacuum, but the Sundiver is moving through it at six hundred times the speed of a rifle bullet, and ram pressure is proportional to gas density and the square of velocity. Sundiver is going to get blown to bits by the rushing gas, and even if it doesn’t, by the time it reaches altitude zero, it’s going to be experiencing the force of nine Space Shuttle solid rocket boosters across its tiny 3.142-square-meter shield. For a 1,000-kilogram spacecraft, that’s a deceleration of 1,200 gees and a pressure higher than the pressure at the bottom of the Mariana Trench. But at the bottom of the trench, at least that pressure would be coming equally from all directions. In this case, the pressure at the front of the shield would be a thousand atmospheres and the pressure at the back would be very close to zero. Atoms of spacecraft vapor and swept-up hydrogen are going to fly from front to back faster than the jet from a pressure washer, and they’re going to play hell with whatever’s left of the spacecraft.

Here’s the closest I could come to a pretty picture of what would happen to Sundiver. Why do my thought experiments never have happy endings?

Sundiver Descent Profile SM WM

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Sundiving, Part 1

Ever since I saw the bizarre, quirky, and entertaining film Sunshine, I’ve been mildly obsessed with the idea of spacecraft flying very close to the Sun. I must give a SPOILER ALERT, but the ship in Sunshine flew right into the Sun. We humans haven’t come anywhere near that close. There’s MESSENGER, the recently-deceased spacecraft that gave us our best-yet view of Mercury. And then there’s Helios 2, which came within 43 million kilometers of the Sun, which is closer to the Sun than Mercury ever gets. (Helios 2 holds another awesome record: the fastest-moving human artifact. At perihelion, it was going over 70 kilometers per second. If you fired an M-16 at one end of a football field at the moment Helios 2 passed the start line (I’m borrowing XKCD’s awesome metaphor here), the bullet would barely have traveled four and a half feet by the time Helios 2 got to the other end. Also, Helios 2 would be far beyond the finish line by the time your brain even registered that it had crossed the start line.)

There are plans to probe the Sun much closer, though. NASA is currently working on Solar Probe+, which I’m really hoping doesn’t get canned next budget cycle. Solar Probe+ will, after half a dozen Venus gravity assists, pass eight times closer to the Sun than Mercury: 5.9 million kilometers, or 8.5 solar radii. I must point out that the original design, which never got a proper name, was much, much cooler. It looked like this

(Source.)

and it was going to make one hell of a trip. It was going to fly out to Jupiter, get a reverse gravity assist to kill its angular momentum, and then plunge down within 4.5 solar radii. Here’s what a sunrise would look like if Earth orbited at 4.5 solar radii:

4-5 Solar Radii

(Rendered, of course, with Celestia.)

See that tiny object in the top-right corner? That’s the Moon, for comparison. Hold me. I’m scared.

Both the original Solar Probe mission and Solar Probe+ had to solve all sorts of brand-new engineering problems. For instance: how do you design a solar panel that can operate hotter than boiling water? How do you pack instruments onto a spacecraft when the shadow of its shield is a cone not much bigger than the shield itself? What the hell do you even build a shield out of, when it has to operate well above 1,000 Kelvin, has to cope with sunlight  3,000 times as intense as what we get on Earth, and has to be mounted on a spacecraft that will, at closest approach, be traveling 291 times faster than a rifle bullet, and will therefore be crashing through solar wind protons and dust grains moving at least as fast as that?

But that’s nothing compared to what I have in mind (to nobody’s surprise). I don’t want to design a probe that can get within 4.5 radii of the Sun. I want a probe that can get closer than one solar radius. I want a space probe that can dive straight into the sun. Not only that, but I want it to be alive and intact when it hits the Sun’s “surface.” This is why NASA will never, ever hire me.

To my surprise, the first problem I have to solve has nothing to do with heat (which will be more than enough to boil a block of iron) or radiation (which will be more than enough to sterilize a cubic meter of that sludge that festers in un-flushed gas-station toilets). The first problem is: How the hell do we get there in the first place?)

I’ll refer you to Konstantin Tsiolkovsky (whose proper Russian name isКонстанти́н Циолко́вский. Why am I telling you that? Because I really like the look of Cyrillic.) Tsiolkovsky is one of those guys who was so ahead of his time he makes you half believe in time travel. He was imagining rockets and space elevators in the freakin’ late 1800s. Before there were even cars, he was thinking about flying to other planets. And he graced us frail mortals with one of the coolest equations in engineering

Tsiol

To put it in less mathematical (and far uglier) terms: the mass ratio R of your rocket (that is, its mass when it’s full of fuel divided by its mass when the tanks are all empty) must be equal to the exponential of your desired change in velocity (delta-V) divided by your effective exhaust velocity (v_e, which is a measure of how efficient your rocket is).

Believe it or not, there’s a reason I’m taking this insane roundabout route to my point. In its orbit around the Sun, the Earth travels about 30 kilometers per second. A spacecraft that just barely manages to escape Earth’s gravity well will be traveling very close to zero speed, relative to the Earth, which means it will also be traveling at around 30 kilometers per second. In order to know how big a booster we’ll need to kill 30 kilometers per second (which will let our probe drop straight down into the Sun), we use 30 km/s as our delta-V. But what’s our exhaust velocity?

Consider the awesome Rocketdyne F-1 engine, five of which powered the Saturn V’s first stage

That’s Wernher von Braun standing by the tail end of a Saturn V first stage, with the five amazing and terrifying F-1 engines behind him. This image fills me with childish glee, because I’ve actually stood exactly where von Braun stood without even knowing it. I’ve seen that very booster. I’ve had my picture taken standing by those mighty engine bells. That’s because that first stage is on display (or at least it was last time I checked) at the U.S. Space and Rocket center in Huntsville, Alabama, which was my favorite place to go on vacation when I was a kid. Suffice to say, those engines are every bit as impressive as they look.

