## Airless Planet Temperature

Some readers may recall that back in June 2010 I tried to build a simple computer simulation model of the Earth and its atmosphere. I started out looking at a square metre on the equator of an airless planet, being warmed throughout the day by sunlight, and radiating heat back into space and conducting it into and out of the ground.

The programme chugged round and round, working out the heat gain from the sun as the sun angle changed every few minutes, and the surface temperature Ts at which this solar heat gain was balanced by radiative heat loss to space, and conductive heat loss into the ground.

After that, I added an atmosphere on top, and a few reflective clouds, and it all started to get complicated as I tried to figure out how the atmosphere might absorb and re-radiate heat from the Sun and the Earth. I ended up studying photon football, and going round and round in circles.

Yesterday, I read about a couple of physicists who’d adopted a similar approach, only using the kind of differential calculus that I can’t do (and which is why I always build simulation models of everything, because the maths is much easier). Like me, they’d started out working out what the mean temperature of an airless planet Earth would be, and came to the conclusion that the usual estimate of 255 degrees Kelvin (-18 degrees C) was way too high. Their figure was about 155 K – about 100 degrees lower than the generally accepted figure. It would be a revolution in physics if they were right. But in the comments under the WUWT presentation of it, a lot of people were very sceptical, and saying that they’d got the maths wrong and the physics wrong and maybe pretty much everything else wrong too.

It wondered if I could maybe adapt my little model to look not just at a single square metre of ground on the equator of the Earth, but at a whole set of square metres of clay running from the equator to the pole, and that way make my own estimate of the mean planet temperature.

Nobody else, as far as I could see, had built a simulation model. They were all doing it with heavy duty maths.

So I worked out a simple approximation for the reduced solar heat gain with increasing latitude, and after a few glitches got the modified code working. It took a few months for the temperatures to settle into a stable cycle.

On the right are my results. The green area shows the  surface temperatures throughout the day, ranging from the equator to the pole. And the mean temperature of the airless planet over the whole day came out at 247 degrees K.

Which is only about 8 degrees lower than the generally accepted figure of 255 K, and not too bad for a bit of a rough approximation. And it’s certainly nowhere near 100 degrees lower.

So I reckon they’ve probably got the maths or the physics wrong too.

But I’m glad they published their results in a public forum. This is how I think science should be done, with people publishing things and other people criticizing it, all freely and openly, and not locked away in university laboratories only talking to like-minded people, and refusing to reveal their data and their methods.

I’ll post a little comment on the WUWT forum, linking to this page of mine, and that will be my rather belated contribution to the debate.

And maybe then I’ll try to polish up the code a bit, and stick an atmosphere back on top of it again, and see if that warms the surface a bit (it needs to be 30-40 degrees warmer to match actual earth temperatures), or whether it just ends up with photons bouncing all over the place again.

smoker
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### 14 Responses to Airless Planet Temperature

1. Tony says:

Nice work Frank but as for fig. 2 I think that personally that you should “hide the decline.” ;-)

2. Gary K. says:

“but at a whole set of square metres of clay running from the equator to the pole, and that way make my own estimate of the mean planet temperature.”

Does clay have the same properties as water?

After all, 70% of the Earth’s surface is water.

• Frank Davis says:

No, it doesn’t. In that respect, my model is unrealistic.

My model planet isn’t tilted at 23 degrees to the ecliptic either, and that’s unrealistic too.

3. Gary K. says:

Unrealistic, but; fun to play around with!!!

At least your are not claiming that the Earth is getting overly warm.

Those people seem to never get around to stating just what the Earth’s ideal temp should be.

• Frank Davis says:

I think, BTW, that water is a lot better conductor of heat than dry clay. I tried out a material with 5 times the conductivity of dry clay, and in my model the earth mean temperature rose to 252 degrees K. Dunno why.

4. “they’d started out working out what the mean temperature of an airless planet Earth would be, and came to the conclusion that the usual estimate of 255 degrees Kelvin (-18 degrees C) was way too high. Their figure was about 155 K – about 100 degrees lower than the generally accepted figure. It would be a revolution in physics if they were right.”

Well yes, but as we have a real life airless Moon right next to us, can’t we just assume that the Earth would be a similar temperature to that as a starting point, and adjust for the fact that water reflects more/less than dark grey sand etc?

• Frank Davis says:

But the moon rotates on its axis once very 28 days, and so its surface is very hot during the day, and very cold at night.

Equally, if the earth or the moon spun on their axes once a minute, they’d have an even temperature (varying with latitude).

• Yes, but wouldn’t the average temperature work out at much the same regardless of how fast a planet is spinning? We know for example that the desert without cloud cover has huge variations in temperature between day and night, it still has an average temperature, is this lower or higher than it would be if the earth turned as slowly as the moon (relative to the sun)?

PS I won’t understand your answer unless it’s as simple as “Makes no difference how fast it spins” or “Faster spin = hotter” or “Faster spin = cooler” etc.

• Frank Davis says:

wouldn’t the average temperature work out at much the same regardless of how fast a planet is spinning?

I used to think it did, but I read a recent discussion somewhere (WUWT probably) that if a planet didn’t spin at all, it would have a higher overall temperature. This was a consequence of the fact that radiative heat loss is proportional to the fourth power of the temperature (i.e. T4 ).

I can’t use my model to check this, because it relies on the planet to be spinning to work properly.

(On second thoughts, with a slight adaptation maybe I can use my model.)

• Frank Davis says:

Imagine the Earth is a coin with one side facing the sun.

The stefan-boltzmann radiation law is that radiative heat loss per unit area = e.s.T4 where e = emissivity (1.0), s = 5.6 x 10-8, and T = surface temperature in degrees Kelvin.

Given a solar constant radiative flux 0f 1360 watts/squ metre, T= 607 degrees K on the side facing the sun. And on the other side it’s 0 degrees K. So the mean surface temperature of the surface is 607/2 K or about 303 degrees K.

5. Pingback: They Just Don’t Know | Frank Davis

6. Going by your actual answers – all things being equal (no atmosphere, same albedo), the average temperature of the earth would be lower than for the moon.

Which still doesn’t make sense to this layman. The cold side, zero K can’t emit any heat and the warm side does, but if radiation is the fourth power of temperature, the hotter the hot side gets (slower rotation) the more heat it radiates so therefore slower rotation = on average a lower temperature.

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