Thursday, November 27, 2014

Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases

The purpose of the recent series of physical proofs is to demonstrate that the greenhouse effect theory is entirely explained by the force of gravity, i.e. "gravity forcing" upon the mass of the atmosphere, rather than "radiative forcing" from greenhouse gases. This alternative "gravity forcing theory" of the greenhouse effect will be demonstrated to be completely independent of greenhouse gas radiative forcing, and compatible with all physical laws and millions of observations, as opposed to the radiative forcing theory.

We will use the ideal gas law, 1st law of thermodynamics, Newton's second law of motion (F = ma), and well-known barometric formulae in this derivation to very accurately determine Earth's surface temperature, the height in the atmosphere at which the effective equilibrium temperature of Earth with the Sun is located, and show that this height is located as expected at the center of mass of the atmosphere on Earth and Titan.

We will show that the mass/pressure greenhouse effect theory can also be used to accurately determine the temperatures at any height in the troposphere from the surface to the tropopause, and compute the mass/gravity/pressure greenhouse effect to be 33.15C, the same as determined from radiative climate models and the conventional radiative greenhouse effect theory. 

1. Conservation of energy and the ideal gas law

We will start once again with the ideal gas law 

PV = nRT (1)

an equation of state that relates the pressure P, volume V, temperature T, number of moles n of gas and the gas law constant R = 8.3144621 J/(mol K)

The properties of gases fall into two categories: 

1. Extensive variables are proportional to the size of the system: volume, mass, energy
2. Intensive variables do not depend on the size of the system: pressure, temperature, density

To conserve energy (and to ensure that no radiative imbalances from greenhouse gases are affecting this derivation) of the mass/gravity/pressure greenhouse effect, we assume

Energy incoming from the Sun (Ein) = Energy out (Eout) from Earth to space

Observations indeed show Ein = Eout = 240 W/m2 (2)

which by the Stefan-Boltzmann law equates to a blackbody radiating at 255 K or -18C, which we will call the effective or equilibrium temperature (Te) between the Sun and Earth. As seen by satellites, the Earth radiates at the equilibrium temperature 255K from an average height referred to as the "effective radiating level" or ERL or "effective radiating height."

2. Determine the "gravity forcing" upon the atmosphere

Returning to the ideal gas law above, pressure is expressed using a variety of measurement units including atmospheres, bars, and Pascals, and for this derivation we will use units in atmospheres, which is defined as the pressure at mean sea level at the latitude of Paris, France in terms of Newtons per square meter [N/m2]

Newtons per square meter corresponds to the force per unit area [or "gravity forcing" upon the atmospheric mass per unit area of the Earth surface]. 

Now let's determine the mass of the atmosphere above one square meter at the Earth surface:

By Newton's 2nd law of motion equation, force (F) is 

F = ma  (3)   where m = mass and a = acceleration

As we noted above, the atmospheric pressure is a force or forcing per unit area. The force in this case is the weight or mass of the atmosphere times the gravitational acceleration, therefore

F = mg  (4) where g is the gravitational constant 9.8 m/s2, i.e. the acceleration due to gravity in meters per second squared.

If we assume that g is a constant for the entire column of the atmosphere above the 1 meter2 area (A) we obtain

m = PA/g = (1.0325 x 10^5 N/m2 )(1 m2 )/(9.8 m/s2 ) = 1.05 x 10^4 kg

thus, the weight of the atmosphere over 1 square meter of the surface is 10,500 kilograms, quite a remarkable gravitational forcing upon the atmosphere.

If m is the mass of the atmosphere and g is the gravitational acceleration, the gravitational force is thus

F = mg (4)

The density (p) is the mass (m) per unit Volume (V), thus,

p = m/V

SI units of pressure refer to N/m2 as the Pascal (Pa). There are 1.0325 x 10^5 Pa per atmosphere (unit). 

Starting again with equation (3) above

F = ma  (3)

F = mg  (4)

F = (PA/g)g = PA  (5)

P = F/A = mg/A = phAg/A = phg (6) 


h=height along either a gas or liquid column under pressure or gravity field
g = gravitational constant
p = density = mass/volume

3. Determine the atmospheric pressures from gravitational forcing, and the height of the effective equilibrium temperature (ERL)

Now we will determine the atmospheric pressures in a gravitational field using (6) above

First let's determine the pressure at the ERL since the temperature must equal the equilibrium temperature of 255K at the ERL.

The pressure is a function of height 

P(h) = ρgh (7)

and the change in pressure dP is related to the change in height dh by 

dP = -ρg dh (8)

The minus sign arises from the fact that pressure decreases with height, subject to an adjustment for density which changes with height. We will determine this adjustment from the ideal gas law. The density is 

ρ = nM/V  (9)

where n is the number of moles, M is the molar mass, and V is the volume. We can obtain n/V from the ideal gas law: 

n/V = P/RT (10)


ρ = MP/RT  (11)

We can now substitute the density into the pressure vs. height formula:

dP = -(MPg/RT)dh  (12)

 dP/P = -(Mg/RT) dh  (13) (the first integral is from 1 to P, second from 0 to h)  

ln(P) = -(Mgh/RT)  (14)

P = e^-((Mgh/(RT))  (15)

We will now determine the height (h) at the ERL where the temperature = the effective equilibrium temperature = 255K, and without use of radiative forcing from greenhouse gases.

