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.
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:
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.
The HS greenhouse equation and quick & dirty explanation below, followed by the derivation from first principles:
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| 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/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)
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:
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?
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/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)
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:
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| 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?
