physics, Space, thought experiment

Hypothetical Nightmares | Black Holes, Part 3

Imagine taking all the mass in the Milky Way (estimated to be around a trillion solar masses) and collapsing it into a black hole. The result wouldn’t be an ordinary black hole. Not even to astrophysicists, for whom all sorts of weird shit is ordinary.

The largest black hole candidate is the black hole at the center of the quasar S5 0014+813, estimated at 40 billion solar masses. In other words, almost a hundred times smaller than our hypothetical hole. As I said last time, as far as astronomical objects go, black holes are a fairly comfortable size. Even the largest don’t get much bigger than a really large star. Here, though, is how big our trillion-sun black hole would be, if we replaced the sun with it:

Galaxy Mass Black Hole.png

(Rendered in Universe Sandbox 2.)

The thing circled in orange is the black hole. When I started tinkering with the simulation, I was kinda hoping there’d be one or two dwarf planets outside the event horizon, so their orbits could at least offer a sense of scale. No such luck: the hole has a Schwarzschild radius of 0.312 light-years, which reaches well into the Oort cloud. That is, the galaxy-mass black hole’s event horizon alone would extend beyond the heliopause, and would therefore reach right into interstellar space. Proxima Centauri, around 4.2 light-years from Earth, is circled in white.

The immediate neighborhood around a black hole like this would be rough. We’re talking “feral children eating the corpse of a murder victim while two garbagemen fight to the death with hatchets over who gets to empty the cans on this street” kind of rough. That kinda neighborhood. No object closer than half a light-year would actually be able to orbit the hole: it would either have to fall into the hole or fly off to infinity.

That is, of course, if the hole isn’t spinning. As I said last time, you can orbit closer to a spinning hole. But I’m going to make a leap here and say that our galaxy-mass black hole isn’t likely to be spinning very fast. Some rough calculations suggest that, if it were rotating at half the maximum speed,the rotational kinetic energy alone would have several billion times the mass of the sun. I’m going to assume there’s not enough angular momentum in the galaxy to spin a hole up that much. I could be wrong. Let me know in the comments.

Spin or no spin, it’s gonna be a rough ride anywhere near the hole. Atoms orbiting at the innermost stable orbit (the photon sphere) are moving very close to the speed of light, and therefore, to them, the ambient starlight and cosmic microwave background ahead of them is blue-shifted and aberrated into a horrifying violet death-laser, while the universe behind is red-shifted into an icy-cold nothingness.

But, as we saw last time, once you get outside a large hole’s accretion disk, things settle down a lot. When it comes to gravity and tides, ultra-massive black holes like these are gentle giants. You could hover just outside the event horizon by accelerating upwards at 1.5 gees, which a healthy human could probably tolerate indefinitely, and which is very much achievable with ordinary rocket engines. The tides are no problem, even right up against the horizon. They’re measured in quadrillionths of a meter per second per meter.

Of course, if you’re hovering that close to a trillion-solar-mass black hole, you’re still going to die horribly. Let’s say your fuel depot is orbiting a light-year from the hole’s center, and they’re dropping you rocket fuel in the form of frozen blocks of hydrogen and oxygen. By the time they reach you, those blocks are traveling at a large fraction of the speed of light, and will therefore turn into horrifying thermonuclear bombs if you try to catch them.

But, assuming its accretion disk isn’t too big and angry, a hole this size could support a pretty pleasant galaxy. The supermassive black hole suspected to lie at the center of the Milky Way makes up at about 4.3 parts per million of the Milky Way’s mass. If the ratio were the same for our ultra-massive hole, then it could host around 200 quadrillion solar masses’ worth of stars, or, in more fun units, 80,000 Milky Ways. Actually, it might not be a galaxy at all: it might be a very tightly-packed supercluster of galaxies, all orbiting a gigantic black hole. A pretty little microcosm of the universe at large. Kinda. All enclosed within something like one or two million light-years. A weird region of space where intergalactic travel might be feasible with fairly ordinary antimatter rockets.

You’ll notice that I’ve skipped an important question: Are there any trillion-solar-mass black holes in the universe? Well, none that we know of. But unlike some of the other experiments to come in this article, black holes this size aren’t outside the realm of possibility.

