The Greenhouse Equation predicts temperatures within 0.28°C throughout entire troposphere without radiative forcing from greenhouse gases

In this continuing series of posts on the greenhouse equation, we will now calculate and plot the tropospheric temperature profile as a function only of the balance of radiative forcing from the Sun and gravitational forcing upon atmospheric mass, and without any contribution from greenhouse gas radiative forcing. We will see that the greenhouse equation reproduces observations and the 1976 standard atmosphere database within 0.28°C at every height from the surface all the way to the ~11,000 meter average height at the top of the troposphere. 

The greenhouse equation in essence determines the temperature based upon the balance of gravitational potential energy and radiative forcing from the only source of energy, the Sun, and is unique for each height from the surface to top of the troposphere. The greenhouse equation maintains the balance of solar/radiative and gravitational/convective equilibrium, and there is one and only one temperature that maintains this balance for each height in the troposphere. 

Note the only radiative forcing considered by the greenhouse equation is from the Sun, and nothing from greenhouse gases. The solar forcing warms the surface, which causes all gases to convect, rise, and expand, and which keeps the atmosphere inflated against the force of gravity, and these forces balance at every local height to maintain a horizontal local equilibrium at that particular height. Note the overall atmosphere is not in equilibrium, which is what drives this whole process of convection/adiabatic lapse rate/and the tropospheric temperature gradient.   

Thus, the opposing forces of convection and gravitation at each local height are in local equilibrium. The solar radiative forcing causes all gases at the surface to warm, rise, and expand against the gravitational force which is constantly pulling all gases back to the surface. Gravitational potential energy increases the higher a gas packet rises and expands, until that gas packet reaches a height where it's temperature is the same as the surrounding air, at which the gravitational potential energy starts to exceed its kinetic energy, then the gravitational potential energy accumulated takes over to make that gas packet subside/fall/compress and warm due to the ideal gas law. Once that gas packet becomes warm enough from compression to once again overcome gravitational potential energy, it starts to rise once again and that whole process continues over and over again ad infinitum. 

This rising/falling and expansion/compression of gas packets is thus driven by two opposing forces:

• Solar radiative forcing to make gases warm, rise, and expand until they cool to the same temperature as the surrounding air, followed by 
• Gravitational forcing pulling the now cooled gases back to the Earth

The balance between these two opposing forces is what generates the entire 33C greenhouse effect. Radiation spectra from greenhouse gases are the result, not the cause, of the temperature gradient in the troposphere that these two dominating, very strong, and opposing forces create. (As an example, in an earlier post, we calculated that the effect of gravity on the atmosphere is creating 10,500 kilograms of gravitational forcing per square meter at the surface.)

The greenhouse equation was derived in the recent series of posts: 

The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively.

and I'll now show the unique numeric solutions perfectly reproduce within 0.28C the temperature at every height in the 1976 Standard Atmosphere database from the surface all the way to ~11,000 meters height, after which the atmosphere at < ~0.223 atmospheres (units) becomes too thin to sustain convection and the equation no longer applies. This is near the transition between the troposphere and tropopause, which is where Robinson & Catling, Nature 2014, demonstrated the transition begins due to loss of convection, not greenhouse gas radiative forcing. Above the tropopause transition level (which is isothermal for several km), increased greenhouse gases cause increased cooling. 

The greenhouse equation solves for the single unique temperature at every height necessary to balance the vertical equilibrium of the atmosphere (since gravitational forcing is vertical only). The atmosphere is in horizontal equilibrium at every height relative to the center of mass at that coordinate. The equation applies as long as the troposphere is capable of sustaining convection, up to ~12,000km in the tropics, but beyond that the atmosphere is too thin to sustain convection and the adiabatic lapse rate, thus the equation is not applicable above this point around 11-12 km average:

