physics, thought experiment

Spin to Win | Black Holes, Part 2

In the previous part of this series, I tried to analyze what it would be like to fly an Apollo Command Module into black holes of various sizes. This time, though, I’m going to restrict myself to a single 1-million-solar-mass black hole. The difference is that, this time, I’m going to let the black hole spin (at 98% of the maximum possible spin, which is pretty average for a fast-spinning hole). But I’m getting ahead of myself. Before I go on, here’s my vehicle:

Apollo 11.jpg

(From the website of the Smithsonian Air and Space Museum)

That’s the actual command module Michael Collins, Buzz Aldrin, and Neil Armstrong took to the Moon (minus the Plexiglas shroud, of course). It would fit in even a medium-sized living room. This time, the crew will consist of me, Jürgen Prochnow (Das Boot Jürgen, naturally. Captain’s hat and all), and Charlize Theron. I was gonna take David Bowie and Abe Lincoln along again, but frankly, I’ve put Bowie through enough, and Lincoln was just so damn grim all the time.

Anyway, back to the subject at hand: the scary monster that is a rapidly-spinning black hole. All the black holes I discussed in the last part were Schwarzschild black holes, meaning they had no spin or electric charge. This black hole, though, is a Kerr black hole: it spins. The spin means this trip is a whole new ballgame. We’re still going to die horribly, of course, but hey, at least it’ll be interesting.

The first difference is that you can get closer to the event horizon of a spinning black hole. For a non-spinning black hole, there are no stable circular orbits closer than one and a half times the radius of the event horizon (the Schwarzschild radius), because in order to be in a circular orbit any closer, you’d have to travel faster than light. For spinning ones, there’s a lozenge-shaped region outside the event horizon called the ergosphere (My first-born daughter will be Ergosphere Sullivan). Objects near a rotating hole (or any rotating mass, to a lesser extent) are dragged along with the hole’s rotation. But inside the ergosphere, though, they’re being dragged along so fast that, no matter what, they can’t stand still. Inside the ergosphere, you have to rotate with the hole, because traveling anti-spinward would require going faster than the speed of light.

Here’s roughly what a free-fall trajectory into our Kerr black hole would look like (looking down at the hole’s north pole):

098-kerr-black-hole-infall

The ergosphere is the gray part. The event horizon is the black part.

Jürgen and Charlize are suspicious of me, but I gave them my word that we’re just orbiting the black hole. To make some observations. For science, and all that. When they’re not looking, I’m gonna hit the retro-rocket and plunge us to our deaths. I feel like there’s a flaw in my reasoning, but I don’t have time for such things.

Even orbits don’t work the way they normally do, near a spinning hole. Orbits around ordinary objects are very close to simple ellipses or circles. But, sitting in our command module, here’s what our orbit looks like (starting from parameters that should have given us a nice elliptical orbit):

Kerr BH Stable Orbit.png

This is because, when we orbit closer to the hole, we get a kick from the spin that twists our orbit around.

From nearby, a non-rotating black hole looks like its name: a black circle of nothingness, surrounded by a distorted background of stars and galaxies. From our orbit around the spinning million-solar-mass hole, though, the picture is much different:

Orbiting a Kerr Black Hole.jpg

(Picture and simulation by Alain Riazuelo.)

In that picture, the hole’s equator rotates from left to right. The reason the horizon is D-shaped is that photons coming from that direction were able to get a lot closer to the horizon, since they were moving in the direction of the rotation. On the opposite side, the horizon is bigger because those photons were going upstream, so to speak, and many of them were pulled to a halt by the spin and then either pulled into the hole, flung away, or pulled into a spinward orbit. Black holes are bullies. Spinning ones say “If you get too close, I’m going to eat you. And if you’re standing within a few arm’s lengths, you have to spin around me, or else I’ll eat you.”