I frequently reference a morbid little cosmology paper titled A Dying Universe. If you’re as warped as I am, you’ll probably enjoy it. It’s a good read, extrapolating, based on current physics, what the universe will be like up to 10^100 years in the future (which they call cosmological decade 100). If you couldn’t guess by the title, the news isn’t good. A hundred trillion years from now (Cosmological Decade 14), so much of the star-forming stuff in galaxies will either be trapped as stellar corpses or will have evaporated into intergalactic space that new stars will stop forming. The galaxies will go dark, and the only stars that shine will be those formed by collisions between high-mass brown dwarfs. By CD 30 (a million trillion trillion years from now), gravitational encounters between stars in the galaxy will have given all the stars either enough of a forward kick to escape altogether, or enough of a backward kick that they fall into a tight orbit around the central black hole. Eventually, gravitational radiation will draw them inexorably into the black hole. By CD 30, the local supercluster of galaxies will consist of a few hundred thousand black holes of around ten billion solar masses, along with a bunch of escaping rogue stars. By this time, the only source of light will be very occasional supernovae resulting from the collisions of things like neutron stars and white dwarfs. Eventually, the local supercluster will probably do what the galaxy did: the lower-mass black holes will get kicked out by the slingshot effect, and the higher-mass ones will coalesce into a super-hole that might grow as large as a few trillion solar masses. Shame that everything in the universe is pretty much dead, so no cool super-galaxies can form. But the long and the short of it is that such a hole isn’t outside the realm of possibility, although you and I will never see one.

The Opposite Extreme

But what about really tiny black holes? In the first post in this series, I talked about falling into a black hole with the mass of the Moon. But what about even smaller holes?

Hobo Sullivan is a Little Black Pinhole

Yeah, I feel like that sometimes. I mass about 131 kilograms (unfortunately; I’m working on that). If, by some bizarre accident (I’m guessing the intervention of one of those smart-ass genies who twist your wishes around and ruin your shit), I was turned into a black hole, I’d be a pinprick in space far, far smaller than a proton. And then, within a tenth of a nanosecond, I would evaporate by Hawking radiation (if it exists; we’re still not 100% sure). When a black hole is this small, Hawking radiation is nasty shit. It would have a temperature of a hundred million trillion degrees, and I’d go off like four Tsar Bombas, releasing over 200 megatons of high-energy radiation. Not enough to destroy the Earth, but enough to ruin the year for the inhabitants of a medium-sized country.

There’s no point in trying to work out things like surface tides or surface gravity: I’d be gone so fast that, in the time between my becoming a black hole and my evaporation, a beam of light would have traveled a foot or two. Everything around me is as good as stationary for my brief lifetime.

A Burial Fit for a Pharaoh. Well, for a weird pharaoh.

Things change dramatically once black holes get a little bigger. A hole with the mass of the Great Pyramid of Giza (around 6 billion kilograms) would take half a million years to evaporate. It would still be screaming-hot: we’re talking trillions of Kelvin, which is hot enough that nearby matter will vaporize, turn to plasma, the protons and neutrons will evaporate out of nuclei, and then the protons and neutrons themselves will melt into a quark-gluon soup. But, assuming the black hole is held in place exactly where the pyramid once stood, we won’t see that. We’ll only see a ball of plasma and incandescent air the size of a university campus or a big football stadium, throbbing and booming and setting fire to everything for a hundred kilometers in every direction. The Hawking radiation wouldn’t inject quite enough energy to boil the planet, but it would probably be enough (combined with things like the fact that it’s setting most of Egypt on fire) to spoil the climate in the long run.

This isn’t an issue if the black hole is where black holes belong: the vacuum of space. Out there, the hole won’t gobble up Earth matter and keep growing until it destroys us. Instead, it’ll keep radiating brighter and brighter until it dies in a fantastic explosion, much like the me-mass black hole did.

Can’t you just buy a space heater like a normal person?

