Thermal simulation

The moon at a given latitude is divided up into longitudinal segments 10 degrees wide. Seven layers of different properties are included, to simulate the known property that the surface is generally composed of a thin layer of badly conducting rubble and dust, over a mostly basalt base. Thermal energy transfer is taken into account between all neighbouring segments, longitudinally and radially, using the conductivities, areas, size and temperature differences between segments. The surface segments receive solar irradiation, and emit according to the Stefan-Boltzmann equation. The temperature change in each segment is found according to the energy transfers, taking into account the density, volume and thermal capacity of each segment. At higher latitudes, a factor of cos( lat) is taken into account, so that the solar irradiation per unit area, and the surface area itself get smaller.

plan view of equatorial latitude

r0=1.737e6 m

Solar irradiation

rotation

incoming radiation thermally

emitted radiation

thermal transfer between layers

Standard lunar profile, 280 K diurnal variation

By adjusting the surface parameters so that the moon is covered by a thin, badly conducting, layer of rubble, the above plot in blue was obtained. This is overlayed with data in red from the Diviner space craft to show a good match in both day and night. To achieve this, the irradiation with angle is not a simple cos( ) function, but rather cos( )1.3. This is discussed at http://tallbloke.wordpress.com/2014/04/18/a-new-lunar-thermal-model-based-on-finite-element-analysis-of-regolith-physical-properties/

Moon temperature vs latitude

Figure from: http://diviner.ucla.edu/science.html Results from this simulation

Plotting the temperature profiles at different latitudes (0, 60 and 75 deg) gives the figure on the right. There are some noticeable differences at higher latitudes, compared to measured values, for example the lat75 curve on the left figure has a shorter night period, but my model does not contain the polar offset angle of 1.54 degrees to the ecliptic. I will ignore such issues for the moment. When the mean temperature is found over all latitudes, the global mean temperature can be calculated by averaging temperature over area. For my simulation of the ‘standard moon’, this is found to be 199 K, close to the accepted value of 197 K. The fact that this is far lower than the moon’s ‘effective temperature’ of 270.6 K, shown for example at http://bartonpaullevenson.com/Albedos.html , therefore presents a problem. This will be addressed in the following slides.

Faster rotation: 25 K variation

The effective temperature is calculated as if the entire surface of the moon receives the solar input uniformly. That the moon spins so that one side receives sunlight, while the other doesn’t is an obvious place to start looking for an explanation. Clearly this non-uniformity of temperature will affect the Stefan-Boltzmann emission, and thus the surface temperature. By reducing the spin period of the moon to 0.01 Earth days, the surface only undergoes a 25 K temperature change. The global temperature goes up from 199 K to 264 K (shown in title of right figure), almost at the effective temperature.

Faster rotation: 2.5 K variation

Increasing the spin speed further so the temperature fluctuation from day to night is only 2.5 K still gives a global temperature of 264.5 K, not quite at the effective temperature.

Moon, very fast rotation, T variation 2.5 K, Lambertian factor = 1.0

The reason the previous simulation did not reach the effective temperature is due to the non-Lambertian coefficient of the surface. It had been found, by simulating the measured properties of the moon, that radiation doesn’t get absorbed as a function of cos( ), but rather as cos( )1.3. This acts to increase the albedo at higher angles. So while a nominal albedo of 0.11 was used in the model, the power of 1.3 increases that a little, and lowers the global temperature. By returning the surface absorption property to Lambertian, and spinning the moon around very fast (0.001 Earth days!) the effective temperature is reached (270.2 K vs 270.6 K, close enough). Alternatively, by retaining the power of 1.3 while reducing the nominal albedo (but maintaining the measured albedo!) will also give the global temperature equal to the effective temperature.

Moon, standard rotation, T variation 280 K, Lambertian factor = 1.0

The effect of the non-Lambertian absorption profile is shown here. The blue line is plotted with cos( ), while the red line is the measured data that is fitted best with cos( )1.3.

For the purposes of this presentation, this is considered a minor difference. The main issue is why the lunar temperature is lower than its effective temperature, and what implications that has for the Earth which is at the same distance from the sun.

To test the implications these results have for Earth, the moon simulation was modified to give properties close to the Earth values. Albedo was changed to 0.306, radius changed to 6.3781e6 m, spin rate set at 1 Earth day. The surface was also modified so that only about 20 K of difference between day and night was produced. According to http://en.wikipedia.org/wiki/Diurnal_temperature_variation , a value of 20 K is a little on the high side, but we can change it later to see its effect. With these parameters, the Earth global average temperature was 253.7 K, very close to the effective temperature of 254.3 K given at http://bartonpaullevenson.com/Albedos.html

Earth, standard rotation, 22 K variation

Earth, faster rotation, 2.4 K variation

Again, by speeding up the Earth spin rate x10, the average edges up to the effective temperature (254.5 K vs 254.3 K, close enough).

Moon and Earth Global T vs Day-Night temperature difference

Earth parameters, Teffective = 254.3 K

The plots on this slide are summary plots for the presentation. The basic result is that a moon/planet with very different day/night temperatures will have a global average significantly below the effective temperature. The moon has about 280 K difference between day and night temperature, so its global temperature is about 70 K below its effective temperature. The Earth has on the order of 20 K (or less) difference between day and night, and so the effective temperature applies. Of course, the Earth also has an atmosphere and internal heating which can bring the measured surface temperature higher than that, but one can take the effective temperature of 254.3 K as a valid ‘baseline’.

measured measured

Moon parameters, Teffective = 270.6 K

Summary

A simulation has been written to take into account physical properties of the moon and the Earth. The simulated daily temperature of the moon matched well with the Diviner data. The moon has a global temperature much lower than its effective temperature due to the high contrast in temperature between night and day, which for the moon is about 280 K. The Earth on the other hand has much less difference between night and day temperature, on the order of 20 K, so it seems that the effective temperature of 254 K can be used as a ‘baseline’ This is modified in practice due to the presence of the Earth’s atmosphere, not simulated here The day-night temperature difference would need to be on the order of 100 K in order to reduce the global temperature to about 5 K below the effective temperature.