The temperature of Daisyworld is calculated using some very basic equations. We start with a basic equation of physics, the Stefan-Boltzmann Law, which states that the rate of energy given off through radiation by an object is proportional to the fourth power of the object's temperature and is described in the following equation:

F = esAT ⁴

where F is the rate of energy flow in Joules/sec (or Watts), e is the emissivity of the object, s is the Stefan-Boltzmann constant, A is the surface area of the object, and T is the temperature of the object in degrees Kelvin. The Stefan-Boltzmann constant has a value of 5.67E-8 Joules/sec m2 K4. The emissivity is a dimensionless number and ranges from 0 to 1; a perfect black body has an emissivity of 1, while very shiny objects have an emissivity of close to 0. Human skin has an emissivity of 0.6 to 0.8.

This means that if we want to know the temperature of some object, say a planet, we just need to know the amount of energy that it emits. In our model, we get the amount of energy emitted by the planet (and therefore the temperature) by assuming that:

Energy Emitted = Energy Absorbed,

which is sometimes called radiative equilibrium. This is a bad assumption to use on a short time scale, like a few hours because we know from experience that planets warm and cool during that kind of a time period, which means that that they cannot be in radiative equilibrium. But, if we are thinking about time scales of many years - something a bit longer than the response time of the planet's heating system - then this is a good assumption to use. By adopting this assumption, we avoid the necessity of employing the kind of model we used in the climate models, where we kept track of the amount of thermal energy stored in the land surface and the atmosphere.

The next question then becomes, how do we find the amount of eneregy absorbed by the planet. This is simply found by knowing that:

Energy Absorbed = Energy Received - Energy Reflected

The task of figuring out the amount of energy received by Daisyworld is made easier by the fact that there are no clouds -- this allows us to say the following:

Energy Received = Solar Luminosity Factor * Solar Flux Constant

Here, the solar luminosity factor varies from 0.6 to 1.8; it is essentially the relative luminosity of the sun. The Solar Flux Constant is set at 917 W/m2. which is less than that of Earth, but remember that on Earth, because of the clouds, the amount of solar energy received by the land surface is quite a bit less than what is received at the top of the atmosphere. So, Daisyworld is quite similar to the Earth as far as solar energy input is concerned.

The energy reflected is simply a matter of the albedo of the planet and the amount of energy received by the surface:

Energy Reflected = Energy Received * Albedo

Thus, if we keep track of the solar luminosity and the albedo of the planet, which will change according to the numbers of the different kinds of daisies, we can easily calculate the temperature of Daisyworld.

8/12