Fun With Logarithms 3: Orders of Magnitude

I started hearing the term “Orders of magnitude” long before I knew what it meant. It’s actually a simple and extremely handy convention. 10 is one order of magnitude larger than 1. 100 is one order of magnitude larger than 10. 0.000001 is six orders of magnitude smaller than 1. Simple.

The best thing about orders of magnitude is that they allow you to get an intuitive grasp on large numbers, which are just about as unintuitive as it gets, and with hardly any math or research necessary. Here’s an example.

I’m 1.9 meters tall (6’3″, if you insist). A peppercorn has a diameter of about 2 millimeters. A millimeter is 1,000 microns. Ordinary bacteria have dimensions of around 1 micron. Therefore, a peppercorn is about 1,000 times larger than a bacterium.

A peppercorn is pretty small, but if you look at it closely, you can still make out details: peppercorns have weird little wrinkles and chips and dust on their surfaces. A peppercorn is an easy object to comprehend. To comprehend the size of a bacterium, go the other direction: imagine an object 1,000 times larger than a peppercorn. That comes out to about 2,000 millimeters, or 2 meters. That means I am to a peppercorn as a peppercorn is to a bacterium. Or, if you want to use me as your basis, a bacterium sitting on a peppercorn is like me standing on the summit of a 1,900-meter mountain. That’s close to the same as me standing on top of El Capitan:

(Source and licensing.)

Anyway.  Bacteria are pretty small. But they’re not as small as viruses, which have dimensions measured in tens or hundreds of nanometers. Twenty nanometers is about eight orders of magnitude smaller than me, so a virus standing next to me is like me standing next to Jupiter.

That’s a lot less intuitive. I know mathematically how big Jupiter is, but it doesn’t make any sense to my gut, and it’s important to take the gut into consideration if you really want to get a feeling for anything. Luckily, you can chain orders of magnitude end-to-end.

A 20-nanometer virus particle is 100 times smaller than a 2-micron bacterium. Therefore, if El Capitan represents me, and I represent a bacterium (which is a weird thought), the virus will be 1.9 centimeters across, or about the size of a wine grape. A virus compared to a person is like a grape on the summit of a mountain. That’s stretching the powers of intuition, but it’s still comprehensible.

Let’s have more fun! A football field (American football or the football that most of the world calls football but Americans call soccer, the proper name of which people really like to argue about for some reason) is about 100 meters from end to end. I stood on many such fields in gym class as a child, so I have a pretty good intuitive grasp of their size. 100 meters is a large distance, but not so large you can’t wrap your head around it, so it’s useful for making even harder order-of-magnitude comparisons.

Most atoms are around 200 picometers across. 1 nanometer is 1,000 picometers, so 100 picometers (ignoring the factor of two, which you’re allowed to do in order-of-magnitude math) is ten orders of magnitude, or ten billion times, smaller than a meter.

A human hair is about 100 microns in diameter, and is another good basis for comparison, since 100 microns is about as small as something can get and still be visible to the average naked eye. 100 microns is 4 orders of magnitude smaller than 1 meter. 100 picometers is 10 orders of magnitude smaller than 1 meter. Therefore, 6 orders of magnitude (or a factor of 1 million) separate the diameter of an atom and the diameter of a hair. It just so happens that a 100-meter football field is 6 orders of magnitude larger than a 100-micron-diameter hair. So an atom in a strand of hair would be in the same proportion as the diameter of that hair compared to the length of a football field.

This same kind of sloppy-but-useful math can be applied to understand astronomical distances, too, up to a point. The Earth has a diameter of about 12,000 kilometers (closer to 12,700, actually, but we’re not being that precise). 12,000 kilometers is 10 million times larger than 1.2 meters. The earth, therefore, is about 10 million times larger than me. Something ten million times smaller than me would be 1.9 microns across, which is the size of a bacterium. Therefore, the bacteria sitting on my skin feel just as small as me standing on the Earth. Which is weird and almost poetic. Almost.

