(Don’t worry. I’ve got one more bit of pixel art on the back burner, and after that, I’ll give it a break for a while.)
This is our solar system. Each pixel represents one astronomical unit, which is the average distance between Earth and Sun: 1 AU, 150 million kilometers, 93.0 million miles, 8 light-minutes and 19 light-seconds, 35,661 United States diameters, 389 times the Earth-Moon distance, or a 326-year road trip, if you drive 12 hours a day every day at roughly highway speed. Each row is 1000 pixels (1000 AU) across, and the slices are stacked so they fit in a reasonably-shaped image.
At the top-left of the image is a yellow dot representing the Sun. Mercury and Venus aren’t visible in this image. The next major body is the blue dot representing the Earth. Next comes a red dot representing Mars. Then Jupiter (peachy orange), Saturn (a salmon-pink color, which is two pixels wide because the difference between Saturn’s closest and furthest distance from the Sun is just about 1 AU), Uranus (cyan, elongated for the same reason), Neptune (deep-blue), Pluto (brick-red, extending slightly within the orbit of Neptune and extending significantly farther out), Sedna (a slightly unpleasant brownish), the Voyager 2 probe (yellow, inside the stripe for Sedna), Planet Nine (purple, if it exists; the orbits are quite approximate and overlap a fair bit with Sedna’s orbit). Then comes the Oort Cloud (light-blue), which extends ridiculously far and may be where some of our comets come from. After a large gap comes Proxima Centauri, the nearest (known) star, in orange. Alpha Centauri (the nearest star system known to host a planet) comes surprisingly far down, in yellow. All told, the image covers just over 5 light-years.
As you might imagine, I’m a big fan of bizarre science fiction. Peter Watts, Charles Stross, Edgar Allan Poe (who was such a good writer that he almost convinced me that you really could travel to the Moon by balloon). Lately, I’ve been reading Hal Clement’s Mission of Gravity, a charming and very well-thought-out book about a massive planet which spins so fast (once every eighteen minutes) that it’s flattened into the shape of a throat lozenge, with a polar gravity somewhere between 250 and 600 gees (instantly lethal to a human explorer), but an equatorial gravity of 3 gees (miserable, but survivable, especially with mechanical support). He called his planet Mesklin. Get your peyote jokes in while you can.
Thinking about Mesklin kindled in me a brief but powerful obsession with fast-rotating planets. And, in the spirit of the thought experiment (and to justify the ungodly amount of time I spent researching the subject), I thought I’d share what I found.
Planets and stars are massive things, so their gravity tries to pull them into spheres. If you think about it, this makes sense: over geological and astronomical timescales, rock flows like liquid. Therefore, it’s not unreasonable, for the sake of simplification, to treat planets like they’re made of an incompressible fluid.
To massively oversimplify things, gravity makes planets spherical because, once they’re spherical, there are no low spots left for the rock to flow into. In a sphere, the weight of the fluid is perfectly and evenly balanced by the pressure it generates in response to compression. Objects like this are said to be in hydrostatic equilibrium, and that’s one of the requirements an object must meet to be a planet according to the International Astronomical Union. (Don’t get too excited, though: this has nothing to do with why they decided Pluto wasn’t a planet.)
But all this talk of spheres and hydrostatic equilibrium ignores one important thing: every planet rotates. Some rotate very slowly. Venus, for instance, rotates so slowly that its day is longer than its year (243 Earth days versus 224 days; I’d hate to see a Venusian calendar). Others rotate really fast: Jupiter rotates once every 9.9 hours. Most of the other equilibrium objects (meaning: planet-like thingies) in the solar system fall in between these two extremes.
This means that none of the planets are actually perfectly spherical. Rotation of an object generates a centrifugal acceleration (and, incidentally, also generates a lot of arguments about the difference between centrifugal and centripetal). Although from the viewpoint of someone standing on the planet, centrifugal accelerations and forces act like regular accelerations and forces, they’re technically “fictitious”: they’re a consequence of the fact that something that’s moving likes to go in a straight line, but in a rotating body, all the bits of mass are being forced to move in a circle by the rotation of the reference frame, and therefore must behave like they have a force acting on them, even though they don’t.
For the Earth and most of the other planets, these centrifugal effects are small. Ignoring other effects, when you’re standing on Earth’s equator, you experience an upward acceleration of about 0.003 gees. That means every kilogram of matter feels 3.4 grams lighter on the Equator than it does at the poles. 3.4 grams is less than the mass of most small coins, so it’s not something that’s going to ruin your day.
It does, however, have an effect. Gravitational acceleration is directed towards the Earth’s center of mass. Centrifugal acceleration is directed away from the earth’s axis of rotation, and therefore opposes gravitational acceleration, the opposition being largest at the equator and smallest at the poles. As a result, the Earth is not quite spherical. It’s very slightly lozenge-shaped (or Skittle-shaped or Smartie-shaped; the technical term is oblate). The difference is small: measured at the equator, the Earth has a radius of 6,378.1 kilometers. Measured at the poles, it has a radius of 6,356.8 kilometers. That’s a difference of 21 kilometers, which is a lot higher than any mountain on Earth, but on the scale of a planet, it isn’t that much. Here, have a visual aid:
Here, the green ellipse represents the cross-section of a perfect sphere with the same volume as Earth. The red ellipse is the cross section of the real Earth. If you look closely, you can see that the red ellipse falls a hair’s-breadth below the green one near the poles.
Saturn is a more extreme case. Saturn has a much larger radius than Earth, and it rotates faster, and centrifugal acceleration is the square of angular velocity (rotation speed) times the radius of the circle in question. So Saturn is flattened a lot more than the Earth. So much so, in fact, that you can see it in photographs:
But I keep getting distracted by the pretty rings and the absolutely horrifying thunderstorm (seriously: that knotty thing in the southern hemisphere is one huge thunderstorm), so here’s a graph:
If you can’t see the difference this time, chances are you’re reading this in Braille, in which case I’m sorry if my alt-text is unhelpful, but that stuff confuses me.