We find the greenhouse equation perfectly reproduces (within 0.28C) the standard tropospheric temperature profile (black=Standard Atmosphere, red=the greenhouse equation). Here's the data calculated by Wolfram Alpha for the greenhouse equation, and the 1976 Standard Atmosphere data:

height (km)	Greenhouse Equation T in K	Std Atmos 1976 T in K
0	288.433	288.15
1	281.933	281.65
2	275.433	275.15
3	268.933	268.65
4	262.433	262.15
5	255.933	255.65
6	249.433	249.15
7	242.933	242.65
8	236.433	236.15
9	229.933	229.65
10	223.433	223.15
11	216.933	216.65
12	210.433	216.65
13	203.933	216.65

1976 Standard Atmosphere:

(Note the greenhouse equation does correct for the change in density with pressure, and the natural logarithmic decay of pressure with altitude)

We annotate the plot which shows the height of the "effective radiating level" or ERL, the height at which the local horizontal equilibrium at that particular height must by definition be where T = equilibrium temperature with the Sun = 255K, at essentially exactly at the center of mass of the atmosphere, which has to coincide with the location where half of the mass is above and half below, where the pressure is one-half the surface pressure after altitude and density correction - both of which are done by the greenhouse equation (and is the reason for the natural logarithms in the equation):

There is nothing in the greenhouse equation which is dependent upon radiative forcing from greenhouse gases, nor greenhouse gas concentrations, nor greenhouse gas absorption/emission spectra, and yet we can perfectly reproduce the temperature within 0.28C anywhere in the troposphere and at the surface. Thus, the absorption and emissions of IR from greenhouse gases are the consequence, and not the cause of the real 33C greenhouse effect. 

Here are some of the numerical solutions from Wolfram Alpha which did the for the one and only unique value T that satisfies the greenhouse equation horizontal equilibrium at a particular height (s) in kilometers above the surface, which is the one and only variable that I changed for each of the these numeric solutions, as you can see from the input and Wolfram Alpha solutions:

Try it yourself: Here's the code for the greenhouse equation to determine the T at height = 0 (i.e. at the surface). I have bolded that 0 in the code below so you can find it. Simply change that variable from 0 to whatever height in kilometers from the surface to calculate the one and only unique local equilibrium at that particular height solution for the greenhouse equation temperature gradient:

T = [1367 (1 - 0.3) / (4*1* (5.6704*10^-8))]^1/4 +((-6.5)*(0-[-[8.31*{[1367 (1 - 0.3) / (4*1* (5.6704*10^-8))]^(1/4)}*log(1/2)]/[9.8*0.029*log(e)]/1000]))

The Greenhouse Equation

A recent series of Hockey Schtick posts

• Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases
• Derivation of the effective radiating height & entire 33°C greenhouse effect without radiative forcing from greenhouse gases
• Why greenhouse gas radiative forcing doesn't explain Earth's energy budget

have derived the entire ~33°C greenhouse effect as a consequence of gravitational forcing rather than radiative forcing from greenhouse gases, and entirely independent of radiative forcing from greenhouse gases. We have also determined the effective radiating height (average) or ERL in the troposphere (where T = the equilibrium temperature of Earth with the Sun), and found the ERL to be located as expected at the center of mass of the atmosphere if the ERL height and temperature are a function of mass/gravity/pressure rather than radiative forcing from greenhouse gases.

We now join the gravitational greenhouse effect to the only source of energy that the Earth receives, the Sun, and show that solar shortwave radiative forcing plus gravitational forcing calculates the Earth's surface temperature, ERL height and temperature, and the entire tropospheric temperature profile perfectly, without any contribution from greenhouse gas forcing, nor dependence on greenhouse gas concentrations, nor dependence upon emission/absorption spectra from greenhouse gases. 

We show that the entire 33°C greenhouse effect that raises Earth's equilibrium temperature with the Sun of -18C or 255K up to +15C or 288K at the surface, and the temperature at any height in the atmosphere from the surface to top of the troposphere (above which the atmosphere is too thin to sustain convection), can be fully explained by the following equation, which I'm calling "the greenhouse equation":

The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively. 

which solves T as a function of mass/pressure/gravity for which none of the variables are dependent upon radiative forcing from greenhouse gases, and for which the only radiative forcing we require to reproduce the entire tropospheric temperature profile is that from the Sun. Note none of the constants and variables on the right side of the greenhouse equation are related to GHG radiative forcing, and temperature does not appear on the right side of the equation and thus it can't be a tautology of temperature.

