astronomy, physics, science, Space, thought experiment

If the Sun went Supernova

I have to preface this article by saying that yes, I know I’m hardly the first person to consider this question.

I also have to add that, according to current physics (as of this writing in December 2017), the Sun won’t ever go supernova. It’s not massive enough to produce supernova conditions. But hey, I’ll gladly take any excuse to talk about supernovae, because supernovae are the kind of brain-bending, scary-as-hell, can’t-wrap-your-feeble-meat-computer-around-it events that make astronomy so creepy and amazing.

So, for the purposes of this thought experiment, let’s say that, at time T + 0.000 seconds, all the ingredients of a core-collapse supernova magically appear at the center of the Sun. What would that look like, from our point of view here on Earth? Well, that’s what I’m here to find out!

From T + 0.000 seconds to 499.000 seconds

This is the boring period where nothing happens. Well, actually, this is the nice period where life on Earth can continue to exist, but astrophysically, that’s pretty boring. Here’s what the Sun looks like during this period:

Pretty much normal. Then, around 8 minutes and 19 seconds (499 seconds) after the supernova, the Earth is hit by a blast of radiation unlike anything ever witnessed by humans.

Neutrinos are very weird, troublesome particles. As of this writing, their precise mass isn’t known, but it’s believed that they do have mass. And that mass is tiny. To get an idea of just how tiny: a bacterium is about 45 million times less massive than a grain of salt. A bacterium is 783 billion times as massive as a proton. Protons are pretty tiny, ghostly particles. Electrons are even ghostlier: 1836 times less massive than a proton. (In a five-gallon / 19 liter bucket of water, the total mass of all the electrons is about the mass of a smallish sugar cube; smaller than an average low-value coin.)

As of this writing (December 2017, once again), the upper bound on the mass of a neutrino is 4.26 million times smaller than the mass of an electron. On top of that, they have no electric charge, so the only way they can interact with ordinary matter is by the mysterious weak nuclear force. They interact so weakly that (very approximately), out of all the neutrinos that pass through the widest part of the Earth, only one in 6.393 billion will collide with an atom.

But, as XKCD eloquently pointed out, supernovae are so enormous and produce so many neutrinos that their ghostliness is canceled out. According to XKCD’s math, 8 minutes after the Sun went supernova, every living creature on Earth would absorb something like 21 Sieverts of neutrino radiation. Radiation doses that high have a 100% mortality rate. You know in Hollywood how they talk about the “walking ghost” phase of radiation poisoning? Where you get sick for a day or two, and then you’re apparently fine until the effects of the radiation catch up with you and you die horribly? At 21 Sieverts, that doesn’t happen. You get very sick within seconds, and you get increasingly sick for the next one to ten days or so, and then you die horribly. You suffer from severe vomiting, diarrhea, fatigue, confusion, fluid loss, fever, cardiac complications, neurological complications, and worsening infections as your immune system dies. (If you’re brave and have a strong stomach, you can read about what 15-20 Sieverts/Gray did to a poor fellow who was involved in a radiation accident in Japan. It’s NSFW. It’s pretty grisly.)

But the point is that we’d all die when the neutrinos hit. I’m no religious scholar, but I think it’d be appropriate to call the scene Biblical. It’d be no less scary than the scary-ass shit that happens in in Revelation 16. (In the King James Bible, angels pour out vials of death that poison the water, the earth, and the Sun, and people either drop dead or start swearing and screaming.) In our supernova Armageddon, the air flares an eerie electric blue from Cherenkov radiation, like this…

(Source.)

…and a few seconds later, every creature with a central nervous system starts convulsing. Every human being on the planet starts explosively evacuating out both ends. If you had a Jupiter-sized bunker made of lead, you’d die just as fast as someone on the surface. In the realm of materials humans can actually make, there’s no such thing as neutrino shielding.