The squashed-ness of an ellipse (or an ellipsoid like the Earth or Saturn) is described by its flattening (also called oblateness), which is the difference between the equatorial and polar radii divided by the equatorial radius. Earth’s flattening is 0.003, meaning its polar radius is 0.3% smaller than its equatorial. Saturn’s flattening is 0.098, allmost a full ten percent.
You may notice that I’ve managed to go almost a thousand words without getting into the hypothetical stuff. Well, fret no longer: what I wanted to know is whether or not there’s a simple formula to calculate flattening from parameters like mass, radius, and rotation rate. The math involved is actually quite tricky. As the planet gets more and more flattened, the matter at the equator moves far enough away from the center of mass for it to experience significantly less gravity than the matter at the poles, which magnifies the flattening effect. Flattening also moves enough of the mass away from the center that you can’t do the usual thing and treat the planet as a dimensionless point. Nonetheless, clever folks like Isaac Newton and Colin Maclaurin worked through these problems in the 18th century, and this is what they got. The flattening of a planet is (approximately)
I know that looks ugly, but trust me, it’s a lot less complicated than the math Newton and Maclaurin had to do to get there.
This formula is, at best, an approximation. It doesn’t give the Earth’s flattening to very high accuracy, because the formula assumes the Earth is equally dense throughout, which is not even close to the truth. Still, it’s a convenient approximation. It tells us, for instance, that if the Earth’s day was only 3 hours long, the Earth would be 27% smaller through the poles than through the equator (a flattening of 0.27). Compared to a spherical Earth, it would look like this:
And there are actual objects out there that are this squashed. The star Altair, for instance, is 1.79 times as massive as our Sun, and while our sun rotates once every 25 days (give or take), Altair rotates once every 8.9 hours. Altair is also close by–only 16 light-years–which means it’s one of the few stars whose surface we’ve actually imaged in any detail. Altair looks like this:
Altair’s oblateness is about 0.25. Just for giggles, here’s what it would look like if the Sun was that oblate:
(This image was made using the wonderful and extremely moddable program Celestia, which is the best planetarium software in the world, and is absolutely free.)
You might think Altair is an extraordinary case, a freak of nature. But as it turns out, a lot of large stars spin very quickly and are very oblate. Vega, which is one of the brightest stars you can see from the Northern Hemisphere, is almost as oblate as Altair. Achernar is even more oblate: it spins so fast that it has a belt of loose gas around its equator; it’s close to the maximum rate at which a star can spin without flying apart.
Now Altair is a pretty weird-looking object, but you know me well enough to know that I like extremes, and while an oblateness of 0.25 is pretty extreme, it’s not super-extreme. (Can you tell I’m a child of the ’90s?) If we spun the Earth faster, could we make it even flatter?
Sure. Up to a point. The mathematicians Carl Jacobi and Henri Poincaré both worked on the problem of fast-rotating self-gravitating fluids. They discovered that the pancake-shaped planet is the most stable configuration as long as its oblateness is less than 0.81, which, for Earth, means a rotation period of about 2 hours. Here’s what that would look like.
This is one weird planet. If the Earth were rotating this fast, Australians would experience a gravity of much less than 1 gee. Hurricanes would have much smaller diameters because of the increased power of the Coriolis effect. If we pretend that, somehow, human evolution proceeded normally in spite of the pancake Earth, cultures that developed in northern Asia, North America, and parts of South America might spend a very long time absolutely convinced that the Earth was flat, because to them, it would very nearly be true. Cultures in India, China, the Middle East, central Africa, and central North and South America, on the other hand, probably wouldn’t know what to think: If you looked north or south, the horizon would be much farther away than if you were looking east and west.
Of course, if the Earth was spinning that fast, it would look nothing like it does today. For one thing, it would probably be a lot colder: for one thing, there would only be a small equatorial belt where the sun could ever pass directly overhead, and everywhere else would get its sunlight at an angle, which is what causes our winters. Human beings as we know them couldn’t live near the poles, where the gravity would approach 8 gees. Fighter pilots can handle that, but that’s only for short periods, and only after a lot of physical conditioning and centrifuge training, and with the help of pressure-equalizing suits. If we stick with the prevailing theory that modern humans began their outward migration in Africa, then they would probably only be able to spread as far north as Spain, Italy, Greece, southern Ukraine, Uzbekistan, India, China, and Japan. Their southward migration would be limited to south Africa, Australia, and possibly New Zealand. Human beings would have to grow a hell of a lot stouter and tougher to survive the trek through the high-gravity regions to cross the Bering Strait into the Americas.
But if you think this is weird, you ain’t seen nothin’ yet. Jacobi, Poincaré, and others that came after them discovered something else: a pancake-shaped body with an oblateness of 0.81 is a maximum. As you keep adding angular momentum to it, it doesn’t just keep flattening out, and its rotation actually slows down. That’s because, above oblateness 0.81, the pancake shape is no longer stable. Above oblateness 0.81, the stable configuration is the so-called Jacobi ellipsoid, which looks like a badly-made rugby ball or a really disturbing suppository:
This would be an even weirder planet to live on than the pancake Earth from before. The tips of the ellipsoid would be the best place to live, since the gravity there would be weakest and they would have the best chance of getting direct sunlight. On the other hand, the bizarre geometry might make for some crazy civilization-ending weather patterns around the tips. Still, that’d be better than the wild centrifugal-Coriolis storms whirling east and west from the planet’s narrow waist. If, once again, human migration starts in Africa, it probably wouldn’t proceed much farther east than Europe and the Middle East, which would not only have the lowest gravity, but would have a better chance of having tolerable weather and getting regular rainfall.
You might be thinking “What does all that have to do with anything?” I get that question a lot. Well, it turns out that there are objects in our own solar system which spin fast enough to distort into this shape. One of the largest is the dwarf planet Haumea, which is larger than Pluto along all but its shortest axis, and is larger than the Moon along its longest. Haumea’s been stretched into this bizarre shape by its rapid rotation: once every 3.9 hours.