But my understanding of 1.9 microns is abstract. (How many times have I said “intuitive” and “abstract” so far? You have my permission to begin a drinking game. Intuitive abstract intuitive abstract intuitive abstract intuitive abstract. Enjoy your evening and try to vomit into the toilet. Intuitive. Abstract.) Let’s use a football field as our basis instead. I am 7 orders of magnitude smaller than the Earth. In order to be 7 orders of magnitude smaller than a football field, an object would have to be 7 – 2 = 5 orders of magnitude smaller than a meter, or 10 microns. Still too small.

El Capitan is about 1,000 meters tall (it’s actually over 2,000, but that’s within the same order of magnitude). That’s 3 orders of magnitude larger than a meter. To be 7 orders of magnitude smaller than El Capitan, an object would have to be 4 orders of magnitude smaller than a meter, or 100 microns, which is the diameter of a hair or a speck of dust. So we are all specks of dust sitting on the mountaintop that is the Earth. (Feel free to punch me for that sentence, if you happen to meet me in the street. Seriously. I’d punch myself right now, except I keep flinching out of the way.)

You can also use this kind of tricky math to build toy models of astronomical systems in your head. For instance: what would the Earth and Moon look like, locked in their orbits, if you saw them from a distance?

Well, the Earth is about 10,000 kilometers across. The moon is about 3,000 kilometers across, or one-third of an Earth diameter. Imagine a really big grape sitting next to a smallish grape, and you’ll have about the right proportions. A coin and the bottom of a drinking glass are also in similar proportions.

The Moon’s semimajor axis is about 300,000 kilometers, so it’s usually separated from the Earth by 30 Earth diameters or 100 Moon diameters. Here’s an experiment you can try at home: set a cup on the floor and line up 98 coins next to it (Not 100, because one and a half coins will be inside the cup and half a coin will be inside the coin representing the moon). Actually, I hate it when people say “Here’s an experiment you can try at home.” I’m gonna be nice and do it for you, although admittedly I’m going to have to switch up and use a coin to represent the Earth, because otherwise, the system would be larger than my floor. But I cheated and did the math and got the proportions right.


Consider that penny. That’s here. That’s home. That’s… No. Sorry. I can’t be sarcastic about Carl Sagan’s Pale Blue Dot speech. Just go watch it. Watch it twice. No smart-assery here: it’s one of the best speeches I’ve heard in my life.


Fun with logarithms 2: A race between a snail and a beam of light.

In my last post, I mentioned how cool logarithmic scales were. A logarithmic scale, for instance, takes two numbers that differ by a factor of one trillion (say 0.001 and 1,000,000,000, which differ by 999,999,999.999) and reduce their difference to a much more manageable number: 12.

Logarithmic scales are handy in the universe we happen to live in, because, as you might have noticed, this universe contains a lot of very tiny things and a lot of ridiculously large things. It contains both bacteria and galaxies, which differ in scale by a factor of 950,000,000,000,000,000,000,000,000. But today, we’re not talking about distances. Today, we’re talking about speeds. Some things in the universe move very quickly. Others move very slowly. Continents drift together (and apart) at speeds of around 1.585×10^-9 meters per second, which is less than the diameter of a DNA helix every second. Light, on the other hand (which, as far as we know, moves as fast as it is physically possible to move) travels at 299,792,458 m/s. This is the perfect place for a logarithmic scale.

Therefore, I present to you, the Bolt, a logarithmic scale named in honor of Usain Bolt, who is the fastest Jamaican in the world. He’s also the fastest human in the world. To get a measurement of an object’s speed in Bolts, divide that speed by our reference speed (1 meter per second, which is about walking speed), and take the base-10 logarithm of the result.

But actually, the Bolt is kind of a large unit. Ironically, we’ve gone from too large a scale (0.0000000015 m/s to 299,792,458 m/s) to too narrow a scale. So let’s take inspiration from the decibel, and multiply the result of our logarithmic calculation by 100, which gives us a measurement in centiBolts (cBo).