The F-1 engines burned liquid oxygen and ultra-high-grade kerosene (which amuses me). They managed a specific impulse (another measure of efficiency which will be very familiar to my fellow Kerbal Space Program addicts) of 263 seconds, for an effective exhaust velocity of 2.579 km/s. Plugging that into our formula, we get a horrifying number: 112,700. That’s right: if we want to kill our orbital velocity relative to the Sun using Saturn V engines, our rocket is going to have to be over a hundred thousand times heavier when full than when empty. That means that, out of the total mass of our rocket when it reaches interplanetary space, only 0.0009% can be anything other than fuel. For comparison, the Saturn V itself had a mass ratio of somewhere around 25, and as far as rockets go, that’s ridiculously large. 112,700 is just dumb, like giving an RPG character a sword the size of an armor plate off a battleship (I’m looking at youFinal Fantasy…).

The problem is that damnable exponent. As we learned in a recent post, as soon as you start putting decent-sized numbers in an exponent, you get ridiculous numbers out the other end.

Lucky for us, there are engines with much higher exhaust velocities. If you have an afternoon to spare, have a wander around Winchell Chung’s unbelievably awesome website Project Rho, which also fills me with childish glee. He’s compiled an amazing compendium of all the facts and equations a lover of science, fiction, and science fiction could ever want. Everything from the exhaust velocities of all the engines physics allows to the number of cubic meters of living space a crewmember needs to stay sane.

According to Project Rho and some of my own research, the NERVA engine (which quite literally produced thrust by passing hydrogen gas over the extremely hot core of a nuclear reactor) managed an exhaust velocity of about 8 km/s in vacuum. (Once again, Kerbal Space Program players will be no stranger to the nuclear rocket’s excellent efficiency and terrible, mosquito-fart thrust.) Putting 8 km/s into the rocket equation, we get a mass ratio of 43. Let’s say our sun-diving spacecraft weighs 1000 kilograms, the miscellaneous equipment weighs 100 kilograms, and for every kilogram of liquid hydrogen, we need a kilogram of fuel tank (that’s a pretty low-ball estimate, surprisingly. I did the math, and now I feel like my brain is frying in my skull. I’ve gotta lay off these side calculations…) Then our spacecraft will mass 46,200 kilograms. That’s surprisingly manageable. Wolfram Alpha tells me you could carry that mass in a 747’s cargo hold. Of course, you have to get that whole mass to Earth escape velocity somehow, which means at least another 92,000 kilograms. Not unmanageable, but pretty out-there.

Besides, there are better options. We could, for instance, use an ion engine. Ion engines are infamous for being absurdly efficient (the one on the Dawn spacecraft that’s currently orbiting Ceres manages 30 km/s exhaust velocity), but having thrusts that make a mosquito fart look like an atom bomb. The thrust of Dawn‘s NSTAR engine is equivalent to the weight of a coin resting on your palm. Thing is, ion engines can keep this thrust up for years at a time. And they have, which Dawn proves (it’s been firing its engine on and off for almost eight years straight). Using an ion engine, we’d need a rocket with a surprisingly sensible mass ratio of 2.72. The NSTAR engine uses xenon as propellant, so let’s say you need 10 kilograms of tank per kilogram of xenon. Even so, we’re only looking at a 1,300 kilogram spacecraft, which is only slightly larger than Dawn itself. So far, Dawn holds the record for the most delta-V expended by any spacecraft engine, at 10 km/s. It’s not too much of a stretch to imagine our sundiver canceling its 30 km/s orbital velocity.

There’s a catch. Remember that mosquito-fart thrust I was talking about? That’s going to give us an acceleration of 70 microns per second per second. My calculus is rusty, so I’ll do the naive thing and just divide 30 km/s by 70 um/s^2. That gives us 13 years. It’s gonna take 13 years for our sundiver to stop. And then it’s still got to fall all the way to the Sun. I’m not that patient.

So why not use the most awesome propulsion system ever designed by human hands. I’m not joking, either. This is, in my opinion, the coolest practical space propulsion concept I’ve ever seen: Project Orion.

If you don’t know, Project Orion was a propulsion system studied in the ’50s and ’60s in the U.S. The propulsion would be provided by nuclear bombs. Nuclear bombs dropped out the back of the ship and detonated once a second. The weirdest part of all this is that, if you ask me (and many other science nerds), Orion actually falls into the “so crazy it might actually work) category. As Scott Manley said, Project Orion is the only interplanetary propulsion system that meets three vital criteria: 1) It provides a decent amount of thrust. 2) It provides that thrust at a reasonable efficiency. and 3) It is based on technologies we already understand. That last one is very important. Maybe we’ll figure out how to build a fusion reactor someday. But, for better or for worse, we already know how to build a nuclear bomb. Not only that, but we know how to make a nuclear bomb direct its energy preferentially in one direction (since, according to Stanislav Ulam, as quoted by Scott Manley, you need to be able to do that in order to build a hydrogen bomb).

An Orion-powered spacecraft has an effective exhaust velocity of 40 km/s. That means we need a spacecraft with a mass ratio of 2.1. There’s a catch, though: the pusher plate in that diagram has to be at least 20 meters across. So, no matter how large or small our spacecraft, we’re going to have to tow that building-sized nuclear shock absorber with us. Let’s say it masses 2,000,000 kilograms (which was about the mass of a fully-loaded space shuttle). We’re looking at 4,200 metric tons of spacecraft, and we have to get all that to escape velocity first.

But this is the impatient way to kill 30 km/s. This is the way I solve problems in Kerbal Space Program, which is always a good sign that it’s not a practical solution.

Funnily enough, the practical solution is very similar to the trajectory in that comic… Instead of trying to kill 30 kilometers per second, we’re going to reach Earth escape velocity, boost ourselves into an elliptical orbit that makes us arrive slightly ahead of Jupiter in its orbit, and use Jupiter’s deep gravity well to sling us backwards along its orbit. A transfer from 1 AU to Jupiter’s distance (5.2 AU) means we’ll only be going 17 km/s when we get there, and a gravity slingshot like I’ve described allows you to change velocity by up to twice the planet’s orbital speed (and for Jupiter, orbital speed is 13 km/s, so we can have an effective delta-V of up to 26 km/s from Jupiter alone (give or take)). We don’t want that much delta-V, since we only want to cancel our 17 km/s velocity, but we can adjust how much of a kick we get simply by changing how close we come to Jupiter. The important thing is that the kick available is at least 17 km/s, which it is, with room to spare.

