astronomy, physics, silly

The Neutronium Necklace

Neutronium Jewel

If you want exotic jewelry, you’ve come to the right place! The Neutronium Necklace has a classic thick-link sterling-silver chain with a striking pendant containing a 26,000,000,000,000-carat brilliant-cut crystal of virgin neutronium, imported directly from J0108.

Care instructions: The pendant’s gravity may attract small objects such as crumbs, grains of sand, and loose paperclips. As the jewel is harder than all known materials, it may be cleaned with a damp cloth, sandblaster, waterjet cutter, high-power laser, or with high explosives. Its setting, however, is sterling silver, and so it should be cleaned separately with a suitable silver polish.

Safety instructions: For your safety, we do not recommend you touch the jewel with bare hands, as tidal forces may cause discomfort, dislocation, or dismemberment. We also strongly recommend against wearing the neutronium necklace in Earth gravity, as its weight will exceed 500,000 tonnes, which may result in neck or back injury or decapitation. For your own safety, and the safety of others, please avoid dropping the necklace, as the jewel will rapidly penetrate the Earth’s crust and be lost. In this situation, the necklace’s warranty will not cover the cost of replacement.

Please note that neutronium is not stable at pressures below 100 megaelectronvolts per cubic femtometer. Exposing the jewel to ambient pressures below this level will void the necklace’s warranty, and may result in a Solar-system threatening explosion exceeding 10 trillion megatons.

Note: As a precaution against theft, black-market resale, and usage by supervillains, demons, or malevolent alien lifeforms, your neutronium jewel is inscribed with an inconspicuous barcode on its rear side. If you wish to have the jewel re-set, please only consult a licensed jeweler who has been certified Not an Evil Psychopath.

Fair Trade Certification: The rough neutronium crystal in your Neutronium Necklace was purchased at fair market value from the neutron-worms of J0108. Mining conditions are certified humane by the RL Forward observatory committee. Please direct all concerns to the RL Forward committee, as the neutron-worms are only capable of communicating via high-energy neutrino beams, which may present a health hazard to untrained civilians.

physics, Space, thought experiment

Houston, We Have Several Problems: Black Holes, Part 1

There aren’t many movies that I loved as a kid and can still stand to watch as a grownup. I remember loving Beauty and the Beast, but I don’t know if I could watch it now, because I’ve developed an irrational hatred of musical numbers. I certainly couldn’t watch that amazingly cheesy monster truck movie where the guy who drove Snakebite drugged the guy who drove Bigfoot. Apollo 13, though, aged well. It’s high on the list of my favorite space movies. Thanks to that movie (and the much weirder, but still pretty damn good Marooned), this vehicle is comfortably familiar to me:


(From Wikipedia.)

That there is the Apollo 15 command-service module. The command module (astronaut container) is the shiny cone at the front. Because I had cool parents, I got to go to the Johnson Space Center as a kid and see an actual command module, in person. As far as spacecraft go, it’s pretty small. If you stole one, you could hide it in your average garage. Now there‘s a good scene for a movie: some mad scientist with a stolen command module in his garage, tinkering with it while 80’s electronic montage music plays. Turning it into a time machine or something.

Once again, I imagine many of you are wondering what the hell I’m talking about. Don’t fret, there’s always (well, almost always) a method to my madness. Earlier today, I got thinking about the classic thought experiment: What would it be like to fall into a black hole. That thought experiment normally involves just dropping some hapless human (usually without even a spacesuit) right into one. You guys know me, though: if I’m going to do a weird thought experiment, I’m going to go into far more detail than necessary. That’s the fun part!

The point is, I’m going to start off my series on black holes by imagining what would happen if you dropped a pristine command module into black holes of various masses. The command module needs a three-person crew, and my crew will consist of myself, a young David Bowie, and Abraham Lincoln. Despite what you may think, no, I’m not on drugs. This is all natural, which, if you think about it, is much scarier.

A Mini Black Hole

As XKCD pointed out, a black hole with the mass of the Moon would not be a terribly dramatic-looking object. Even accounting for gravitational lensing, it would be next to impossible to see unless you were close to it, or you were looking really hard, or you had an X-ray/gamma-ray telescope. Nobody knows for sure whether black holes with masses this small actually exist. They might possibly have formed from the compression of the high-density plasma that filled the early universe, but so far, nobody’s spotted their telltale radiation.

But me, Bowie, and Lincoln are about to see one, and far too close.

At 1,700 kilometers, we’ve already gone from flying to falling. There’s nothing exciting happening, though, because as physicists always say, from a reasonable distance, a black hole’s gravitational field is no different from the gravitational field of a regular old object of the same mass. The tidal acceleration (which is the real killer when it comes to black holes) amounts to less than a millionth of a gee. Detectable, but only just.

At 100 kilometers, the tides are noticeable. Lincoln’s top-hat (which he foolishly left untethered. There wasn’t time to explain space flight and free-fall to a 19th-century politician) is slowly migrating to the end of the cabin. The tidal acceleration (the difference in pull between the center of the CM and either of its ends) is similar to the gravity of Pluto. The command module is stretching, but no more than, say, a 747 flexes in flight. Only detectable with fancy things like strain gauges.

At 10 km, we’re really motoring. Traveling at 31 kilometers per second, we’ve broken the record set by Apollo 10 for the fastest-moving humans. Bowie’s crazy Ziggy Stardust hair is getting misshapen by differential acceleration, which by now amounts to almost a full gee. Anything untethered (Lincoln’s hat, Bowie’s guitar, my cup of coffee, et cetera) is stuck at either end of the cabin.