Let’s work an example before I get to the list, which is the fun part. The land speed record for a garden snail (which record, apparently, you can only challenge at the World Snail Racing Championships in Congham, England) is 0.002752 meters per second. To get the speed in centiBolts, we calculate 100 * log10((0.002752 m/s) / (1 m/s)), which works out to -256.035 centiBolts. Now that you know how the math is done, let’s compute the centiBolt rating for some ridiculously low and ridiculously high speeds!

With a helium-neon laser and a Michelson interferometer, you could (probably) measure a change in distance of 300 nanometers over the course of an hour, which is 0.00000000009 m/s or -1005.61 centiBolts.

Continents drift apart (or together) at a speed of about 3 centimeters per year, or -902.2 cBo.

Because it loses orbital energy by stretching the Earth (making tides) and slowing Earth’s rotation, the Moon’s orbit is very gradually getting larger. The moon, therefore is receding at about 3.8 cm/year, or -891.9 cBo.

Human hair grows at about 15 cm/year, or -832.3 cBo.

Bamboo grows at a rate of about 14 microns per second (meaning it can grow several feet in a day), giving it a speed of -485.4 cBo.

Jakobshavn Isbræ glacier, in Greenland, reaches a maximum speed of 12,600 meters per year, which comes out to -339.8 cBo.

As I mentioned before, the land speed record for a garden snail is 0.00275 m/s or -256.07 cBo.

A wind speed of 1 mile per hour (MPH) would just barely be detectable by the drift of smoke. It wouldn’t even move weather vanes. It gets a speed rating of -34.97 cBo.

1 m/s, being our reference point, gets a rating of 0 cBo.

Usain Bolt, for whom we named this unit, ran at an average speed of 10.438 m/s when he broke the 100 m world record. He was running at 101.9 cBo…

…but his maximum speed was 12.42 m/s, or 109.41 cBo.

In the United Sates, most non-residential roads have a speed limit of 45 MPH. In my experience, no matter the speed limit, people usually drive around 50 MPH, which is 134.9 cBo.

I once drove my car at 105 MPH (don’t tell anybody). I was moving at 167.2 cBo.

The Bugatti Veyron, the world’s fastest street-legal production car, can get up to 267.856 MPH (431.072 km/h), or 207.825 cBo.

The fastest wheel-driven car on record (as of May 2014) got up to 403.10 MPH (648.73 km/h), which is a terrifying 225.58 cBo.

The land speed record (again, as of May 2014, set by the ThrustSSC, the first land vehicle to break the sound barrier) is 760.343 MPH (1223.657 km/h), or 253.1356 cBo.

The F-22 raptor can supposedly reach Mach 2 (the actual top speed is probably classified), which is 277.093 cBo.

The awesome-looking (and sadly retired) SR-71 Blackbird could manage 2,193.2 MPH, or 299.1 cBo.

The even more awesome rocket-powered X-15 could do 4,160 MPH (6,695 km/h), or 326.9 cBo.

At this point, we’re moving from the realm of really fast aircraft to the realm of really slow spacecraft.

The International Space Station orbits at 7.656 km/s, which is 388.4 cBo. It moves so fast, that it only takes 14 milliseconds to travel its own length. Curious about what 14 milliseconds sounds like? So was I. It sounds like this: https://soundcloud.com/hobo-sullivan/the-iss-passes. Which is just barely long enough for my ears to recognize as an actual sound.

The human speed record (relative to the Earth) was set by the crew of Apollo 10, who reached 24,791 mph, or 404.5 cBo. At these speeds, which are frankly quite ridiculous, the spacecraft was covering 1 kilometer every 90 milliseconds. To give you an idea how fast that is, here’s a 90-millisecond tone: https://soundcloud.com/hobo-sullivan/apollo-10-travels-1-kilometer.

The light gas gun at Sandia Labs (which, it has been noted, is essentially a high-tech BB gun) can fire projectiles at a silly 36,000 MPH, or 420.7 cBo. At these speeds, a small plastic projectile can make a hand-sized crater in a block of aluminum.