T = temperature at height (s) meters above the surface, thus at the surface s = 0

s = height in meters above the surface to calculate the temperature T, thus at the surface s=0
S = the solar constant = 1367 W/m2, derivation here
ε = emissivity = 1 assuming Sun and Earth are blackbodies
σ = the Stefan-Boltzmann constant = 5.6704 x 10-8 W m-2 K-4
g = gravitational acceleration = 9.8 m/s^2

m = average molar mass of the atmosphere = 29g/mole = 0.029kg/mole
α = albedo = 0.3 for earth

C = Cp = the heat capacity of the atmosphere at constant pressure, ~ 1.5077 average for Earth

P = surface pressure in the unit atmospheres, defined as = 1 atmosphere for latitude of Paris
R = universal gas constant = 8.3145 J/mol K
e = the base of the natural logarithm, approximately equal to 2.71828

As shown by the prior posts listed above, all of the components of this entire gravitational "greenhouse equation" were first derived from the ideal gas law, the First Law of Thermodynamics, the Stefan-Boltzmann equation, Newton's second law of motion (F = ma), and well-known barometric formulae, without ever once introducing any variables dependent upon radiative forcing from greenhouse gases. 

In a prior post we determined surface pressure by the relation:

P = e^-((Mgh/(RT))  (15) where T = T at height (s) (or (h) in the prior post)

for which we substitute for P in the greenhouse equation above (note s = h from the prior post) to yield:

After plugging in the numerical values, Wolfram Alpha solves the greenhouse equation to find Earth's surface (where the height (s)=0) the temperature T is equal to 288.433K or 15.28C, which is the same as determined from satellite measurements:

Note the 1000 in the numeric solution is the conversion factor between meters and kilometers

We can now use the greenhouse equation for many other purposes including determining the effect of a change in solar activity on the expected Earth surface temperature. If we increase the solar constant in the above numerical solution by 1 W/m2 from 1367 to 1368 W/m2, we find an increase in surface temperature from T=288.433K to 288.486K, an increase of 0.056C (this includes the division by 4 to convert solar insolation from a flat disk to a sphere):

Note Wolfram Alpha is solving for T, providing a unique solution of T as a function of the mass/gravity/pressure forcing at each height (s) in meters above the surface (s = 0 at the surface). Each solution for T as a function of height (s) perfectly reproduces the observed tropospheric temperature profile at each height (s), from the surface to the ERL located at the atmospheric center of mass and then on up to the top of the troposphere. 

Wolfram Alpha solves for the unique temperature (T) that satisfies the greenhouse equation for a given height (s)  with P substituted by e^-((Mgh/(RT)) (15) above, showing there is a unique solution of T for every value of height (s), with no T on the right-hand side of the solution for T, proving the greenhouse equation is not a tautology:

We have also demonstrated why the atmospheric mass/gravity/pressure theory of the greenhouse effect also perfectly explains the observed greenhouse effect on 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 (Titan: 98.4% Nitrogen, 0.1% hydrogen, and only 1.5% greenhouse gas methane compared to Earth's 78.09% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide).

Note this equation would not be expected to hold for planets with thick, opaque cloud tops, such as Venus, which in addition to heating from the bottom up due to gravity/pressure, is also heated from top down by absorption of sunlight at the TOA and probable conduction of heat downward from the opaque cloud tops. It also cannot be applied to planets with thin atmospheres such as Mars, which has a surface pressure of only ∼0.006 bar. In addition, the ideal gas law and barometric equations are only true if the heat capacity C in the greenhouse equation stays constant while the temperature changes, which is true for N2 and O2 (more than 99% of our atmosphere and 98.4% of Titan's atmosphere), but CO2 does not. Since CO2 comprises 96.5% of the Venus atmosphere, the heat capacity C (Cp) in the greenhouse equation would have to be adjusted as a function of temperature.

I welcome all suggestions and refutations of the "greenhouse equation." There can be only one valid theory of the 33C greenhouse effect, since if both the gravitational and greenhouse gas radiative forcing theories had merit, the Earth would be at least 33C warmer than the present. 

Quick and dirty explanation of the Greenhouse Equation and theory

Here's a quick and dirty explanation of the greenhouse equation, which I hope will be helpful to understand the theory behind it. 

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)

where

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:

New paper finds increased CO2 or methane will have 'essentially no effect' upon global temperature or climate

A new paper by USC Professor Emeritus of Geology, Dr. George Chilingar (with three co-authors), finds that increasing levels of the greenhouse gases CO2 & methane will have "essentially no effect" upon global temperatures or climate. 

The authors utilize a one-dimensional adiabatic model of climate to demonstrate that the entire tropospheric temperature profile of the atmosphere on both Earth and Venus may be mathematically derived solely on the basis of atmospheric pressure/mass and solar activity, confirmed by observations on both planets, despite vast differences in atmospheric composition and mass/pressure on Earth and Venus. The paper corroborates the 33C Maxwell/Clausius/Carnot greenhouse theory and thereby excludes the alternative 33C Arrhenius radiative greenhouse theory.

Excerpts:

"The writers investigated the greenhouse effect using their adiabatic model, which relates the global temperature of troposphere to the atmospheric pressure and solar radiation. This model allows one to analyze the global temperature changes due to variations in mass and chemical composition of the atmosphere. Even significant releases of anthropogenic carbon dioxide and methane into the atmosphere do not change average parameters of the Earth’s heat regime and have no essential effect on the Earth’s climate warming. Moreover, based on the adiabatic model of heat transfer, the writers showed that additional releases of CO2 and CH4 lead to cooling (and not to warming as the proponents of the conventional theory of global warming state) of the Earth’s atmosphere. The additional methane releases possess a double cooling effect: First, they intensify convection in the lower layers of troposphere; Second, the methane together with associated water vapor intercept part of the infrared solar irradiation reaching the Earth. Thus, petroleum production and other anthropogenic activities resulting in accumulation of additional amounts of methane and carbon dioxide in the atmosphere have practically no effect on the Earth’s climate."

Physically, an explanation of the cooling effect of the atmosphere with the high content of “greenhouse gases” is the high efficiency of the convective heat transfer from the planet’s surface to the lower stratosphere, from which this heat is rapidly dissipating into the outer space through radiation. As the greenhouse gases absorb the Earth’s heat radiation in the lower layers of troposphere, its energy transforms into the heat oscillations of the gas molecules. This, in turn, leads to expansion of the gas mixture and its rapid ascent to the stratosphere where the heat excess is lost through radiation into the outer space.  

To replace these volumes of the warm air, the already cooled air descends from the upper troposphere. As a result, the global average atmospheric temperature slightly decreases. One particular consequence of it is that with an increase in the carbon dioxide and methane contents in troposphere the convective mass exchange of the atmospheric gases must substantially accelerate. Thus, it is not out of the question that the intensification of synoptic processes in Earth troposphere (but not temperature increase) may be a result of the carbon dioxide and other “greenhouse gases” accumulation."

The primary equation of the paper [2] is similar to the 'greenhouse equation' described in a recent series of posts on the 33C Maxwell/Clausius/Carnot greenhouse theory. 

The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively.

The primary differences between Chilingar et al equation [2] and the 'greenhouse equation' are:

1. Chilingar et al introduce a correction for solar insolation based on the Earth's precession angle of 23.44 degrees 

2. Chilingar et al assume an Earth surface temperature of 288K or 15C, whereas the HS 'greenhouse equation' only assumes the equilibrium temperature of the Earth with the Sun (255K or -18C) & atmospheric mass/pressure to derive the surface temperature, as well as that of the entire troposphere, replicating the 1976 US Standard Atmosphere. 