So we’re getting 17 km/s for free. (Not really: the energy change is always balanced perfectly between the change in velocity of the spacecraft and the (infinitesimal, but nonzero) change in velocity of the planet, as a result of their mutual gravitation.) To put it better: we’re getting 17 km/s without having to fire our engines. But we do have to fire our engines to get to Jupiter in the first place. If we do a standard Hohmann transfer,

we’ll need a delta-V of 16 km/s. If we use a NERVA engine (which I’m choosing because it’s a sensible middle-ground between the pathetic efficiency of the NSTAR and the a-little-too-much awesomeness of Orion), we can do that using a spacecraft with a mass ratio of 7. If we use Project Rho’s mass for a NERVA engine and assume 10 kilograms of tank per kilogram of hydrogen, we end up with a 17,100-kilogram interplanetary rocket. You could get that in to low Earth orbit using either a Saturn V or the much cooler-looking (but, unfortunately, more deadly) Soviet N1. By the time you get to low Earth Orbit, you’re already traveling at 7.67 kilometers per second, and to reach escape velocity only takes 3.18 km/s more. The rocket involved in launching 17,000 kilograms’ worth of interplanetary stage plus 3.18 km/s worth of Earth-escape engine is probably going to be among the largest ever constructed, but it’ll probably be no bigger than the Saturn V, the N1, or the Space Shuttle.

But as I said, I’m not a patient man. How long is it going to take to get to the Sun? The time to launch and reach escape velocity are negligible. The Hohmann transfer to Jupiter is not, requiring 2 years and 8 months. The fall inwards from Jupiter needs another 2 years and 1 month, for a total of 4 years 9 months. A lot better than the 13 years it was going to take us just to stop from Earth orbit.

And that’s where I’m going to end Part 1. Our Sundiver has launched from Earth on a skyscraper-sized rocket a little bigger than a Saturn V, entered low Earth orbit, boosted to escape velocity with its upper stage, made the transfer to Jupiter, done its swing-by, and fallen the 780 million kilometers to the Sun. As it reaches an altitude of 1 solar radius from the Sun’s surface, it’s traveling at 438 kilometers per second, which is 0.146% of the speed of light and six times faster than Helios 2. Remember how, at the beginning, I said the heat shield and the radiation weren’t the first problem? Well, now that we’re only 1 solar radius above the Sun’s surface, we can no longer ignore them. But I’ll leave that for Part 2.

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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:

Bio-linac

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…

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Approaching Infinity

One of the cool (and terrifying) things about math is that it’s almost a trivial task to construct a number which is not only larger than any number a human being will ever be able to use, but is also larger than any number that occurs in the Universe, even if you measure its mass in electron masses or its volume in Planck volumes.

The average human’s mathematical circuits are not that hard to overload. If I give you a deck of one hundred photographs and give you one hour to memorize all of them, you might very well be able to do it, but odds are you’ll miss some details. If I ask you to remember a 500-digit number, unless you’re a savant (like Daniel Tammet, who once recited pi to over 22,000 digits from memory, and who allegedly has a distinct mental image for every integer from 1 to 10,000), you’ll need some sort of fancy technique to do it. When it comes to counting objects, human beings don’t need very many numbers. I am one person. You (the reader), and I are two people having a sort of conversation. When I’m talking to a friend and somebody annoying butts in, that’s three people. If I have three apples, and I can ford a river to get to a tree with four apples, at the cost of dropping the ones I already have, I’ll do it. Numbers like three, five, and seven show up in most of the world’s myths and superstitions. Occasionally, you’ll get to seven or nine or even eleven, but rarely much farther than that. On a basic hunter-gatherer level, one hundred is a bit excessive. It’s only science, mathematics, and economy that have made a hundred of anything meaningful.

Take the number one trillion. That’s 10^12, or 1,000,000,000,000. (According to the American number scale, anyway.) It’s a big number. Draw a square. Divide it with ten vertical and ten horizontal lines. Divide each of those boxes with ten pairs of lines. Do this eight more times, and you’ve got a trillion squares. I should, of course, point out that, if you’re working on regular letter-size paper, by the time you get to a trillion boxes, the lines will be so close together that a virus will take up more than one square. Even if you drew your grid in the heart of Asia, where there’s a nice big squarish landmass 3,780 kilometers on an edge (stretching from the coast of China to the Caspian Sea along the east-west axis and from Siberia to the Himalayas on the north-south axis), the squares would be the size of a small closet.

But I already talked about a trillion at length in a previous post. A trillion green peas would just about fit on a football field (for most reasonable definitions of “football”). It’s a lot, but it’s a sensible, comprehensible number.

And a trillion is the largest number you’ll see mentioned frequently in serious astronomy, although there’s also the pleasant-sounding number “ten sextillion.” It sounds like something Lewis Carroll would’ve come up with. Ten sextillion is 10,000,000,000,000,000,000,000. That’s how many stars there are in the visible universe, according to Carl Sagan’s estimation. If you took the heart of Asia from before and divided it into ten sextillion squares, the lines would be separated by less than a hair’s breadth: about 30 microns. Cramped lodgings even for an amoeba.

But with nothing but digits and a few symbols, we can effortlessly construct numbers so massive that there’s no sensible way to describe how massive they are. Consider one trillion again. One trillion is 10^12. That’s the number 10 to the 12th power: 10 multiplied by itself 12 times. Here’s a number that will hurt your head: 10^(10^12). Simple: just take 10, and multiply it by itself one trillion times. I thought I’d be able to actually copy-and-paste that number, but it turns out that, in a 12-point font, I’d need almost 218 million pages (printed both sides). That’s a whole library’s worth of dictionary-sized books, just to hold the digits of a number I described using ten characters a second ago. If you divided the diameter of the observable universe into 10^(10^12) pieces, the distance between them would be 999,999,999,938 orders of magnitude smaller than the Planck length, which is just about the smallest length that makes sense, according to our current physics.