The Helios 2 solar probe holds the record for the fastest human-made object. When it swung by the sun (getting slightly closer to the Sun than Mercury), it hit 157,100 MPH, or 484.7 cBo.

But, as all physics geeks know, 157,100 MPH isn’t all that fast. It’s only 0.00023 times the speed of light, or 0.00023 c (as science-fiction authors put it). There are much faster things in this crazy-ass universe.

Like, for instance, the brightest component of a lightning bolt, the return stroke, which, according to some measurements, reaches 220,000,000 MPH, or 0.3281 c, or 799.3 cBo.

The Large Hadron Collider can accelerate protons much faster. It can get them up to 670,615,282 MPH, 0.999999991 c, or 847.7 cBo.

Which is not far from the maximum speed any object can attain, the speed of light itself: 670,616,629 MPH, 299,792,458 meters per second, or 847.7 cBo.

Sorry, Mr. Bolt. You’re going to have to train harder. You’re still running 29,979,246 times slower than you could be. Are you even trying? Come on!


Decibels of DEATH!

Ear Protection

When I see the word “decibel,” I think two things. First, I think “Noise.” Then, I think “Oh god, decibels confuse the hell out of me…”

Well, I think I finally understand the decibel. It’s kind of a weird unit, but it’s also nifty, and it showcases one of the coolest things in mathematics: the logarithm.

Here’s how you compute the decibel-level of a sound. First, you figure out the acoustic power of that sound, probably using a microphone (or using a sound engineer who has a microphone and understands better than I do the difference between “acoustic power” and “sound amplitude.”) The acoustic power tells you the maximum pressure the sound wave exerts on things (say, your eardrums). Most of the time, you measure that sound pressure in pascals. Take that sound pressure and divide it by 20 micro-pascals. 20 micro-pascals is a semi-arbitrary reference point. It’s about the sound pressure where a 1000-Hertz sine wave first becomes audible to a human ear. It’s not a lot of pressure. The pressure 10 meters underwater is about twice what it is at sea level, which means the overpressure is about 1 atmosphere (1000 hectopascals. I’d like to note that Hectopascal would be a good name for a movie villain.) Well, the depth of water it would take to get an overpressure of 20 micropascals is 2 nanometers, which is about the diameter of a strand of DNA. Did you know human ears were that sensitive? I didn’t.

Anyway, you can use decibels to express a wide range of noise levels without using too many digits (Because, let’s face it, we all start zoning out after you get beyond about six digits, give or take.) To get the decibel number, you divide your sound pressure by 20 micropascals, take the base-10 logarithm of that, and multiply the result by 20. For example: a sound pressure of 20 micropascals gives you 20 * log10(20/20) = 20 * log10(1) = 20 * 0 = 0 dB.

With a title like Decibels of Death, you knew this article was going to be all about extremes. The quietest officially-measured place in the world is the anechoic chamber at Orfield Laboratories. It’s a room encased in a foot-thick concrete vault. The room itself sits on I-beams which are on springs to isolate external vibration. The inside of the room is full of wedge-shaped foam blocks which prevent echoes and dampen the sound from outside even further. The Guinness Book of World Records measured the sound level in the Orfield anechoic chamber at -9.4 decibels. That works out to a sound pressure level of 6.8 micropascals. To produce an overpressure that small, you’d only need a layer of water 0.612 nanometers thick. At that point, it’s less a puddle and more a molecular stack. That’s pretty damn quiet.

It’s actually intolerably quiet, apparently. The longest anybody’s ever spent in the chamber is 45 minutes, according to that Daily Mail article I linked above. I’ve heard stories about people who freaked out in the chamber because, all of a sudden, they can hear their heartbeats. And some people have auditory hallucinations when deprived of sound long enough, which probably makes the Orfield chamber even scarier.

