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V3.0, November, 2007

Stars generate huge amounts of energy through the process of nuclear fusion. Our own sun, an unremarkable medium-sized
star, produces a total power E of about 3.9×10^{26}W. This power is radiated into space uniformly in all directions.
Fundamental physical laws tell us that the intensity of this radiation decreases as the inverse square of the distance from the sun.
The solar constant S_{o} is defined as the average power per unit area of solar radiation falling on the surface of an imaginary sphere of radius R around the sun:

S_{o} = E/(4π R^{2}) = 1370 W/m^{2}

where R is the average Earth/sun distance, about 150,000,000,000 m. The solar "constant" actually fluctuates a little and the energy Earth receives varies by about ±3% with the yearly cycle in the Earth/sun distance:

S_{max} = S_{o}/(1 - e)^{2} = S_{o}/(0.983)^{2} = 1417 W/m^{2}

S_{min} = S_{o}/(1 + e)^{2} = S_{o}/(1.017)^{2} = 1324 W/m^{2}

where e is the eccentricity (a measure of departure from a circle) of Earth's orbit around the sun. Earth’s eccentricity varies slowly (with periods of hundreds of thousands of years). Currently, Earth's eccentricity is about 0.017 and Earth is closest to the sun in early January (yes, really), so this is when maximum solar radiation reaches Earth. The minimum amount of radiation is received about six months later.

From space, Earth looks like a circular disk of radius r (area = πr^{2}).
Some of the
solar energy S_{o} striking this
disk is absorbed and some is reflected back to space:

E_{absorbed} = S_{o}πr^{2}(1 - α)

where α is the fraction of solar energy reflected back to space (the albedo). Earth's
albedo is about 0.30.
The solar energy absorbed by Earth is then re-emitted as thermal radiation from the
entire Earth's surface (area = 4πr^{2}) according
to Planck's law of blackbody radiation:

E_{emitted} = 4πr^{2}T^{4}σ

where T is the temperature of the Earth/atmosphere system, in Kelvins, and σ is the Stefan-Boltzmann
constant, 5.67×10^{-8} W/(m^{2}K^{4}). Averaged over time, the absorbed energy
must equal total emitted energy (conservation of energy), so:

4σT^{4} = S_{o}(1 - α)

Solving for T:

T = [S_{o}(1 - α)/(4σ)]^{1/4} = 255K

0°C = 273K, so the temperature at which the Earth/atmosphere system is in radiative equilibrium is about -18°C! This is very cold from a human perspective -- far below the freezing point of water. But, the average Earth surface temperature is actually about 16ºC, certainly a more pleasant value from a human perspective. The greenhouse effect accounts for the difference of about 34ºC. Although the temperature of the Earth/atmosphere as viewed from space must still be -18ºC, greenhouse gases (primarily water vapor) warm the lower atmosphere by absorbing some emitted radiation and re-radiating it back to Earth's surface.

A very simple way of accounting for the greenhouse effect is to modify the radiative balance equation:

4σT^{4}(1 – x) = S_{o}(1 - α)

where x is a "greenhouse factor" that provides a measure of how much radiation is absorbed by greenhouse gases in the atmosphere and re-radiated down to Earth's surface, rather than out to space. For x=0, there is no greenhouse effect. For Earth, a value of x=0.4 produces an equilibrium temperature for the Earth/atmosphere system of about 16°C. When x approaches 1, a planet experiences a "runaway greenhouse effect," such as is observed on Venus. In this equation, T is now the temperature of the lower atmosphere, in Kelvins, which is roughly the surface temperature of the planet.

In the "build your own planet" model, you can specify the fraction surface coverage for land, water, ice/snow, and clouds for your planet, and the albedo for each category. Cloud cover is assumed to reduce each of the surface types by the same amount. For example, if 30% of the planet is land and the average cloud cover is 50%, this reduces the land cover to 15%.

In the spreadsheet model, you can graph temperature as a function of greenhouse parameter, planet/sun distance, and cloud cover. Within each set of calculations, you must specify the values for the x-axis of the graph. These columns are colored blue. Do this either by entering the values manually or by using a combination of manual entry and formulas, as noted in the spreadsheet. For values held constant during the calculation (surface types, for example), you can enter one set of values in the top row, colored light aqua. These values are automatically copied to the remaining rows.

If you want to modify the spreadsheet, it is a good idea to save a copy of the original file for future reference.