It’s easy as pie to create a scarier number. I’ll do it right now! (10^12)^(10^12). That is, (1,000,000,000,000)^(1,000,000,000,000). Multiply a trillion by itself a trillion times. This is where things not only get horrifying and migraine-inducing, but where they start to get strange: (10^12)^(10^12) isn’t really all that much larger than 10^(10^12). A trillion to the trillionth power is only 10^(12,000,000,000,000), or ten to the twelve trillionth power. That’s because of the way exponents work: the twelve in the first exponent gets multiplied by the trillion in the second exponent. Simple.

Don’t worry, though. With hardly any work, we can construct a function which will generate numbers as scary as you like with almost no effort.

Let’s say that the function M[1](a,b) is just a * b. Simple multiplication (which is really just adding a to itself b times, or vice versa). Let’s extend the concept by saying that M[2](a,b) is a^b, or a multiplied by itself b times. There’s no reason we can’t define M[3](a,b). It would just be a nested series of M[2] being applied over and over, exactly b times. For example, M[3](2,8) is 2^(2^(2^(2^(2^(2^(2^2)))))). You know you’ve wandered into the weird part of mathematics when you get a headache just from dealing with the damn parentheses…

There is no way to write M[3](2,8) out. As a matter of fact, there’s no way to write out its number of digits (which, after all, is only ten times smaller than the number itself). Here’s the closest I can get to writing M[3](2,8). Prepare for an absolutely horrific number-salad. I PROMISE this is the only time in this article that I’m going to do this:

[HUGE NUMBER GOES HERE]

If you’re mad at me right now, I understand. But if it makes you feel better, trying to work out that formatting has both given me a headache and made me physically nauseous. And I still screwed it up.

Nope. Couldn’t do it. It was just too damn hard to look at. Suffice to say, most of the article would have been digits, if I’d pasted that ugly bastard in.

But even the M[3](2,8) thing is unwieldy. We need better notation. Thankfully, Donald Knuth (who, at the age of nineteen, created an entire system of measurement based partly on the thickness of Mad magaizne, issue 26) provided a more elegant solution.

(I should, at this point, mention that the enormous number I copy-and-pasted above was so big that it was making the WordPress text editor lag, so I had to copy it into a Notepad file so that I can continue writing. You’ll never see it, but up above, I’ve written “[HUGE NUMBER GOES HERE]”. I have a headache.)

Knuth’s up-arrow notation is just like my M-notation, but it’s (slightly) easier on the eyes. No easier on the brain, though. In Knuth notation, a ^ b is replaced by a↑b, that is, a multiplied by itself b times. For example: 2↑8 is 256.

Things get scary very, very fast. a↑↑b is defined as a↑(a↑(…↑a)), where a is repeated a total of b times, with all the associated symbols. That’s too damn abstract for me, so let’s compute 2↑↑3. That’s 2↑2↑2, or 2^(2^(2)), which is only 16. 2↑↑4 is 2^(2^(2^(2))), or 65,536.

We can go further, though, although my headache is telling me to stop. a↑↑↑b is just a↑↑a↑↑a…↑↑a, with a repeated b times. 3↑↑↑2 is 3↑↑3 or 3^(3^(3)), which is about seven and a half trillion. 3↑↑↑3, on the other hand, is a number so large that I can’t express it in decimal notation. Hell, I can’t even express it using exponentials or up-arrows. It’s equal to 3↑↑3↑↑3, which is equal to 3↑↑(3^(3^3)), which is equal to 3↑3↑…↑3, where there are over seven trillion threes. That means a tower of exponents seven trillion threes tall. My word processor tells me that a superscript is 0.58 times the size of a regular letter, and by the time we get to the 7.6 trillionth three, it’ll be infinitely smaller than a proton.

That’s the level we’re at. Even trying to describe the typography of this ridiculous number is impossible.

What about a↑↑↑↑b? Well, 3↑↑↑↑2 is 3↑↑↑3, which we just saw was the most horrible thing in the world. 3↑↑↑↑3 is 3↑↑↑(3↑↑↑3). That is to say, 3↑↑↑3↑↑↑…↑↑↑3, with 3↑↑↑3 threes.

But I’m not letting you get off that easy. Let’s say that a↑[c]b means a↑…↑b, with c arrows in total. So a^b would be a↑[1]b. 3↑↑↑3 would be 3↑[3]3.

You know what I’m going to do. I can’t stop myself. If I knew any Medusas, I’d be a statue by now, because I wouldn’t be able to resist sneaking a peek.

There’s no turning back. It’s too late for you now. Too late for me.

Consider the number 3[3↑↑↑3]3. That’s 3↑…↑3, with seven trillion arrows. Think of the endless eternities of parentheses and arrows and evaluations, and that wouldn’t even get you close to the number of digits in this horror. Let’s call this horror X.

Now consider 3↑[X]↑3. Call it Y.

I imagine that my punishment in Number Hell will be evaluating 3↑…↑3, with Y arrows. And that’s infinitely smaller than 3↑[3↑…↑3 with Y arrows]↑3.

I’m not exaggerating for dramatic effect: I am genuinely smelling rotten eggs right now. I think I might have given myself a stroke. But before the aphasia sets in, let me introduce you to the Devil Incarnate: the Ackermann Function.

The Ackermann Function is the kind of thing they must’ve tortured Winston Smith with in 1984. It’s the reason some mathematicians walk around with that horrified thousand-yard stare. It’s an honest-to-goodness nightmare.

The Ackerman function is dead-simple. You write it A(a,b), for positive integers a and b. Here’s how you evaluate it.

If = 0, then A(a,b) = b+1

If a > 0 and b = 0, then A(a,b) = A(a-1,1)

If > 0 and b > 0, then A(a,b) = A(a-1,A(a,b-1)).