(Incidentally, if you read about the movie’s background, the black hole in Interstellar was spinning at something like 99.999999% of the maximum rate. Its horizon would have been D-shaped like the picture above, from up close. From a distance, it would have looked…well, it would have looked like it did in the movie. They got it right, because they hired Kip motherfucking Thorne, Mr. Black Hole himself, to help write their ray-tracing code.

Speaking of Interstellar, the fact that you can get so much closer to a spinning black hole than a non-spinning one (providing you’re orbiting spinward) means you can get much deeper into its time-dilating gravity well. That means, as long as the tides aren’t strong enough to kill you, you can experience much bigger timewarps. The only way to get the same timewarp from a non-rotating black hole is to apply horrendously large forces to hover just outside the horizon. It’s much more practical to do in the vicinity of a spinning hole. Well, I mean, it’s no less practical than putting a Command Module in orbit around a black hole.

According to the equations from this Physics Stack Exchange discussion, as Jürgen, Charlize, and I zip around the hole close to the innermost stable orbit, time is flowing upwards of four times slower than it is for observers far away. I’m gonna keep us in orbit for a week, to lull my crewmates into complacency, so I’ll have the element of surprise when I try to kill us all. Well, we think it’s a week. Everyone outside thinks we’re orbit for a month and change

Then, without warning, I flip us around, turn on the engines, and take us into the hole. Jürgen fixes me with those steely blue eyes and that pants-shittingly intimidating face he was doing all through Das Boot. Charlize spends fifteen seconds trying to reason with me, then realizes I’m beyond all help and starts beating the shit out of me. Did you see Fury Road? She can punch. Neither of them can do anything to stop me, though: we’re already seconds from death.

But because this is a big black hole, the tides are gentle, at least outside the horizon. They’re stronger than the tides the Moon exerts on the Earth (which are measured in hundreds of nanometers per second per second), but they’re not what’s going to tear us to pieces.

What’s going to tear us to pieces is frame-dragging. Let’s go back to the metaphor of the whirlpool. The water moves much faster close to the center than it does far away. Because your boat is a physical object with a non-zero size, when you get really close, the water on one side of your boat is moving significantly faster than the water on the other side, because the near side is significantly closer than the far side. This blog hasn’t had any horrible pictures recently. Here’s one to explain the frame-dragging we experience:

Horrible Frame Dragging.png

In this picture, the capsule orbits bottom-to-top, and the hole rotates clockwise (this is the opposite of the view in the orbit plots; in this picture, we’re looking at the hole from the bottom, looking at its south pole; the reason has nothing to do with the fact that I screwed up and drew my horrible picture backwards).

Space closer to the hole is moving faster than space farther from the hole. The gradient transfers some of the hole’s angular momentum to the capsule, which is bad news, because that means the capsule starts spinning. It spins in the opposite direction of the hole (counter-clockwise, in the Horrible Picture (TM)).

I say the spin is bad news because, from the research for “Death by Centrifuge“, I know that things get really messy and horrible if you’re in a vehicle that rotates too fast.

Here’s a fun fact: Neil Armstrong came perilously close to death on his first spaceflight. During Gemini 8, while Armstrong and crewmate Dave Scott were practicing station-keeping and docking maneuvers with an uncrewed Agena target vehicle, the linked spacecraft started spinning. Unbeknownst to them, one of the Gemini capsule’s thrusters was stuck wide-open. Thinking it was the Agena causing the problem, they undocked. That’s when the shit really hit the fan, though I think Armstrong probably described it more gracefully. A video is worth a thousand words: here’s what it looked like when they undocked. Before long, the capsule was tumbling at 60 revolutions per minute (1 per second), wobbling around all three axes.

Did you ever spin in a circle when you were a kid? I did. Did you ever try it again as an adult, just to see what it was like? I did. I spent the next fifteen minutes lying in the grass (because I couldn’t tell which way was down) wondering if I should just go ahead and puke. Human beings don’t handle rotation well. According to this literature survey (thanks to Nyrath of Project Rho for helping direct me to it; it was hell to try and find a proper paper otherwise), average people do okay spinning at 1.7 RPM. At 2.2 RPM, susceptible people will probably start puking everywhere. At 5.44 RPM, ordinary tasks become stressful, because, thanks to the Coriolis effect (that troublemaking bastard), things like limbs, bodies, and inner-ear fluid don’t move normally, which plays hell with coordination. Also, it makes you puke. At 10 RPM, even the tough subjects in the study were seriously distressed.

