Lab - Thermal Modeling

Ge151: Spring 2011

Thermal Modeling of Planetary Surfaces & Thermal Inertia

Introduction to Thermal Modeling

A. The Physics

The fundamental physical law behind modeling the temperature of planetary surfaces is the law of conduction:

F = -k dT/dz

This law states that the flux of energy conducted in a certain direction is proportional to the temperature gradient in that direction. The constant of proportionality, k, is the conductivity, expressed in W/mK.

The flux for a given temperature difference is dependent on the material. When this law is applied in a one-dimensional problem (conduction through an infinite slab, e.g.) along with the conservation of energy, one can derive the heat diffusion equation:

dT/dt = (k/(rho*c)) * d²T/dz²

The density, rho, and the heat capacity, c, describe the storage of energy within the slab.

As with other diffusion equations, the diffusion depth as a function of time can be approximately derived by inserting finite values in the above equation and re-arranging the terms:

d = sqrt (Kt)

where K is the product k/(rho*c), the thermal diffusivity and t is time.

Using the conductivity (2.5 W/mK), the density (2700 kg/m³, and the heat capacity (875 J/kgK) of granite, find the depth in the Earth's crust that has not yet realized that the last ice age (10,000 yrs ago) is over. If the ice age began 100,000 yrs ago, what is the depth which hasn't yet realized that it has begun?

B. Application to Planetary Surfaces

Most planetary surfaces are heated during the day by sunlight and lose energy during the night by radiation from the surface. This cyclical heating produces an oscillation of temperature that decays with depth. To first order, the depth of this thermal wave can be estimated by using the period of the insolation cycle as the characteristic time in the above thermal skin depth calculation.

Find the diurnal thermal 'skin depths' for Mars (regolith), the Moon (lunar regolith), and Europa using the values listed below. 'Skin Depth' is usually defined as the depth at which the variations drop off by a factor of 1/e. Also find the seasonal skin depth for Mars (the penetration depth of the seasonal temperature change). Assume the product of rho and c is 10⁶ J/m³K for all planets and that k(mars)=0.112, k(moon)=0.002, and k(Europa) = k(water ice) = 2.2 W/mK.

Conductivity

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It can be determined from lab experiments that the product of the heat capacity and the density is close to 10⁶ J/m⁶K for most geologic materials. Therefore most of the variation in thermal properties between planetary surfaces is due to differences in conductivity.

The strict definition of thermal conductivity includes only the transfer of thermal (vibrational) energy along the crystalline structure of a rock or along the contacts between grains. In practice, other effects are included to produce an effective conductivity.

A. Temperature Dependence

Two important additional sources of thermal conduction should be considered when working with planetary surfaces. The first is important at high temperatures for granular surfaces. On Mercury, where temperatures can reach 700K, the energy transferred between individual grains by radiation can be greater than that transferred by contact conduction. The effective conductivity was empirically found to be:

k = A + BT³

where A represents the contact conductivity and the BT³ term describes the radiative component. Note that now the conductivity is temperature dependent.

Plot the contact, radiative, and total conductivity vs. temperature for Mercury from 0 to 700 K. Assume A=0.001 W/mK and B=3.19x10^-11 W/mK⁴

Consider a surface, like Mercury, with a temperature-dependent conductivity. Think about the conduction of energy during the day and night. How would the temperature vs. depth on Mercury differ from the temperature profile on Mars, for example? What are the consequences of a temperature dependent conductivity?

B. Atmospheric Effects

The second effect is important for Mars and other planets with substantial atmospheres. In these cases energy is transferred between grains by gas molecules. This effect becomes important when the mean free path of the main atmospheric constituent becomes smaller than the spacing between grains.

What is the mean free path of nitrogen on the Earth? What is the mean free path of C0₂ on Mars? Use the ideal gas equation P=N kT, where P is the pressure, N is the number density and k is the Boltzmann constant. T is the mean near-surface atmospheric temperature, 288 K for Earth and 215 K for Mars. From the number density, estimate the mean free path (the distance a molecule can travel before hitting another one). How do your results compare to the size of typical dust or sand grains?

On Mars, (effective) conductivities are derived from observed temperature variations. These conductivities are then converted to regolith particle sizes. Based on the previous question, how does this conversion work? Earth and Venus also have atmospheres, but this effect doesn't operate, why not?

Thermal Modeling in Action

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I've put a demonstration of a thermal model of Mars' surface in the Ge151 account. Login on a UNIX machine and go to the /home/ge151/lab1 directory.

Using IDL:

Type 'idl' to start IDL. Then type:

.run temp

Using MATLAB:

Type 'matlab' to start MATLAB. At the command prompt, type:

temp

The program displays the temperature vs. depth for Mars at 20N latitude over one day, and the surface temperature variation over one day. Make sure that the window remains active (in IDL, click on it; in MATLAB, put the cursor over it) and follow the instructions in it.

From the above program, do the following things:

Roughly plot (just hand sketch) the diurnal maximum and minimum temperature vs. depth.

Estimate the diurnal skin depth (where the surface temperature variation is attenuated by a factor of e).

Plot the phase of the temperature variation vs. depth: on plot of depth vs. time of day, i.e. record the time at which each depth experiences its maximum temperature.

When do the maximum and minimum surface temperatures occur? Why are they not simply at noon and midnight?

Describe the shape of the diurnal surface temperature variation and explain the different parts of it.

Qualitatively Understanding Thermal Inertia

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As we discussed in class thermal inertia is a catchall quantity that describes how difficult it is to change the temperature of a surface. A high thermal inertia surface will be cooler during the day and warmer during the night relative to a low thermal inertia surface.

In the same directory (/home/ge151/lab1) type the command:

In IDL:

plot_tes,lat=lat,lon=lon,ti=ti

In MATLAB:

plot_tes

This will read thermal inertia values, which have been previously derived by a sub-surface thermal conduction model similar to the one which produced the results for the previous section.

Take a look at the interpolated map that appears. Which regions stand out the most as having high thermal inertias? Suggest reasons why. (If you need help finding names for Martian regions, there are geological maps laid out in the basement computer lab, South Mudd, room 056. Google Mars is also a useful tool!)

The pathfinder-landing site was about 20^o north of the equator at a longitude of about 30^o W. What do the thermal inertias look like in this region? How does that relate to the surface seen in the lander's cameras?

Plot a histogram of thermal inertias with the command:
In IDL:

plot,50+indgen(500),histogram(ti),ps=10

In MATLAB:

hist(zti2(:),50)
Ignoring the fake spike at a thermal inertia of 0, what does the majority of mars look like thermal inertia-wise? What can you say from that about things like dust coverage etc? If you want to look at a specific range of values on the map in matlab, you can run plot_tes again and type (for values of 0 to 100, for example):
caxis([0 100])

Plot zonally averaged thermal inertia vs. latitude or vs. longitude using the IDL programs 'lat' and 'lon'. (In MATLAB type 'plot(yt,mean(zti2')); axis([-60 60 0 400])' for latitude and 'plot(xt,mean(zti2)); axis([-180 180 0 300])' for longtitude.) Do you see interesting trends? What about on visual inspection of the 'plot_tes' map?

Finally, check out Google Mars. This is a thermal infrared mosaic of the surface taken from the THEMIS instrument onboard the Mars Odyssey spacecraft. In this view, warm places are bright, and cool places are dark. Knowing what you now know about the thermal inertia of Mars, do you think these images were taken during the day or night on Mars?

That's it! Email your TA if you have questions.

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