One of the things I teach in my undergraduate courses is how
to do order-of-magnitude calculations.

An order-of-magnitude calculation (otherwise known as
“Fermi-problem”) is a method to estimate quantitative answers for complex
problems by combining smart logic with pre-calculus arithmetic. No calculators allowed! But scrwaling on napkins is encouraged.

Here’s one of my favorite order of magnitude questions
because 1. It’s not difficult 2. But it’s an interesting Earth problem 3. With gobs
of science-y richness at its center. I do it in all of my undergraduate and
graduate geophysics classes.

The problem is this: Given the following map of surface heat
flux, make an order-of-magnitude of the total surface heat flux coming from the
Earth’s interior.

Map of Earth's Surface Heat Flux From Davies and Davies (2010) via Wikipedia |

Here’s how I break it down for an undergraduate class:

1. What quantity is mapped here and what are the units?

2. What are the lowest values and where are they?

3. What are the highest values and where are they?

4. What is the average value of heat flux for the surface of
the Earth?

5a. optional—how does this value compare with an
incandescent light bulb (I can joke here about how this question will be
obsolete soon)

5b. optional—how does this value compare with our own (human)
energy output? (appeal to the ergometers machine at the gym here)

5c. optional—how does this value compare with the solar
heat flux?

6. Now that we have an average value for heat flux, what
other information do we need to get the total heat?

7. How does one estimate Earth’s surface area? (crowd-source for formula for surface area of sphere –remind students that this is a good formula to memorize. Crowd-source for Earth’s radius of Earth. Encourage students to use iphones/internst for this step.)

7. How does one estimate Earth’s surface area? (crowd-source for formula for surface area of sphere –remind students that this is a good formula to memorize. Crowd-source for Earth’s radius of Earth. Encourage students to use iphones/internst for this step.)

8. Pretend that students are already perfectly competent to
calculate order of magnitude surface areas once they have the values of radii
and formula. Suggest that they round up to 1 sig digit on the radius and
suggest that 4 * pi =10.

9. Then remind students to deal with units.

10. remind students that numbers with 10^12 have prefix “Tera”

11. Students should get an answer that is roughly 50
terawatts.

12. Watch them smile when they realize that the five or ten
minutes that we have spent on this problem gets them fairly close to “accepted”
values ranging from 44 to 47 TW.

****

Next up—where the map above comes from, another teaching opportunity for the concept of diffusion, why the total heat flux is
important, and how and why scientists argue about it.