But let’s pretend we can ignore the neutrinos. We can’t. They contain 99% of a supernova’s energy output (which is why they can kill planets despite barely interacting with matter). But let’s pretend we can, because otherwise, the only spectators will be red, swollen, feverish, and vomiting, and frankly, I don’t need any new nightmares.

T + 499.000 seconds to 568.570 seconds (8m13s to 9m28.570s)

If we could ignore the neutrino radiation (we really, really can’t), this would be another quiet period. That’s kinda weird, considering how much energy was just released. A typical supernova releases somewhere in the neighborhood of 1 × 10^44 Joules, give or take an order of magnitude. The task of conveying just how much energy that is might be beyond my skills, so I’m just going to throw a bunch of metaphors at you in a panic.

According to the infamous equation E = m c^2, 10^44 Joules would mass 190 times as much as Earth. The energy alone would have half the mass of Jupiter. 10^44 Joules is (roughly) ten times as much energy as the Sun will radiate in its remaining 5 billion years. If you represented the yield of the Tsar Bomba, the largest nuclear device ever set off, by the diameter of a human hair, then the dinosaur-killing (probably) Chicxulub impact would stretch halfway across a football field, Earth’s gravitational binding energy (which is more or less the energy needed to blow up the planet) would reach a third of the way to the Sun, and the energy of a supernova would reach well past the Andromeda galaxy. 1 Joule is about as much energy as it takes to pick up an egg, a golf ball, a small apple, or a tennis ball (assuming “pick up” means “raise to 150 cm against Earth gravity.”) A supernova releases 10^44 of those Joules. If you gathered together 10^44 water molecules, they’d form a cube 90 kilometers on an edge. It would reach almost to the edge of space. (And it would very rapidly stop being a cube and start being an apocalyptic flood.)

Screw it. I think XKCD put it best: however big you think a supernova is, it’s bigger than that. Probably by a factor of at least a million.

And yet, ignoring neutrino radiation (we still can’t do that), we wouldn’t know anything about the supernova until nine and a half minutes after it happened. Most of that is because it takes light almost eight and a quarter minutes to travel from Sun to Earth. But ionized gas is also remarkably opaque to radiation, so when a star goes supernova, the shockwave that carries the non-neutrino part of its energy to the surface only travels at about 10,000 kilometers per second. That’s slow by astronomical standards, but not by human ones. To get an idea of how fast 10,000 kilometers per second is, let’s run a marathon.

At the same moment, the following things leave the start line: Usain Bolt at full sprint (10 m/s), me in my car (magically accelerating from 0 MPH to 100 MPH in zero seconds), a rifle bullet traveling at 1 kilometer per second (a .50-caliber BMG, if you want to be specific), the New Horizons probe traveling at 14 km/s (about as fast as it was going when it passed Pluto), and a supernova shockwave traveling at 10,000 km/s.

Naturally enough, the shockwave wins. It finishes the marathon (which is roughly 42.195 kilometers) in 4.220 milliseconds. In that time, New Horizons makes it 60 meters. The bullet has traveled just under 14 feet (422 cm). My car and I have traveled just over six inches (19 cm). Poor Usain Bolt probably isn’t feeling as speedy as he used to, since he’s only traveled an inch and a half (4.22 cm). That’s okay, though: he’d probably die of exhaustion if he ran a full marathon at maximum sprint. And besides, he’s about to be killed by a supernova anyway.

T + 569 seconds

If you’re at a safe distance from a supernova (which is the preferred location), the neutrinos won’t kill you. If you don’t have a neutrino detector, when a supernova goes off, the first detectable sign is the shock breakout: when the shockwave reaches the star’s surface. Normally, it takes in the neighborhood of 20 hours before the shock reaches the surface of its parent star. That’s because supernovas (at least the core-collapse type we’re talking about) usually happen inside enormous, bloated supergiants. If you put a red supergiant where the Sun is, then Jupiter would be hovering just above its surface. They’re that big.