Armstrong and Scott were spinning six times faster. When you spin, your brain loses the ability to compensate for movements of the eyes: you lose the ability to stabilize the image on your retinas, and the world wobbles and jumps. That’s bad news, especially if, for instance, you’re stuck inside a metal can which is spinning way too fast, and the only way you can stop it spinning way too fast (so that you don’t die) is by focusing your eyes on buttons and moving your Coriolis-afflicted hands to press them. Armstrong was an especially tough, calm dude, and he managed it, even though both men were starting to have serious vision problems. He did what any good troubleshooter would do: he switched the thrusters off and then on again (more or less). That saved the mission.

Now, I don’t know how fast the black hole will spin us, because the math is very complicated. But considering it’s a black hole we’re dealing with, probably pretty fast. Like I said, nothing about black holes is subtle. At 10 RPM, I throw up. My vomit describes a curved Coriolis-arc through the cabin and splatters on the wall. Jürgen doesn’t throw up until 20 RPM (after all, he’s a seafaring submarine captain). Charlize doesn’t throw up until 25,because she’s a badass.

At 60 RPM, I’m already screaming my head off, hyperventilating, and desperately regretting my decision to plunge us into a black hole. I try to hit buttons (pretty much at random), but I can’t get my fingers to go where I want them, and I press all the wrong ones. Jürgen is trying to calm me down and telling me he wants proper damage reports, but in my panic, I’ve forgotten all my German. Charlize has written both of us off and is trying to re-orient us and thrust away from the horizon, but it’s already too late.

At 60 RPM, the centrifugal acceleration on the periphery of the CM is already over 7 gees. There’s probably a bit of metal creaking, but nothing too serious. Because the crew couches are only a foot or so from the center of mass, we only experience an acceleration of 1.2 gees. For the moment, our main problem is that we’re punching and/or throwing up on each other.

At 120 RPM, the command module is starting to complain. Its extremities experience 28 gees. Panels slam shut. A cable pops loose and causes a short that trips the circuit breakers and kills our power. Even in our couches, we’re feeling almost 5 gees. I’m making a face like this:

gloc-face-735x413.jpg

(Source.)

At 200 RPM, the heat shield, experiencing 77 gees of centrifugal acceleration, cracks and flies off. It’s possible that the kick it gets from leaving our sorry asses behind, combined with the kick from being in the hole’s ergosphere (sounds dirty) is enough to slingshot the fragments to safety. Kind of a moot point, though, since I only care about the human parts, and all of those have blacked out at 13 gees.

Somewhere between 200 RPM and 500 RPM, the hull finally tears open. The bottom dome is flung off, letting important things like our air, our barf bags, and possibly our crew couches, fly out. Not that it matters: at 83 gees, we’ve all got ruptured aortas, brain hemorrhages, and we’re all in cardiac arrest.

At 1000 RPM (16.7 revolutions per second), the command module is flung decisively apart. Thanks to conservation of angular momentum, all the pieces are spinning pretty fast, too. Jürgen, Charlize, and I, are very dead, and in the goriest of possible ways: pulped by centrifugal force, and then shredded as we were spun apart.

The fragments closer to the horizon appear to accelerate ahead of us. The parts farther away fall behind. Most of the fragments fall into the hole. Spaghettification takes a while: once again, a more massive hole has weaker tides near the horizon.

As for what happens as we fall into the really nasty part of the hole (because it was sunshine and jellybeans before…), physics isn’t sure. The simplest models predict a ring-shaped singularity, rather than a point-shaped one. Some models predict that the ring singularity might act as an actual usable wormhole to another universe. But it’s also possible that effects I don’t pretend to understand (which have to do with weird inner horizons only rotating holes have) blue-shift the infalling light, creating a radiation bath that burns our atoms into subatomic ash. Either way, we’re not going to be visiting any worlds untold.

