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.

Standard
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)

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

Pixel Solar System

pixel-solar-system-grid

(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.

Standard
geology, image, science

Pixel Earth 2

1 Radian Wedge Pixel Art.png

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!

Standard
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.crust-1-px-eq-1-km-numbered-large

 

Legend:

  1. The International Space Station at perigee.
  2. The aurora borealis.
  3. 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.
  4. 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.
  5. 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.
  6. The streak denotes the range of altitudes at which meteors glow.
  7. The streak denotes the altitudes at which the 2013 Chelyabinsk meteorite glowed. The starburst denotes the approximate altitude at which it exploded.
  8. The altitude at which the Space Shuttle Columbia stopped sending telemetry and began its final breakup.
  9. On a less sad note: the altitude from which Felix Baumgartner began his famous skydive.
  10. The top of the troposphere (where weather happens); the beginning of the stratosphere; the top of thunderstorms in middle and tropical1 latitudes.
  11. 10,000 meters: the altitude at which passenger airplanes cruise.
  12. The summit of Mt. Everest.
  13. The Challenger Deep (over 10,000 meters deep).
  14. The deepest active mining operation: 4,000 meters, at the Mpomeng gold mine in South Africa.
  15. The deepest human beings have ever drilled: 12 kilometers at the Kola Superdeep Borehole, in Russia.
  16. 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.
  17. 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.
  18. 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.
  19. Very deep magma chambers.
  20. 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.
  21. 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.)

Standard
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.

Standard
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.

Standard