It’s starting to get cold here in North Carolina. Much as I love the cold, I’ve been forced to turn my heater on. But, you know, electric heating is kinda inefficient, and this house isn’t all that well insulated. I wonder if I could heat the house using Hawking radiation instead…

Technically, yes. Technically in the sense of “Yeah, technically the equations say yes.” Technically in the same way that you could technically eat 98,000 bacon double cheeseburgers at birth and then go on a 75-year fast, because technically, that averages out to 2,000 Calories per day. What I mean is that while the numbers say you can, isolated equations never take into account all the other factors that make this a really terrible idea.

A black hole with the mass of a very large asteroiod (like Ceres, Vesta, or Pallas) would produce Hawking radiation at a temperature of 500 Kelvin, which is probably too hot to cook with, but cool enough not to glow red-hot. That seems like a sensible heat source. Except for the fact that, as soon as you let it go, it’s going to fall through the floor, gobble up everything within a building-sized channel, and convert that everything into superheated plasma by frictional effects as it falls into the hole. And except for the fact that if you’re in the same neighborhood as the hole, you’ll simultaneously be pulled into it at great speed by its gravity, and pulled apart into a bloody mass of fettuccine by tidal forces. And except for the fact that, as the black hole orbits inside the Earth, it’s going to open up a kilometer-wide tunnel around it and superheat the rock, which will cause all sorts of cataclysmic seismic activity, and ultimately, the Earth will either collapse into the hole, or be blasted apart by the luminosity of the forming accretion disk, or some combination thereof.

Back to the Original Extreme

But there’s one more frontier we haven’t explored. (I was watching Star Trek yesterday.) That is: the biggest black hole we can reasonably (well, semi-reasonably) imagine existing. That’s a black hole with a mass of around 1 x 10^52 kilograms: a black hole with the mass of the observable universe. Minus the mass of the Earth and the Sun, which make less of a dent in that number than stealing a penny makes a dent in Warren Buffett’s bank account.

The hole has a Schwarzschild radius of about 1.6 billion light-years, which is a good fraction of the radius of the observable universe. Not that the observable universe matters much anymore: all the stuff that was out there is stuck in a black hole now.

For the Earth and Sun, though, things don’t change very much (assuming you set them at a modest distance from the hole). After all, even light needs over 10 billion years to circumnavigate a hole this size. Sure, the Earth and Sun will be orbiting the hole, rather than the former orbiting the latter, but since we’re dealing with gravitational accelerations less than 3 nanometers per second per second, and tides you probably couldn’t physically measure (4e-34 m/s/m at the horizon, and less further out, which falls into the realm of the Planck scale), life on Earth would probably proceed more or less as normal. The hole can’t inflict any accretion-disk horror on the Sun and Earth: there’s nothing left to accrete. Here on Earth, we’d just be floating for all eternity, living our lives, but with a very black night sky. If we ever bothered to invent radio astronomy, we’d probably realize there was a gigantic something in the sky, since plasma from the Sun would escape and fall into a stream orbiting around the hole, but we’d never see it. What a weird world that would be…

Then again, if the world’s not weird by the end of one of my articles, then I’m really not doing my job…

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Addendum, Cars, physics, Space, thought experiment

Addendum: A City On Wheels

While I was proofreading my City on Wheels post, I realized that I’d missed a golden opportunity to estimate just how heavy a whole city would be. When I was writing that post, I wanted to use the Empire State Building’s weight as an upper limit, because I was pretty sure that would be enough space for a whole self-sufficient community. Trouble is, the weight of buildings isn’t usually known. The Empire State Building’s weight is cited here and there, but never with a very convincing source. I couldn’t figure out a way to estimate its weight that didn’t feel like nonsense guesswork. That’s why I used the Titanic’s displacement as my baseline.

The reason estimating the mass of a building was so tricky is that, generally, buildings are far form standardized. Yeah, a lot of houses are built in similar or identical styles, but even if you know their exact dimensions, converting that into a reasonably accurate weight turns into pure guesswork, because you don’t know what kind of wood was used in the frame, how much moisture the wood contained, how many total nails were used, et cetera. But, just now, I realized something. There is a standardized object that represents the shape, size, and weight of a dwelling pretty well: the humble shipping container.

