Note, this physical derivation is absolutely not suggesting the ~33C greenhouse effect doesn't exist. On the contrary, the physical derivation and observations demonstrate the 33C greenhouse effect does exist, but is explained by a different mechanism not dependent on radiative forcing from greenhouse gases. Also note, it is impossible for both explanations of the greenhouse effect to be true, since the global temperature would have to increase by an additional 33C (at least) above the present. You cannot have it both ways. We will show how the mass/gravity/pressure theory causes the temperature gradient and that the emission spectra of greenhouse gases seen from space are a consequence rather than the cause of that temperature gradient.
This derivation uses very wellknown physical principles and barometric formulae possibly first described by the great physicist Maxwell in 1872, who demonstrated that the atmospheric temperature gradient and greenhouse effect are due to pressure from Earth's gravitational field, not radiative forcing. Maxwell makes no mention of any influence of radiation as the cause of the temperature gradient of the atmosphere, but rather relates temperature at a given height to pressure. He discusses the convective (dominated) equilibrium of the atmosphere in his book Theory of Heat, pp. 330331:
"...In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure raised to the power (γ1)/γ, or 0,29..."Twenty four years later, Arrhenius devised his radiative forcing theory of the greenhouse effect, which unfortunately makes a huge false assumption that convection doesn't dominate over radiativeconvective equilibrium in the lower atmosphere, and thus Arrhenius completely ignored the dominant negativefeedback of convection over radiative forcing in his temperature derivations. Johns Hopkins physicist RW Wood completely demolished Arrhennius' theory in 1909, as did other published papers in 1963, 1966, 1973, (and others below), but it still refuses to die given its convenience to climate alarm.
We now know from Robinson & Catling's paper in Nature 2014 (and others) that radiativeconvective equilibrium on all planets with thick atmospheres in our solar system (including Earth of course) is dominated by convection/pressure/lapse rate in the troposphere up to where the tropopause begins at pressure = 0.1bar. When P < 0.1 bar, the atmosphere is too thin to sustain convection and radiation from greenhouse gases takes over to cause cooling of the stratosphere and above.
Since Maxwell's book was published in 1872, many others have confirmed that the greenhouse effect is due to atmospheric mass/pressure/gravity, rather than radiative forcing from greenhouse gases, including Hans Jelbring, Connolly & Connolly, Nikolov & Zeller, Mario BerberanSantos et al, Claes Johnson and here, Velasco et al, Giovanni Vladilo et al, Heinz Thieme, Jacques Henry, Stephen Wilde, Alberto Miatello, Gerhard Gerlich and Ralf D. Tscheuschner, Verity Jones, William C. Gilbert & here, Richard C. Tolman, Lorenz & McKay, Peter Morecombe, Robinson and Catling, and many others, so this concept is not new and preceded the Arrhenius theory.
Step 1: Derivation of the dry adiabatic lapse rate from the 1st Law of Thermodynamics and ideal gas law:
First, the basic assumption can be adopted that the atmosphere, in hydrostatic terms, is a selfgravitating system in constant hydrostatic equilibrium due to the balance of the two opposing forces of gravity and the atmospheric pressure gradient, according to the equation:
dP/dz =  ρ × g (1)
where ρ is the density (mass per volume) and g the acceleration due to gravity. This equation, from a mathematical point of view, can be derived by considering the hydrostatic equilibrium function as a system of partial derivatives depending on P and ρ and considering all three spatial dimensions:
∂P/∂x = ρ × X, ∂P/∂y = ρ × Y, ∂P/∂z = ρ × Z (2)
As, within a fluid mass in equilibrium, pressure and density does not vary along the horizontal axes (X and Y), the related partial derivatives equal zero. But, in the remaining vertical dimension, the partial derivative is nonzero, with density and pressure varying inversely as a function of fluid height (density and pressure decrease with increasing height relative to the bottom) and, considering gravitational force as a constant connected to the measure of density, thus equation (2) can be derived.
For a precise calculation involving the valid parameters, the 1st Law of Thermodynamics can be used:
Δ U = Q – W (3)
where U is the total internal energy of the system, Q its heat energy, and W the mechanical work the system is undergoing. Applying this relationship to Earth's atmosphere, yields:
U = C(p)T + gh (4)
where U is the total energy of atmospheric system in hydrostatic equilibrium and equal to the sum of the thermal energy (kinetic plus dissipative and vibrorotational), the specific heat C(p) multiplied by the temperature T plus the gravitational potential energy, with gravitational force g at height h of the gas. In this case, because the force of gravity has a negative sign as the system is undergoing work, the potential energy ( g × h) can be equated to the mechanical work (W) that the system undergoes in the 1st Law of Thermodynamics.