An upcoming post will join the mathematics of these two equations to explain the entire temperature profile of the atmosphere from the surface to the edge of space at 100+ km geopotential altitude, without incorporating 'radiative forcing' from CO2. 

 

Why the effective radiating level (ERL) is always located at the center of mass of the atmosphere & not controlled by greenhouse gas concentrations

The Arrhenius radiative greenhouse effect proponents, having abandoned "back-radiation" from greenhouse gases as the explanation of the greenhouse effect, now claim global warming is instead due to an increase of the "effective radiating height" or "effective radiating level" [ERL] of greenhouse gases in the atmosphere. So the theory goes, an increase of CO2 levels will cause longwave (~15 micron) infrared emissions from CO2 to occur from colder heights in the atmosphere, and since colder blackbodies emit less radiation, more radiation will allegedly be "trapped" by the colder CO2 "blackbody" in the fabled tropospheric "hot spot" & unable to escape to space. 

In contrast, the competing 33C gravito-thermal greenhouse effect of Maxwell, Clausius, Carnot, Boltzmann, Feynman, Poisson, Helmholtz, et al, shows that the "effective radiating level" or ERL is fixed at the center of mass (COM) of the atmosphere.

As we can see in Fig 1a, the observed ERL or "emission level for OLR (Outgoing Longwave Radiation)" global average is right around 500 millibars or 0.49 atmospheres ~ 0.5 atmospheres, exactly at the center of mass of the entire atmosphere as predicted in the HS greenhouse equation below. 

The HS greenhouse equation "triangulates" the geopotential height of the 255K ERL at the center of mass using:

1. The center of mass (COM) of the atmosphere where P=0.5 atmospheres (after density correction), i.e. exactly one-half of the surface pressure
2. The adiabatic lapse rate = -(gravitational acceleration constant g)/(heat capacity at constant pressure Cp)
3. The equilibrium temperature of Earth with the Sun = 255K

all of which are essentially constants in the atmosphere, and without any knowledge of the surface temperature, greenhouse gas concentrations, or Arrhenius "radiative forcing" from greenhouse gases.  

Why use the center of mass of the atmosphere in calculation of the gravito-thermal greenhouse effect? Because the force of gravity by Newton's Second Law is F = ma = mg, and for a system of particles like our atmosphere, one must determine the center of mass in applying Newton's 2nd Law F = mg to the force of gravity. 

Thus, since the height of the ERL is fixed at the COM, and the COM is essentially a constant, the height of the ERL will not change, regardless of greenhouse gas concentrations.

In addition, in the longwave infrared band of Earth’s thermal radiation, the only band in which CO2 absorbs and emits is centered at ~15 microns. The kinetic temperature of the surrounding atmosphere and the CO2 molecules has nothing to do with the fact that CO2 emits at a fixed ~15 microns in the longwave IR due to its fixed molecular structure bending transitions. The entire atmosphere surface to space is warmer than the CO2 “equivalent partial blackbody” fixed band-emitting temperature of 193K at ~15 microns.

Also, absorption followed by emission of a photon by CO2 only takes microseconds, and all the bouncing around at the speed of light between greenhouse gases in the atmosphere only delays the average photon a few milliseconds on its way from the surface to space. Thus, the only "slowing of cooling" or "heat trapping" by CO2 absorption/emission is a few milliseconds and easily reversed and erased over a 12 hour night. 

Addition of more CO2 increases the few milliseconds delay by adding a few more milliseconds, but once again is easily reversed and erased over a 12 hour night.

More importantly, increased CO2 increases radiative surface area, which increases radiative loss to space. That’s why increased CO2 cools the stratosphere through thermosphere, and troposphere as well as I’ve shown.

And even more importantly, the probability of CO2 transferring heat by collisions with N2/O2 in the troposphere is about 2 orders of magnitude higher than emitting a photon, which increases convective cooling.

