## Global Warming on the Moon?

Posted by Jeff Id on December 6, 2009

We’ll in my recent post on why some fraction of global warming is absolutely real, I made the comment that if people wanted to see whether the blackbody equations presented by Lucia hold up reasonably well to a no-atmosphere planet in earth levels of sunlight they could simply look at the moon. There has been a bit of discussion on it so I did the work myself.

First I had to find the lunar albedo it turns out that the value is 0.078. I had no idea the moon was dark material, blogging is fun when you learn stuff like that. Link for the data is here.

So using the amazingly simple approximations by Lucia, I plugged in the same values of solar energy as earth. After all the moon is closer to the sun and farther from the sun exactly half of the time. I calculate an average temperature of 273K (according to the above).

The average day temperature and average night temperature are given at this link. It’s in degrees C so we need to add

is 273+107 = 380K in the day and 273-153 = 120K at night for a net average of 250K. This is 23C cooler than a simple calculation would demonstrate and if the same result was made on a planet with an atmosphere we would give have the impression that there was 23 C created by greenhouse gasses. However, the lunar surface has a great deal more variance in temperature than the earth which creates substantially more error in the simple calculation.

If someone is interested in working the math to take into account non-linearity of temperature distributions, it would make an interesting post. It’s Sunday though, and I think some fun is in order.

## P Gosselin said

How about global warming in a bottle?

Just how stupid do they think we are!!

h/t EIKE Germany

http://news.bbc.co.uk/2/hi/science/nature/8394168.stm

## P Gosselin said

I just watched it again.

I can’t take this iron-clad stupidity anymore! Our poor kids!

This confirms that the Beeb has gone over the edge. They’re out to lunch, completely. Like total kooksville! I need a beer…

## Jeff Id said

#I’ve seen the experiment before on mythbusters. She did it with two bulbs and the one on the right is substantially brighter. I believe that while CO2 captures more heat from this kind of experiment, the test was rigged. Why not use the same lamp source and space both equally.

## Jeff Id said

Either way it’s mind sucking tripe.

## DeWitt Payne said

Jeff Id,

I’ll put my spreadsheet back in order again and post the graphs and data for a moon-like body. I’ve already posted the zero heat capacity lower limit on average temperature on a different thread. It’s not difficult since you can assume zero horizontal thermal conductivity. The trick is to get the surface heat capacity just high enough to get the correct night time minimum temperature.

## Bill said

I don’t think this is a matter of temperature distributions. If you assume the moon doesn’t rotate and calculate the surface temp (so the incoming solar for each sq meter equals the radiated long wave), the temperature is 389, fairly close to the daylight value.

The nightside temp is going to depend on the thermal conductivity of the surface and how much heat is stored there.

## DeWitt Payne said

Bill,

The moon does rotate. The rotational period is the same as the orbital period so we only see one face. The maximum temperature of 386 K (albedo 0.078 and solar constant 1368 W/m2) will only be seen at the lunar equator with sun directly overhead (local lunar noon), assuming a low surface heat capacity. Everywhere else will have a lower temperature because the angle of incidence of solar radiation is not perpendicular to the surface. As the surface heat capacity increases, peak temperature occurs at a later time and the maximum temperature will be less than 386 K. Minimum temperature will occur just before local sunrise and will indeed be strongly dependent on surface heat capacity. If the rotational axis were exactly perpendicular to the solar orbital plane and the surface heat conductivity were exactly zero, the temperature at the lunar poles would be 2.73 K. I don’t think either condition holds for our moon.

## THE TWILIGHT SAGA: NEW MOON – Trailer said

[…] Global Warming on the Moon? « the Air Vent […]

## Mike said

So in other words, if the difference between calculated and measured temperatures is 23 degrees, it is on the same order of magnitude as the supposed greenhouse effect, which means the simple approximation is not adequate. Back to square one.

## Jeff Id said

#9 I believed there were effects due to the T^4 nonlinearity because the temperature varies so wildly on the moon. On the earth these effects are not nearly of the same magnitude. I’ve not verified it but if Bill in #6 is right, a half moon is a good approximation – this makes sense in that the moon rotates over 2 weeks which is plenty of time to reach temp and cool down.

## Jeff Id said

Day changes to night in 2 weeks I should say.

## DeWitt Payne said

Ok.