So -9.4 dB is quiet enough to make you crazy. 0 dB is the threshold of hearing. 10 dB is about the quietest environment you or I will ever experience, and that’s only if we don’t breathe too loud. 25 dB is a very quiet room. According to a funky app I’ve got on my smartphone, the noise level at this desk is 51 dB. The EPA (the US environmental agency) recommends your everyday environment not exceed 70 dB. 85 dB can cause hearing damage over the long-term. 130 dB is painful. 150 dB can rupture your eardrums. This is what I was talking about earlier: logarithmic scales allow you to convert numbers orders of magnitude apart into nice numbers with low digit counts, which makes it easier to compare them side-by-side. When we started out, back at -9.4 decibels, the pressures were so low they were almost impossible to measure. Now, they’re so high they’re doing organ damage.

And speaking of organ damage… The strength of a blast wave is measured by its overpressure, just like the strength of a sound wave. In this fascinating and unnerving paper, some doctors report the effects of 62,000-pascal blast waves on rats. They speak of “minimal to mild alveolar hemorrhages,” as though there were such a thing as a mild case of bleeding fucking lungs. The upshot of all this is that, although 150 dB may burst your eardrums, 189.9 decibels (which is the decibel equivalent of 62,000 pascals overpressure) can actually damage your guts.

But if you’re catching a 190-decibel blast, you’ve got more serious things to worry about than bleeding lungs. Yes, really. There are a lot of reports of soldiers who have been hit by blasts from roadside bombs and car bombs and other such nasty things. Some of these soldiers, although they didn’t hit their heads on anything and nothing hit them in the head, developed serious cognitive problems: difficulty concentrating and short-term memory loss, enough to pretty much spoil their day-to-day lives. In another experiment, rats exposed to a blast overpressure of 20 kilopascals (180 decibels) experienced similar symptoms, and when they were dissected, had lots of dying brain cells. Which is all really pretty damn sad.

As it turns out, there’s actually technically a maximum sound pressure, at least if you want an undistorted sound wave. A pure tone has the shape of a sine wave: the pressure rises a certain amount above atmospheric, drops in a graceful sinusoidal curve, falls that same amount below atmospheric pressure, returns to atmospheric pressure, rinse and repeat. The thing about these kinds of sine waves is that, after their maximum overpressure, they have to drop that far below atmospheric pressure. And if the sound pressure of your sine wave happens to be greater than atmospheric pressure, that can’t happen: pressure is a number that doesn’t go any lower than zero, which is a vacuum. So a sine wave with a sound pressure larger than 1013 hectopascals (1 atmosphere) will sound all right when the pressure goes up, but will get cut off (“clipped,” the sound-engineer people call it) when it goes down. (And when I say “Will sound all right” I mean “Will rupture your aorta, destroy your lungs, tear your limbs off, and knock your house down,” as we learned from nuclear tests.) The maximum for unclipped sound, therefore, is 194.1 decibels.

But since we’re already blowing everything up, why worry about a little distortion? You know the Barrett M82? The big-ass .50-caliber sniper rifle? That one from that movie The Hurt Locker? The big scary one? Well, when that thing fires, its cartridge sees a blast wave of 265.5 decibels, which is just one more reason not to live in a rifle barrel.

You would experience 270 decibels if you were standing about 100 meters (350 feet) from a 1-megaton nuclear bomb when it went off. I use “experience” loosely here, since you wouldn’t have long to enjoy the racket before you were spread over an alarmingly large area.

Now, let’s say you were standing on the surface of a star just as it went supernova. Well, you’d be exposed to a blast pressure of something in the (very rough) neighborhood of 476 decibels, which I’m pretty sure the EPA would classify as “potentially hazardous.”

As it turns out, there’s a maximum pressure that is still physically meaningful, at least according to our current understanding of physics. It’s called the Planck Pressure, and it’s very large. It’s the kind of pressure you get inside black holes. It’s the kind of pressure the universe experienced (we think) right after the Big Bang. The Big Bang had a noise rating of 2,367.3 decibels. The explosion that set the current universe in motion had a pressure which can be quantified in five significant digits.

That’s what I mean about logarithmic scales being awesome. They turn unimaginable cosmic numbers into nice, manageable, comprehensible numbers. You’d better believe I’m going to be playing with logarithmic scales again soon. Which sounds way dirtier than I intended.