Once again, I’ve killed myself and two much cooler people. At least I only did it once this time. In the next and final part, I’m going to spend a little more time playing around with far less realistic black holes. (WARNING! Don’t actually play with black holes. If you have to ask why, then you skipped to the end of both articles.)

Standard
math, short, thought experiment

Short: Probabilities

For this thought experiment, let’s equate a probability of 1 (100% chance, a certainty) with the diameter of the observable universe. The diameter of the observable universe is about 93 billion light-years (because, during the 13.8 billion years since it started, the universe has been steadily expanding). With this analogy, let’s consider some probabilities!

According to the National Weather Service, your odds of being struck by lightning this year (if you live in the US, that is) are 1 in 1,042,000. Less than one in a million. One part in a million of the diameter of the universe is 93,000 light-years, which is far enough to take you outside the Milky Way, but on a cosmic scale, absolutely tiny.

The odds of winning the jackpot with a single ticket in the U.S. Powerball lottery are around 1 in 292 million. That’s like 318 light-years set against the diameter of the universe. 318 light-years is a long way. Even so, it’s an almost-reasonable distance. Most of the brighter stars you see in the night sky are closer than that. That’s almost the Sun’s neighborhood. Compared to the entire universe. Maybe that’s why they say the lottery is for suckers…

The odds of being struck by lightning three times in your lifetime are, mathematically, 1 in 1,000,000,000,000,000,000. The actual odds are even lower, since there’s a non-zero chance that you’ll be killed by a lightning strike, making getting another impossible. If your odds of dying in a lightning strike are 10%, then your odds of surviving are 9/10, and your odds of surviving the first two so you can get the third are (1 in a million) * (9/10) * (1 in a million) * (9 in 10) * (1 in a million), or about 81 in one hundred million trillion.That’s 81 in 100,000,000,000,000,000,000. That’s roughly the diameter of the Earth-moon system compared to the diameter of the universe.

The odds of putting 100 pennies in a cup, shaking them up, and scattering them so they all land flat, and then having every single coin come up heads, are 1 in 1, 267, 650, 600, 228, 229, 401, 496, 703, 205, 376. That’s the diameter of a grain of sand compared to the entire universe. Literally.

Get a standard deck of cards. Take out the jokers and the instructions. Shuffle the deck and pick a card at random. Do this 25 times. The odds of picking the jack of clubs every single time are like a proton compared to the visible universe.

If you pick 43 letters at random, the odds of forming the string

actisceneielsinoreaplatformbeforethecastlef

(that is, the first 43 letters of Hamlet) are as small as one Planck length (which is the smallest unit of distance that ever gets used in actual physics) compared to the visible universe. For reference, a Planck length is ten million trillion times smaller than a proton, which is itself a trillion times smaller than a grain of salt.

Incidentally, if you assembled random 43-letter strings, you would have to do it

32, 143, 980, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000

times to have a 99% chance of producing the first 43 letters of Hamlet in one of them. But a human bard did it in, at most, a couple hundred tries. Isn’t that weird? More probability stuff (and black hole stuff) to come!

Standard
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:

Apollo_CSM_lunar_orbit.jpg

(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:

scr00004

(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:

GravityPotential.jpg

(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:

Fig1.png

(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.

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Cars, physics, Space, thought experiment

A Toyota in Space

I talk all the time about the weird nerdy epiphanies I had as a kid. One of those epiphanies involved driving a car around on the outside of a space station. I realized that the car would have to bring along its own air supply, because an internal combustion engine can’t run on vacuum. I know that sounds obvious, but when you consider I was like nine years old at the time, it’s almost impressive that I figured it out. Almost.

