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, 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



  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.)

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.

biology, math, science, short, statistics, thought experiment

Short: Immortality Math

There are people out there who are quite seriously trying to make human beings immortal. It sounds like something from a bad 1970s pulp comic, but it’s true. Of course, when serious people say “immortal,” they’re not talking Highlander. They’re talking biological immortality, sometimes called by fancy names like “negligible senescence”: the elimination of death by aging. Whether we can (or should) ever achieve biological immortality is a question I’ll leave to people smarter than me, but either way, biological immortality doesn’t mean full immortality. It just means that you can no longer die from, say, a heart attack or cancer or just generally wearing out. You can still quite easily die from things like falls, car accidents, or having Clancy Brown chop your head off with a sword.

There are a number of organisms out there which are either believed or known to be biologically immortal, or at the very least, nearly so. These include interesting but relatively simple organisms like hydras and jellyfish, but also more complex organisms like the bristlecone pine (many living specimens of which are confirmed to be over 1,000 years old, and one of which is over 5,000 years old), and the lobster. (Technically, though, the lobster isn’t really immortal, since they most molt to heal, and each molt takes more energy than the last, until the molts grow so energy-intensive they exhaust the lobster to death.) For the record, the oldest animal for which the age is well-established was a quahog clam named Ming Hafrun, who died at 507 years old when some Icelandic researchers plucked it out of the water.

If a human was made biologically immortal, how long could they expect to live before getting hit by a bus or falling down the stairs (or getting stabbed in the neck by Christopher Lambert)? That’s actually not too hard to estimate. According to the CDC (see Table 18), there were 62.6 injury-related deaths per 100,000 Americans, in 2014. With a bit of naive math (I’m not adjusting for things like age, which probably inflates that statistic a fair bit, since older people are at a higher risk of falls and similar) that means the probability of death by accident is 0.000626 per year, or roughly 0.06%. Knowing that, it’s almost trivial to compute the probability of surviving X years:

probability of surviving X years = (1 – 0.00626)^X

This formula is based on one of my favorite tricks in probability: to compute the probability of surviving, you do the obvious and convert that to the probability of not-dying. And you can take it one step further. At what age would 90% of a biologically-immortal group still be alive? All you have to do is solve this equation for N:

0.9 = (1- 0.00626)^N

which is no trouble for Wolfram Alpha a math genius like me: a biological immortal would have a 90% chance of surviving 168 years. Here are a few more figures:

  • A 75% probability of living up to 459 years.
  • A 50% probability of living up to 1,107 years.
  • A 25% probability of living up to 2,214 years.
  • A 10% probability of living up to 3,677 years.
  • A 5% probability of living up to 4,784 years.
  • A 1% probability of living up to 7,354 years.
  • A one-in-a-thousand chance of living 11,031 years.
  • A one-in-a-million chance of living 22,062 years.

For reference, the probability of a member of a population surviving (in the US, in 2012, including death by biological causes) doesn’t drop below 75% until around age 70. To put it in slightly annoying media jargon: if we’re biologically immortal, then 459 is the new 70.

biology, physical experiment, science, short, Weird Food

Real Mad Science 2: Mead

Mad Science Mead.png

I used to be pretty fond of booze. My favorite libations were Johnnie Walker Black Label, cheap supermarket Moscato, this horrible fluorescent-blue fruity cognac stuff called Hpnotiq (yes, really), and White Russians. But, towards the end of my college career, I made a nasty error: I got careless with Jägermeister. Jäger, exactly like plutonium or nitroglycerin, is very unforgiving of carelessness. The next day was among the ugliest in my life, and I’ve hardly touched hard liquor since then.

That’s actually a good thing, since it nudged me towards drinking less stomach-scorching things like proper decent wine and beer (and alcoholic ginger beer). And, since I’ve been in a mad science mood lately, I decided I’d take advantage of Amazon’s “Black Friday is happening some time during this month money money money” sale and pick up a little proper brewing equipment.

Several words of warning. 1) Home brewing comes with risks. You could get nasty unwanted yeasts or bacteria or mold that turn your brew toxic. To that end, I sterilized my equipment with a cheap and easy (and slightly nostril-stinging) potassium metabisulfite-citric acid wash. 2) Booze can get you into trouble if you don’t treat it with respect, and it’ll get you into a lot of trouble if you’re under drinking age. 3) Home brewing is illegal some places.

Now that I’ve made it painfully obvious that I’m trying not to get sued, it’s time to make mead! Mead is a fermented honey beverage favorited by Norsemen, English Majors, Beowulf, and pretty much everybody in Skyrim. Here’s how I made it:

I added two cups of honey (about 475 mL) to a saucepan. Because it was a really cold day, I added some water to make the honey less viscous. (Filtered well water, mind you.) To make sure the yeast had vitamins and minerals that pure honey might not provide, I added a generous handful (roughly 1 cup, or 100 – 150 grams) of cranberries, along with a modest handful of raisins. They were just what I had lying around. For flavor, I added about a teaspoon of cinnamon. I brought the whole mixture to a boil. I checked the temperature with an instant-read meat thermometer (never use the proper tool for the job, I always say). Once it reached 212° Fahrenheit (100° Celsiusigrade), I started a ten-minute timer. There are probably nasty unwanted things like weird bacteria, wild yeasts, mold, and microscopic politicians in the fruit and maybe the water and honey, so I figured ten minutes at a boil would heat everything enough to kill them.

Before I started boiling the mixture, I had prepared my brewing gear according to reasonable sanitary standards. Into a 1 gallon (3,750 mL) glass carboy (moonshine jug, as I’m sure many people call them), I added half a gallon (roughly 2,000 mL) of clean filtered well water. To that, I added two teaspoons of powdered potassium metabisulfite and one tablespoon of granulated food-grade citric acid. The reaction produces sulfur dioxide, which kills germs. (Don’t smell the jug while the chemicals are sitting in there: sulfur dioxide really burns the nose…) Just before it was time to pour the fruit-honey-water mixture into the carboy, I gave the carboy and the airlock (which keeps dust and other potential germ-carrying stuff from falling into the carboy during fermenting) a rinse with clean water. I let the boiled mixture cool and then added it slowly to the carboy, which I’d warmed in the oven. I didn’t want to risk temperature differentials shattering the glass. Luckily, there were no problems. I bloomed 2 grams of distiller’s active dry yeast in a cup of warm water with two tablespoons of sugar dissolved in it, then added the bloomed yeast to the carboy. I topped it up almost to the top with clean water, then added the airlock, filled the airlock with water, and gave the jug a gentle shake to get things going.

That was yesterday. Today, when I took the picture, the mead was bubbling merrily away. It’s eating sugar and making things like carbon dioxide and ethanol. The airlock produces a bubble once every five or ten seconds, which tells me the fermentation’s going well. I’m not sure how long you’re supposed to leave mead, but I guess I’ll wait until the bubbling stops, which will tell me I’ve got a bunch of dead yeast drowned in ethanol. I suppose I’ll make the tasting of the mead part of my weird food series…

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


(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!