The Sun is much smaller, and so it only takes a couple minutes for the shock to reach the surface. And when it does, Hell breaks loose. There’s a horrific wave of radiation trapped behind the opaque shock. When it breaks out, it heats it to somewhere between 100,000 and 1,000,000 Kelvin. Let’s split the difference and say 500,000 Kelvin. A star’s luminosity is determined by two things: its temperature and its surface area. At the moment of shock breakout, the Sun has yet to actually start expanding, so its surface area remains the same. Its temperature, though, increases by a factor of almost 100. Brightness scales in proportion to the fourth power of temperature, so when the shock breaks out, the Sun is going to shine something like 56 million times brighter. Shock breakout looks something like this:

But pretty soon, it looks like this:

Unsurprisingly, this ends very badly for everybody on the day side. Pre-supernova, the Earth receives about 1,300 watts per square meter. Post-supernova, that jumps up to 767 million watts per square meter. To give you some perspective: that’s roughly 700 times more light than you’d be getting if you were currently being hit in the face by a one-megaton nuclear fireball. Once again: However big you think a supernova is, it’s bigger than that.

All the solids, liquids, and gases on the day side very rapidly start turning into plasma and shock waves. But things go no better for people on the night side. Let’s say the atmosphere scatters or absorbs 10% of light after passing through its 100 km depth. That means that, after passing through one atmosphere-depth, 90% of the light remains. Since the distance, across the Earth’s surface, to the point opposite the sun is about 200 atmosphere-depths, that gives us an easy equation for the light on the night side: [light on the day side] * (0.9)^200. (10% is approximate. After searching for over an hour, I couldn’t find out exactly how much light the air scatters, and although there are equations for it, I was getting a headache. Rayleigh scattering is the relevant phenomenon, if you’re looking for the equations to do the math yourself).

On the night side, even after all that atmospheric scattering, you’re still going to burn to death. You’ll burn to death even faster if the moon’s up that night, but even if it’s not, enough light will reach you through the atmosphere alone that you’ll burn either way. If you’re only getting light via Rayleigh scattering, you’re going to get something like 540,000 watts per square meter. That’s enough to set absolutely everything on fire. It’s enough to heat everything around you to blowtorch temperatures. According to this jolly document, that’s enough radiant flux to give you a second-degree burn in a tenth of a second.

T + 5 minutes to 20 minutes

We live in a pretty cool time, space-wise. We know what the surfaces of Pluto, Vesta, and Ceres look like. We’ve landed a probe on a comet. Those glorious lunatics at SpaceX just landed a booster that had already been launched, landed, and refurbished once. And we’ve caught supernovae in the act of erupting from their parent stars. Here’s a graph, for proof:

(Source. Funnily enough, the data comes from the awesome Kepler planet-hunting telescope.)

The shock-breakout flash doesn’t last very long. That’s because radiant flux scales with the fourth power of temperature, so if something gets ten times hotter, it’s going to radiate ten thousand times as fast, which means, in a vacuum, it’s going to cool ten thousand times faster (without an energy source). So, that first bright pulse is probably going to last less than an hour. But during that hour, the Earth’s going to absorb somewhere in the neighborhood of 3×10^28 Joules of energy, which is enough to accelerate a mass of 4.959×10^20 kg. to escape velocity. In other words: that sixty-minute flash is going to blow off the atmosphere and peel off the first 300 meters of the Earth’s crust. Still better than a grisly death by neutrino poisoning.

T + 20 minutes to 4 hours

This is another period during which things get better for a little while. Except for the fact that pretty much everything on the Earth’s surface is either red-hot or is now part of Earth’s incandescent comet’s-tail atmosphere, which contains, the plants, the animals, most of the surface, and you and me. “Better” is relative.

It doesn’t take long for the shock-heated sun to cool down. The physics behind this is complicated, and I don’t entirely understand it, if I’m honest. But after it cools, we’re faced with a brand-new problem: the entire mass of the sun is now expanding at between 5,000 and 10,000 kilometers per second. And its temperature only cools to something like 6,000 Kelvin. So now, the sun is growing larger and larger and larger, and it’s not getting any cooler. We’re in deep dookie.

