In the first post of this series we derived the following equation from the 1st law of thermodynamics and ideal gas law to calculate the temperature at any height in the troposphere:
T = Te + (lapse rate)*(h - he) (1)
T = calculated T at height (h), or (s) used in equation below
Te = equilibrium temperature with the Sun (a constant)
lapse rate = -g/Cp = -gravity/heat capacity at constant pressure
he = height at the average "effective radiating level" or ERL, where T= equilibrium Te with the Sun
Since we are calculating the gravitational greenhouse effect on the mass of the atmosphere, in order to conserve energy, one-half of the gravitational potential energy of the atmosphere has to be above the center of mass and one-half below. This point is at 1/2 of the surface pressure after a logarithmic adjustment for pressure and density with altitude. Since the surface pressure in atmosphere units is by definition = 1 atmosphere, in the greenhouse equation log(P/2) = log(1/2) below.
Again to conserve energy, the equation has to balance the local vertical equilibrium (since gravity is a vertical forcing vector) at every given local height the gravitational potential energy with the opposing thermodynamic energy of convection. Thus, the center of mass where log(P/2) must be the balance point where the two opposing forces balance, and this same point must also be at the equilibrium temperature with the Sun and near the mid-point of the adiabatic lapse rate.
We previously calculated he to be located at the h where P=log(Ps/2) and the same point where Te = 255K = equilibrium temperature with the Sun.
The gravitational acceleration constant (g) appears twice in the equation, since to calculate the gravitational forcing we used Newton's second law of motion F=ma, which applied to the atmosphere is F=mg. The second use of the gravitational constant is from the dry adiabatic lapse rate.
Here's the greenhouse equation and my quick and dirty notes on how the components I just discussed enter into equation (1) above to calculate the temperature T at any height including the surface, as well as the entire 33C greenhouse effect, and without ever once using any radiative forcing whatsoever from greenhouse gases: