In addition, the program also uses Stefan's law to determining the blackbody temperature of an object at equilibrium.
P * (1-Albedo) = s * T^4
|Power||W/m2||Power Albedo Temp|
|Albedo||% Reflected||Power Albedo Temp|
|Temperature||°K °C °F||Power Albedo Temp|
P * (1-Albedo) = s * T4 P Power - W/m2 s Stefan's Constant = 5.67 x 10-8 W m-2 K-4 T Temperature in degrees Kelvin
For thin (low mass) items in a vacuum, equilibrium may be reached in a few minutes.
On the Moon (no atmosphere and 28 day rotation), the maximum temperature is within a few degrees of what is predicted. However, on the dark side the minimum predicted temperature is never reached ... it is not even close. This indicates that objects heat up faster than they cool down. This asymmetry means that Stefan's law can be used to predict maximum temperatures, but is fairly worthless at computing the minimum temperature of a rotating body. It also means that using this equation to compute the expected temperature from the average energy will always give the wrong results.
1,361 W/m2 at TOA - 67*4 W/m2 absorbed by the stratosphere = 1,093 W/m2 at the surface Which can produce a temperature of 211°F (albedo = 0%)
For items on the Earth's surface (think roads), the predicted temperature (211°F) is never reached because of convection - the air gets hot and carries some heat away. Working with black asphalt samples, I was able to measure a maximum temperature of about 135°F. Assuming that the peak power at the surface is 1,093 W/m2 and an albedo of 4%, the expected temperature is 204°F. This difference is due to convection.
For a closed car (i.e. no convection), on a hot summer's day, the peak temperature (inside the car) is also about 135°F. (Yes, I have measured this.) Using this value (in the calculator above) allows computation of the effective albedo.
135°F -> 674 W/m2 (1093 - 674)/1093 = 0.38 (Albedo = 38%)
Note: This is why solar water heaters work in Canada.
To simplify using the calculator above, buttons are provided for TOA (1,361 W/m2) and TOA minus the energy absorbed in the stratosphere (1,093 W/m2).
TOA/4 * (1 - Albedo) = 340.25 * 0.70 = 238 W/m2 -> -18°C, -1°F (TOA-Strat)/4 * (1 - Albedo) = 273.25 * 0.70 = 191 W/m2 -> -32°C, -25°F
In addition, the albedo of sand, ocean, deserts, and people varies considerably.
At equilibrium, the rate of absorbing energy is equal to the rate of emitting energy. However, in a system where the energy input is pulse width modulated (the Earth spins), the difference in these rates will cause the average temperature to be different than the value computed via Stefan's law.