An earlier post also provides nine additional reasons why the effective radiating level (ERL) is always located at the center of mass of the atmosphere & not controlled by greenhouse gas concentrations.

Thus, the false notion that global warming is instead due to an increase of the "effective radiating height" or "effective radiating level" [ERL] of greenhouse gases in the atmosphere is effectively disproven.

The HS greenhouse equation and quick & dirty explanation below, followed by the derivation from first principles:

The "Greenhouse Equation" calculates temperature (T) at any location from the surface to the top of the troposphere as a function of atmospheric mass/gravity/pressure and radiative forcing from the Sun only, and without any radiative forcing from greenhouse gases. Note the pressure (P) divided by 2 in the greenhouse equation is the pressure at the center of mass of the atmosphere (after density correction), where the temperature and height are equal to the equilibrium temperature with the Sun and ERL respectively.

http://hockeyschtick.blogspot.com/2014/11/quick-and-dirty-explanation-of.html

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 (note weight is not the same as mass and is in physical definitions of mass, length, time-2) or mass of the atmosphere times the gravitational acceleration, therefore

F = mg  (4) where g is the gravitational constant 9.8 m/s^2, 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) 

where 

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)

thus 

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?

Debunking Myths & Strawmen about the Gravito-Thermal Greenhouse Effect & Radiative Greenhouse Effect

This post will be continuously updated with a list of all posts concerning the gravito-thermal greenhouse effect, the derivation and use of the greenhouse equation of the gravito-thermal greenhouse effect, as well as numbered responses to common objections.

This is in lieu of constantly repeating information in responses to new comments here & elsewhere, to link to the numbered list below referring to a specific post which addresses the argument in question for or against the two competing 33C greenhouse effect theories (because one and only one of these greenhouse theories can be correct, otherwise Earth would be at least 33C warmer than present): 

1) The Arrhenius radiative greenhouse effect theory (the catastrophic man-made CO2 global warming theory)

vs.

2) The Maxwell gravito-thermal greenhouse effect theory

1]  The Greenhouse Equation

2] How Gravity continuously does Thermodynamic Work on the atmosphere to control pressure & temperature

3] Why Greenhouse Gases Don't Affect the Greenhouse Equation or Lapse Rate (debunks claim that greenhouse gases are necessary for convection or a lapse rate to occur or that greenhouse gas radiative forcing can affect the lapse rate)

4] Quick and dirty explanation of the Greenhouse Equation and theory

5] The Greenhouse Equation predicts 1% change in cloud cover changes global temperature by 1°C

6] Why the atmosphere is in horizontal thermodynamic equilibrium but not vertical equilibrium (debunks claims that the gravito-thermal greenhouse effect assumes thermodynamic equilibrium in all three x, y, and z planes).

7] The Greenhouse Equation predicts temperatures within 0.02°C throughout entire troposphere without radiative forcing from greenhouse gases

8] Why increased water vapor decreases the lapse rate by half to cause surface cooling of up to 25.5C

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

10] Derivation of the entire 33°C greenhouse effect without radiative forcing from greenhouse gases

11] French scientist explains why the greenhouse effect is primarily due to atmospheric mass/gravity/pressure

12] Modeling of the Earth’s Planetary Heat Balance with an Electrical Circuit Analogy

13] Why can't radiation from a cold body make a hot body hotter?

Answers these queries:

Can radiation from a cold body increase the temperature of a warmer body?

Are the Stefan-Boltzmann and Planck Laws applied correctly in calculating the greenhouse effect?

How can radiation from a cold body not be thermalized [cause an increase in temperature] of a warmer body?

How does quantum mechanics explain why a cold body can't make a warm body warmer still?

How do photons "know" how to do this?

Does water vapor warm or cool the planet?

Do clouds warm or cool the planet?

Why are cloudy nights warmer?

Do clouds cause 25% of the radiative greenhouse effect theory as claimed?