Now, you have to re-define oblateness when you’re working with scalene ellipsoids like these. Because each of their three axes are a different length, there are actually three separate oblatenesses. The one we’re concerned with, though, is the oblateness of the cross-section taken parallel to the longest axis. We’re interested in this because it turns out that, just like the symmetrical lozenge-shaped Maclaurin spheroid became unstable beyond oblateness 0.81, the suppository-shaped Jacobi ellipsoid becomes unstable beyond a long-axis oblateness of 0.93. At this point, the Earth would be shaped a little like a torpedo or a cartoon cigar.
But what would happen as you pumped more angular momentum into it? Until now, we’ve been able to get a pretty good approximation by assuming that the Earth is a zero-viscosity fluid of uniform density. But the results of exceeding the Jacobi limit depend strongly on Earth’s material properties. Simulations of such fast-spinning fluids have been performed, and depending on how they compress under pressure, sometimes they deform into asymmetric cones before splitting in two, and sometimes they lose mass from their tips, which carries away angular momentum and slows their rotation until they return to the Jacobi equilibrium.
There is reason to believe that the Earth would take the fission route rather than the mass-shedding route. The Earth, after all is a solid object made of fairly strong stuff. Stars that spin too fast (like Achernar) tend to lose mass, since gas doesn’t hold together too well. Planets, on the other hand, tend to break into smaller planets.
There’s more evidence to believe that the Earth would fission: similar things have happened to other objects in our solar system. Here’s a picture of a weird-looking object:
You could be forgiven for thinking that this was an X-ray picture of a bone from an extinct squirrel. In fact, it’s an asteroid: 216 Kleopatra, which is 217 kilometers long and about 91 in diameter. It’s what’s known as a contact binary: a pair of objects orbiting so close together that they touch. The theory is that, a long time ago, Kleopatra was hit by a glancing blow from another asteroid, which broke it apart and gave it so much angular momentum that it couldn’t even pull itself into a Haumea suppository shape. Contact binaries are right on the border between “One object spinning fast” and “Two objects orbiting very close together.” It might look something like this:
This would be an even weirder place to live than the others. Picture this: you step outside your house one summer evening. The sun has already set, but the other lobe of the world reflects enough light to give the yard a pleasant glow. You can’t see the waist of the world from here, but you know it’s there. At this very moment, planeloads of businesspeople are flying vertically up through the waist to the continents on the other side, and cargo ships are sailing up the massive waterfall that connects the two halves of the planet. As the shadow of your half of the world creeps across the opposite half, you see city lights coming on on the other side. Weird, right?
What’s weirder is that, as in the case of the absurdly oblate objects we looked at before, contact binaries are not all that rare. Many asteroids are contact binaries. The comet 67P/Churyumov-Gerasimenko (which, as of this writing, is being explored by the Rosetta spacecraft) might be a contact binary. Weirder still, there are contact binary stars. It seems strange that an object like a star could survive direct contact with another star, but contact binary stars actually form a whole class, called W Ursae Majoris variables, after W Ursae Majoris, a double star which looks like a peanut.
Contact binary stars don’t behave like any other kind of star. That’s partly because they’re essentially one star with two cores, which means the whole surface tends to stay at the same temperature when it normally wouldn’t; it’s also partly because of the huge angular momentum you get when two stars orbit that close together.
The universe can be a really weird place, but I have to admit, for all its messiness, it can be pretty aesthetically-pleasing, too. I mean, look at all those pretty ellipses up there. They’re smooth and curved like eggs (or parts of human anatomy, if you stretch your perverse imagination). And they’re proof that we humans can be clever when we work at it. Isaac Newton figured out his flattening formula using math and physics he helped invent, and this was in a time when hardly anybody had indoor plumbing. Now, we’ve got Rosetta getting ready to drop a lander on a comet, and next year (2015), we’re going to get our first up-close looks the giant dwarf planets Pluto and Ceres. We’ve got plenty of flaws, don’t get me wrong, but we can be pretty cool sometimes.
(And speaking of cool, someone made an amazing animation of the transition from spherical Earth to Maclaurin spheroid to Jacobi Ellipsoid to Poincare pear:
When I was little, there was always that one kid on the playground who thought he was clever. We’d be drawing horrifying killer monsters (we were a weird bunch). I would say “My monster is a thousand feet high!” Then Chad would say “My monster is a mile high!” Then I would say “Nuh-uh, my monster is a thousand miles high!” Then Taylor would break in, filling us with dread, because we knew what he was going to say: “My monster is infinity miles high!” There would then follow the inevitable numeric arms race. “My monster is infinity plus one miles high!” “My monster is infinity plus infinity miles high!” “My monster is infinity times infinity miles high!” Our shortsighted teachers hadn’t taught us about Georg Cantor, or else we would have known that, once you hit infinity, pretty much all the math you do just gives you infinity right back.
But that’s not what I’m getting at here. As we got older and started (unfortunately) to care about money, the concept of “infinite money” inevitably started coming up. As I got older still and descended fully into madness, I realized that having an infinite amount of printed money was a really bad idea, since an infinite amount of mass would cause the entire universe to collapse into a singularity, which would limit the number of places I could spend all that money. Eventually, my thoughts of infinite wealth matured, and I realized that what you really want is a machine that can generate however much money you want in an instant. With nanomachines, you could conceivably assemble dollar bills (or coins) with relative ease. As long as you didn’t create so much money that you got caught or crashed the economy, you could live really well for the rest of your life.
But that’s not what I got hung up on. I got hung up on the part where I collapsed the universe into a Planck-scale singularity. And that got me thinking about one of my favorite subjects: weird objects in space. I’ve been mildly obsessed with creating larger and larger piles of objects ever since. Yes, I do know that I’m weird. Thanks for pointing that out.
Anyway, I thought it might be nice to combine these two things, and try to figure out the largest pile of money I could reasonably accumulate. My initial thought was to make the pile from American Gold Eagle coins, but I like to think of myself as a man of the world, and besides, those Gold Eagles are annoyingly alloyed with shit like copper and silver, and I like it when things are pure. So, instead, I’m going to invent my own currency: the Hobo Sullivan Dragon’s Egg Gold Piece. It’s a sphere of 24-karat gold with a diameter of 50 millimeters, a mass of 1,260 grams, and a value (as of June 29, 2014) of $53,280. The Dragon’s Egg bears no markings or portraits, because when your smallest unit of currency is worth $53,280, you can do whatever the fuck you want. And you know what? I’m going to act like a dragon and pile my gold up in a gigantic hoard. But I don’t want any arm-removing Anglo Saxon kings or tricksy hobbitses or anything coming and taking any of it, so instead of putting it in a cave under a famous mountain, I’m going to send it into space.