Simple rules. Not simple to apply. For instance, A(2,2) = A(1,A(2,1)) = A(1,A(1,A(2,0))) = … a horrifying mess of parentheses that ultimately gets you to 7. At least it’s a sensible number. So is A(3,2). It’s 29. A(4,2), on the other hand, is over 19 thousand digits long. When I typed Ackermann(4,4) into WolframAlpha, it actually told me “(too large to represent).” It’s always nice when a computation engine built by one of the masters of symbolic computation says “Hell with this. I give up.”

You know how evil I am. You know what I’m going to do. You know how psychotic and depraved I’ve become after looking at unfathomable numbers for an hour.

The Number of the Devil isn’t 666. It’s Ackermann(666↑↑↑↑↑↑666,666↑↑↑↑↑↑666).

Sleep well. I know I won’t.

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Addendum: How Many Novels Can There Be?

A savvy commenter pointed something out about my post “How Many Novels Can There Be?” In a nutshell, they said that word count isn’t everything. That seems obvious, at first: Ernest Hemingway wrote a very short story called “A Very Short Story” (I know, I know… damn you, Hemingway…) that’s in the realm of five or six hundred words long. It’s about love and loss and drinking, and while I haven’t read enough Hemingway to decide if his reputation is deserved, I have to admit, I’m impressed. Meanwhile, there are novels like Slither, which tops a hundred pages and says little more than “The author is very horny and wants to write about vagina parasites.” I think. It was too awful to finish, I’ll admit.

So if that’s obvious, why am I mentioning it? Why am I wasting all these words re-hashing “Quality before quantity”? (And putting myself firmly in the Slither category at the same time.) The first reason is that I like the clicky sound my keyboard makes. The second is that the fact that word count isn’t everything tells you something profound about human minds.

Consider this: a novel which consists of the sentence “It’s been a long time, and I’m still waiting for Godot.” 4,546 times is not only grammatically valid, but it technically has a storyline, a point of view, a narrator, and two characters (the narrator and whoever the hell Godot is). But even I, crazy though I am, wouldn’t want to sit down and read that sentence four and a half thousand times. I think even a Zen monk would start swearing and kicking things before the end.

The number of possible fifty-thousand-word novels is something like 1.382 x 10^114,701. My hunch is that, even if the universe doesn’t collapse or rip itself apart with whatever the hell dark energy is, there simply isn’t enough time and energy before the heat death to write that many novels. Even so, when you consider how many novels we humans could have written, our actual output is a lot less. Part of the reason is that not everybody cares to read or write novels, and for a lot of history, only the wealthy were even taught to read and write. But there’s another side to it.

Even a dim human mind (like mine!) is a remarkable machine for making judgments. Give a computer a series of pictures of a human face. We’ve done a lot of research into computer perception, so it might be able to detect the emotions on that face. But unless it was specifically programmed to do it, by a human, it won’t bother to string those emotions together and say “Hey! These are all pictures of the same guy! And he’s making a face like someone just put him in front of a computer and kicked him in the gonads!”

The same applies to novels. Narratively, this is a valid ending to a novel: “…and then Sarah walked out the door. I never saw her again. I miss her dearly. Also, everything I’ve been telling you over the last 350 pages was a lie I made up because I hate you.” From a grammatical viewpoint, it’s valid. The sentences follow each other and make sense together. They’re the conclusion of a story arc, obviously. And they absolutely ruin the story that went before them. Only the snobbiest readers will get any satisfaction out of a snarky ending like that, and frankly, I think those people need a kick in the shins to straighten them out.

So, when faced with the task of writing something (be it a novel, a sentence, an epitaph, a text message, or exciting bathroom graffiti (I really do need to call that George; he promised he could show me a good time. I hope he has cake.), a human being can pick from the thousands of possible words that might come next and decide which one matches the idea in their head. Not only that, but they can pick from the trillions upon trillions of possible sentences that follow the previous one. (There are something like 7.594×10^45 possible twenty-word sentences, using the same math as in the last post. That’s seven quadrillion quadrillion quadrillion.) And even beyond that, a uman being can look at the sentence “Also, everything I’ve been telling you over the last 350 pages was a lie I made up because I hate you.” and decide that it kind of massively ruins whatever story they were trying to write about Sarah and the narrator. And readers can detect, almost in an instant, that this sentence, which is grammatically valid, fits with the story, and completes it, is a really, incredibly stupid thing to put at the end of a novel.

Here’s the point I’ve been drunkard’s-walking my way towards: You could call a human being a machine, but we’re machines for producing meaning. The Champernowne constant is an irrational number: you can’t write it out to perfect accuracy using a finite number of symbols like you can with 42 or 9.1 or 5.3279 (I know, the repeat bar goes on the top. My formatting options are limited here.) Champernowne’s constant starts out 0.123456789101112131415… You can write out a formula that will produce Champernowne’s number perfectly (it’s not a terrifying formula, but it is damned ugly). Or, you can do the sensible thing and say “You get Champernowne’s number by sticking all the integers greater than zero together, end-to-end: 0 becomes 0.1, 0.1 becomes 0.12, 0.12 becomes 0.123. Repeat for all eternity, because apparently, you’ve been sent to Hell.”

But the creepy thing about Champernowne’s number is that it’s what mathematicians call “normal.” That means that every digit is equally likely. Which means that every pair of digits is equally likely, and so is every triplet of digits, and so on. Which means there’s a tiny, but non-zero probability that, if you pick a random stretch of Champernowne’s number, it will contain the opening lines of Hamlet, or a decimalized .wav file of that really awful Yoko Ono song that’s just her making a horrible squeaking sound, or a version of the Mona Lisa where she has three breasts and eight hundred eyes. Basically, if you look far enough downstream in Champernowne’s number, you’ll find every possible string of finite length. Perfect copies of every poem, play, novel, and dirty magazine that has ever been or will ever be published, plus all the ones that we won’t get to. Somewhere in Champernowne’s number (and possibly in pi, although nobody seems to know for sure whether or not pi is actually normal; frankly, I think he’s a bit weird, myself), you will find the most beautiful picture it is possible to create. Maybe it’s so beautiful that it causes violent epileptic seizures the moment anybody looks at it. (Maybe it’s a picture of a parrot.)