Now that I’m older, I realized “Hey! I can actually figure out how much air I’d need to bring with me!” Conveniently, the worldwide craze for automobiles (some say they’ll replace the horse and buggy. I think that’s a pretty audacious claim, sir.) means that all sorts of vital statistics about gasoline engines are known. For instance: the air-fuel ratio. It’s as simple as it sounds. It’s the mass of air you need to burn 1 mass unit of fuel. The “ideal” ratio is 15:1: combustion requires 15 grams of air for every gram of fuel burned. Of course, if you’ve watched Mythbusters, you’ll know that stoichiometric (ideal) mixtures of air and fuel detonate, often violently. You don’t actually want that happening in a cylinder. You want subsonic combustion: deflagration, which is rapid burning, not an actual explosion. Supersonic combustion (detonation) produces much higher temperatures and pressures. At best, it’s really rough on the. At worst, it makes the engine stop being an engine and start being shrapnel. So, in practice, mixtures like 14:1 and 13:1 are more common. I’ll go with 14:1, although I freely admit I don’t know much about engines, and might be talking out my butt. No change there.

Either way, we now know how many mass units of air the engine will consume. Now, we need to know how many mass units of fuel the engine will consume. There are lots of numbers that tell you this, but for reasons of precision, I’m using one commonly used in airplanes: specific fuel consumption (technically, brake specific fuel consumption). The Cessna 172 is probably the most common airplane in the world. It has a four-cylinder engine, just like my car, though it produces 80% more horsepower. Its specific fuel consumption, according to this document, is 0.435 pounds per horsepower per hour. The Cessna engine produces 180 horsepower, and my car produces 100, so, conveniently, I can just multiply 0.435 by 100/180 to get 0.242 pounds per horsepower per hour. Assuming I’m using 50% power the whole way (I’m probably not, but that’s a good upper limit), that’s 50 horsepower * 0.242, or 12.1 pounds of gasoline per hour.

So, we know we need 12.1 pounds of gasoline per hour, and from the air-fuel ratio, we know we need 169.4 pounds of air per hour. That’s all fine and dandy, but I’m not sure how much room 169.4 pounds of air takes up. Welders to the rescue! According to the product catalog from welding-gas supplier Airgas, a large (size 300) cylinder of semiconductor-grade air has a volume of 49 liters, and the air is stored in that bottle at about 2,500 PSI. (I don’t know what you actually do with semiconductor-grade air, but it’s got the same ratio of gases as ordinary air, so it’ll do.) At room temperature, the bottled air is actually a supercritical fluid with a density 1/5th that of water. Therefore, each cylinder contains about 10 kilograms (22 pounds) of air. Much to my surprise, even when it’s connected to an air-hungry device like an internal combustion engine, a single size-300 cylinder could power my car for over seven and a half hours.

But you guys know me by now. You know much I like to over-think. And I’m gonna do it again, because there are a lot of things you have to consider when driving a car in a vacuum that don’t come up when you’re driving around in air.

Thing 1: Waste heat. This is a major issue for spacecraft, which live in a vacuum (unless you’ve really screwed up). The problem is that there’s only one good way to expel waste heat in a vacuum: radiation. Luckily, the majority of automobile engines are already radiator-cooled. Normally, they depend on heat flowing from the engine to the cooling water, into the metal fins of a radiator, and into the atmosphere. In vacuum, the cooling will run engine-water-radiator-vacuum. The engine produces 100 horsepower at maximum, which is about 75 kilowatts. A radiator operating at the boiling temperature of water radiates about 1,100 watts per square meter, for  a total area of 68 square meters, which means a square 27 feet (8.2 meters) on a side. You could play tennis on that. Luckily, the radiator is two-sided, which cuts the radiator down to a square 19 feet (5.8 meters) on a side. It’s still going to be larger than my car, but if I divide it into ten fins, it would only be absolutely ridiculous, rather than impractically ridiculous. That’s already my comfort zone anyway.

