slight error in the last post.
"s is Stefan–Boltzmann constant". It should be theta. The theta symbol keeps getting changed to "s"
While I was looking to correct that error I was going through the wikipedia pages on the subject to try and find the page I quoted from. Fact is it was on quora.com
quora.com
However I did come across this... It's most interesting how Stefan determined an acurate temperature of the Sun. [Some of the equation details got lost on the copy and past please refer to the link below]
en.wikipedia.org
It was an exciting time for science.
Temperature determination of the Sun
With his law Stefan also determined the temperature of the Sun's surface. [8] He inferred from the data of Jacques-Louis Soret (1827–1890) [9] that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle as the Sun. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. Stefan surmised that 1/2 of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.
Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K (the modern value is 5778 K [10]).
and regarding the temperature of the Earth... Please do note my underlining and emboldment regarding the forth power. Therefore any deviation caused by a temperature increase (or decrease) will tend to correct unless the Earth changes it radiation characteristic or the Sun changes it's temperature. Clouds and ice cover tend to be a driving factor (albedo) and water vapour and in particular clouds account for majority of the greenhouse effect.
Effective temperature of the Earth
Similarly we can calculate the effective temperature of the Earth T? by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation (Earth's own production of energy being small enough to be negligible). The luminosity of the Sun, L?, is given by:
At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the irradiance (received power per unit area) is given by
Because the Stefan–Boltzmann law uses a fourth power, it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed, close to the steady state where:
This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.
The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving 255 K (-18 °C). [14] [15]
The above temperature is Earth's as seen from space, not ground temperature but an average over all emitting bodies of Earth from surface to high altitude. Because of the greenhouse effect, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.
In the above discussion, we have assumed that the whole surface of the earth is at one temperature. Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it. This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through. When the sun is at the zenith and the surface is horizontal, the irradiance can be as high as 1120 W/m2. [16] The Stefan–Boltzmann law then gives a temperature of
or 102 °C. (Above the atmosphere, the result is even higher: 394 K.) We can think of the earth's surface as "trying" to reach equilibrium temperature during the day, but being cooled by the atmosphere, and "trying" to reach equilibrium with starlight and possibly moonlight at night, but being warmed by the atmosphere.
I think the public needs to be made aware of this process, and these temperature calculations before anyone should be running off with truly amazing calculations of what 400 ppm of CO2 can do. |