At 5 km, we’re all shrieking in pain. The stray objects at the ends of the cabin are starting to get smashed. We’re all being stretched front-to-back (since that’s how you sit in a command module). The tides are pulling the command module like a rubber band, but a command module’s a compact and sturdy thing, so apart from some alarming creaking of metal, and maybe some cracking in the heat shield, it’s still in one piece.

All three of us die very quickly not long after the 2.5 km mark. Since the university still won’t let me use their finite-element physics package (well, more accurately, they won’t let me in the physics department…), I can only conjecture what will kill us, but it’ll either be the fact that the blood in our backs is being pulled toward the black hole much harder than the blood at our fronts (probably turning us very nasty colors and causing lots of horrible hemorrhages), or the fact that the command module has been stressed beyond its limits and sprung a leak. I’d wager the viewports would shatter before anything else happened, as their frames start to bend out of shape.

Time dilation hasn’t really kicked in, even by 500 meters, so an observer at a safe distance would see the CM crumple in real-time. The cone collapses like an umbrella being closed. Fragments of broken glass, shards of metal, control panels, shattered heat shield, and pieces of a legendary rocker, a melancholic President, and an idiot are spilling out of the wreckage.

By 10 meters, the command module is no longer falling straight down. It’s started falling inward, compressing side-to-side even as it stretches top-to-bottom. The individual components and fragments cast off from the wreck stretch and pull apart. Metal panels tear in two. Glass shards crack. Bits of flesh tear messily in half. The fragments divide and divide and divide. As they approach the event horizon, they explode into purple-white incandescent plasma: because the atoms are falling inward towards a point, they slam sideways into each other at high speeds. All 13,000 pounds of command module and crew either help bulk up the black hole, or form a radiant accretion disk denser than lead and smaller than a grape.

A 5-Solar-Mass Black Hole

In black hole thought experiments, the starting-point is usually a 1-solar-mass black hole. That makes sense: as far as astronomical objects go, the Sun is nicely familiar. But like I said before, scientists aren’t sure if there are any 1-solar-mass black holes anywhere in the Universe. As far as we know, all black holes in this mass range form from stars, and supernova leftovers smaller than something like 3 solar masses can still be supported by things like radiation pressure, degeneracy pressure, and the fact that atoms don’t like other atoms too close to them. In practice, there aren’t any known black hole candidates smaller than about 3 solar masses. There are a couple around 5 solar masses, and 5 is a nice round number, so that’s what I’m going with. We’re resetting this weird-ass experiment and dropping me, Bowie, and Lincoln into a stellar black hole.

At 380,000 kilometers (the distance from the Earth to the Moon), the tidal acceleration is detectable, but not noticeable. Good thing we’re free-falling (I should’ve made Tom Petty the third crewmember…), because if we were held stationary (say, by a magic platform hovering at a fixed altitude above the hole), we’d be flattened by a lethal 469-gee acceleration. Good thing, too, that we’re in an enclosed spacecraft: if the black hole has an accretion disk orbiting around it, we’re probably close enough that an unshielded human would be scalded to death by its heat, light, and X-rays. For our purposes, though, we’ll assume this black hole’s been floating through interstellar space for a long time, and has already cannibalized its own accretion disk, rendering it almost dark.

As we reach 10,000 km from the black hole, the same thing happens that happened with the mini black hole. David Bowie, who has gotten out of his seat to have a second tube of strawberry-banana pudding, finds it difficult to climb back to his couch against the gravity gradient. He’s being gently pelted by loose objects. Lincoln is just sitting in his couch looking very grave. I’m screaming my head off, so I miss all of this.

At 5,000 kilometers, the command module starts to creak. David Bowie is now stuck upside-down at the top of the cabin. I, having lost my shit and tried to open the hatch to end it quickly, have fallen to the bottom and broken my coccyx. Lincoln is still sitting in his couch looking very grave. We’re experiencing a total acceleration of over a million gees. If we tried to maintain a constant altitude, that million gees would turn us and the command module into a sheet of very thin and very gory foil. We’re moving almost as fast as the electrons shot from the electron gun of an old CRT TV.

Between 5,000 and 1,000 kilometers, the command module starts popping apart. It’s not as fast as the last time. First, the phenolic plastic of the heat shield cracks and pulls free of the insulation beneath. Then, the circular perimeter of the cone starts to crumple and wrinkle. (True story: the command module was built with crumple zones, just like a car, so that it didn’t pulp the astronauts too much when it hit the ocean at splashdown.) Not long after, the pressure hull finally ruptures, spewing white jets of gas and condensation in all directions like a leaky balloon. Then, the bottom of the pressure hull bursts. Think of a sledgehammer hitting a sheet of aluminum foil. All the guts of the command module spill out: wires, seats, guitars, apples (I was hungry), tophats, shoes, hoses, spare spacesuits, screaming idiots (I fell out). We’re moving 10% of the speed of light.

By 100 kilometers, the command module has spaghettified into a long stream of debris. The individual metal parts, although badly warped by being torn from their mountings, are mostly holding together, though they’re really starting to stretch. Anything softer is shattering/pulping/shredding. The black hole’s event horizon is the largest object in the sky: a fist-sized black disk of nothingness surrounded by a very pretty mandala of distorted stars and galaxies. It looks something like this:


(Screenshot from the unbelievably awesome (and free) program Space Engine.)

We can’t see it, though: we’re all dead.