Thing 2: Materials behave differently in a vacuum. Everything behaves differently under vacuum. Water boils away at room temperature. Some of the compounds in oil evaporate, and the oil stops acting like oil. Humans suffocate and die. To prevent that last one, I’m going to have to beef up my car’s cabin into a pressure vessel. And since I’m doing that, I’ll go ahead and do the same to the engine bay, so that I don’t have to re-design the whole engine to work in hard vaucuum. I’ll make the two pressure vessels separate compartments, because carbon monoxide in a closed environment is bad and sometimes engines leak.

I’ll also have to put a one-way valve on the exhaust pipe, because my engine is designed to work against an atmospheric pressure of 1 atmosphere, and I feel like working against no pressure at all would cause trouble. I’m also going to have to change the end of my exhaust pipe. I’ll seal it off at the end and drill lots of small holes down the sides, to keep the exhaust from acting like a thruster and making my car spin all over the place.

Thing 3: Lubrication. A car’s drivetrain and suspension contain a lot of bearings. There are bearings for the wheels, the wheel axles, the steering linkages, the universal joints in the axles, the front and rear A-arms… it just goes on and on. Those bearings need lubrication, or they’ll seize up and pieces will break off, which you very rarely want in engineering. Worse, in vacuum, metal parts can vacuum-weld together if they’re not properly protected. We can’t enclose and pressurize every bearing and joint. That would make my car too bulky, for one. For two, there would still have to be bearings where the axles came out of the pressurized section, so I’ve gotta deal with the problem sooner or later. Luckily, high-vacuum grease is already a thing. It maintains its lubricating properties under very high vacuum and a wide range of pressures, without breaking down or gumming up or evaporating. We’ll need built-in heaters to keep the grease warm enough to stay greasy, but that’s not too big a hurdle.

Thing 4: Tires. My car’s owner’s manual specifies that I should inflate my tires to 35 psi (gauge). I’ll have to inflate them to a higher gauge pressure in vacuum, since they’ll have almost no pressure working against them. If I don’t, they’ll be under-inflated, and that’ll make them heat up, and in vacuum, that goes from a minor problem to a potentially fatal tire-melting and tire-bursting disaster. Actually, I think I’ll eliminate that risk altogether. I’ll do what most rovers do: I’m getting rid of pneumatic tires altogether. Because my car’s going to be fast, heavy and have a human passenger, I can’t do what most rovers have done and just make my wheels metal shells. I need some cushioning to stop from rattling myself and my car to pieces.

nasa_apollo_17_lunar_roving_vehicle

That’s Gene Cernan driving the Lunar Roving Vehicle (the moon buggy). It’s about five times lighter than my car, but it proves that airless tires can work at moderate speed. Michelin is also trying to design airless rubber tires for military Humvees, and while they don’t absorb shocks quite as well as pneumatic tires, they can’t puncture and explode like pneumatic tires. So I’m going with some sort of springy metal tire, possibly just composed of spring-steel hoops or something like that.

Thing 4: Fuel. If I was sensible, I’d have chucked the whole idea of powering a vacuum-roving Toyota with a gasoline engine. (Actually, I’d have chucked the whole idea of a vacuum-roving Toyota and started from scratch…) We know I’m not sensible, so I’m going to demand that my Lunar Toyota run on gasoline. 10,000 liters of gasoline (I like to mix units, like an idiot) will let me drive 42,500 kilometers. Enough to go around the Moon’s equator three (almost four) times. You might think that carrying a small tanker’s worth of gasoline to the Moon is an impossible feat, but when you consider that the mass of my car (about 1,000 kilograms) plus the mass of all that gasoline (7,300 kilograms) plus tankage is less than the weight of the Apollo Command-Service Module and the Lunar Module, not only does the Apollo program seem that much more audacious and impressive, but it becomes possible to talk sensibly (sort of…) about putting my car, my air tanks, and a lifetime supply of gasoline on the Moon. That also takes care of…

Thing 5: Getting my car on the Moon. We can just use a Saturn V, or wait for the engineers to finish building the Falcon Heavy or Space Launch System. Lucky for me, the rocket scientists have already solved the problem of landing a heavy vehicle, too: the ballsy sky-crane landing used during the Curiosity rover’s descent would almost certainly work just fine for my car, since it’s only 200 kilograms heavier than Curiosity. The fuel and air can just be landed under rocket power, or by expendable airbags.