31-shipping-container-house-01-850x566

You may notice that that’s not a shipping container. It’s a bunch of shipping containers put together to make a rather stylish (if slightly industrial-looking) house. Building homes out of shipping containers is a big movement in the United States right now. They’re cheaper than a lot of alternatives, and they’re tough: shipping containers are built to be stacked high, even while carrying full loads. For example:

cscl_globe_arriving_at_felixstowe_united_kingdom

The things are sturdy enough that they far exceed most building codes, when properly anchored. Their low price, their strength, and the fact that they’re easily combined and modified, has made them popular as alternative houses.

Because different shipping containers from different manufacturers and different countries often end up stacked together, they all have to be built to the same standard. Their dimensions, therefore, are standardized, which is good news for us. I re-imagined the rolling city as a stack of shipping containers approximately the size of the Titanic, with their long axes perpendicular to the ship’s long axis. You could fit two across the Titanic‘s deck this way, and 110 along the deck, and if you stacked them 20 high, you’d approximate the Titanic’s shape and volume. To account for the fact that the people living in these containers are going to have furniture, pets, physical bodies, and other inconvenient stuff, I’ll assume that each container would have twelve pieces of the heaviest furniture I could think of: the refrigerator.

Amazon is a great thing for this kind of estimation, because from it, I learned that an ordinary Frigidaire is about 300 pounds. Multiply that by twelve, add the mass of the container itself (3.8 metric tons each), round up (to keep estimates pessimistic), and you get 6 metric tons per container. Considering that a standard 40-foot intermodal container (which is the standard I worked with) can handle a gross weight (container + cargo) of over 28 metric tons, we’re nowhere near the load limit for the containers. There are 4,400 containers in all, for a total mass of 26,400 metric tons. Increase the mass by 25% to account for the weight of the nuclear reactor, chassis, and suspension, and we get 33,000 metric tons. That’s still a hell of a lot, but it’s only just over half of the 50,000 tonnes we were working with before.

As you might remember, I wrote off the Titanic-based city on wheels as probably feasible, but requiring a heroic effort and investment. But using the shipping container mass, which is 1.5-fold smaller, I think it moves into the “impressive but almost sensible mega-project” category, along with the Golden Gate Bridge, the Burj Khalifa, the Great Pyramid of Giza, and Infinite Jest.

Another note: There’s one heavy, mobile object whose weight I didn’t mention in the City on Wheels post: the Saturn V rocket. I did mention the Crawler-Transporter that moved the Saturn V from the Vehicle Assembly Building to the launchpad, however. And the weight of the fully-loaded Saturn V gives us an idea of how massive an object a self-propelled machine can move: 3,000 tonnes. Because, to nobody’s surprise, NASA knows the weight of every Apollo rocket at liftoff. Because it’s mildly (massively) important to know the mass of the rocket you’re launching, because that can make the difference between “rocket in a low orbit” and “really dangerous and expensive airplane flying really high until it explodes with three astronauts inside.”

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physics, Space, thought experiment

The Moon Cable

It was my cousin’s birthday. In his honor, we were having lunch at a slightly seedy Mexican restaurant. Half of the people were having a weird discussion about religion. The other half were busy getting drunk on fluorescent mango margaritas. As usual, me and one of my other cousins (let’s call him Neil) were talking absolute nonsense to entertain ourselves.

“So I’ve got a question,” Neil said, knowing my penchant for ridiculous thought experiments, “Would it be physically possible to tie the Earth and Moon together with a cable?” I was distracted by the fact that the ventilation duct was starting to drip in my camarones con arroz, so I didn’t give the matter as much thought as I should have, and I babbled some stuff I read about space elevators until Neil changed the subject. But, because I am an obsessive lunatic, the question has stuck with me.

The first question is how much cable we’re going to need. Since the Earth and Moon are separated, on average, by 384,399 kilometers, the answer is likely to be “a lot.”

It turns out that this isn’t very hard to calculate. Since cable (or wire rope, as the more formal people call it) is such a common and important commodity,  companies like Wirerope Works, Inc. provide their customers (and idiots like me) with pretty detailed specifications for their products. Let’s use two-inch-diameter cable, since we’re dealing with a pretty heavy load here. Every foot of this two-inch cable weighs 6.85 pounds (3.107 kilograms; I’ve noticed that traditional industries like cabling and car-making are stubborn about going metric). That does not bode well for the feasibility of our cable, but let’s give it a shot anyway.