Based on this equation, the atmosphere's "average" temperature change can be found for any point with the system in equilibrium; for now and for simplicity, weather phenomena and disturbances at specific locations are not considered because, with the system in overall hydrostatic and macroscopic equilibrium, any local internal, microclimatic perturbation by definition triggers a rebalancing reaction. In fact, to calculate the energy change of the system in equilibrium (here U is constant) as a function of temperature and height change, differentiation yields:
dU = 0 = C(p)dT + gdh,
which becomes:
dT/dh = g/C(p), or dT = (g/C(p))dh. [Dry adiabatic lapse rate equation]
This is a splendid equation, describing precisely the temperatures’ distribution of a gas (as the air of Earth’s atmosphere) in hydrostatic equilibrium between the 2 forces of the lapserate (preventing the collapse of the atmosphere at the Earth’s surface) and gravity (preventing the escape of the atmosphere in the void of space).
In other words, temperature variation (dT) is a function of altitude variation (dh), whose solution at any point of height (h°) and for any temperature (T°), can be found by integrating as follows:
∫dT = g/C(p) × ∫dh (5)
and whose solution is:
T  T° = g/C(p) × (h  h°) (6)
where:
T – T° = ∆ T (or dT) = Interval of temperatures
g = Newton’s gravitational constant = 6.67 × 10^11 N (m/Kg)^2
h – h° = ∆ h (or dh) = Space interval (vertical) in the atmosphere
Cp = heat capacity at constant pressure
Step 2: Determine the height at the center of mass of the atmosphere
We are determining the temperature gradient within the mass of the atmosphere using a linear function of atmospheric mass (the lapse rate), therefore the equilibrium temperature is located at the center of mass. The "effective radiating level" or ERL of planetary atmospheres is located at the approximate center of mass of the atmosphere where the temperature is equal to the equilibrium temperature with the Sun. The equilibrium temperature of Earth with the Sun is commonly assumed to be 255K or 18C as calculated here. As a rough approximation, this height is where the pressure is ~50% of the surface pressure. It is also located at the approximate halfpoint of the troposphere temperature profile set by the linear adiabatic lapse rate, since to conserve energy in the troposphere, the increase in temperature from the ERL to the surface is offset by the temperature decrease from the ERL to the tropopause.
Fig 1. From Robinson & Catling, Nature, 2014 with added notations in red showing at the center of mass of Earth's atmosphere at ~0.5 bar the temperature is ~255K, which is equal to the equilibrium temperature with the Sun. Robinson & Catling also demonstrated that the height of the tropopause is at 0.1 bar for all the planets in our solar system with thick atmospheres, as also shown by this figure, and that convection dominates over radiativeconvective equilibrium in the troposphere to produce the troposphere lapse rates of each of these planets as shown above. R&C also show the lapse rates of each of these planets are remarkably similar despite very large differences in greenhouse gas composition and equilibrium temperatures with the Sun, once again proving pressure, not radiative forcing from greenhouse gases, determines tropospheric temperatures. 
Step 3: Determine the surface temperature
For Earth, surface pressure is 1 bar, so the ERL is located where the pressure ~0.5 bar, which is near the middle of the ~10 km high troposphere at ~5km. The average lapse rate on Earth is 6.5C/km, intermediate between the 10C/km dry adiabatic lapse rate and the 5C/km wet adiabatic lapse rate, since the atmosphere on average is intermediate between dry and saturated with water vapor.
Plugging the average 6.5C/km lapse rate and 5km height of the ERL into our equation (6) above gives
T = 18  (6.5 × (h  5))
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))
Ts = 18 + 32.5
Ts = 14.5°C or 288°K
which is the same as determined by satellite observations and is ~33C above the equilibrium temperature with the Sun.
Thus, we have determined the entire 33C greenhouse effect, the surface temperature, and the temperature of the troposphere at any height, entirely on the basis of the 1st law of thermodynamics and 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.
The greenhouse gas water vapor does have a very large negativefeedback cooling effect on the surface and atmospheric temperature by reducing the lapse rate by half from the 10C/km dry rate to the 5C/km wet rate. Increased water vapor increases the heat capacity of the atmosphere Cp, which is inversely related to temperature by the lapse rate equation above:
dT/dh = g/Cp
Plugging these lapse rates into our formula for Ts above:
Ts = 18  (10 × (0  5)) = 32C using dry adiabatic lapse rate
Ts = 18  (5 × (0  5)) = 7C using wet adiabatic lapse rate [fully saturated]
showing a cooling effect of up to 25C just from changes in the lapse rate from water vapor. Water vapor also cools the planet via evaporation and clouds, and which is confirmed by observations. Water vapor is thus proven by observations and theory to be a strong negativefeedback cooling agent, not a positivefeedback warming agent as assumed by the overheated climate models to amplify warming projections by a factor of 35 times.
What about CO2? At only 0.04% of the atmosphere, CO2 contributes negligibly to atmospheric mass and only slightly increases the heat capacity Cp of the atmosphere, which as we have shown above, is inversely related to temperature. CO2 would thus act as a cooling agent by slightly increasing troposphere heat capacity. Increased CO2 also increases the radiative surface area of the atmosphere to enhance outgoing radiation to space, analogous to putting a larger heat sink on your microprocessor which increases radiative surface area and convection to cause cooling.