But Champernowne’s number (or possibly pi, or a random number generator or a finite number of monkeys) will only crank something meaningful out eventually. A human being is able, from the time they start speaking, to put together a meaningful string of symbols on the first try. We can pick through that ocean of possibilities (which is as close to infinite as we’re ever likely to get), and decide that we want to say “Will you marry me?” rather than “Will you celery celery?” Unless you’re aphasic or have a very specific case of Tourette’s syndrome, you probably won’t even consider the second option. It’s out there, in the realm of possibility. But you know that it’s not a valid sentence, and, unless your lover has a celery fetish, is a piss-poor proposal.

And the fact that, starting from age two or three, almost all of us have learned enough to chart a precise course through an infinity of confusion and nonsense, is pretty cool.

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The Physics of Dragonfire

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…

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How to Survive 100 Gees (Maybe…)

In a previous post, I discussed some of the gory things that would happen if you put me into a centrifuge and spun it up until I was experiencing an acceleration of 1,000,000 gees.

Now, I’m a science-fiction buff, so I’m all about imagining wild new technologies, but frankly, if I tried to handwave a way to protect myself against 1,000,000 gees, I’d be going pretty far towards the fantasy end of the science fiction-fantasy scale. I’d be far to the right of Firefly, beyond Star Trek, past Star Wars. Hell, I’d probably be closer to Lord of the Rings than to Star Wars.

But I set myself a challenge: figure out a way that a human being could survive 100 gees. That’s 980.665 m/s^2. That’s the acceleration of the Sprint missile, the scary awesomeness of which I’ve talked about before. Here’s the video of it again, because if you haven’t seen it, you should:

I’m obligated to remind you that the tracking shot in that video is played at actual speed. This bastard could accelerate from liftoff to Mach 10 in 5 seconds.

So let’s pretend I’ve invented a handwavium-burning rocket engine that can accelerate my capsule at 100 gees for, say, an hour. In my previous post (and based on experimental data, mostly from race-car crashes), we decided that 100 gees applied over more than a second would be more than enough to kill me. The problem, as I said, is the fact that I have blood, and at 100 gees, even if I was supine (on my back, the ideal position for tolerating accelerations when traveling in the back-to-front direction), that blood would collect at the bottom of my body, rupturing blood vessels and starving the upper parts of oxygen. My heart would almost certainly stop within seconds, either from pure mechanical strain, from the effects of pressure differentials, or because my ribcage caved in and turned it into carne asada.

But I’ve devised some absurd ways to get around this. The first is to put my acceleration couch into a sealed steel coffin (let’s face it, I’m gonna end up in a coffin one way or another; might as well save everybody the cleanup). The coffin will be filled with saline that approximates, as close as possible, the density of my body. Let’s say the coffin is an elliptical cylinder long enough for my body, 1 meter across the short axis, and 2 meters across the long axis. And let’s say I’m positioned so that I’m as close to the top of the tank as possible. (The tank has to be filled right to the brim. If it isn’t, the undulations of the surface will probably be more than enough to kill me.) Let’s say no part of me is deeper than 50 centimeters. At 50 centimeters’ depth, the hydrostatic pressure from my blood would be fifteen times the blood pressure that qualifies as an instant medical emergency: over 3,000 mmHg. More than enough to burst every capillary in my back, and probably the rest of the bottom half of my body.

But when I’m floating in saline that’s very close to the density of my body, the problem all but disappears. In the previous post, my capillaries only burst because they were experiencing a blood-related hydrostatic pressure (sounds like a weather forecast in Hell) of 4.952 bars (just over 5 atmospheres), with only 1.000 bars to oppose it. Things flow from areas of high pressure to low pressure. In this case, that probably means my blood flowing from the high-pressure capillaries through the slightly-lower-pressure skin and out onto the low-pressure floor of the centrifuge.

Suspended in saline, the story is different. The saline exerts 4.952 bars of hydrostatic pressure, exactly (or very nearly, I hope) opposing the pressure exerted by the blood, therefore meaning my heart doesn’t have to work itself to death trying to get blood to my frontal organs.

Speaking of organs, though, my lungs are the next problem. While I’m suspended in saline, they’ll be filled with air. Normally, I like my lungs being filled with air. It keeps me from turning blue and making people cry and then bury me in a wooden box. But air, being so light, doesn’t produce nearly the hydrostatic pressure that saline does, and so there’s nothing to keep my lungs from collapsing. Here’s a brief picture of how the lungs work: your chest is a sealed cavity (if it’s not, you’d better be in the ER or on your way there). The diaphragm moves down when you inhale. This increases the volume of the chest cavity. The lungs are the only part that can expand in volume, since they’re full of gas. That lowers the pressure, which draws air in. Ordinary human lungs weigh something in the neighborhood of 0.5 kilograms. So we know the diaphragm can cope with the weight of 1 kilo. At 100 gees, though, that rises to 100 kilos. Not even Michael Phelps’s diaphragm could make the lungs expand against that much weight.

But fret not! There’s (kind of) a solution! It’s called liquid ventilation, and it’s one of those cool sci-fi things that’s a lot realer than you might think. Instead of breathing gas, you breathe liquid. Normally, that’s bad news (remember the blue and the wooden box and the sad from before?). But certain liquids (for example, perfluorodecalin, a slightly scary-looking fluorocarbon) happen to be very good at dissolving oxygen. Good enough that people and animals have been kept alive while getting part (or, in a few cases, all) of their oxygen from liquid.

There is, however, one snag. Perfluorodecalin is denser than water. Its density is 1.9 g/cc. If the frontmost parts of my lungs are 5 centimeters from the top of the tank, then the hydrostatic pressure from the saline is 0.450 bars. 20 centimeters deeper, at the back of my lungs, the hydrostatic pressure from the saline is 2.450 bars. Meanwhile, the pressure from that heavy column of perfluorodecalin (plus the pressure from the water on top of it) is 4.180 bars. That’s almost a two-fold pressure differential. More than enough to blow out a lung. You might be able to overcome this problem by mixing one part perfluorodecalin with three or four parts high-grade inert mineral oil, but not being a chemist, I can’t guarantee that it’ll end well.