Now, a single Dragon’s Egg is already valuable enough for a family to live comfortably on for a year, or for a single person to live really comfortably. But I’m apparently some kind of ridiculous royalty now, so I want to live better than comfortably. As Dr. Evil once said, I want one billion dollars. That means assembling 18,769 Dragon’s Eggs in my outer-space hoard. Actually, now that I think about it, I’m less royalty and more some kind of psychotic space-dragon, which I think you’ll agree is infinitely cooler. 18,769 Dragon’s Eggs would weigh in at 23,649 kilograms. It would form a sphere with a diameter of about 1.46 meters, which is about the size of a person. Keen-eyed (or obsessive) readers will notice that this sphere’s density is significantly less than that of gold. That’s because, so far, the spheres are still spheres, and the closest possible packing, courtesy of Carl Friedrich Gauss, is only 74% sphere and 26% empty space.
You know what? Since I’m being a psychotic space-dragon anyway, I think I want a whole golden planet. Something I can walk around while I cackle. A nice place to take stolen damsels and awe them with shining gold landscapes.
Well, a billion dollars’ worth of Dragon’s Eggs isn’t going to cut it. The sphere’s surface gravity is a pathetic 2.96 microns per second squared. I’m fascinated by gravity, and so I often find myself working out the surface gravity of objects like asteroids of different compositions. Asteroids have low masses and densities and therefore have very weak gravity. The asteroid 433 Eros, one of only a few asteroids to be visited and mapped in detail by a space probe (NEAR-Shoemaker), has a surface gravity of about 6 millimeters per second squared (This varies wildly because Eros is far from symmetrical. Like so many asteroids, it’s stubbornly and inconveniently peanut-shaped. There are places where the surface is very close to the center of mass, and other places where it sticks way up away from it.) The usual analogies don’t really help you get a grasp of how feeble Erotian gravity is. The blue whale, the heaviest organism (living or extinct, as far as we know) masses around 100 metric tons. On Earth, it weighs 981,000 Newtons. You can also say that it weighs 100 metric tons, because there’s a direct and simple equivalence between mass and weight on Earth’s surface. Just be careful: physics dorks like me might try to make a fool out of you. Anyway, on Eros, a blue whale would weigh 600 Newtons, which, on Earth, would be equivalent to a mass of 61 kilograms, which is about the mass of a slender adult human.
But that’s really not all that intuitive. It’s been quite a while since I tried to lift 61 kilograms of anything. When I’m trying to get a feel for low gravities, I prefer to use the 10-second fall distance. That (not surprisingly) is the distance a dropped object would fall in 10 seconds under the object’s surface gravity. You can calculate this easily: (0.5) * (surface gravity) * (10 seconds)^2. I want you to participate in this thought experiment with me. Take a moment and either stare at a clock or count “One one thousand two one thousand three one thousand…” until you’ve counted off ten seconds. Do it. I’ll see you in the next paragraph.
In those ten seconds, a dropped object on Eros would fall 30 centimeters, or about a foot. For comparison, on Earth, that dropped object would have fallen 490 meters. If you neglect air resistance (let’s say you’re dropping an especially streamlined dart), it would have hit the ground after 10 seconds if you were standing at the top of the Eiffel Tower. You’d have to drop it from a very tall skyscraper (at least as tall as the Shanghai World Financial Center) for it to still be in the air after ten seconds.
But my shiny golden sphere pales in comparison even to Eros. Its 10-second fall distance is 148 microns. That’s the diameter of a human hair (not that I’d allow feeble humans on my golden dragon-planet). That’s ridiculous. Clearly, we need more gold.
Well, like Dr. Evil, we can increase our demand: 1 trillion US dollars. That comes out to 18,768,769 of my golden spheres. That’s 23,649,000 kilograms of gold. My hoard would have a diameter of 14.6 meters and a surface gravity of 29.6 microns per second squared (a 10-second fall distance of 1.345 millimeters, which would just barely be visible, if you were paying close attention.) I am not impressed. And you know what happens when a dragon is not impressed? He goes out and steals shit. So I’m going to go out and steal the entire world’s economy and convert it into gold. I’m pretty sure that will cause Superman and/or Captain Planet to declare me their nemesis, but what psychotic villain is complete without a nemesis?
It’s pretty much impossible to be certain how much money is in the world economy, but estimates seem to be on the order of US$50 trillion (in 2014 dollars). That works out to 938,438,439 gold balls (you don’t know how hard I had to fight to resist calling my currency the Hobo Sullivan Golden Testicle). That’s a total mass of 1.182e9 kilograms (1.182 billion kilograms) and a diameter of 54 meters (the balls still aren’t being crushed out of shape, so the packing efficiency is still stuck at 74%). 54 meters is pretty big in human terms. A 54-meter gold ball would make a pretty impressive decoration outside some sultan’s palace. If it hit the Earth as an asteroid, it would deposit more energy than the Chelyabinsk meteor, which, even though it exploded at an altitude of 30 kilometers, still managed to break windows and make scary sounds like this:
This golden asteroid would have a surface gravity of 0.108 millimeters per second squared, and a 10-second fall distance of 0.54 centimeters. Visible to the eye if you like sitting very still to watch small objects fall in weak gravitational fields (and they say I have weird hobbies), but still fairly close to the kind of micro-gravity you get on space stations. I can walk across the room, get my coffee cup, walk back, and sit down in 10 seconds (I timed it), and my falling object would still be almost exactly where I left it.
Obviously, we need to go bigger. Most small asteorids do not even approach hydrostatic equilibrium: they don’t have enough mass for their gravity to crush their constituent materials into spheres. For the majority of asteroids, the strength of their materials is greater than gravitational forces. But the largest asteroids do start to approach hydrostatic equilibrium. Here’s a picture of 4 Vesta, one of the other asteroids that’s been visited by a spacecraft (the awesome ion-engine-powered Dawn, in this case.)