Much to my surprise, we wouldn’t have to dig up all of North America to get the iron for our mega-cable. It would have a mass of 3,919,000,000 kilograms. I mean, 3.918 billion is hardly nothing. I mean, I wouldn’t want to eat 3.919 billion grains of rice. But when you consider that we’re tying two celestial bodies together with a cable, it seems weird that that cable would weigh less than the Great Pyramid of Giza. But it would.

So we could make the cable. And we could probably devise a horrifying bucket-brigade rocket system to haul it into space. But once we got it tied to the Moon, would it hold?

No. No it would not. Not even close.

The first of our (many) problems is that 384,399 kilometers is the Moon’s semimajor axis. Its orbit, however, is elliptical. It gets as close as 362,600 kilometers (its perigee, which is when supermoons happen) and as far away as 405,400 kilometers. If we were silly enough to anchor the cable when the Moon was at perigee (and since we’re tying planets together, there’s pretty much no limit to the silliness), then it would have to stretch by 10%. For many elastic fibers, there’s a specific yield strength: if you try to stretch it further than its limit, it’ll keep stretching without springing back, like a piece of taffy. Steel is a little better-behaved, and doesn’t have a true yield strength. However, as a reference point, engineers say that the tension that causes a piece of steel to increase in length by 0.2% is its yield strength. To put it more clearly: the cable’s gonna snap.

Of course, we could easily get around this problem by just making the cable 405,400 kilometers long instead of 384,399. But we’re very quickly going to run into another problem. The Moon orbits the Earth once every 27.3 days. The Earth, however, revolves on its axis in just under 24 hours. Long before the cable stretches to its maximum length, it’s going to start winding around the Earth’s equator like a yo-yo string until one of two things happens: 1) So much cable is wound around the Earth that, when the moon his apogee, it snaps the cable; or 2) The pull of all that wrapped-up cable slows the Earth’s rotation so that it’s synchronous with the Moon’s orbit.

In the second scenario, the Moon has to brake the Earth’s rotation within less than 24 hours, because after just over 24 hours, the cable will have wound around the Earth’s circumference once, which just so happens to correspond to the difference in distance between the Moon’s apogee and perigee. Any more than one full revolution, and the cable’s gonna snap no matter what. But hell, physics can be weird. Maybe a steel cable can stop a spinning planet.

Turns out there’s a handy formula. Torque is equal to angular acceleration times moment of inertia. (Moment of inertia tells you how hard an object is to set spinning around a particular axis.) To slow the earth’s spin period from one day to 27.3 days over the course of 24 hours requires a torque of 7.906e28 Newton-meters. For perspective: to apply that much torque with ordinary passenger-car engines would require more engines than there are stars in the Milky Way. Not looking good for our cable, but let’s at least finish the math. Since that torque’s being applied to a lever-arm (the Earth’s radius) with a length of 6,371 kilometers, the force on the cable will be 1.241e22 Newtons. That much force, applied over the piddling cross-sectional area of a two-inch cable, results in a stress of 153 quadrillion megapascals. That’s 42 trillion times the yield strength of Kevlar, which is among the strongest tensile materials we have. And don’t even think about telling me “what about nanotubes?” A high-strength aramid like Kevlar is 42 trillion times too weak. I don’t think even high-grade nanotubes are thirteen orders of magnitude stronger than Kevlar.

So, to very belatedly answer Neil’s question: no. You cannot connect the Earth and Moon with a cable. And now I have to go and return all this wire rope and get him a new birthday present.

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biology, Dragons, thought experiment

Even More Dragonfire

Because I like dragons and I can’t help myself. Don’t worry. This one won’t be nearly as long as my usual posts about dragons. If anything, it’s more to show the thought process that goes into my thought experiments.

