Can You Estimate That?

“How many piano tuners are there in Boston?” That was the first question on our first problem set of freshman physics at MIT. The question was not really about pianos or the people who tune them. It was a way to get us to make estimates based on facts that we know.

The first step in any estimate is to recognize that there is such a number, even if it is very big and even if it cannot be known with precision. Mr. Spock in Star Trek would get annoyed with humans for not being precise. But his approach is actually not scientifically helpful most of the time.

What is far more useful than a Mr. Spock answer is an “order of magnitude” estimate. Meaning that you know how many zeroes are in the answer. Suppose I ask you how much the Earth weighs. Start with the fact that you know more than you think and that you probably have a starting point for the answer.

Do you know how big the Earth is? If not, you may know how long it takes to fly across the country and how fast planes go. The continental U.S. has four time zones. There are 24 time zones. So, the U.S. spans about 1/6 of the Earth at this latitude. Most Americans know it is about 3,000 miles across the U.S. Meaning it is about 6 x 3000 = 18,000 miles around at this latitude.

This is when we can start playing the order of magnitude game. Instead of 18,000 miles, round it off to 20,000. An order of magnitude (10,000) with one digit of precision. The Earth is actually about 24,000 miles around at the equator.

You also should know something about weights and volumes. A pint of water is a pound, and a liter of water is a kilogram. The Earth is made of rock. Rocks sink in water. Maybe you would guess that rock is ten times heavier than water.

The rest of the estimate is just making conversions and simple calculations. There is a physics joke with the punch line “imagine a spherical cow.” If you don’t remember the formula for the volume of a sphere, “imagine a cubical Earth.” If the Earth is 20,000 miles around and it is a cube, then imagine a cube that is 5,000 miles on a side.

It is good to switch to metric units at this point. 5,000 miles is about 8,000 km. It is also helpful to switch to exponential notation. 8,000 km becomes 8 x 10^3 km or 8 x 10^6 meters. The volume of this cube would be 8x8x8 x 10^6 x 10 ^6 x 10^6 which is 512 x 10 ^ 18 cubic meters. A cubic meter of water weighs a metric ton which is why the metric system is handy! And a metric ton is about like a familiar ton within an order of magnitude!

So, you now know that the Earth weighs about 512 x 10^18 tons if it were water. But we guessed rock weighs 10 times as much, so it comes to 512 x 10^19 tons. Or 5 x 10 ^ 21 tons. A Google search shows this is about right to within an order of magnitude!

Why does this matter? As a good citizen in a modern technological world, we need to be able to ask good questions and recognize adequate answers. Suppose someone proposed that a solution to the Climate Crisis is to move the Earth further from the Sun with some rockets. You now have the tools to make an estimate of how much energy it would take to do that. And you would know this is an absurd proposition.

Looking at things in terms of orders of magnitude also can put things in perspective without having to know a lot of Mr. Spock level of detail. Fossil fuels are an accumulation of carbon from 100 million years of living things that have died and been buried in rock. Suppose I told you that I was going to put 100 million years’ worth of accumulated carbon from living things into the atmosphere over the course of about 100 years. Would that give you a sense that this might not turn out well?

Making such estimates can become a fun habit. You might want to learn a few numbers that make some estimates easier. Like the interesting fact that all chemical fuels from donuts to diesel fuel have about 100 calories of energy per ounce. To within an order of magnitude! Have fun!