It is wellknown that CO2 and ozone are the primary cooling agents of the stratosphere up to the thermosphere, but even the warmist proponents are unable to agree on a coherent explanation why CO2 would assume the opposite role of a warming agent in the troposphere. As the mass/gravity/pressure greenhouse theory shows, and just like water vapor, CO2 also acts to cool the troposphere, and the rest of the atmosphere by increasing radiative surface loss and outgoing radiation to space.
Millions of weather balloon observations confirm that there is no greenhouse gasinduced "hot spot" in the midupper troposphere, which is the alleged "fingerprint of AGW." The 2nd law of thermodynamics principle of maximum entropy production also explains why such a "hot spot" will not form. However, observations do show a cooling of the stratosphere over the satellite era, which would be consistent with increased CO2 increasing outgoing radiation to space. Observations also show an increase of outgoing longwave radiation to space over the past 62 years, which is entirely consistent with increased outgoing radiation from greenhouse gases and a decrease of "heat trapping", the opposite of AGW theory.
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 and thus pressure 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?
Update: The atmospheric center of mass assumption in step 2 above also appears to be applicable to Titan, the closest Earth analog with a thick atmosphere in our solar system. For Titan, the surface temperature is 94K, equilibrium temperature with the Sun is 82K, and surface pressure is 1.47 bar.
Thus, the center of mass of the atmosphere is located at ~1.47/2 = ~0.74 bar, which observations show is where Titan's atmospheric temperature is ~82K, the same as the equilibrium temperature with the Sun. I have added the notations in red to Robinson and Catling's graph below:
3. We have already shown that temperature is a function of pressure, and radiance and emission spectra from greenhouse gases are in turn a function of temperature, not the other way around.
What about CO2? At only 0.04% of the atmosphere, CO2 contributes negligibly to atmospheric mass and only slightly increases the heat capacity Cp of the atmosphere, which as we have shown above, is inversely related to temperature. CO2 would thus act as a cooling agent by slightly increasing troposphere heat capacity. Increased CO2 also increases the radiative surface area of the atmosphere to enhance outgoing radiation to space, analogous to putting a larger heat sink on your microprocessor which increases radiative surface area and convection to cause cooling.
It is wellknown that CO2 and ozone are the primary cooling agents of the stratosphere up to the thermosphere, but even the warmist proponents are unable to agree on a coherent explanation why CO2 would assume the opposite role of a warming agent in the troposphere. As the mass/gravity/pressure greenhouse theory shows, and just like water vapor, CO2 also acts to cool the troposphere, and the rest of the atmosphere by increasing radiative surface loss and outgoing radiation to space.
Millions of weather balloon observations confirm that there is no greenhouse gasinduced "hot spot" in the midupper troposphere, which is the alleged "fingerprint of AGW." The 2nd law of thermodynamics principle of maximum entropy production also explains why such a "hot spot" will not form. However, observations do show a cooling of the stratosphere over the satellite era, which would be consistent with increased CO2 increasing outgoing radiation to space. Observations also show an increase of outgoing longwave radiation to space over the past 62 years, which is entirely consistent with increased outgoing radiation from greenhouse gases and a decrease of "heat trapping", the opposite of AGW theory.
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 and thus pressure 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?
Update: The atmospheric center of mass assumption in step 2 above also appears to be applicable to Titan, the closest Earth analog with a thick atmosphere in our solar system. For Titan, the surface temperature is 94K, equilibrium temperature with the Sun is 82K, and surface pressure is 1.47 bar.
Thus, the center of mass of the atmosphere is located at ~1.47/2 = ~0.74 bar, which observations show is where Titan's atmospheric temperature is ~82K, the same as the equilibrium temperature with the Sun. I have added the notations in red to Robinson and Catling's graph below:
Update 2: Some still claim the ERL is set by radiative forcing, but this is false because:
1) A purely radiative model of the atmosphere without convection sets the ERL too high, at ~7km instead of ~5 km where observations show it is located. At 7km altitude, observations show the temperature to be 242K instead of the equilibrium temperature of 255K.
Solely radiative "greybody" model of the atmosphere without convection shown in blue. I have added in red the actual temperatures from the US Standard Atmosphere calculator. Note how the purely radiative model is up to 20K hotter, e.g. at the top of the troposphere, than the observations show. This is because convection dominates and "shortcircuits" radiative forcing in the troposphere to cause cooling.
2. Observations show no correlation between tropopause height and troposphere temperature 19802004:

That's because, as Robinson and Catling have shown, the height of the troposphere and the adiabatic lapse rate that extends from the 0.1 bar tropopause all the way to the surface is controlled by pressure, not temperature nor radiative forcing from greenhouse gases:
"At higher pressures [P > 0.1 bar], atmospheres become opaque to thermal radiation, causing temperatures to increase with depth and convection to ensue. A common dependence of infrared opacity on pressure, arising from the shared physics of molecular absorption, sets the 0.1 bar tropopause"
3. We have already shown that temperature is a function of pressure, and radiance and emission spectra from greenhouse gases are in turn a function of temperature, not the other way around.