Let’s dispense with the notion of dragonfire hotter than the surface of the sun, and with biologically-produced antimatter. Let’s pretend that dragons are made of fairly ordinary flesh. They breathe fire from their mouths (naturally), so they’re going to have to be careful not to burn their tongues off. Let’s assume they have funny saliva glands that mist their mucous membranes to stop them getting scalded off by direct contact with hot air and fire. There’s still thermal radiation to deal with.

According to NOAA (who usually talk about weather, but have, in this case, started talking about fire), exposure to thermal radiation at an intensity of 10 kilowatts per square meter will cause severe pain after 5 seconds and second-degree burns (nasty blisters) after 14 seconds. With that in mind, I want to find out how hot dragonfire can be before its thermal radiation is too much for a dragon’s mouth to handle.

Well, let’s assume a dragon’s mouth is a cylinder 1 meter long and 30 centimeters in diameter. Multiply the circumference of that cylinder by its length to get its surface area (minus the ends), and then multiply the area by 10 kilowatts per square meter to get the maximum radiant power that can reach the mucous membranes. The result: 9.425 kilowatts. Now, let’s model the jet of fire as a cylinder (again, without ends) 1 centimeter in diameter and 1 meter long. That cylinder can’t emit more than 9.425 kilowatts as radiant heat. Divide 9.425 kilowatts by the cylinder’s surface area. To stay below 9.425 kilowatts, the jet of flame can’t emit at an intensity higher than 300 kilowatts per square meter. Apply the Stefan-Boltzmann law in reverse to get an estimate of what temperature gas radiates at 300 kilowatts per square meter. That comes out to a disappointing 1,517 Kelvin, which is cooler than the average wood fire.

I’m not satisfied with that, so I’m going to cheat. Sort of. I’m going to assume that the dragon has a bone in its fire-spewing orifice that acts like a supersonic rocket nozzle, which allows it to emit a very narrow, fast-moving stream of burning gas. The upshot of this is that the jet becomes narrower than that of a pressure washer: 1 mm in diameter throughout its transit through the mouth. That’s a bit more encouraging: 2,697 Kelvin, about the temperature of a hydrogen-air flame (which means we can just use hydrogen as the fuel). It’s still nowhere as hot as I want it to be, but I don’t think Sir Knight is going to be walking away from this one.

We could, of course, push the temperature up by taking into account the fact that the dragon’s mouth isn’t a perfect blackbody, and reflects some of the radiation, but like I said, this isn’t a full post. I just wanted to show you guys how I flesh out an idea.

Stay safe out there. And don’t try to breathe fire.

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biology, Dragons, physics, thought experiment

Dragon Metabolism

As you might have noticed, I have a minor obsession with dragons. I blame Sean Connery. And, because I can never leave anything alone, I got to wondering about the practical details of a dragon’s life. I’ve already talked about breathing fire. I’m not so sure about flight, but hell, airplanes fly, so it might be possible.

But I’ll worry about dragon flight later. Right now, I’m worried about metabolism. Just how many Calories would a dragon need to stay alive? And is there any reasonable way it could get that many?

Well, there’s more than one type of dragon. There are dragons small enough to perch on your shoulder (way cooler than a parrot), and there are dragons the size of horses, and there are dragons the size of cathedrals (Smaug again), and there are, apparently, dragons in Tolkein’s universe that stand taller than the tallest mountains. Here’s a really well-done size reference, from the blog of writer N.R. Eccles-Smith:

dragon-size-full-chart

The only downside is that there’s no numerical scale. There is, however, a human. And, if you know my thought experiments, you know that, no matter what age, sex, or race, human beings are always exactly 2 meters tall. Therefore, the dragons I’ll be considering range in size from 0.001 meters (a hypothetical milli-dragon), 1 meter (Spyro, number 3, purple in the image) to 40 meters (Smaug, number 11), and then beyond that to 1,000 meters, and then beyond to the absolutely ludicrous.

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Uncategorized

A Toyota on Mars (Cars, Part 1)

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

2007_toyota_yaris_9100

(Image from RageGarage.net)

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

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

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

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

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

torque_convertor_ford_cutaway1

(Image from dieselperformance.com)

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

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

cutaway20axle20differential20diff-1

(Image from topgear.uk.net)

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

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

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

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

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

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

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

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

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

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

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

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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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