By 25 kilometers, we’re just a stream of fine dust hurtling towards the event horizon at close to the speed of light. For an observer at a great distance, our disintegration proceeds in slow motion, both from the massive speed at which we’re traveling, and from the time-dilating effects of extreme gravity.

As we scream through 14.8 kilometers, we’ve almost reached the event horizon. The individual atoms the command module used to be made of are accelerating apart, spraying the whole CM into a narrow stream of plasma. Outside observers, though, just see the incandescent dust slow to a halt, change color from electric-arc purple to brilliant blue-white to the color of the sun to the color of hot steel to red-hot to black. What happens when we hit the singularity not long after is anybody’s guess. By definition, at a singularity, the equations you’re working with just quit making sense.

Sagittarius A*

There’s a very massive and very dense thing at the center of the Milky Way. It has about 3.6 million times the mass of the Sun, and because there’s a star (poetically called S0-102) that orbits pretty close to it (relatively speaking, anyway: its closest-approach distance is still larger than the distance from the Sun to Pluto), we know it has to be quite small, and therefore quite dense. According to our current understanding of physics, any mass like that would inevitably collapse into a black hole no matter what. The short version: it’s probably a black hole. (Note 1: as of this writing in November 2016, radio astronomers have finally committed to using a gigantic virtual telescope to take a picture of the actual event horizon in 2017, which is awesome) (Note 2: Though the actual mechanism for their formation isn’t known, some astrophysicists have done simulations suggesting that they formed from super-massive stars in the early universe. These days, the largest stars are a few hundred solar masses, with the largest stars for which we have firm evidence weighing around 120 solar masses. That’s massive, but not super-massive. These super-massive primordial stars contained thousands of solar masses. The one in that article massed 55,500. Some may have exceeded a million.)

Because a black hole (or, at least, its event horizon) is as compact as you can make anything, black holes tend to be really small compared to normal objects of similar mass. A Moon-mass black hole would look like a black dust-grain. An Earth-mass one would look like a pea. A Sun-mass one (and remember, there’s a lot of stuff in the Sun) would be the size of a small town. The 5-solar-mass hole we considered a second ago would be the size of a city. Sagittarius A*, though, containing so much mass, is actually a proper astronomical-sized object: 15 times larger than the Sun. If some deity with a really sick sense of humor replaced the Sun with Sgr A*, it would hang in the sky, a little smaller than a fist at arm’s length. We would also all be dead. For many reasons: no sunlight, radiation from the accretion disk, and the fact that we’d be orbiting so fast that grains of dust would heat the upper atmosphere lethally hot.

At 1 AU (the average distance from the Earth to the Sun), Sgr A*’s event horizon is much larger in the sky than the Moon. We’re accelerating at 1800 gees, but we don’t feel it, because we’re in free-fall. The tidal acceleration is minuscule: less than a micron per second per second. Once again, we’re pretending the black hole has no accretion disk, because if it did, its radiation would probably have incinerated us by now.

By the time we pass through Mercury’s orbit (0.387 AU) (assuming we actually are in this nightmarish black-hole solar system), we’re going one-third the speed of light, accelerating at over 15,000 gees. The tides are still detectable only by specialized instruments.

By 0.1 AU, we’re moving three-quarters the speed of light. David Bowie is singing “Space Oddity”, because I’ve smuggled a durian fruit onboard and threatened to cut it open if he doesn’t. Lincoln is starting to get sick of this shit, but this just gives him that same grave expression he has in all his photographs. The tides are detectable by instruments, but probably not by our human senses. Hitting a stationary dust particle the size of a bacterium unleashes a burst of light as bright as a studio flash.

At 0.071 AU, we pass through the event horizon without even realizing it. Falling into a black hole intact is a little like having a gigantic black bag closed around you: the event horizon already covers more than half of the sky, thanks to the fact that the black hole bends light toward the horizon. The sky shrinks into an ever-diminishing circle in a black void. The circle grows brighter and bluer with every passing second.

By 0.05 AU, stray objects are drifting to the ends of the cabin again: the tides are finally picking up. David Bowie is holding me down and punching me repeatedly, because he’s sick of me resurrecting him and killing him over and over. Lincoln is letting him do it, because frankly, I’ve cracked his statesman’s patience with my bullshit. We’re only 30 seconds from the singularity.

Even at 0.01 AU, the tides aren’t stretching the capsule. The effects might be palpable, but they’re nothing compared to what we’ve already been through in the previous experiments. We’re riding down a shaft of blue-shifted light, concentrated into a point straight overhead: everything that’s fallen into the hole recently can’t help but curve inward, until it’s falling almost ruler-straight towards the singularity.

At one Sun radius from the singularity, the differential acceleration is approaching one gee. Things are starting to get uncomfortable. David Bowie has stopped punching me, because he’s fallen to the top of the capsule again. Lincoln, though, still has the strength to pick up a pen and stab me in the sternum. He’s cursing at me, and as I start to bleed to death, I observe that Lincoln is much more creative with his swears than I would have given him credit for.

At 150,000 km, we start to get woozy on account of the blood in our bodies pooling in all the wrong places. We narrowly avoid a collision with Matthew McConaughey in a spacesuit, who has gone from muttering about quantum data to describing the peculiar aging patterns of high-school girls.