Plugging in numbers of M = 29 grams/mole (0.029 kg/mole) as average molar mass for atmosphere, g = 9.8 m/s^2, Pressure = 0.50 atmospheres at the approximate center of mass of the atmosphere, R=8.31, and T=Te=255K effective equilibrium temperature we obtain:

0.50 atmosphere P at the ERL= e^-((.029*9.8*5100)/(8.31*255))

So the height of the ERL set by gravity forcing is located at 5100 meters and is where T=Te=255K and pressure = 0.5 atmospheres, right at the center of mass of the atmosphere as we predicted from our gravity forcing hypothesis. 

4. Determine the temperatures at any location in the troposphere, and the magnitude of the mass/pressure greenhouse effect

Now that we have solved for the location of the ERL at 5100 meters, we can use the adiabatic lapse rate equation to determine all troposphere temperatures from the surface up to the ERL at 255K and then up to the top of the troposphere. The derivation of the lapse rate equation from the ideal gas law and 1st law of thermodynamics is described in this post, thus will not be repeated here, except to mention that the derivation of the lapse rate 

dT/dh = -g/Cp where Cp = heat capacity of the atmosphere at constant pressure

is also completely independent of any radiative forcing from greenhouse gases, greenhouse gas concentrations, emission/absorption spectra from greenhouse gases, etc., and is solely a function of gravity and heat capacity of the atmosphere. 

Plugging the average 6.5C/km lapse rate and 5100 meter or 5.1 km height of the ERL we determine above into our derived lapse rate equation (#6 from prior post) gives

T = -18C - (6.5C/km × (h - 5.1km)) 

Using this equation we can perfectly reproduce the temperature at any height in the troposphere as shown in Fig 1. At the surface, h = 0, thus temperature at the surface Ts is calculated as

Ts = -18 - (6.5 × (0 - 5.1)) 

Ts = -18 + 33.15C (gravity forced greenhouse effect)

Ts = 15.15°C or 288.3°K at the surface

which is exactly the same as determined by satellite observations and is 33.15C above the equilibrium temperature -18C or 255K with the Sun as expected.

Thus, we have determined the entire 33.15C greenhouse effect, the surface temperature, and the temperature of the troposphere at any height, and the height at which the equilibrium temperature with the Sun occurs at the ERL entirely on the basis of the Newton's 2nd law of motion, the 1st law of thermodynamics, and the ideal gas law, without use of radiative forcing from greenhouse gases, nor the concentrations of greenhouse gases, nor the emission/absorption spectra of greenhouse gases at any point in this derivation, demonstrating that the entire 33C greenhouse effect is dependent upon atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases. Also note, it is absolutely impossible for the conventional radiative theory of the greenhouse effect to also be correct, since if that was the case, the Earth's greenhouse effect would be at least double (66C+ rather than 33C). 

In essence, the radiative theory of the greenhouse effect confuses cause and effect. As we have shown, temperature is a function of pressure, and absorption/emission of IR from greenhouse gases is a function of temperature. The radiative theory tries to turn that around to claim IR emission from greenhouse gases controls the temperature, the heights of the ERL and tropopause, and thus the lapse rate, pressure, gravity, and heat capacity of the atmosphere, which is absurd and clearly disproven by basic thermodynamics and observations. The radiative greenhouse theory also makes the absurd assumption a cold body can make a hot body hotter,disproven by Pictet's experiment 214 years ago, the 1st and 2nd laws of thermodynamics, the principle of maximum entropy production, Planck's law, the Pauli exclusion principle, and quantum mechanics. There is one and only one greenhouse effect theory compatible with all of these basic physical laws and millions of observations. Can you guess which one it is?

Note the gravity forcing greenhouse theory also perfectly predicts the height of the ERL and surface temperature of Titan, the closest Earth analog in our solar system, and the only planet other than Earth with an atmosphere comprised of mostly non-greenhouse gases. The theory would not apply to any planets with thin atmospheres such as Mars which is unable to sustain significant convection. In the odd case of Venus, which I will pursue next, the atmospheric temperatures will likely be as determined by the mass/pressure theory plus additional warming from conduction downward from the thick opaque cloud top of the atmosphere, but this work is in progress.


  1. Thanks for all your work on this topic. I have enjoyed your posts immensely.

    ~ Mark

  2. MS, when are you ready to put this out on the open, like WUWT? It would be interesting to see such theory debated with a wider group of participants

    1. Very soon, I'm working on one or two more posts first to further refine the explanation of the theory, then I'll be ready to send it out to whoever wishes to post it for widespread discussion.

