A few years ago, I thought I would pursue a career in cosmology. To prepare for such a trajectory, most of my electives during my first few years of college involved astrophysics. My interest has since switched to biophysics. The transition was less difficult than I expected. Many of the basic ideas used in astrophysics carry over the biophysics, including the use of order of magnitude estimations.
Some particular students complain that order of magnitude estimations aren't accurate or precise enough. That we must know every fundamental constant to at least 17 places after the decimal or else we're failing as scientists. In reality, order of magnitude estimates are a good tool to give you an idea of what scale you're working with. Is the Earth 100, 1000, 1000000000 kilometers in radius? Are there hundreds, millions, trillions of bacteria in my gut? What is a reasonable amount of concrete to order for my construction project? 10 pounds, 100 pounds, more?
Working with orders of magnitude also allow you to simplify your calculations so you can focus on how the variables interact with each other. What happens when the radius of a marble falling through a liquid is very big, orders of magnitude bigger than the other variables like the viscosity and density of the liquid? What if it is small? In what range does the density of the liquid affect the marble's behavior more than any other variable? How do I scale these variables to apply them to the problem of a submarine moving through water? You don't need to calculate precise constants or results until you're ready to look for small details, confirm the numerical prediction of a theory, or design and build whatever you're planning. Of course, if you're an engineer, please, please don't stop at order of magnitude estimations when designing your bridge. But for astrophysicists and biophysicists, order of magnitude calculations can provide useful insight into the problem at hand.
For a quick example of an order of magnitude problem, I will estimate how many human cells and how many bacteria cells are in the human body solely from prior knowledge. However my prior knowledge includes some things I learned in class. To even out the playing field, I will just say that there are 2 kilograms of bacteria in the average human gut and an E. coli cell has a volume 1 cubic micrometer. Now we can begin.
Start by thinking about the average mass of an adult human. Are we ~10 kilograms, ~100 kg, ~1000 kg? Our answer is in the 100 kg range. We can get more accurate than that if we want to, but it probably won't make a difference. Trying using 70 kg instead of 100 kg if you want to check for yourself.
At some point in elementary school, you probably learned that about 2/3 of the human body is made up of water. That leaves us with 1/3 of 100 kg or 33 kg worth of human cells in the human body. Note that I didn't bother subtracting out the 2 kg of bacterial cells that I mentioned at the start of the problem. That's because 100 - 2 = 98 which is more or less equal to 100. Once again, if you don't trust me, I encourage you to substitute 98 for 100 and see how much the answer changes.
I don't know the mass of the average human cell, but it's not too difficult to make a reasonable guess. First I will estimate lower and upper bounds for the size of a human cell. I said earlier that an E. coli cell is about 1 cubic micrometer. Human cells are larger than E. coli cells, so 1 cubic micrometer is my lower bound. For the upper bound, I will consider the volume of an onion skin cell, since I remember looking at them under microscopes in high school. Let's say they have a volume of 100 cubic micrometers. Then we can take the square root of the product of our bounds to get a sort of weighted average. The result of sqrt[(10^-19) cubic meters * (10^-12) cubic meters] is about 10^-16 cubic meters.
We have an estimate for the volume of the average human cell. By assuming that the cell is mostly water and therefore has the density of water, we can determine the mass. Multiply our volume by the density of water to get (10^-16 cubic meters) * (1000 kg / cubic meter) = 10^-13 kg. Then to find the number of human cells in the body we multiply like so: (33 kg) * (1 human cell / 10^-13 kg) = 10^14 human cells in the body, approximately. Check the results using 98 kg and 70 kg instead of 100 kg. Doesn't make much of a difference does it?
To find the number of bacteria cells in the average human body, first find the mass of an E. coli cell using the same assumption as above that its density is equal to that of water. The result: 10^-19 cubic meters * (1000 kg / 1 cubic meter) = 10^-16 kg. Then 2 kg * (1 bacteria / 10^-16 kg) leads to 10^16 bacteria in the human body. There are more bacterial cells than human cells in the human body!
Searching online for answers about how many human cells and how many bacterial cells are in the body gives values close to the ones estimated here. Where "close" means possibly one of two orders of magnitude off. But what's a couple of orders of magnitude between friends, eh?