At 50,000 km, moving very, very nearly the speed of light, the command module finally starts to disintegrate. Seconds later, we spaghettify, just as before, and strike the singularity. And, once again, we run afoul of the fact that physicists have very little idea what happens that deep in a gravity well. For reference, at 100 meters from the singularity (and ignoring relativistic effects and pretending we can use the Newtonian equation for tides down here), the differential acceleration is measured with twenty-digit numbers. If the capsule were infinitely rigid and didn’t spaghettify, by the time its bottom touched the singularity, the tides would be measured in 25-digit numbers. If, somehow, we’d survived our trip to the singularity, we’d be accelerating so fast that, thanks to the Unruh effectempty space would be so hot we’d instantly vaporize.

And here’s where physics breaks down. If I’m reading this paper right, the distance between any point and the singularity is infinity, because space-time is so strongly curved near it.

Imagine that space is two-dimensional. It contains two-dimensional stars with two-dimensional mass. That curves two-dimensional space into a three-dimensional manifold. The gravity well (technically, the metric) around a very dense (but non-black-hole) looks roughly like this:


(From Wikipedia.)

If you measure the circumference of the object, you can calculate its diameter: divide by two times pi. But when you measure its actual diameter, you’ll find it’s larger than that, because of the way strong gravitation stretches spacetime. In the case of a black hole, spacetime looks more like this:


(From the paper cited above.)

The cylindrical part of the trumpet is (if I’m understanding this correctly) infinitely long. The “straight-line” distance through the black hole, on a line that just barely misses the singularity, is much larger than you’d expect. But the distance through the black hole, measured on a line that hits the singularity is infinite. All lines that hit the singularity just stop there.

But, to be honest, I really don’t know what it’d be like down there. Nobody does. The first person to figure out what gravity and particle physics do under conditions like that will probably be getting a shiny medal from some Swedes.


Carousel planet.

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:

(Image courtesy of JPL/NASA.)

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)

(5/4) * [((2 * pi) / (rotation period))^2 * (planet’s radius)^3] / [Newton’s gravitational constant * planet’s mass]

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:

(Image courtesy of the University of Michigan.)

Altair’s oblateness is about 0.25. Just for giggles, here’s what it would look like if the Sun was that oblate:

Oblate Sun

(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.

Oblate Earth

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:

Scalene Earth

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.


(Source and licensing.)

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:

Contact Binary Earth

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:


A great big pile of money.

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.)

(Image courtesy of NASA via Wikipedia.)

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:

Gold Tectonics

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:

Benard Cells

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?


A piece of a neutron star.

In the previous article, I talked about neutron stars, and like pretty much everybody else who’s ever tried to describe a neutron star’s absurd density, I explained that a piece of a neutron star the size of a 500-micron grain of sand would weigh as much as a small cargo ship.

That’s the kind of scientific example I like: it uses to comprehensible objects to illustrate something that would otherwise be pretty much impossible to visualize. I know the mass of a cargo ship isn’t exactly intuitive, but it’s more intuitive than saying 2e17 kilograms per cubic meter.

But one thing such examples gloss over is just how hard it is to pack this much mass so close together. In order to reach the pressures and temperatures necessary for fusion, we need the mass of the entire Sun, and that still only compresses the Sun’s core to 1.5e5 kilograms per cubic meter. It takes a truly massive star that has no way to maintain its internal temperature to compress matter the rest of the way and keep it there.

Neutron stars are supported almost entirely by neutron-degeneracy pressure, which, to seriously oversimplify matters, is a product of the fact that neutrons don’t like to occupy the same quantum state, and therefore don’t like to be brought too close together. It produces a lot of pressure. Enough to support 1.38 solar masses or more against a surface gravity of 100 billion gees. It also means that if we got really literal and took an actual piece out of a neutron star, it would not end well.

Let’s say we teleported a cube of neutron fluid, at a density of 2e17 kilograms per cubic meter and 3,000,000 Kelvin, into the air a meter over an empty field on Earth. The pressure exerted by all those neutrons packed too close together is complicated to calculate, but would probably be in the neighborhood of 5e33 pascals, or about 5e18 atmospheres. That’s a million trillion times higher than the pressure during the detonation of a hydrogen bomb.

That’s a lot of energy in one place, but as we’ve learned while trying to kill humanity with a BB gun and contemplating killer asteroids, even when you deposit a ridiculous amount of energy into a small volume, if there’s enough matter around it, eventually, it’ll be converted into a more ordinary form. This is another way of saying that, up to a certain limit, all explosions are going to behave a lot like scaled-up nuclear explosions.

But a whole lot of interesting shit is going to happen very rapidly before we get to that point. First, our grain of neutronium (which, admit it, sounds way cooler than “neutron superfluid,” cool as that one is) will expand rapidly. This will cause its pressure to decrease, and so it’ll be a lot like ascending through the layers of a neutron star, moving from outer core to crust. When the pressure drops low enough, many of the neutrons will decay into protons, emitting electrons and neutrinos. Neutrinos are infamous for carrying off energy, and also for refusing to interact with things. They might heat the ground below them by a few fractions of a degree, but considering that they pass right through the Earth without difficulty, they’re probably not important, except in the fact that they’ll cool the nuclear matter down.

Now, our grain of neutronium is a slightly larger grain of protons and neutrons all mashed together. Without the surrounding pressure to force them unnaturally close, the protons will naturally repel each other. They’ll still be attracted to each other and to the neutrons via the strong force, but once again, without the ridiculous pressures provided by the bulk of a neutron star, their clustering will be limited by the short range of the strong force. That is to say, they’ll stop being a soup of nucleons and go back to being atomic nuclei.

