Average surface temperature of Earth

Question

I had a question in my school exam. Will the average surface temperature of the Earth be lower or higher, if there was no atmosphere?

Now, the answer expected is "The avg temp will be lower, because of no greenhouse effect."

Isn't this an extremely myopic view of the whole situation? Places where it's night, the temperature will be sub-zero, and places where it's daytime, the temperature will be unbearably high, somewhat like on the moon, right? So the thing is, what's the correct answer? I know my education board is not something to be trusted, so please don't worry about their answer.

Answer 1

It is actually very easy to consider this. We don't need any models. we are blessed with a sizable rock at exactly 1 AU from the Sun, a rock with no trace of any atmosphere at all:

Moon (Source: Wikimedia Commons)

Temperatures on the moon vary from 70K to 390K. Average temperatures, depending on location, vary from 130K at the poles to 220K at the equator (through Wikipedia). Note that due to non-linearities, this is not the same as the equilibrium temperature (255K on Earth). Indeed, as Edgat Bonet points out in his answer, for the same equilibrium temperature, average temperatures will be lower if thermal amplitudes are higher. And they couldn't be much higher than they are on the moon, with its long days and complete lack of atmosphere.

So, something that is different between Earth and Moon causes the Moon to be a lot colder. The most obvious difference is the lack of an atmosphere. In principle, the lack of oceans and clouds and trees also contributes, but since none of those can exist without an atmosphere, it is clear that a planet with an atmosphere is hotter than the same planet without — through both direct and indirect effects.

Note that the lack of the atmosphere is not the only difference between the Earth and the moon. There is internal heat generation, tidal heating, and many other effects. None of those effects come close to the effect of the atmosphere for heating the Earth.

Answer 2

Simple black body radiation theory (adapted from http://www.atmos.washington.edu/2002Q4/211/notes_greenhouse.html):

The earth receives a certain amount of heat per unit area from the sun - this amount is about 1370 W/m$^2$ for parts of the earth facing the sun when there is no atmosphere. But the earth presents a "disk" with area $\pi R^2$ to the sun, when it is in fact a sphere with an area of $4\pi R^2$. This means that on average the power received per unit area over 24 hours is 1/4 of that, or about 340 W.

Now some of the sun's power is reflected by earth - about 30%. The power left to get rid of by "gray body radiation" is about 240 W/m$^2$.

Using Stefan-Boltmann, with again an emissivity of 0.7, we solve for the "equivalent" temperature of Earth:

$$\Phi = \epsilon \sigma T^4\\ 240 = 0.7 \cdot 5.67 \cdot 10^{-8} T^4\\ T = 278 K$$

If you ignore the emissivity, the value would be even lower - 255 K. This is well below the temperature of Earth (mean value 288 K), suggesting that the atmosphere is indeed helping keep the temperature up. The effect of the atmosphere on surface temperature is much more extreme on Venus (100's of degrees K), where the layer of carbon dioxide is very thick.

Of course all kinds of simplifying assumptions went into the above. For example, the poles will be significantly colder when there is no atmospheric convection to bring heat from the lower latitudes - and the temperature fluctuations between day and night could be enormous. The hotter surface during the day will emit relatively more power (because of that 4th power) so the mean would be lower than the "effective" temperature I calculated above. Either way - the atmosphere absorbs more strongly in the IR than in the visible, and this means that it is easier for sunlight to get in, than for emitted thermal radiation to get out. This helps keep the planet warm. Your school board, in this case, had the right answer.

But the bottom line is - the atmosphere makes earth livable. Changing the composition will change the conditions on earth. But the balance is precarious. Let's not disturb it if we can help it.