Assuming the exploding sun is expanding at 5,000 km/s, it only takes two and a quarter minutes to double in size. If it’s fallen back to its pre-supernova temperature (which, according to my research, is roughly accurate), that means it’s now four times brighter. Or, if you like, it’s as though Earth were twice as close. Earth is experiencing the same kind of irradiance that Mercury once saw. (Mercury is thoroughly vaporized by now.)

In 6 minutes, the Sun has expanded to four times its original size. It’s now 16 times brighter. Earth is receiving 21.8 kilowatts per square meter, which is enough to set wood on fire. Except that there’s no such thing as wood anymore, because all of it just evaporated in the shock-breakout flash.

At sixteen and a quarter minutes, the sun has grown so large that, even if you ignored the earlier disasters, the Earth’s surface is hot enough to melt aluminum.

The sun swells and swells in the sky. Creepy mushroom-shaped plumes of radioactive nickel plasma erupt from the surface. The Earth’s crust, already baked to blackened glass, glows red, then orange, then yellow. The scorched rocks melt and drip downslope like candle wax. And then, at four hours, the blast wave hits. If you thought things couldn’t get any worse, you haven’t been paying attention.

T + 4 hours

At four hours, the rapidly-expanding Sun hits the Earth. After so much expansion, its density has decreased by a factor of a thousand, or thereabouts. Its density corresponds to about the mass of a grain of sand spread over a cubic meter. By comparison, a cubic meter of sea-level air contains about one and a quarter kilograms.

But that whisper of hydrogen and heavy elements is traveling at 5,000 kilometers per second, and so the pressure it exerts on the Earth is shocking: 257,000 PSI, which is five times the pressure it takes to make a jet of abrasive-laden water cut through pretty much anything (there’s a YouTube channel for that). The Earth’s surface is blasted by winds at Mach 600 (and that’s relative to the speed of sound in hot, thin hydrogen; relative to the speed of sound in ordinary air, it’s Mach 14,700). One-meter boulders are accelerated as fast as a bullet in the barrel of a gun (according to the formulae, at least; what probably happens is that they shatter into tiny shrapnel like they’ve been hit by a gigantic sledgehammer). Whole hills are blown off the surface. The Earth turns into a splintering comet. The hydrogen atoms penetrate a full micron into the surface and heat the rock well past its boiling point. The kinetic energy of all that fast-moving gas delivers 10^30 watts, which is enough to sand-blast the Earth to nothing in about three minutes, give or take.

T + 4 hours to 13h51m

And the supernova has one last really mean trick up its sleeve. If a portion of the Earth survives the blast (I’m not optimistic), then suddenly, that fragment’s going to find itself surrounded on all sides by hot supernova plasma. That’s bad news. There’s worse news, though: that plasma is shockingly radioactive. It’s absolutely loaded with nickel-56, which is produced in huge quantities in supernovae (we’re talking up to 5% of the Sun’s mass, for core-collapse supernovae). Nickel-56 is unstable. It decays first to radioactive cobalt-56 and then to stable iron-56. The radioactivity alone is enough to keep the supernova glowing well over a million times as bright as the sun for six months, and over a thousand times as bright as the sun for over two years.

A radiation dose of 50 Gray will kill a human being. The mortality rate is 100% with top-grade medical care. The body just disintegrates. The bone marrow, which produces the cells we need to clot our blood and fight infections, turns to blood soup. 50 Gray is equivalent to the deposition of 50 joules of radiation energy per kilogram. That’s enough to raise the temperature of a kilo of flesh by 0.01 Kelvin, which you’d need an expensive thermometer to measure. Meanwhile, everything caught in the supernova fallout is absorbing enough radiation to heat it to its melting point, to its boiling point, and then to ionize it to plasma. A supernova remnant is insanely hostile to ordinary matter, and doubly so to biology. If the Earth hadn’t been vaporized by the blast-wave, it would be vaporized by the gamma rays.