14] Okulaer on "Why atmospheric radiative greenhouse warming is a chimaera"

15] Why the ideal gas law, gravity, & atmospheric mass explain the entire 33C greenhouse effect

16] Why Earth's climate is self-regulating & independent of man-made CO2

17] New paper demonstrates climate models don't even have the 'basic physics' of the greenhouse effect correct


Short summary of the 33C gravito-thermal greenhouse effect 

The ~33C gravito-thermal greenhouse effect, first described by the great physicist Maxwell in 1872 by the barometric Poisson Relation, describes the temperature gradient from the 220K tropopause all the way down to the 288K Earth surface. As we have shown, the "average temperature" within the distribution of this quasi-linear [lapse rate] temperature gradient of the troposphere matches the energy input from the Sun, thus conserving energy:

(288K + 220K)/2 = 254K ~ 255K = Equilibrium temperature with the Sun (located at center of mass of atmosphere)

288K (at surface) - 255K (at center of mass of the atmosphere) = 33C gravito-thermal greenhouse effect

Thus fulfilling the 1st law requirement of conservation of energy. (Before someone comments, I know you can't properly average temperatures, and that temperature is not a direct proxy for heat energy because calorimetry requires the mass, specific heats, heats of fusion and vaporization, and all phase changes be accounted, but use of temperature as a proxy of heat is done for illustrative purposes and simplification of the explanation)

The temperature (a rough proxy for heat energy) distribution on either side of equilibrium temperature with the Sun Te = 255 is approximately an equal distribution around the center of atmospheric mass in the ~middle of the troposphere at ~5100 meters, thus conserving energy and placing Te = 255K at the center of mass (where P=1/2 of surface pressure) of the gravito-thermal greenhouse effect.

Common myths & strawmen arguments & rebuttals:

18] Myth: The Arrhenius radiative greenhouse theory is incontrovertible "basic physics"

Rebuttal: Twenty-six years before Arrhenius devised his radiative greenhouse theory, the greatest physicist in history on the topics of heat and radiation, James Clerk Maxwell, said that the gravito-thermal greenhouse effect is what creates the atmospheric temperature gradient, not radiation from greenhouse gases. The Arrhenius radiative greenhouse theory makes at least three huge incorrect physical assumptions:

1) cold bodies can make much hotter bodies much hotter (in violation of the 2nd law of thermodynamics which says transfer of any heat from cold to hot would cause an impossible decrease of entropy, since the second law requires total entropy to always increase),

2) that radiation dominates over convection in the troposphere (disproven by countless papers and observations),

3) gravitational forcing upon atmospheric mass (as described by the barometric formulae) does not create the temperature gradient in the troposphere (disproven by Maxwell, the barometric formulae, and millions of observations).

Thus, all of the major physical assumptions of the Arrhenius radiative greenhouse effect are false. One and only one 33C greenhouse theory, either gravitational or radiative, can explain the entire 33C greenhouse effect; you cannot have it both ways and both cannot have merit, otherwise the Earth would be 33C warmer than at present. In addition the "Maxwell theory" is the only one compatible with the 18-26 year "pause" or "hiatus" of global warming along with a 20% increase in CO2 levels.


19] Myth: “Every pressurised container would be hot if pressure increases temperature”

Rebuttal: Let's use a bicycle tire analogy. When you use a pump to pressurize the tire it gets hotter for awhile, but then cools to ambient room temp by the tire convecting that heat to the atmosphere.

Second situation is we have a leaky tire that we have to keep pumping to maintain the same pressure, so that tire remains hotter as long as compression of the air is a continuous process.

The atmosphere is only analogous to the second situation, since air packets are continuously warming at the surface then rise/expand/cool until equilibrium with surrounding air in upper atmosphere, then due to gravitational potential energy these air packets have accumulated then fall/compress/warm down to the surface.

This is how the troposphere temperature gradient from 220K-288K is entirely controlled by these barometric processes and perfectly predicted by the barometric formula.