You’re probably saying “Hobo, that’s not very fucking spherical.” Well first of all, that’s a pretty damn rude way to discuss asteroids. Second of all, you’re right. That’s partly because of its gravity (still weak), partly because its fast rotation (once every 5 hours) deforms it into an oblate spheroid, and partly because of the massive Rheasilvia crater on one of its poles (which also hosts the solar system’s tallest known mountain, rising 22 kilometers above the surrounding terrain). But it’s pretty damn spherical when you compare it to ordinary asteroids, like 951 Gaspra, which is the shape of a chicken’s beak. It’s also large enough that its interior is probably more similar to a planet’s interior than an asteroid’s. Small asteroids are pretty much homogenous rock. Large asteroids contain enough rock, and therefore enough radioactive minerals and enough leftover heat from accretion, to heat their interiors to the melting point, at least briefly. Their gravity is also strong enough to cause the denser elements like iron and nickel to sink to the center and form something approximating a core, with the aluminosilicate minerals (the stuff Earth rocks are mostly made of) forming a mantle. Therefore, we’ll say that once my golden asteroid reaches the same mass as 4 Vesta, the gold in the center will finally be crushed sufficiently to squeeze out the empty space.
It would be convenient for my calculations if the whole asteroid melted so that there were no empty spaces anywhere. Would that happen, though? That’s actually not so hard to calculate. What we need is the golden asteroid’s gravitational binding energy, which is the amount of energy you’d need to peel the asteroid apart layer by layer and carry the layers away to infinity. This is the same amount of energy you’d deposit in the asteroid by assembling it one piece at a time by dropping golden balls on it. A solid gold (I’m cheating there) asteroid with Vesta’s mass (2.59e20 kg) would have a radius of 147 kilometers and a gravitational binding energy of about 1.827e25 Joules, or about the energy of 37 dinosaur-killing Chicxulub impacts. That’s enough energy to heat the gold up to 546 Kelvin, which is less than halfway to gold’s melting point.
But, you know what? Since I don’t have access to a supercomputer to model the compressional deformation of a hundred million trillion kilograms of close-packed gold spheres, I’m going to streamline things by melting the whole asteroid with a giant draconic space-laser. I’ll dispense with the gold spheres, too, and just pour molten gold directly on the surface.
You know where this is going: I want a whole planet made of gold. But if I’m going to build a planet, it’s going to need a name. Let’s call it Dragon’s Hoard. Sounds like a name Robert Forward would give a planet in a sci-fi novel, so I’m pleased. Let’s pump Dragon’s Hoard up to the mass of the Earth.
Dragon’s Hoard is a weird planet. It has the same mass as Earth, but its radius is only 66% of Earth’s. Its surface gravity is 22.64 meters per second squared, or 2.3 earth gees. Let’s turn off the spigot of high-temperature liquid gold (of course I have one of those) for a while and see what we get.
According to me, we get something like this:
The heat content of a uniform-temperature sphere of liquid gold depends on its volume, but since it’s floating in space, its rate of heat loss depends on surface area (by the Stefan-Boltzmann law). The heat can move around inside it, but ultimately, it can only leave by radiating off the surface. Therefore, not only will the sphere take a long time to cool, but its upper layers will cool much faster than its lower layers. Gold has a high coefficient of thermal expansion: it expands more than iron when you heat it up. Therefore, as the liquid gold at the surface cools, it will contract, lose density, and sink beneath the hotter gold on the surface. It will sink and heat up to its original temperature, and will eventually be displaced by the descent of cooler gold and will rise back to the surface. When the surface cools enough, it will solidify into a solid-gold crust, which is awesome. Apparently, my fantasies are written by Terry Pratchett, which is the best thing ever. I’ve got Counterweight Continents all over the place!
Gold is ductile: it’s a soft metal, easy to bend out of shape. Therefore, the crust would deform pretty easily, and there wouldn’t be too many earthquakes. There might, however, be volcanoes, where upwellings of liquid gold strike the middle of a plate and erupt as long chains of liquid-gold fountains. It would behave a bit like the lava lake at Kilauea volcano, in Hawai’i. See below:
What a landscape this would be! Imagine standing (in a spacesuit) on a rumpled plain of warm gold. To your right, a range of gold mountains glitter in the sun, broken here and there by gurgling volcanoes of shiny red-hot liquid. Flat, frozen puddles of gold fill the low spots, concave from the contraction they experienced as they cooled. To your right , the land undulates along until it reaches another mountain range. In a valley at the foot of this range is an incandescent river of molten gold, fed by the huge shield volcano just beyond the mountains. Then a psychotic space-dragon swoops down, flying through the vacuum (and also in the face of physics), picks you up in his talons, carries you over the landscape, and drops you into one of those volcanoes.
Yeah. It would be something like that.
As fun as my golden planet is, I think we could go bigger. Unfortunately, the bigger it gets, the more unpredictable its properties become. As we keep pouring molten gold on it, its convection currents will become more and more vigorous: it will have more trapped heat, a larger volume-to-surface area ratio, and stronger gravity, which will increase the buoyant force on the hot, low-density spots. Eventually, we’ll end up with convection cells, much like you see in a pot of boiling water. They might look like this:
Those are Rayleigh-Bénard cells, which you often get in convective fluids. I used that same picture in my Endless Sky article. But there, I was talking about supercritical oxygen and nitrogen. Here, it’s all gold, baby.
Eventually, the convection’s going to get intense enough and the heat’s going to get high enough that the planet will have a thin atmosphere of gold vapor. If it rotates, the planet will also develop a powerful magnetic field: swirling conductive liquid is believed to be the thing that creates the magnetic fields of Earth, the Sun, and Jupiter (and all the other planets). This planet is going to have some weird electrical properties. Gold is one of the best conductors there is, second only to copper, silver, graphene, and superconductors. Therefore, expect some terrifying lightning on Dragon’s Hoard: charged particles and ultraviolet radiation from the Sun will ionize the surface, the gold atmosphere (if there is one) or both, and Dragon’s Hoard will become the spherical terminal in a gigantic Van de Graaff generator. As it becomes more and more charged, Dragon’s Hoard will start deflecting solar-wind electrons more easily than solar-wind protons (since the protons are more massive), and will soak up protons, acquiring a net positive charge. It’ll keep accumulating charge until the potential difference explosively equalizes. Imagine a massive jet or bolt of lightning blasting up into space, carrying off a cloud of gold vapor, glowing with pink hydrogen plasma. Yikes.