And that’s the end of the line. There’s a reason astronomers were so shocked to discover planets orbiting pulsars: pulsars are born in supernovae, and how the hell can a planet survive one of those?

engineering, math, physics, thought experiment

The Treachery of Plumb-Lines

I’m pretty sure that’s my most pretentious article title to date, but really, the only pretentious thing about it is that it’s a Rene Magritte reference, because if you read it literally, that’s exactly what this article is about.

Imagine two skyscrapers. Both start from ordinary concrete foundations 100 meters by 100 meters, and each will be 1,000 meters high, when finished. We’ll call the first skyscraper Ruler, and the second skyscraper Plumb, for reasons I’ll explain.

Ruler is built exactly according to architectural specifications. Every corner is measured with a high-grade engineer’s square and built at precisely 90 degrees. Importantly, Ruler is constructed so that every floor is precisely 10 meters above the previous one, and every floor is 100 meters by 100 meters. This is done, of course, using a ruler. Because it’s kept so straight and square at every stage, Ruler is a very straight, square building.

Plumb, on the other hand, is kept straight and square using one of the oldest tricks in the architect’s book: the plumb-bob. True story: plumb-bobs are called that because, back in the day, they were almost always made of lead, and the Latin for lead is plumbus (or something like that; I took Latin in high school, but the teacher got deathly ill like two weeks in, so I never learned much). A well-made and well-applied plumb-bob is an excellent way to make sure something is absolutely vertical.

The builders of Plumb do use a ruler, but only to mark off the 10-meter intervals for the floors. They mark them off at the corners of the building, and they make sure the floors are perfectly horizontal using either a modified plumb-bob or a spirit level (which is largely the same instrument).

One might assume that Plumb and Ruler would turn out to be the exact same building. But anybody who’s read this blog knows that that’s the kind of sentence I use to set up a twist. Because Plumb was kept straight using plumb-bobs, and because plumb-bobs point towards the center of the Earth, and because the 100-meter difference between the east and west (or north and south walls) gives the bobs an angle difference of 0.009 degrees, Plumb is actually 11 millimeters wider at the top than at the bottom. Probably not enough to matter in architectural terms, but the difference is there.

Not only that, but Plumb’s floors aren’t flat, either, at least not geometrically flat. The Earth is a sphere, and because Plumb’s architects made its floors level with a spirit level or a plumb-bob, those floors aren’t geometrically flat: they follow the spherical gravitational equi-potential contours. Over a distance of 100 meters, the midpoint of a line across the Earth’s surface sits 0.2 millimeters above where it would were the line perfectly, geometrically straight. This difference decreases by the time you reach the 100th floor (the top floor) because the sphere in question is larger and therefore less strongly curved. But the difference only decreases by around a micron, which is going to get swamped out by even really small bumps in the concrete.

“Okay,” you might say, “so if you blindly trust a plumb-bob, your building will end up a centimeter out-of-true. What does that matter?” Well, first of all, if you came here looking for that kind of practicality, then this blog is just gonna drive you insane. Second, it doesn’t matter so much for ordinary buildings. But let’s say you’re building a 2,737-meter-long bridge (by total coincidence, the length of the Golden Gate Bridge). If you build with geometric flatness in mind, your middle pier is going to have to be 14.7 centimeters shorter than the ones at the ends. That’s almost the length of my foot, and I’ve got big feet. It’s not a big enough difference that you couldn’t, say, fill it in with concrete or something, but it’d certainly be enough that you’d have to adjust where your bolt-holes were drilled.