So it wasn’t all that insane for my nine-year-old self to imagine driving an ordinary street car around on the Moon. That is, from the point of view of fueling and aspirating (ventilating? aerating? Providing air to, is what I mean…) the engine and the passenger. But the physics of driving around in vacuum and/or under low gravity pose another challenge, and that challenge is interesting enough to get a post of its own. Watch this space!

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Addendum, Cars, physics, Space, thought experiment

Addendum: A City On Wheels

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

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

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

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

cscl_globe_arriving_at_felixstowe_united_kingdom

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

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

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

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

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

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Addendum, Space, thought experiment

Addendum: The Moon Cable

Reader Dan of 360 Exposure, pointed out something that I completely neglected to mention, regarding cable strength. Not only would the Moon Cable be unable to connect the Earth and Moon without breaking (either by being stretched, or by winding around the Earth and then being stretched), but it couldn’t even support its own weight.

There’s a really cool measurement used in engineering circles: specific strength. Specific strength compares the strength of a material to its weight. It’s often measured in (kilonewtons x meters) / kilograms. But there’s another measurement that I like better: breaking length. Breaking length tells you the same thing, but in a more intuitive way. Breaking length is the maximum length of a cable made of the material in question that could dangle free under 1 gee (9.80665 m/s^2) without the cable’s own weight breaking it.

Concrete’s breaking length is only 440 meters. Oak does better, at 13 kilometers (a really bizarre inverted tree. That’d make a good science-fiction story). Spider silk, which has one of the highest tensile strengths of any biological material, has a breaking length of 109 km (meaning a space-spider could drop a web from very low orbit and snag something on the ground. There’s a thought.) Kevlar, whose tensile strength and low density make it ideal for bullet-proof vests, has a breaking length of 256 kilometers. If you could ignore atmospheric effects (you can’t) and the mass of the rope (you can’t), you could tie a Kevlar rope to a satellite and have it drag along the ground. Zylon is even better. It’s a high-tensile synthetic polymer with a higher tensile strength than Kevlar, and a larger breaking length: 384 kilometers. You could attach a harpoon to a Zylon rope and use it to catch the International Space Station (no you couldn’t).

And, funnily enough, specific strength is one of those things that has a well-established upper limit. According to current physics, nothing (made of matter, magnetic fields, or anything else) can have a breaking length longer than 9.2 trillion kilometers. This is demonstrated in this paper, which I could get the gist of but which I can’t vouch for, because I understand the Einstein Field Equations about as well as I understand cricket, or dating, or the politics of Mongolian soccer. But the long and the short of it is that it’s not possible, according to current physics, to make anything stronger than this without violating one of those important conservation laws, or the speed of light, or something similar.

Not that we were ever going to get there anyway. The strongest material that has actually been produced (as of this writing, July 2016) is the colossal carbon tube. Think of a tube made of corrugated cardboard with holes in it, except that the cardboard and the corrugation is made of graphene. Colossal carbon tubes have a breaking length of something like 6,000 km (remember, this is under constant gravity, not real gravity). And that’s theoretical. So we’re not building a giant ISS-catching harpoon any time soon.

You might have noticed that I skipped over the one material that I was actually talking about in the Moon Cable post: steel. There’s a reason for that. I want to leave the big punch in the gut for the very end. For dramatic purposes. Ordinary 304 stainless steel has a pitiful breaking length of 6.4 km. Inconel (which is both surprisingly tough and amazingly heat-resistant, and is often used in things like rocket combustion chambers) only does a little better, at 15.4 km. There’s no handwaving it: you can’t attach the Moon to the Earth with a metal cable.

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physics, Space, thought experiment

The Moon Cable

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

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

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

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

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

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

No. No it would not. Not even close.

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

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

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

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

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

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