These nuclei will start out quite heavy, but the falling pressure will cause them to rapidly fission and give off a lot of radiation. There’ll be a lot of gamma rays, a lot of stray protons and neutrons, a lot of alpha particles, and probably a lot of beta decays producing protons and electrons from neutrons. It’d take a particle physicist to tell you exactly what elements to expect in the fallout, but I’d wager it’d mostly be lead, iron, hydrogen, and helium, with a smattering of lighter and heavier elements.

By now, we’re dealing with energies too low for massive neutrino emission, so the only way this expanding sphere of plasma can lose energy is by emitting traditional electromagnetic radiation and by expanding. It is now, for all intents and purposes, an extremely hot and extremely small version of a nuclear fireball.

How big would the fireball ultimately get? That depends on a lot of things: first, on how much energy was initially contained in our deadly granule. Second, on how much of that energy got carried off by the snobbish non-interacting neutrinos. It’s hard to be certain how much potential energy would have been in the grain to start with, but I’ve read that the neutron degeneracy pressure of neutronium is one third of its mass density. E = m * c^2, so mass density is just energy density. One third of the energy density of our grain of neutronium comes out to about 7.5e23 joules, which is of the same order of magnitude as the Chicxulub impact. So, even though we’re dealing with a very exotic explosion, we know that it’s not the kind of explosion that’s going to blow off the entire atmosphere or boil all the oceans. And actually, since so much energy is likely to be lost to neutrinos (neutrinos carry off 99% of the energy in supernovae, which considering that they still shine as bright as 10 billion suns, is horrifying to contemplate), it could be an almost-ordinary thermonuclear explosion. But, because I don’t know exactly how much energy we’re losing to neutrinos here, I’m going to assume the whole 7.5e23 joules is going to get deposited in the atmosphere.

Using this number, we can estimate the relevant parameters by using the excellent Impact Effects program, written by some very nifty folks. This program is, as far as I’m concerned, justification enough for the existence of the Internet all by itself. By assuming a stony asteroid 12 kilometers across, impacting perpendicular to the ground at 22 kilometers per second, we get an impact energy in the right ballpark.

The fireball would grow to massive proportions. As we learned from nuclear tests, hot plasma is pretty much completely opaque to radiation, since it’s got electrons flying around loose, and since photons like to bounce off of electrons. An initial burst of gamma rays would escape, but much of the radiation from our exploding grain of neutronium would be trapped in the plasma bubble, bouncing around while the bubble expanded at high speed. This bubble would reach a radius of 95 kilometers, reaching vertically to near the edge of space and pushing a massive shockwave out in front of it. Anything that happened to be caught within the fireball wouldn’t be destroyed. It wouldn’t even be vaporized. It would be flash-ionized into hot plasma. But, once the bubble had expanded to 95 kilometers in radius, it would finally have cooled enough to de-ionize and become transparent to ordinary radiation again.

This is very briefly good news for the people in the surrounding area, since it means they’re not going to get smacked in the face with a wall of plasma at 5000 degrees. Then, it becomes very bad news, since there’s a lot of thermal energy in that fireball that can now suddenly escape. The fireball would be visible from 1,100 kilometers away, and possibly farther, if you’re unlucky enough to be in an airliner or on a mountain. And if this fireball is visible to you, that pretty much means you’re dead. We’re looking at flash-fires and third-degree burns for five hundred miles in every direction.

About an hour later, the people at 1,200 kilometers, for whom the fireball was below the horizon, would stop being lucky: the blast wave would arrive, bringing overpressures of almost 2 atmospheres (enough to blow down just about any building) and wind speeds of 610 miles an hour (enough to blow down just about any building).

But the disaster would only just be beginning. Here, the peculiar origin of the explosion would make itself apparent: there would likely be a lot of radioactive fallout, and it would be made of peculiar isotopes generated in a flash when those protons and neutrons were separating into nuclei again. Not only that, with all the ionizing radiation, there would be even more nitric oxide in the plume than usual. Imagine if you will a pancake-shaped incandescent cloud hundreds of kilometers across–the size of a country. This cloud glows from within a larger, dark-red cloud of nitric oxide, ozone, iron, lead, and radioactive dust. Over the course of hours, the upper half of the cloud collapses downward as it cools, while the other half rises buoyantly upward. Within days, there’s a sheet of opaque vapor thousands of kilometers across, trapped in the stratosphere, blowing with the winds, fed from below by a firestorm of a kind not seen for 65 million years. Smoke and dust circle the planet within weeks. Temperatures drop far below freezing. People and animals are poisoned by toxic gases. With the sunlight blotted out, plants die. People starve. There’s a mass extinction. Only the hardiest species survive. After the dust settles out and the climate rebounds, new creatures populate the Earth. The only reminder of the catastrophe is a thin layer of exotic elements, and a crater 160 kilometers across and 2 kilometers deep. Perhaps if Earth ever spawns another species that spawns paleontologists, they will think the crater came from an asteroid impact. But it didn’t. It was created by an object the size of a grain of sand.

So take this as a grim warning: Under no circumstances should you take a useful scientific analogy so literally that you actually remove a piece of an exotic compact star and transport it to a planet. And they say you can’t learn anything from psychotic bloggers!


The weather in hell.

Neutron stars are horrifying things. They’re born in supernova explosions, which can shine with the light of 10 billion suns for weeks on end. But even after the fireworks, they’re still scary as hell. A neutron star compacts over 1.44 times the sun’s mass into a sphere about 20 kilometers across (about the size of a city). Apart from black holes, they’re the most extreme objects we know about (There may actually be more extreme variants, but none have been conclusively observed, and thank goodness: what we’re dealing with is scary enough.).