What’s the moral of this story? It’s an old moral that probably seems fairly ridiculous, but is nonetheless true: we live on the surface of a sphere. And, when it comes down to it, that’s just kinda fun to think about.

astronomy, physics, short

Weight of the World

According to this report, the Earth’s mass (M_⊕) is

5,972,190,000,000,000,000,000,000 kilograms

You might notice that there are an awful lot of zeros in that number. That’s because the report doesn’t actually directly specify the Earth’s mass. Like a lot of astronomical papers, it instead uses the Earth’s gravitational parameter, which is the Earth’s mass multiplied by the Newtonian gravitational constant. You see, when it comes to gravity, the force is ultimately determined by the gravitational parameter, rather than directly by the mass. As a result, the gravitational parameter is, as a rule, known to much higher accuracy than the mass. Newton’s gravitational constant is hard to measure, since it’s so tiny, so the report only gives it to six significant digits. So six significant digits is what I gave for the Earth’s mass.

I imagine you’re wondering why the hell I’m talking about all this. Well, I was thinking about planets, whose masses are very often measured in Earth masses. That made me wonder what the mass of say, a person, is, compared to the mass of the Earth. So, without further nonsense, here’s my big list of random objects measured in Earth masses. (I probably need to come up with a better name.)

2.78045 × 10^-51 M_⊕: Hydrogen atom.

1.13926 × 10^-24 M_⊕: a dumbbell

2.279 × 10^-23 M_⊕: me

1.674 × 10^-22 M_⊕: my car

7.023 × 10^-20 M_⊕: the International Space Station

9.878 × 10^-16 M_⊕: the Great Pyramid of Giza

1.671 × 10^-12 M_⊕ : Comet 67P/Churyumov-Gerasimenko

8.620 × 10^-7 M_⊕(not quite a millionth): The Earth’s atmosphere

4.470 × 10^-5 M_⊕ : asteroid 4 Vesta.

1.590 × 10^-4 M_⊕ : asteroid 1 Ceres (the largest in the solar system)

2.344 × 10^-4 M_⊕ (two ten thousandths and change): the Earth’s oceans

0.00219 M_⊕: Pluto

0.0123 M_⊕: the Moon

0.0552 M_⊕: Mercury

0.107 M_⊕: Mars (I always forget how small Mars actually is…)

0.815 M_⊕: Venus (Venus was my second-favorite planet as a kid, after Pluto, which was still a planet back then)

1.000 M_⊕: Earth (Might as well stick it in the list…)

10 M_⊕: Planet Nine (Lower bound. If it exists.)

14.536 M_⊕: the mass of Uranus (I still think it’s funny…)

17.148 M_⊕: Neptune

95.161 M_⊕: Saturn

317.828 M_⊕: Jupiter

332,949 M_⊕: the Sun (1 solar mass, 1 M_☉. Guess who finally learned how to do subscripts!)

26,600 M_⊕: the mass of TRAPPIST-1, which is significant for being one of the smallest stars ever observed, for having seven rocky planets, and for having three planets in its habitable zone. If there’s radio-communicating life on one of them, and we send a message right now, some of you might still be alive if we get the response. Not me. I’d be 98, and I suspect I’m gonna fall into a vat of curry or something stupid like that before then.

672,600 M_⊕: Sirius A, the brightest star in the sky (besides the Sun, obviously)

710,850 M_⊕: Vega, a fairly bright nearby star distorted into a lozenge shape by its rapid rotation.

1,270,000 M_⊕: Alcyone, the brightest star in the Pleiades

2,830,000 M_⊕: UY Scuti, a likely candidate for the largest known star as of March 2017. It’s around 1,700 times the diameter of the Sun, and if you placed it where the Sun is, it’d engulf Jupiter and come close to engulfing Saturn.

3,862,000 M_⊕: Betelgeuse, the bright reddish star on the shoulder of Orion (cue Rutger Hauer.) It’s also an enormous, lumpy star. If you put it where the Sun is, it’d reach at least as far as the orbit of Mars.