For a spherical airless body with a synoptic period of 29.5 days, an albedo of 0.078, a solar constant of 1368 W/m2 and a sky temperature of 2.72 K with the surface heat capacity adjusted to give a minimum temperature at dawn of 91 K or -182 C similar to that measured at the Apollo landing site in Jeff’s link. The equatorial temperature over the course of a lunar day looks like this. I’ve included a plot of the surface temperature at the equator for a zero heat capacity surface for comparison.

For the model, the average temperature over the course of a lunar day at the equator is 219.6 K or -53.6 C. Max temp is 386.2 K and minimum is 91.1 K. The global average temperature is 207.2 K. The average temperature for a zero heat capacity surface at the equator is 165.5 K. A superconducting and therefore isothermal body would have a temperature of 273 K. A surface with infinite heat capacity but zero horizontal thermal conductivity would have an average temperature of 269.6 K. For a planar surface with perpendicular insolation on one face with zero thermal conductivity so that the back side would be 2.72 K and the front 386.2 (albedo 0.078) would have an average temperature of 194.5 K. All these different temperatures are the result of the non-linearity of the Stefan-Boltzmann equation.

For the Earth, the horizontal thermal conductivity and surface heat capacity are high enough that the superconducting body approximation is not too far off, probably no more than 3 degrees low. Surface heat capacity minimizes the diurnal temperature variation and thermal conductivity minimizes both latitudinal and longitudinal variation.

## Jeff Id said

#12 Cool. I like that you plotted the zero heat capacity surface. When you say “global average temperature” how is this calculated?

## Bill said

Dewitt,

Great job. I realized that the moon does rotate. Two weeks of sunlight aught to be long enough for the surface to come to equilibrium as you showed. Surprised that it took so long to get there.

Working your model in reverse, one could estimate the soil heat capacity if one knows the moon surface temps when it is unilluminated.

One might note that the reflection from the earth may not be negligible, especially for the unlit condition where a bright earth competes with the 4K background.

Again, great analysis.

## DeWitt Payne said

Jeff Id,

I calculated the temperature at latitudes from 0 to 90 at 5 degree intervals reducing the insolation by cos(lat). Then I calculated the fractional area of each latitude band and multiplied that factor by the average of the temperatures for the band boundaries and summed the results. Because the both the temperature and the band area are functions of cos(lat), the equatorial region dominates the average temperature. I’ve made a plot of average temperature by latitude. The steep drop between 85 and 90 is unrealistic, but it represents less than 0.5% of the total area. In fact, the average temperature at 89.5 degrees latitude is still 87.9 K.

Radiation, emitted and reflected, from the Earth should make the side facing the Earth warmer than the other side. I’m not sure by how much, but I don’t think it’s a lot.

## DeWitt Payne said

For the temperature vs. time, I calculated the insolation at 0.5 degree intervals of rotation (3540 seconds of time/step) and assumed a sky temperature of 2.7 K for when the sun was below the horizon. Then the temperature at the end of a step was [initial heat content (initial temp/heat capacity) + joules in – joules out]/heat capacity. Joules in is the sun angle corrected insolation multiplied by the time/step and joules out is the radiated power from the SB equation multiplied by the time/step. For simplicity emissivity was assumed to be 1.0. Initial temperature at time zero was adjusted to equal final temperature. Iteration was only necessary for very low peak insolation, 89.5 degrees latitude for example.

## Carrick said

Jeff:

By comparison, the Earth is 0.3. Imagine what a full moon would be like on the Earth if it were 4 times brighter.

## Mark T said

You’ve never seen any of the moon dust/rocks at some science museum, Jeff? It’s pretty dark. That the moon looks almost white to us is certainly misleading.

Carrick, I have a hard enough time sleeping as it is, let’s not make me think of things that will only make it worse. 😉

Mark

## Joel Heinrich said

Why not take some look at actual measured data? From the Lunar Surveyor Program (pages 193ff)

## DeWitt Payne said

Bill,

On the slow warm up: I’m using a well-mixed surface layer heat capacity. That’s a crude approximation. I really should be using a diffusive surface layer, but the math is considerable more complicated. A diffusive surface layer model would warm up faster when the sun comes up and cool faster when it goes down. The Lunar Surveyor Program temperature calculations linked above uses a diffusive surface model.