Everything about a neutron star is horrifying. Their surfaces broil at temperatures of over 100,000 Kelvin, which is twice as hot as the hottest stars. The young ones can get hotter than 1,000,000 Kelvin. Red-hot metal emits red light because most hot objects emit a so-called blackbody spectrum, with the most intense wavelength (color) of light depending on temperature. Iron near its melting point (1811 Kelvin) emits strongest at a wavelength of 1.6 microns, which is in the near-infrared (which is not the kind of infra-red the Predator used to hunt Arnold Scwharzenegger). To our eyes, the iron would look bright red-orange. The sun, with a blackbody temperature of about 5800 Kelvin, emits most strongly in the blue-green part of the spectrum (it looks yellow on Earth partly because of atmospheric scattering and partly because the human eye is a lot less sensitive to indigo and violet than it is to blue, yellow, orange, and red). At 100,000 to 1,000,000 Kelvin (and above), neutron stars would have about the same purplish-blue color as lightning bolts or the most powerful electric arcs, and they would emit most of their light in the deadly far-ultraviolet and really deadly X-ray. If you replaced the Sun with a 1,000,000-Kelvin neutron star, the Earth would only receive five times less energy than it does from the Sun. That’s ridiculous, considering all that energy is coming from an object the size of downtown Tokyo. (Speaking of which, that gives me an idea for a new Godzilla movie…) Of course, almost all of that energy would be in the x-ray and ultraviolet portions of the spectrum, and therefore would blow off the ozone layer and kill us all. But what would this blog be if I didn’t kill humanity every article?

But let’s handwave the radiation away (in true Star Trek fashion) and say you were able to get close to the neutron star.

Sorry. You’re still dead. Say you replaced Earth with a 2-solar-mass neutron star. If you were in the International Space Station, you’d probably survive the tidal forces, but they would be enough that if you oriented yourself feet-down, you’d feel a noticeable and unpleasant stretch: your head and feet would experience a difference in acceleration of about 0.4 gees. If you were orbiting at 4,000 kilometers, you’d experience a very painful 1.5-gee stretch (and, incidentally, you’d be completing one orbit every 3 seconds, which is ridiculous). By the time you got within 1,000 kilometers of the neutron star, your feet and your head would experience a difference in acceleration of 100 gees, more than enough to pull half your blood to your feet and the other half to your brain, stopping your heart and giving you lethal cerebral hemorrhages at the same time.

But as bad as things are in the neutron star’s neighborhood, they’re even worse on its surface. This is where the “Hell” part comes from. I’ve read through a few papers on neutron star structure (and skimmed many more). It’s hard to get a straight answer on what neutron stars are like on the inside (partly because it’s really hard to simulate the imponderable conditions near the core), but here’s neutron star structure as  I understand it:

Neutron Star Structure

At the top is the atmosphere, crushed to a thickness of between 10 centimeters and 10 meters (definite numbers are hard to find) by a gravity of 200 billion gees. Most neutron star atmospheres are pure hydrogen or pure helium, but sometimes, because of the insane pressures and temperatures, the atoms undergo fusion, leaving behind a carbon atmosphere. Below that is a crust that, thanks to the way nuclei get smashed together in supernovae, is mostly iron ions, their electrons wandering around with little regard for the nuclei.

The uppermost layers of the crust are made of almost-normal mater, albeit crushed to insane densities. But as you get deeper, the weight of the crust above puts extreme pressure on the nuclei. Suddenly, nuclei that have half-lives measured in seconds on Earth become as stable as ordinary gold or lead, because the pressure keeps decay products from escaping. The higher the pressure, the easier it is for extra neutrons to slip into nuclei. The nuclei get heavier and heavier and closer and closer together as you go down. Then, a few hundred meters below the surface, you encounter the “neutron drip,” where neutrons start leaking out of nuclei and roaming free. (On Earth, loose neutrons decay with a half-life of about 10 minutes. In the neutron star, once again, the pressure makes them stable.) This region gives us one of the coolest scientific terms ever devised: “nuclear pasta.”

Say it. Nuclear pasta. That sounds like something from an awesomely shitty ’70’s superhero movie. But, according to our current physics, it’s a real thing.

Because like charges repel, a nucleus wants to fly apart, since it’s full of chargeless neutrons (which don’t attract or repel anything) and positive protons (which fiercely repel each other). The strong nuclear force (or Strong Force, which sounds like the name of a badly-translated kung fu movie) provides the glue that holds nuclei together. At small distances, it’s far stronger than the electromagnetic force that causes like charges to repel (thus the name). But it decays very quickly with distance: beyond about a millionth of a nanometer, it gets vanishingly weak. Nuclei that are too large for the strong force to span their whole diameter tend to be unstable.

For this reason, nuclei also repel each other when brought close together. But deep in a neutron star’s crust, the pressure begins to overwhelm the repulsion. The protons still repel each other, but now there are places where they’re forced so close together that the strong force starts winning out. The nuclei grow oblong, and then they fuse into long tubes of protons and neutrons, like subatomic sausage (Subatomic Sausage. Another good band name.). This is the eponymous “pasta phase.” As you go down and the pressure goes up, these tubes get closer together and adjacent ones merge into two-dimensional sheets. This is called the “lasagna phase.” If you go on Google Scholar and type in “nuclear pasta”, you can find an actual peer-reviewed university-supported scientific paper that uses the phrase “lasagna phase” with a straight face. That makes me smile.