33,295,000 M_⊕: the larger component of Eta Carinae, an enormous, extremely bright, angry multiple star that’s so massive and so hot that it’s vomiting its own guts into space and making a pretty nebula in the process.

38,622,000 M_⊕: the poetically-named NGC 3603-A1. With 116 times the Sun’s mass, this is the largest star (as of March 2017, blah blah blah) whose mass is known with any certainty. There are other stars predicted to be more massive, but while their masses are estimated from models of stellar evolution, NGC 3603-A1’s mass is inferred from the orbital period of it and its binary companion, which is much more precise and less guess-y.

2.331 × 10¹⁵ M_⊕: the mass of the Small Magellanic Cloud, one of the Milky Way’s small galactic neighbors.

2.830 × 10¹⁷ M_⊕: the mass of our Milky Way galaxy (roughly).

4.994 × 10¹⁷ M_⊕: the mass of the Andromeda galaxy (roughly).

1.647 × 10²⁸ M_⊕: mass of ordinary matter in the observable universe (atoms and other familiar stuff) (very roughly)

3.349 × 10²⁹ M_⊕: mass of the observable universe, including weird stuff like dark matter and dark energy (very roughly)

astronomy, image, pixel art, science, short, Space, Uncategorized

Pixel Solar System

(Click for full view.)

(Don’t worry. I’ve got one more bit of pixel art on the back burner, and after that, I’ll give it a break for a while.)

This is our solar system. Each pixel represents one astronomical unit, which is the average distance between Earth and Sun: 1 AU, 150 million kilometers, 93.0 million miles, 8 light-minutes and 19 light-seconds, 35,661 United States diameters, 389 times the Earth-Moon distance, or a 326-year road trip, if you drive 12 hours a day every day at roughly highway speed. Each row is 1000 pixels (1000 AU) across, and the slices are stacked so they fit in a reasonably-shaped image.

At the top-left of the image is a yellow dot representing the Sun. Mercury and Venus aren’t visible in this image. The next major body is the blue dot representing the Earth. Next comes a red dot representing Mars. Then Jupiter (peachy orange), Saturn (a salmon-pink color, which is two pixels wide because the difference between Saturn’s closest and furthest distance from the Sun is just about 1 AU), Uranus (cyan, elongated for the same reason), Neptune (deep-blue), Pluto (brick-red, extending slightly within the orbit of Neptune and extending significantly farther out), Sedna (a slightly unpleasant brownish), the Voyager 2 probe (yellow, inside the stripe for Sedna), Planet Nine (purple, if it exists; the orbits are quite approximate and overlap a fair bit with Sedna’s orbit). Then comes the Oort Cloud (light-blue), which extends ridiculously far and may be where some of our comets come from. After a large gap comes Proxima Centauri, the nearest (known) star, in orange. Alpha Centauri (the nearest star system known to host a planet) comes surprisingly far down, in yellow. All told, the image covers just over 5 light-years.

geology, image, science

Pixel Earth 2

A slightly more comprehensive version of the previous post. Once again, each pixel is 1 kilometer deep. The pixels at sea level (the thin green line near the top) are 1 kilometer wide, corresponding to a total width of 6,371 kilometers at sea level or an angular width of 1 radian, or 57 degrees. There’s an increasing horizontal distortion as you go towards the inner core (orange), which becomes infinite at the very bottom row.

In this picture, you’ll find Krubera Cave, the Burj Khalifa, the Kola Superdeep Borehole, a typical thunderstorm, Mt. Everest, a typical volcano, a subduction zone, an airliner at cruising altitude, and the International Space Station. Try and find them: it’s like a badly-drawn Where’s Waldo!

geology, image, physics, science, short

Pixel Earth 1

I present you: a scale model of the Earth’s surface, from an altitude of 400 kilometers down to a depth of 300 kilometers. At this scale, every pixel is 1 km by 1 km.