So you know what? Screw the lungs! Let’s just fill ’em with saline! (I wonder how many frustrated respiratory therapists have screamed that in their offices…) Instead, I’m going to get my oxygen the SCIENCE way! That is, by extracorporeal membrane oxygenation. Now, while it might sound like something that would involve a seance and a lot of ectoplasm, ECMO is also a real technology. It’s a last-ditch life-saving measure for people whose hearts and/or lungs aren’t strong enough to keep them alive. ECMO is the ultimate in scary life-support. Two tubes as thick as your finger are inserted into the body through a big incision. One into a major artery, and one into a major vein. The one in the vein takes blood out of the body and passes it through a machine that diffuses oxygen into the blood through a membrane and removes carbon dioxide. The blood’s temperature and pressure are regulated, and usually blood thinners like heparin are added to stop the patient’s blood sludging up from all the foreign material it’s in contact with. Then a pump returns it to the body, well-enough oxygenated to keep that body alive.

There are several hundred problems with using ECMO for ventilation under high G forces. One of them is, of course, that I have to have a tube rammed up my aorta. Another other is that I’d probably have to be anesthetized the whole flight. And yet another is the fact that those tubes and fittings are likely to be significantly more or significantly less dense than saline, and might therefore have enough residual weight (after the effects of buoyancy) to either pierce through my abdomen and into my spine, or float up and pull all my guts out.

So breathing is a problem. But let’s do another hand-wave and say we’ve invented a special polymer that can hold enough oxygen to keep me alive and can dissolve in saline without adding too much density and doesn’t destroy my lungs in the process. There are still problems.

The balance between the hydrostatic pressure exerted by the weight of my body and the pressure exerted by the weight of the saline means there’s either no pressure gradient between body and tank, or the gradient is survivable. But, although most of pressure’s effects depend on pressure differences, some of them depend on absolute pressure. One of those effects is nitrogen narcosis. Air is mostly nitrogen (78% by volume). Nitrogen, being a fairly inert gas, isn’t too important in respiration. But when it’s under high enough pressures, more and more of it starts to dissolve into the bloodstream. If this happens, and then the pressure suddenly falls, it bubbles back out of the bloodstream and you get the horrifying affliction known only as the bends. (Actually, it has lots of different names. Damn. Spoiled my own drama again…) But even if you don’t have sudden pressure drops, when the pressure gets above 2 bars, all that extra dissolved nitrogen starts to interfere with brain function. Since the maximum pressure I’ll be experiencing is about 4.910 bars, Wikipedia’s handy table tells me I’ll probably be feeling a bit drunk and clumsy. When you’re accelerating at 100 gees on top of a magic super-rocket, you really don’t want to be drunk and clumsy.

And it turns out divers who have to work underwater where everything can kill you don’t want to be drunk and clumsy either. But they can’t solve the problem by just breathing pure oxygen. In a normal atmosphere, oxygen’s partial pressure is 0.210 bars. Breathing 100% oxygen at the surface means breathing 1.000 bars. While it’s probably not good long-term, when you’re flying on a super-rocket, “not good long-term” means you can worry about all the other things that are about to kill you.

However, at high pressures, pure oxygen becomes toxic. In a slightly worrying paper from the British Medical Journal, some scientists described how they exposed volunteers to pure oxygen at a pressure of 3.6 atmospheres (about 3.6 bars). Some experienced troubling symptoms like lip-twitching, nausea, vomiting, and fainting after as little as 6 minutes. Even their toughest subject only lasted 96 minutes before suffering “prolonged dazzle” and “severe spasmodic vomiting.” If I was breathing 100% oxygen (in magic-liquid form, of course), some parts of my body would be much higher than that, so I’d be in serious trouble.

But those clever divers have figured out a way around this, too. Sort of. Instead of breathing pure oxygen at depth, they breathe blends of gas containing oxygen, nitrogen, and an inert gas like helium. This means that, for a given pressure, the partial pressure of oxygen will be lower than it would be in an oxygen-nitrogen or pure-oxygen mixture. That saves the diver from oxygen toxicity.

Of course, when you go deep enough, the nitrogen becomes an issue. Very deep divers sometimes breathe a gas mixture called heliox, which is just oxygen and helium (I recognize heliox from reading Have Space-Suit–Will Travel as a pimply, lonely adolescent). Helium has a much smaller narcotic effect than nitrogen. Since I’m going to be experiencing as much as 4.91 bars (let’s call it 5, to be safe), I need to adjust the mixture so that the partial pressure of oxygen stays around 0.210 bars. That means I’ll be breathing a mixture of 96% helium and 4% oxygen.

Because I’m a geek, I know that my lungs can inflate comfortably to 3 liters (they max out at 4 liters). That means, at 4% oxygen, I’ll be getting 120 milliliters of oxygen per breath. When I’m exercising or under severe strain (say, for example, when I’m trapped in a metal coffin at the top of a rocket accelerating at 100 gees), I need 2.2 liters of oxygen per minute. To get that much oxygen when each inhale gets me 120 milliliters requires a respiratory rate of 36 breaths per minute. That’s awfully fast for an adult, and when you consider that I’m either breathing magic low-density fluorocarbons or magic oxygenated saline, that’s a lot of work for my lungs to do, and a lot of wear and tear.

So my conclusion is that, sadly, I won’t be able to strap myself to a speeding infinite-fuel Sprint missile for an hour. But all this math makes me think that it’s probably possible to protect the human body against milder accelerations (say 10 gees) for long periods, using the same techniques. Any fighter pilots who want to climb into a saline coffin and breathe Fluorinert, let me know how it turns out.

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How Many Novels Can There Be?

I like reading. I like writing. When you’ve been writing for a while, you start to get really obsessed with word counts. Anybody you talk to about publishing something you’ve written will want to know your word count. For short fiction, you sometimes get paid by the word. And the number of words in the thing you’ve written determines whether it counts as a short story, a novella, a novel, as War and Peace, or as an encyclopedia.

