In the last section, we've already discussed the role of albedo as reflecting incoming radiation back into space. On a planet like Mercury, assuming that the albedo is the same all across the planet might be somewhat reasonable - but what of Earth where we have ice in the polar regions, a mixture of oceans, forest and sand desert otherwise? We have already discussed that, dependent on axis tilt, different latitudes of the planet receive different amounts of energy - so would it not matter that their albedo differs too?

There are more concepts we need in order to compute temperatures - one of them is thermal inertia. If a surface with a given albedo is exposed to incoming radiation, we can compute its radiative equilibrium temperature. That is the temperature for which absorbed energy equals radiated energy (all warm surfaces radiate in the IR). But that temperature is not reached immediately, because materials can store energy. It takes time to heat them to equilibrium, but when the radiation is taken away, it also takes time to radiate off the stored energy. This is the thermal inertia.

For the amount of energy that a surface can store, two properties are relevant. One is the heat capacity, i.e. the amount of thermal energy per kilogram the substance can store. The second is the total amount of material that participates in storing energy.

For irradiated surfaces, we need to define something like an active depth. When the sunlight falls on a rock, it is absurd to assume that a 100 m thick rock layer will be warmed - only the outer part of the rock is affected. When light falls on water, in a sense more water participates because water is liquid and mixes. In a rock layer, only diffusion of thermal energy can transport heat deeper into the material, in water in addition different transport mechanisms can play a role.

In addition, we need to be careful when dealing with slowly rotating planets where a rotation period may be as long or even longer than the orbital period. In these cases, the problem is not only one of latitude, but irradiation becomes a function of both latitude and longitude.

## Albedo maps

To take account of the fact that albedo can be regionally different, material maps can be defined. First, a material definition file needs to be created where for every material albedo, heat capacity and active thermal depth are specified.

 material name Water albedo -1 heat_capacity 4.2 depth 1.0 material name Rock albedo 0.07 heat_capacity 2.6 depth 0.1 material name Sand albedo 0.2 heat_capacity 2.0 depth 0.1 material name Ice albedo 0.6 heat_capacity 4.2 depth 0.2 end

This file is implicitly indexed, with the first material having the index 0 and numbered from there.

Then, a second file contains a material map in which the material index is written to the surface element defined by lat and long. The row and column numbers here must match what has been defined as elements_lat and elements_lon in the planet definition before.

Here is a (rather coarse-grained) material map of Earth:

 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 1 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 0 1 2 2 1 0 0 0 0 1 0 0 0 0 2 2 2 0 1 1 0 0 0 0 0 0 1 0 0 0 0 2 2 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Finally, the names of the two files need to be specified in the main configuration file. The map is part of the planet block whereas the materials list is part of the config block.

 config materials materials.cfg planet materials_file material_map_earth.cfg

(The alert reader will have noticed that water has been assigned an albedo of -1. This is due to the fact that the albedo of water is highly angle-dependent: Wather mostly absorbs incoming light when it meets the surface under a large angle, but it mostly reflects when the angle is shallow. Assigning this -1 value requests this angular dependent albedo value from the code.)

## An example

The following configuration file (along with the material map above and the materials list) is a scenario for Earth without an atmosphere.

 config materials materials.cfg input star name Sun mass 1.0 T_surf 5778.0 planet name Earth mass 1.0 radius 1.0 semimajor_au 1.0 eccentricity 0.03 axis_tilt 22.0 dec_offset 270.0 rot_period_h 24.0 mean_albedo 0.3 elements_lat 12 elements_lon 16 materials_file material_map_earth.cfg evolution evolve_d 365 timestep 1000.0 plot_surface file earth_irradiation.dat var_z total_irradiation end

Running this file and comparing with a mean albedo of 0.3 reveals the following:

 Irradiation of Earth with albedo map vs. mean albedo

Generally the amount of energy that is absorbed is now locally different, compared with the mean albedo we see less energy deposited in th epolar regions because of the high albedo of ice. In this comparison, there is more energy found in total with the albedo map. The reason for this is chiefly clouds - the mean albedo of Earth includes cloud cover, but the map materials do not.

We will discuss this point later when talking about atmospheres.

## Slow rotation

A good example for a slowly rotating world and its complications is the case of Mercury. mercury has a 3:2 spin orbit resonance which means that the planet turns three times around every two orbits.

In such a case, one has to study irradiation for two orbits to get back to the original situation, otherwise one gets the wrong results. Likewise, the energy deposition has a pronounced two-hump structure in longitude - there are two 'hotspots' divided by two cold valleys on Mercury, which is a very unfamiliar arrangement of 'climate zones' for someone who is used to Earth.

Continue with Temperature.

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