In this lesson, we’re going to do what physicists like to do – we’re going to over-simplify the Earth and do a “thought experiment”.

So, we’re going to imagine that the Earth doesn’t have an atmosphere and we’ll work out what temperature it “should be”. This builds on the lesson on blackbodies.

First, the Sun is sending light towards the Earth. The Sun is very hot and emitting light in all directions, but the amount of energy coming directly towards the Earth (the solar irradiance) is 1360 W / m^{2} (ish – we’ll come back to how we measure this later). But that’s the light coming towards the Earth and, of course, half the Earth doesn’t get hit at any one moment (it’s the night time) and towards the poles, that 1360 W gets spread over a much bigger area of Earth.

To understand that consider 1 square metre rings in a row in front of the Earth (top picture): over the equator the light going through those rings forms a circle on the Earth; but over the poles, it would be spread over a much bigger ellipse. So – the average power falling on a square metre of Earth’s surface at any one time (averaged over the whole Earth) is about 1360 / 4 = 340 W / m

^{2}(watts per square metre). That’s like having 4-old fashioned lightbulbs on every square metre of the Earth.

Now, the Earth can’t get hotter and hotter and hotter! It will reach an “equilibrium” where the heat in equals the heat out. (Equilibria are very common in physics). The way it releases energy into space is via its own blackbody radiation. You may remember from our lesson on blackbodies that everything that is hotter than absolute zero radiates energy with a blackbody curve. And that is true of the Earth too. As the temperature of the Earth is quite low (compared to the Sun!), it will radiate most of its blackbody radiation in what we call the “thermal infrared” (long wavelengths).

We can work out the total power of the blackbody by working out the area underneath that curve. There’s a simple calculation there. The total power of a blackbody in a square metre of its surface emitted into space is sigma times Temperature to the power 4. (σT^{4}. Sigma (σ) is the Stefan-Boltzmann constant and is 5.670 367 × 10^{-8} W m^{-2} K^{-4 }.

If the Earth were perfectly black at both short wavelengths (visible, near infrared – the wavelengths the Sun emits) and at long wavelengths (thermal infrared – wavelengths the Earth emits), then we could write:

Incoming power in a square metre = outgoing power in a square metre

340 = σ × T^{4}

So the temperature = 278 K = 5 ºC.

(To do this calculation yourself, remember that the Stefan-Boltzmann constant can be written with the decimal place moved 8 places, so 0.000 000 056 704 and to get from Temperature to the power 4 is something to Temperature is something you can press the square root button twice)

If, as is more realistic, the Earth has an average reflectance in the visible of 30% (so it reflects about 30% of the light from the sun straight back to space and absorbs 70%) but it is still perfectly black in the thermal infrared (not unreasonable), then

Incoming power in a square metre = outgoing power in a square metre

70% × 340 = σ × T^{4}

So Temperature = 254 K = -18 ºC

Now, we have made A LOT of approximations here. The actual average reflectance of the Earth might not be quite 30%, and it’s not quite perfectly black in the thermal infrared – but the basic picture holds. If the Earth had no atmosphere at all, the average temperature across the whole world would be something close to -15 ºC to -18 ºC.

And just in case you think I’ve pulled the wool over your eyes, I thought I’d find out the average temperature of the moon – after all, it’s about the same distance from the sun as us and it’s about the same sort of reflectance. So I searched the internet for “average temperature of the moon” and found an answer here: https://socratic.org/questions/what-is-the-average-surface-temperature-of-the-earth-s-moon

I was most amused that it said:

“You could take an average of the mean maxima and minima to get a mean surface temperature of -23 °C, but it wouldn’t be very meaningful.”

I beg to differ – that is very meaningful – as it’s very close to what I calculated with my overly simplistic thought experiment. (The moon reflectance is a bit different and it’s fraction pointing towards the sun is perhaps a bit different and the average temperature is not necessarily the average of the minimum and maximum temperatures)

Of course, the reason that the average temperature of the Earth is not that cold – and is nearly 35 ºC hotter – is that we do have an atmosphere. But you might be surprised to know that if our atmosphere was 100% oxygen, nitrogen and argon (and not 99.9 % oxygen, nitrogen and argon) it would still be -18 ºC. I’ll explain why in the next lesson.

## 3 thoughts on “Lesson 4: the temperature of the Earth”