After Dragon’s Hoard surpasses Jupiter’s mass, weird things will begin happening. Gold atoms do not like to fuse. Even the largest stars can’t fuse them. Therefore, the only things keeping Dragon’s Hoard from collapsing altogether are the electrostatic repulsion between its atoms and the thermal pressure from all that heat. Sooner or later, neither of these things will be enough, and we’ll be in big trouble: the core, compressed to a higher density than the outer portions by all that gold, will become degenerate: its electrons will break loose of their nuclei, and the matter will contract until the electrons are squeezed so close together that quantum physics prevents them from getting any closer. This is electron degeneracy pressure, and it’s the reason white dwarf stars can squeeze the mass of a star into a sphere the size of a planet without either imploding or exploding.
The equations involved here are complicated, and were designed for bodies made of hydrogen, carbon, and oxygen instead of gold (Those short-sighted physicists never consider weird thought experiments when they’re unraveling the secrets of the universe. The selfish bastards.) The result is that I’m not entirely sure how large Dragon’s Hoard will be when this happens. It’s a good bet, though, that it’ll be somewhere around Jupiter’s mass. This collapse won’t be explosive: at first, only a fraction of the matter in the core will be degenerate.As we add mass, the degenerate core will grow larger and larger, and more and more of it will become degenerate. It will, however, start to get violent after a while. Electron-degenerate matter is an excellent conductor of heat, and its temperature will equalize pretty quickly. That means that we’ll have a hot ball of degenerate gold (Degenerate Gold. Add that to the list of possible band names.) surrounded by a thin layer of hot liquid gold. Because of the efficient heat transfer within the degenerate core, it will partly be able to overcome the surface-area-versus-volume problem and radiate heat at a tremendous rate. The liquid gold on top, though, will have trouble carrying that heat away fast enough, and will get hotter and hotter, and thanks to its small volume, will eventually get hot enough to boil. Imagine a planet a little larger than Earth, its surface white-hot, crushed under a gravity of 500,000 gees, bubbles exploding and flinging evaporating droplets of gold a few kilometers as gaseous gold and gold plasma jet up from beneath. Yeah. Something like that.
But in a chemical sense, my huge pile of gold is still gold. The nuclei may be uncomfortably close together and stripped of all of their electrons, but the nuclei are still gold nuclei. For now. Because you know I’m going to keep pumping gold into this ball to see what happens (That’s also a line from a really weird porno movie.)
White dwarfs have a peculiar property: the more massive they are, the smaller they get. That’s because, the heavier they get, the more they have to contract before electron degeneracy pressure balances gravity. Sirius B, one of the nearest white dwarfs to Earth, has a mass of about 1 solar mass, but a radius similar to that of Earth. When Dragon’s Hoard reached 1.38 solar masses, it would be even smaller, having a radius of around 3000 kilometers. The stream of liquid gold would fall towards a blinding white sphere, striking the surface at 3% of the speed of light. The surface gravity would be in the neighborhood of 2 million gees. If the gravity were constant (which it most certainly would not be), the 10-second fall distance would be 2.7 times the distance between Earth and moon. Now we’re getting into some serious shit.
Notice that I specified three significant figures when I gave that mass: 1.38 solar masses. That is not by accident. As some of you may know, that’s dangerously close to 1.39 solar masses, the Chandrashekar limit, named after the brilliant and surprisingly handsome Indian physicist Subrahmanyan Chandrasekhar. (Side note: Chandrashekhar unfortunately died in 1995, but his wife lived until 2013. Last year. She was 102 years old. There’s something cool about that, but I don’t know what it is.) The Chandrasekhar limit is the maximum mass a star can have and still be supported by electron degeneracy pressure. When you go above that, you’ve got big trouble.
When ordinary white dwarfs (made mostly of carbon, oxygen, hydrogen, and helium) surpass the Chandrasekhar limit by vacuuming mass from a binary companion, they are unable to resist gravitational contraction. They contract until the carbon and oxygen nuclei in their cores get hot enough and close enough to fuse and make iron. This results in a Type Ia supernova, which shines as bright as 10 billion suns. It’s only recently that our supercomputers have been able to simulate this phenomenon. The simulations are surprisingly beautiful.
I could watch that over and over again and never get tired of it.
Unfortunately, even though it’s made of gold (which, as I said, doesn’t like to fuse), the same sort of thing will happen to Dragon’s Hoard. When it passes the Chandrasekhar limit, it will rapidly contract until the nuclei are touching. This will trigger a bizarre form of runaway fusion. The pressure will force electrons to combine with protons, releasing neutrinos and radiation. Dragon’s Hoard will be heated to ludicrous temperatures, and a supernova will blow off its outer layers. What remains will be a neutron star, which, as I talked about in The Weather in Hell, is mostly neutrons, with a thin crust of iron atoms and an even thinner atmosphere of iron, hydrogen, helium, or maybe carbon. Most or all of the gold nuclei will be destroyed. The only thing that will stop the sphere from turning into a black hole is that, like electrons, neutrons resist being squeezed too close together, at least up to a limit.
But you know what? That tells us exactly how much gold you can hoard in one place: about 1.38 solar masses. So fuck you, Taylor from kindergarten! You can’t have infinity dollars! You can only have 0.116 trillion trillion trillion dollars (US, and according to June 2014 gold prices) before your gold implodes and transmutes itself into other elements! So there!
But while I’m randomly adding mass to massive astronomical objects (that’s what space dragons do instead of breathing fire), let’s see how much farther we can go.