Every year, I participate in National Novel-Writing Month. Unless, you know, I don’t feel like it. But I’ve participated more years than not, and I’ve produced a surprising number of novels. Every single one of them terrible, but that’s not NaNoWriMo’s fault. The goal in NaNoWriMo is to write a novel of at least 50,000 words in 30 days. And I got to thinking: how many novels that length are there?

Well, in the English language, there are somewhere between 100,000 and 1,000,000 words. But you’ll be able to understand 95% of everything written in English by knowing only the 3,000 most common ones. After all, even though it’s a valid word, people generally don’t go around calling each other antipodean anymore.

The question is: “How many 50,000-word novels are possible, using mostly the 3,000 most common words?” The naive answer is to allow each word to be any of those 3,000, which means the number of possible novels is 3,000^(50,000). That’s 1.155 x 10^173,856. You’ll be happy to know that this number is so large that, when I tried to copy and paste it into this article, it crashed my browser.

Of course, this will include novels that consist entirely of the sentence “Anus anus anus anus anus!” over and over again, which is so avant-garde it makes me want to go pee on Samuel Beckett. The list will also contain more coherent, although still somewhat dubious works, like Stuart Ashen’s peerless desk reference, Fifty-Thousand Shades of Grey. But Fifty-Thousand Shades of Grey is actually constructed of coherent sentences. (Well, one coherent sentence, at least…) Most of the novels in this ridiculously long list will be more along the lines of “Him could carpet but also because you die but but the but the but the butt.”

We’re working from a flawed assumption: that a text is just a bunch of words stuck together. But unless you’re James Joyce (or, to a lesser extent, Stephanie Meyer), that’s not how it works. A novel is a bunch of words stuck together in a particular way. Although “that that” is grammatically valid (even though it looks weird on the page), “the the” isn’t, and “centipede cheese carpet muffin” is the kind of thing I say when I haven’t been getting enough sleep.

We’ve been working from the assumption that any word is equally likely to follow any other word. That is, that all word-pairs are equally likely. They’re not. “Our way” is a lot more common than “our anus,” for instance. Naively, the probability of any two-word combination is (1 / 3,000)^2, or 1 in 9,000,000. To put it another way, there are 9,000,000 two-word pairs, 25,000 of which would make up our nonsensical novel. It’d be much closer to reality to assume that, on average, there are only 50 words that make sense after a given word (the number will be much higher (in the thousands, I’d imagine), for words like “the”, and lower for words like “hoist.”) So, in reality, there are only 150,000 two-word combinations that make sense.

We could extend this to three-word combinations, but there are two problems with that: 50,000 isn’t evenly divisible by three, and that repeating decimal will drive me crazy. More importantly, the longer your word-block, the more words become possible at the end, until you’re getting close to 3,000 possibilities again. For example: “The” could be followed by any noun in our 3,000-word list. “The man” must be followed by a verb, the start of an adjective phrase (example: “The man I met last summer“), or something like that. “The man talked” will likely be followed by a word like “to” or “about.” But there’s an enormous range of things that the man could be talking to or about, so pretty much any noun or participle is fair game, bringing the number of possibilities back up into the thousands again.

So how many novels can there be? Well, the upper bound is probably (as we’ve seen), (3,000 * 50)^25,000, which is 1.912 x 10^129,402. That’s still a number so large there’s no name for it, but it’s smaller than our first number by almost fifty thousand orders of magnitude, which is something.

But let’s take it one step further. To simplify the math, I’m going to skip right to four-word combinations. And let’s say that any two-word combination forms the start of a phrase, and that the third word in the phrase can only be one of 10 words, on average. And, to take into account the fact that the number of choices start rising again with a long enough phrase, let’s say the fourth word can be any one of 500 words. The number of possible 50,000-word novels is now (3,000 * 50 * 10 * 1,000)^(12,500), or 1.382 x 10^114,701. So we’ve chopped off another ten thousand orders of magnitude. Still, that’s a big number. And, although I don’t have the math or linguistics background to prove it, I’m guessing that’s pretty close to the number of actual, sensible novels you could construct with 50,000 words: it takes into account the rough structure of the English language. This is related to the idea of a Markov Chain, which is a mathematically-formal way of saying “where you’re likely to go next depends on where you’re at now.”

For your amusement, I’m going to back up this post, and try to copy and paste (3,000 * 50 * 10 * 1,000)^(12,500) just below. If you see a horrific salad of numbers, you’ll know it worked. If you see an apology, you’ll know it crashed my browser again. Wish me luck!

Sorry. It didn’t work. Browser crashed again. But that’s probably good news for you, the reader, since, when I pasted the number of possible sensible novels into my word processor, it produced a document 32 pages long consisting of nothing but digits in 12-point Helvetica. I think that’d make most people’s eyes bleed. Or explode. Or sprout wings and fly away.

The moral of this story is: don’t worry about machines taking over the writing of novels. If a computer could output one word of its current novel every Planck time (which is generally agreed to be close to the shortest time interval that makes sense in our physics), the time it would take would be larger than the current age of the universe. And that’s an understatement. It would actually be so much larger than the current age of the universe, that if I were to express it as a multiple (in the same way I say 10^24 is a trillion trillion times larger than 1), then I’d have to write out the word “trillion” 9,558 times just to express it. If I allow the convention that 1 googol googol is (10^100) * (10^100), or 10^200 times bigger than 1, then I’d need to write “googol” over 1,100 times. There is simply no good way to express the size of this number. It’s 10^110,000 times larger than the age of the universe in Planck times, the diameter of the observable universe in Planck lengths, and the number of particles in the universe.

Boy oh boy. I started out talking about novels, and now I’m getting into numbers that trip the circuit breakers in my brain. Math can be scary sometimes. And you wanna know the scariest thing? There are numbers, like Graham’s number and the outputs of the Ackerman function for inputs larger than (6,6), that make the number of possible novels look exactly like zero by comparison, for any practical definition.

…I need to go lie down now. Although I’m probably going to come back later and talk about really enormous numbers, because part of my brain seems to want me to have a stroke.

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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…

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