Uncategorized

# 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:

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.

Standard