Wednesday, October 5, 2016

The Earth is out of Equilibrium; That is why it is warming

(sigma)(T_e)^4 is the radiation entering the system.  It is the solar constant adjusted by the planet’s albedo (amount of energy from the sun it reflects into space). sigmaT_e^4 is constantly warming the surface, and sigmaT_e^4 is constantly being radiated to the atmosphere and emitted back to the surface meaning sigmaT_e^4 is doubled as the energy per second per square meter warming the surface of the earth. For the temperature of the earth to be constant, as much energy as enters the system must leave the system.  That amount is sigmaT_e^4. This is called an equilibrium state.  If the earth is out of equilibrium, then it is either warming or cooling.  Since satellite measurements measure that the earth is receiving one watt per square meter more than it is losing, this means the earth is out of equilibrium and is warming.  This can be attributed to excess CO2 in the atmosphere, a powerful green house gas that started existing in excess amounts during the beginning of the industrial age.

Let me elaborate:

sigmaT_e^4=(S_0/4)(1-a)

S_0 is the solar constant.  It is the amount of energy per second per square meter (watts per square meter) from the sun at earth orbit.  a is the albedo of the earth which is about 0.3 (that is to say the earth reflects 30% of the radiation from the sun back into space meaning, 1-a = 0.7 is the 70% that reaches the earth.

S_0 is calculated by reducing the energy the sun emits by the inverse square law, so that the radiation that reaches the earth is given by:

(L_0)/4(pi)r^2 = 3.9E26/4(3.141)(1.5E11)^2 = 1,370 watts/meter^2

Where L_0 is the energy per second emitted by the sun which is 3.9E26 Joules per second and r is the earth’s orbital radius which is 1.5E11 meters.

T_e is the annual average temperature of the earth.

sigma is the steffan-boltzmann constant = 5.67E-8 with all that you can determine the annual average temperature of the surface of the earth. you will get a value of about 86 F, (30 C) which is higher than the annual average, which is about 15 C. However in everything said here we have not taken into account cooling by convection (energy lost from evaporization of the ocean to form clouds, and the energy lost when that precipitates and returns to the ocean as rain.)

The calculation must be done in Kelvin.  That can then be converted to Fahrenheit and Celsius.

Twice sigmaT_e^4 is the annual average temperature of the earth (sigmaT_e entering from the sun and sigmaT_e radiated back from the atmosphere) where T_s is the annual average temperature of the surface of the earth, is given by 2sigmaT_e^4. So T_s=2^1/4T_e = 1.189T_e. Finally from above sigmaT_e^4 = (S_0/4)(1-a)=(1370/4)(1-0.3)=239.75 and T_e^4 = 239.75/5.67E-8=4228395062. Take the fourth root of that with your calculator: And T_e=255 degrees Kelvin. Now we have T_e. We said a minute ago that T_s = 1.189T_e so T_s=1.189(255)=303 degrees kelvin. That would be the annual average temperature of the earth without taking convection into account. 

Let's convert that to Celsius: degrees C = degrees K - 273 degrees C =303-273=30 degrees centigrade.

Why do we say sigma T sub e to fourth (sigmaT_e^4) is the watts per square meter of radiation? The answer is if you do an experiment you will find the flux (watts per square meter) of a radiating body are proportional to the fourth power of the temperature and sigma is the constant of proportiionality that allows you to set the two equal to one another.