Legend:

The International Space Station at perigee.
The aurora borealis.
The greatest altitude at which human beings have died: cosmonoauts Georgy Dobrovolsky, Vladislav Volkov, and Viktor Patsayev died just before the reentry of Soyuz 11, when the explosive decoupling of the descent module opened an oxygen seal in the cockpit.
The highest altitude reached by the Air Force’s X-15, which still holds the speed record for a crewed aircraft, and which was among the first crewed vehicles to cross into space.
The official edge of space: the Kármán line, at around 100 kilometers altitude. Above this line, you have to move faster than orbital velocity for wings to provide usable lift, so you might as well just orbit.
The streak denotes the range of altitudes at which meteors glow.
The streak denotes the altitudes at which the 2013 Chelyabinsk meteorite glowed. The starburst denotes the approximate altitude at which it exploded.
The altitude at which the Space Shuttle Columbia stopped sending telemetry and began its final breakup.
On a less sad note: the altitude from which Felix Baumgartner began his famous skydive.
The top of the troposphere (where weather happens); the beginning of the stratosphere; the top of thunderstorms in middle and tropical latitudes.
10,000 meters: the altitude at which passenger airplanes cruise.
The summit of Mt. Everest.
The Challenger Deep (over 10,000 meters deep).
The deepest active mining operation: 4,000 meters, at the Mpomeng gold mine in South Africa.
The deepest human beings have ever drilled: 12 kilometers at the Kola Superdeep Borehole, in Russia.
The deepest confirmed location in a natural cave: 2 km, in Krubera Cave, in Abkhazia, Georgia (the Eastern European Georgia, not the American one.) The cave very likely goes deeper.
Volcanic magma chambers. Contrary to popular belief, most of the mantle is a plastic solid (like very, very stiff Silly Putty), rather than molten. Magma is the exception. The magma chamber that feeds Hawai’i’s volcanoes is on the shallow end of the spectrum. The magma chamber underneath the Yellowstone Caldera (which provides heat for Yellowstone’s famous geysers) sits at around 25 to 35 kilometers deep. We have actual rough maps of it. It’s awesome.
The Mohorovičić discontinuity (or Moho; no, not the KSP one): the official boundary between crust and mantle. It can be as shallow as 5 kilometers deep (beneath the seafloor) and 90 kilometers deep (under mountains); it averages 35 kilometers deep.
Very deep magma chambers.
The end of the asthenosphere, a region of rock made weak and squishy (relatively speaking) by the enormous temperature and pressure. This starts beneath the solid crust (the lithosphere). Its boundary isn’t well-defined.
A hot plume in the upper mantle. Droplets (well, droplet-sized compared to the whole Earth) of lower-melting-point material rise through the mantle to fill magma chambers.

(I should point out that I’m not a geologist. If I’ve made a mistake, please let me know. You won’t hurt my feelings. I’d rather admit I’m wrong than put out a misleading graphic.)

image, short, Space

Earth versus Sun

Earth vs Sun at 1 AU.png

Nothing too special here: just a size comparison between the Earth and the Sun. The only difference from the usual ones, is that I’ve based their relative sizes on their angular diameters. For the Sun, I computed the angular diameter at a distance of 1 AU (which is how we see it here on Earth). For the Earth, I computed the angular diameter at a distance of 1 AU minus the diameter of the Sun. In other words, the Earth appears as large as it would if it were sitting at the point on the Solar surface nearest us. This is how the Earth would look as a very unfortunate close-transiting planet.

To paraphrase Carl Sagan: that little blue blob is home. That’s us. Everything that’s ever happened to you happened there.

Now consider that compared to the Sun…

Earth vs Sun Closeup.png

Here’s a closeup of the same image, showing the Earth compared to the weird convection granules on the Sun’s surface.

Both images are from NASA. The Solar image is from the Solar Dynamics Observatory (HMI intensitygram, February 7th, 2016), and the Earth-disk image is from the GOES earth-observing satellite.

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 hits 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.

Sublime Curiosity

A zoo for thought experiments.

Tag Archives: earth