Deeper down, parts of the sheets come into contact, and you end up with a weird latticework of nuclear-matter tubes called the “gyroid phase.” Below that is a phase where you have cylindrical holes surrounded by nucleus-stuff, a sort of negative of the pasta phase. Then you get spherical holes. Then, the electromagnetic repulsion just gives up entirely, and the neutron-star matter reaches the density of an atomic nucleus, which is so huge (2 x 10^17 kilograms per cubic meter) that a piece the size of a grain of sand (a cube 500 microns on an edge) would weigh as much as a small cargo ship (25,000 metric tons). Of course, without the pressure provided by a whole neutron star to compress it, that grain would expand rapidly, and you would be spread over a large area. So don’t go removing pieces of neutron stars and carrying them around. Ain’t safe.

This is the outer part of the core. So far, the electrons have pretty much been minding their own business, ignoring the pained cries of the protons and neutrons as they were squeezed unnaturally close (the bastards). Deeper down, the pressures get so high that the electrons and protons combine to form neutrons, releasing a neutrino. Below a certain depth, it’s just a soup of neutrons crammed shoulder-to-shoulder. This soup has some weird properties, which we’ll get to later.

Below the neutron fluid, physicists aren’t quite sure what happens. Funnily enough, we here on Earth don’t have a lot of experience with soups of pure atomic-nucleus fluid. It’s possible that, in the depths, quarks could leak out of the individual neutrons, or even weirder stuff could happen, but we just don’t know. Most diagrams of neutron star structures either just put a question mark at the center or list a bunch of exotic particle names and then put a question mark. I won’t even attempt to guess what’s going on down there. I’ve got hellish weather to talk about.

But would there even be weather on a neutron star? That’s hard to say. Neutron stars have powerful magnetic fields, which could very well hold the plasma in the atmosphere in place, or at least make it awfully hard for it to move around. But, if you think about it, the temperature difference between the bottom and the top of a neutron star atmosphere is between 900,000 and 2,500,000 Kelvin, which is several thousand times the 350-Kelvin temperature difference that drive’s Earth’s weather. I’ll be working under the assumption that the atmosphere of a neutron star is able to circulate. If anybody has better information than me, feel free to set me straight.

In the last article, I talked about “scale height,” which is a nifty number that tells you how high you have to go for the atmosphere on a planet to be e times (about 2.7 times) less dense. Because their gravity is so monstrous, even the superheated plasma in a neutron star’s atmosphere is crushed tight against the surface. The Earth’s scale height is about 8,500 meters. The scale height in a neutron star’s atmosphere (depending on whether it’s made of hydrogen, helium, carbon, or something else, and depending on temperature) can range from a fraction of a millimeter to a few centimeters. On Earth, if you go up two scale heights, you’re higher than most airliners ever fly. If you go up 12 scale heights, you’re in space. I couldn’t find any really reliable numbers on the surface density of neutron-star atmospheres, but let’s assume it’s the same as the density of the sun’s core: 150,000 kilograms per cubic meter. The gravity is so strong that even the deepest neutron star atmospheres reach outer-space densities within half a meter. If I could stand on a neutron star (which, I will remind you, is a bad idea), the atmosphere would just about reach my knees. Right before I evaporated and then collapsed into a one-atom-thin layer of plasma.

Lucky for us, since the density profile of an atmosphere is exponential, many of its features will scale nicely from more familiar examples, like the atmospheres of stars and planets. Here, I’m pretty much making shit up, but the shit I’m making up is informed by some real-world knowledge. It’s also based on a few assumptions. I’m assuming, for one thing, that our neutron star is a pulsar, a neutron star spun up like a top by the infall of matter from a binary companion. I’m further assuming that this neutron star has about the same rotation rate as PSR J1748-2446ad, the fastest-spinning pulsar yet discovered (as of June 2014). It spins 716 times a second. Let’s imagine there was a single spot on the pulsar that emitted radio waves (in reality, there’s one at each magnetic pole, but one spot is usually looks brighter to us on Earth). If you could hear the signal it emitted, it would sound about like this (TURN DOWN YOUR SPEAKERS! It’s not a pretty sound.):

716-Hertz Pulsar

Even for an object like a pulsar, which is small in astronomical terms, spinning 716 times per second is ridiculous. It means that the matter at the equator is moving at 15% of the speed of light. It also means the Coriolis effect is gonna come kick some ass.

For those of you who find the Coriolis effect as confusing as I once did, here’s a brief explanation. Imagine you roll a ball inward from the edge of a frictionless spinning carousel. In the reference frame of the ground, the ball just travels straight to the center, across the other side, and off the edge. But if you were rotating with the carousel, the ball wouldn’t appear to travel in a straight line. This is because, according to you, the ball has both the inward velocity the outside observer sees and a radial velocity tangent to the circle’s circumference, since it’s not moving and the carousel is. This radial velocity is at its highest at the carousel’s edge, so as the ball approaches the center, it’s moving faster than the carousel’s surface, since the inner parts have lower radial velocities. Therefore, it appears to curve across the carousel as though acted upon by a force, pass through the center, and curve back out, describing a (roughly) semicircular trajectory. I know that’s the dunderhead layman’s explanation, but since I’m a dunderheaded layman, what do you expect?