  3. There are many who don't accept the mass/gravity induced greenhouse effect on the basis that although compressed air warms up during the pumping process it cools down when the pumping stops and so they think gravity cannot be heating the atmosphere via compression.

    What they miss is that the continuous arrival of solar energy into the system is providing a continuous pumping up of the atmosphere against the continuous force of gravity and so the necessary heat energy is being replenished constantly.

    In effect there is a closed adiabatic energy loop between surface and atmosphere which is maintained with its own discrete energy content for as long as the sun keeps shining.

    That discrete adiabatic energy loop allows radiative energy in to match radiative energy out indefinitely.

    Furthermore, the thermal effect of any radiative capability within the atmospheric gases is simply negated by an adjustment in the adiabatic loop (which involves conduction and convection).

    That is why the Gas Laws work with no term for the radiative characteristics of the atmospheric gases.

    Good number work from MS, by the way.

  4. This may be of interest in that it overlaps with the work of MS:

  5. In your explanation you implicitly assume that the atmosphere absorbs incoming short wave radiation and not the surface of the earth. Isn't the atmosphere transparent to short wave radiation?
    And, you'll always find a planet's black body temperature (255°K in the earth's case) somewhere in its atmosphere. If you then take the planet's actual lapse rate (6.5°K/km) and extrapolate down to the surface, its just tautology that you get the actual surface temperature. You are not refuting the greenhouse effect...

    1. "In your explanation you implicitly assume that the atmosphere absorbs incoming short wave radiation and not the surface of the earth."

      Completely false. This derivation does not include the solar forcing derivation, which is upcoming in my next post and definitively proves your interpretation is false.

      "Isn't the atmosphere transparent to short wave radiation?"

      Mostly, although there is some absorption by Argon which I've been told has some overlapping (weak) lines and a bit by scattering from water vapor droplets and a bit from GHGs too, so I've read but not independently confirmed.

      "And, you'll always find a planet's black body temperature (255°K in the earth's case) somewhere in its atmosphere."

      Of course, but these series of posts show the height of the ERL is at the center of mass of the atmosphere, because gravitational forcing, not GHG forcing sets the level of the ERL.

      "If you then take the planet's actual lapse rate (6.5°K/km) and extrapolate down to the surface, its just tautology that you get the actual surface temperature. You are not refuting the greenhouse effect."

      You apparently have not understood the point that the entire GHE and lapse rate are determined by gravity, not GHG RF. Please read my next post also for additional proof.

    2. Its just that normally one assumes that incoming solar energy is absorbed at the earth surface (except from some reflection) and then warms the air above. Don't you assume that the incoming solar energy is absorbed by the whole atmosphere (more specifically by the center of mass of the atmosphere) and then warms the air and surface below? Otherwise I don't see how your "gravitational forcing" could work...

  6. According to AGW theory, only greenhouse gases radiate to space in LWIR absorption bands. What proponents of the theory seem to forget is that a good absorber is also a good radiator. According to AGW theory all other gases in our atmosphere are inert in terms of LWIR radiation. I would think then that the real heat trapping gases are the non greenhouse gases because they do not radiate to space. According to the theory here, how is heat energy radiated to space in LWIR absorption bands? I understand the idea of heat energy flowing up through the troposphere primarily by convection but how is the energy radiated to space?

    1. Yes, you're right. GHGs increase radiative surface area to enhance radiative cooling to space. It's analogous to putting a larger heat sink on your microprocessor, which cools more due to a larger radiative surface area (and convection).

      Additional reasons why this is true and N2 and O2 the true heat "trapping" gases are in this excellent essay:

  7. And here:

    Published by Stephen Wilde December 5, 2012

  8. Dear Sir, you should be using the adiabatic formulae? Your derived pressure lapse rate is for constant temperature. The principle is sound, however the adiabatic formula cannot have T = 255 at the centre of mass of the column of atmosphere. Although, I understand you have used the average temperature in the isometric formula. I am convinced that the +33C is due to gravitational redistribution of kinetic energies of molecules. I am still working on the problem in my own time. Observations from radiosondes actually give a better fit to pressure with height using the isometric equation you describe, than with the adiabatic formula, this is a bit confusing for me and I am still working on it. The radiosonde data I have looked at shows the average temperature (weighted by mass of course) of the atmospheric column is equal to the OLR measured by satellite - within a couple of percent. I got the OLR data from KNMI. I am looking for a more rigorous demonstration - i.e. resisting the urge to use the isometric equation - that can demonstrate the temperature distribution with height as a function of surface temperature, and account for that 33K difference. Kind regards, Luke

  9. I want to add that, by definition, there can be no temperature lapse rate in an isothermal atmosphere. The relation T=T(0)-(g/Cp)z is derived from the adiabatic formulae. Regards, Luke

    1. Luke,
      I am using the adiabatic and barometric formula. Any atmosphere in any gravitational field must produce a temperature gradient or lapse rate, and as you said a Boltzmann distribution of local kinetic energies due to the force of gravity F = ma = mg.