The answer is: Nobody’s exactly certain. The Chandrasekhar limit is based on pretty well-understood physics, but the physics of neutron-degenerate matter at neutron star pressures and temperatures (and in highly curved space-time) is not nearly so well understood. The Tolman-Oppenheimer-Volkoff limit (Yes, the same Oppenheimer you’re thinking of.) is essentially a neutron-degenerate version of the Chandrasekhar limit, but we only have the TOV limit narrowed down to somewhere between 1.5 solar masses and 3 solar masses. We’re even less certain about what happens above that limit. Quarks might start leaking out of neutrons, the way neutrons leak out of nuclei in a neutron star, and we might get an even smaller, denser kind of star (a quark star). At this point, the matter would stop being matter as we know it. It wouldn’t even be made of neutrons anymore. But to be honest, we simply don’t know yet.
Sooner or later, though, Karl Schwarzschild is going to come and kick our asses. He solved the Einstein field equations of general relativity (which are frightening but elegant, like a hyena in a cocktail dress) and discovered that, if an object is made smaller than its a certain radius (the Schwarzschild radius), it will become a black hole. The Schwarzschild radius depends only on the object’s mass, charge, and angular momentum. Dragon’s Hoard, or rather what’s left of it, doesn’t have a significant charge or angular momentum (because I said so), so its Schwarzschild radius depends only on mass. At 3 solar masses, the Schwarzschild radius is 8.859 kilometers, which is only just barely larger than a neutron star. Whether quark stars can actually form or not, you can bet your ass they’re going to be denser than neutron stars. Therefore, I’d expect Dragon’s Hoard to fall within its own Schwarzschild radius somewhere between 3 and 5 solar masses. Let’s say 5, just to be safe. There are suspected black holes with masses near 5.
That’s the end of Dragon’s Hoard. The physics in the center gets unspeakably weird, but the gold-spitting space dragon doesn’t get to see it. He’s outside the event horizon, which means the collapse of his hoard is hidden to him. He just sees a black sphere with a circumference of 92.77 kilometers, warping the images of the stars behind it. It doesn’t matter how much more gold we pour into it now: it’s all going to end up inside the event horizon, and the only noticeable effect will be that the event horizon’s circumference will grow larger and larger. But fuck that. If I wanted to throw money down a black hole, I’d just go to Vegas. (Heyo!) Dragon’s Hoard isn’t getting any more of my draconic space-gold.
But one last thing before I go. Notice how I suddenly went from saying Schwarzschild radius to talking about the event horizon’s circumference. That’s significant. Here’s a terrible picture illustrating why I did that:
Massive objects create curvature in space-time. Imagine standing at the dot on circle B, in the top picture. If you walk to the center-point along line A, you’ll measure a length a. If you then walk around circle B, you’ll get the circle’s circumference. You’ll find that that circumference is 2 * pi * a. The radius is therefore (circumference) / (2 * pi) But that only holds in flat space. When space is positively curved (like it is in the vicinity of massive objects), the radius of a circle will always be larger than (circumference) / (2 * pi). That is to say, radius C in the bottom picture is significantly longer than radius A in the top one, and longer than you would expect from the circumference of circle D.
In other words, the radius of a massive object like a star, a neutron star, or a black hole, differs from what you would expect based on its circumference. The existence of black holes and neutron stars has not actually been directly confirmed (because they’re so small and so far away). It is merely strongly suspected based on our understanding of physics. The existence of spacetime curvature, though, has been confirmed in many experiments.
Imagine you’re standing in a field that looks flat. There’s a weird sort of bluish haze in the center, but apart from that, it looks normal. You walk in a circle around the haze to get a better look at it. It only takes you fifteen minutes to walk all the way around and get back to your starting point. The haze makes you nervous, so you don’t walk straight into it. Instead, you walk on a line crossing the circle so that it passes halfway between the haze and the circle’s edge at its closest approach. Somehow, walking that distance takes you twenty minutes, which is not what you’d expect. When you walk past the haze at a quarter-radius, it takes you an hour. When you walk within one-eighth of a radius, it takes you so long you have to turn back and go get some water. Each time, you’re getting closer and closer to walking along the circle’s radius towards its center, but if you actually tried to walk directly into its center, where the haze is (the haze is because there’s so much air between you and the stuff beyond the haze, which is the same reason distant mountains look blue), you would find that the distance is infinite.
That’s how black holes are. They’re so strongly-curved that there’s way more space inside than there should be. The radius is effectively infinite, which is why it’s better to talk about circumference. As long as the black hole is spherically symmetric, circumferences are still well-behaved.
But the radius isn’t actually infinite. When you consider distance scales close to the Planck length, Einstein’s equations butt heads with quantum mechanics, and physicists don’t really know what the fuck’s going on. We still don’t know what happens near a black hole’s central singularity.
Incidentally, the Planck length compared to the diameter of an atom is about the same as the diameter of an atom compared to the diameter of a galaxy. The Universe is a weird place, isn’t it?
I’ve been a nerd pretty much my whole life, so as a kid, I wondered nerdy things. I wondered, for instance, if it was possible to have an endless sky: clouds below you and clouds above. As a slightly older kid, I realized that skies don’t work that way. A sky is just an atmosphere, just a layer of gas wrapped around a solid (or liquid) planet.
Now, though, with a fair bit of obscure physics knowledge under my belt, I can finally decide not only whether or not an all-sky planet is possible, but if it is, what that planet would be like. I knew there had to be some good things about being a grownup.
Because I want my planet to be all sky, I’m going to build it from dry air and water vapor (the water vapor is so we can have clouds). Can you build a planet out of nothing but air? Well-informed intuition suggests you can: after all, Jupiter and the Sun are mostly hydrogen, and hydrogen is less dense and therefore harder to squeeze together than air. But as it turns out, we can get more precise. We can ask how large a cloud of air would have to be to collapse into a planet. This number is called the “Jeans length” or “Jeans radius” (and thanks to XKCD for making me aware of this formula). For air at Earth surface density (1.2 kilograms per cubic meter) and room temperature (68 Fahrenheit, 20 Celsius, or 293.15 Kelvin), the Jeans length is 35,400 kilometers, which is just over half the equatorial radius of Jupiter. You might think that would mean that the silly planet known as Endless Sky would be quite massive. In fact, it’s only got three times the mass of the moon.
Unfortunately, this manageably-small gas planet wouldn’t collapse very much under its own gravity. Its gravity would be weak, and as it collapsed, compression would heat it up and probably evaporate most or all of it. It’s important to note that the Jeans equation was intended to be used on nebulae, not inexplicable pockets of air floating in space for no reason. I guess Sir James Jeans just wasn’t thinking ahead. (Incidentally, Sir James Jeans would be a really pretentious name for a line of denim pants).
But no matter the details, what would Endless Sky actually be like? This is where the fun begins.
As anybody who’s ever looked into hydrodynamics knows, fluids are an enormous pain in the ass to deal with. They’re always swirling around and compressing and carrying pressure waves and expanding and contracting. You can simulate fluid behavior using the Navier-Stokes equations, which are frightening:
(From the Wikipedia article.)
We’ve got partial derivatives and dot products and divergence operators all over the fucking place. And to actually turn these equations into a computer program, you need a whole set of conservation equations, as well as initial conditions and boundary values. These equations are complicated because things like air and water are swirly and their mass moves around all the time. So you’d think the equation for the pressure in an atmosphere would be horrifying. Actually, it’s not. It’s
(pressure at the surface) * exp(-1 * (altitude) / (scale height))
The scale height is the altitude at which the pressure is e times smaller than it is at the surface (where e is about 2.71). This means that pressure corresponds quite closely to altitude. The scale height on Earth is (roughly) 8500 meters, so at an altitude of 8.5 kilometers, the pressure is 1/e atmospheres (0.37 atmospheres or 37 kilopascals). Because this dependency is pretty stable, you can also say that, if you’re measuring a pressure of 37 kilopascals, you’re at an altitude of 8.5 kilometers.
We can use this nifty formula to figure out what our Endless Sky will be like. Before I get to the amusing pictures, here’s a caveat: I’m assuming constant Earth surface gravity throughout the atmosphere, which is inaccurate. That’s why this is a thought experiment, and why I majored in English instead of physics.
But here’s roughly, what the atmosphere of Endless Sky would look like:
You can learn two things from this picture right away: First, how thin tropospheres are, which tend to be where human beings and similar organisms hang out. Second, don’t buy cheap graph-paper notebooks, because the paper can apparently detect when you’re trying to tear it out gently and violently self-destructs.
Because I decided to make the parameters of my endless sky pretty much the same as Earth’s atmosphere, it all looks pretty familiar from the 1 atmosphere level to the 0.00001 atmosphere level (where we pass the Kármán line and go into space). Because Endless Sky has an oxygen atmosphere and (I’ve just decided) a sun, it’ll have an ozone layer much like Earth’s. I always wondered why thunderstorms grow to a certain height and then stop and spread out. It turns out to be because, below a certain level (called the tropopause), the higher you go, the colder the air gets. But above this level, and in fact throughout the stratosphere, the air actually gets warmer as you go higher. This is partly because of the ozone layer. Thunderstorms happen when hot, moist air rises into the lower-pressure air above, expands, and its water condenses. But this can only happen if it’s warmer than the surrounding air, and that pretty much becomes impossible at the tropopause, since the air above is already warmer.
Basically, if you were flying around Endless Sky in a hot-air balloon, and you sat so that the gondola obscured the horizon, it would be easy to think you were flying around Earth: bright blue skies, fluffy white clouds (no happy trees, though, unfortunately).
But if you looked down, you would see something horrifying. Namely, you would see no ground. Depending on atmospheric conditions below you, you might get different kinds of exotic clouds, but these would be smaller than the ones you see on Earth, because the higher pressure wouldn’t allow them to expand as much.
Looking into the sky below you would probably be quite a lot like looking into a deep, clear ocean: it would grow a deeper and darker blue. The blue is due to Rayleigh scattering, which (combined with the spectrum of light the Sun puts out) is why the sky above is blue. The darkness is because less of the sunlight would make it through all that air.
You might think that the pressure would just keep going up and up, and the air would just get denser and denser as you got deeper. On most planets, solid, liquid, and gaseous alike, temperatures tend to go up as you go down. In the case of rocky planets, this is partly because of radioactive decay. But no matter what kind of planet, there’s internal heat left over from its formation. Therefore, it gets pretty hot down there. And when the temperature and pressure rise above the so-called critical point of the gases involved, something magical happens: the gases don’t quite liquefy, but they don’t remain gaseous, either. They sort of forget what they are, and become supercritical fluids. Supercritical fluids are amazing. They can be as dense as water (or denser), but they compress and expand like a gas and fill their containers. Here’s a video from awesome YouTuber Ben Krasnow showing you what supercritical carbon dioxide looks like:
Nitrogen’s critical point is lower than oxygen’s, so at a depth of about 128 kilometers (below the 1-atmosphere level), you would encounter a broiling opalescent sea of semi-liquid nitrogen containing a lot of dissolved oxygen. Looking down, you would see city-sized Bénard cells, which look approximately like this:
(Image courtesy of NOAA, the US government atmospheric science people, who are pretty neat.)
They probably wouldn’t be quite that orderly, though. But there would be nothing but broiling opalescent clouds rising and falling as far as the eye could see, twisted into peculiar shapes or into alien jet streams by Coriolis forces.
Being denser, the supercritical ocean would attenuate light much faster than the gaseous part of the atmosphere. You would see deep, dark blue down to the opalescent layer, and then nothing. But, farther down still (about 200 kilometers deep), the sky would no longer be endless: the pressure would pass 88,000 atmospheres, which is the pressure at which oxygen solidifies into pretty blue crystals. This means there would be another layer of weather on Endless Sky: down there, the crust of oxygen and nitrogen “ice” would evaporate into the supercritical fluid above, rise, and cool, driving powerful convection currents and stirring the deep, and where it fell and cooled, the supercritical fluid would condense into oxygen and nitrogen snow. Perhaps it would form enormous dune fields like the ones we see on Earth and, amazingly, on Saturn’s moon Titan:
(Earth dunes on top. Titan dunes (probably made of water ice) on the bottom. Image courtesy of NASA, via Wikipedia.)
I hope my eight-year-old self is vindicated. He’s probably dancing around like a lunatic, the hyperactive little freak…