The Coriolis effect is why low-pressure systems swirl anti-clockwise over Earth’s Northern hemisphere (and clockwise over the Southern hemisphere): the Earth is a sphere, so as you move towards the poles, you’re closer to the Earth’s axis of rotation. The faster the rotation of the body in question, the stronger the Coriolis effect and the tighter the circulation. Since our pulsar is spinning so damn fast, the circulation will be very tight, and since the bottom of the atmosphere is so much hotter than the top, the motion will be quite violent. Here’s my guess at what the pulsar’s atmosphere will look like:

Neutron Star Surface Weather Small

Here, I’m calling upon the concept of inertial circles. The radius of an inertial circle is given by:

(speed of the moving fluid) / (2 * angular velocity of planet * sin(latitude, with the poles being plus or minus 90 degrees and the equator 0))

An inertial circle is the path a body would take on a planet’s surface under the influence of the Coriolis effect alone. On Earth, if you assume a wind speed of 100 MPH (about 44 meters per second), then the inertial circle at a latitude of 45 degrees has a radius of about 480 kilometers, which is about right for a hurricane. I’ll make the very naive assumption that the winds on a neutron star will scale up in proportion to the increase in the temperature difference (from about 320 Kelvin (and 44 meters per second) on Earth to as much as 2,500,000 Kelvin on a neutron star). Neutron star winds will therefore have speeds on the order of 1,700 kilometers per second (Dorothy’s not going to make it to Oz in this tornado. Forget an F-5. We’re looking at an F-68000.) At a latitude of 45 degrees, a neutron star hurricane will have a radius of about 250 meters:


Imagine standing up to your knees in glowing gas. Spreading out around you is a brilliant electric-purple hurricane the size of a football stadium. At its center, it has an eye a few meters across, which feeds down into a needle-thin funnel with gas swirling at 0.1% of the speed of light.

I would guess that our pulsar wouldn’t have features like jet streams. For one thing, pretty much any movement in the atmosphere is going to be twisted into a circle by the Coriolis effect and turn into a cyclone. For another thing, the magnetic field would probably put the brakes on big swathes of moving fluid. A neutron star, having such a violent, hot atmosphere, would have hellish weather, and its surface would be paved with hurricanes. I imagine it would look something like this:

Neutron Star Weather

(Yes, I mixed up North and South and scribbled them out. Yes, I do know that I’m an idiot.)

The magnetic poles of neutron stars are often not aligned with the geographic poles (also true of Earth!), so there would probably be spots where the emerging magnetic field, all bunched up and concentrated, would stop the gas from moving much at all, even if the field was weak enough to let it move around everywhere else. These are also the spots where material tends to fall onto neutron stars, so they would have perpetual hot high-pressure systems (much like Louisiana or North Carolina). I took the liberty of adding magnetically-dampened high-pressure cyclones around these poles, and putting hurricanes everywhere else.

But this isn’t the only possible weather on a neutron star. Wherever you have fluid, gravity, and a density and/or temperature gradient, you can have weather-like phenomena. They happen in Earth’s atmosphere, they happen in Earth’s oceans, they happen on Venus, and they happen on the Sun. I’ve just spent quite a while talking about the weather on the surface of a neutron star, but there could also be weather in the interior. Beneath the crust, where the protons and electrons combine and it’s almost all neutrons, the material stops being solid and becomes a neutron liquid. But it’s not just any liquid. It’s a neutron superfluid. Superfluids are weird shit. They behave more or less like liquids, except that they have zero viscosity. Bizarre, but true. Water has a non-zero viscosity: as your pipe gets smaller, it gets harder and harder for water to move at a given velocity. Viscosity is pretty much the internal friction within the moving fluid. Viscosity determines the minimum size a stable vortex can have. Water has a viscosity of about 8.9 x 10^-4 pascal-seconds. But superfluids like liquid helium-4 have a viscosity of zero. The viscosity isn’t just very small, it’s actually zero. Superfluids can slip through any hole (that sounds dirty), and because of capillary action (which allows a wet spot on a paper towel to spread out), and because there’s no viscous friction to oppose it, they can crawl up and out of containers.

That’s all awesome, but, for my money, one of the coolest things about superfluids is what happens when they start swirling. Normally, when you rotate a container of fluid, the fluid starts to rotate with the container. Essentially, the whole container of fluid becomes one giant vortex.

This doesn’t happen in superfluids. No matter how large or small the rotating container, the superfluid forms lots and lots of extremely tiny vortices, their number depending only on the spin rate. If spin a glass of water at 1 revolution per minute, you’ll get one big, slow vortex. If you fill the same glass with superfluid helium-4 and rotate it at 1 RPM, you’ll get thousands of them. Don’t know what the hell I’m talking about? Here’s a truly beautiful video of swirling superfluid helium in action:

And here’s a terrible picture I drew of the same phenomenon:


This is basically quantum mechanics acting on a large scale, which often happens when things get cold enough. Since superfluid helium has zero viscosity, when it has to form vortices, the vortices are infinitely small, or rather, as small as the fact that the helium is made of atoms will allow them to be. This is called “quantization of vortices,” and is extremely weird, and most likely also happens in the superfluid interiors of neutron stars. These tiny vortices will be oriented along the axis of rotation, so they’ll be parallel to the crust near the equator and perpendicular near the poles (with additional changes depending on the magnetic field and whether the rotation is  faster in some regions than in others, which is usually how it works out). So if you look at it from the bottom up, you get knee-deep plasma tornadoes the size of football stadiums. And if you look at it from the top down, you get a field of weird nuclear pasta crawling with trillions and trillions of microscopic tornadoes piercing a neutron sea. I’ve said it before and you know I’ll say it again: astronomy is awesome.


What’s the most interesting place in the universe?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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