A Critical Lesson from the NASA Earth Energy Budget

The essence of the argument that an increase in the concentration of carbon dioxide, an infrared-active or greenhouse gas, causes the Earth's surface to become warmer lies in a radiation-dominant viewpoint in the transport of energy between the surface and the atmosphere.  This viewpoint is depicted in the NASA energy budget shown below:

If the Earth's surface is simply viewed as a black body radiator at temperature T as though it were isolated in space with no other energy loss mechanisms, then the rate of energy loss by means of radiation would be

P = 398.2 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, and T = 289.5 K

Since the Earth's average surface temperature is usually said to be about 288 K, this means that the Earth's surface is assigned an emissivity of 1.02, making it a super black body radiator.  Real objects generally have an emissivity less than 1 and never more than 1.

Let us note a result discussed in my post mgh, Not Just Greenhouse Gases, Provides a Warm Earth because I do not want this critical observation to be lost among other observations.  I want to discuss its implications more here.

Of the 398.2 W/m² of infrared radiation emitted from the surface in this NASA Earth energy budget, 358.2 W/m² is absorbed by the atmosphere.  The atmosphere only absorbs infrared radiation at the wavelengths that water vapor and carbon dioxide absorb, aside from very minor absorption by other infrared-active gases.  It does not act like a black body absorber and it does not act like a black body emitter.  But we can establish upper and lower bounds on what the atmosphere is capable of doing if we treat it as though it were a black body radiator.  When dealing with complex physics problems, it is always good to know when the answers you provide are within the physically possible bounds.

Doing so, and taking Ta to be the temperature of the black body absorber in the atmosphere and Ts the temperature of the surface, we have 

σ (Ts)⁴ -  σ (Ta)⁴ = 358.2 W/m² ,

which is the maximum electromagnetic radiation the absorber can absorb from the higher temperature emitter black body.  We know that for the NASA Earth energy budget that the first term on the left is 398.2 W/m² , albeit with an emissivity of 1.02, so we calculate that 

Ta = 163.0 K.

Now this is a very interestingly low temperature.  There is no temperature this low in the U.S. Standard Atmosphere Table of 1976.  The temperature with altitude drops approximately linearly in the troposphere, stabilizes at the minimum temperature of 216.65 K in the tropopause, and then increases with altitude through the stratosphere until there is essentially no atmosphere left to do any absorbing of infrared photons.  At 86 to 90 km altitude, in the mesopause, there is another temperature minimum of about 186.9 K in that table, though other sources say the temperature can get as low as 173 K.  These very low mesopause temperatures are caused by strong cooling due to carbon dioxide radiation of heat to space.  At 1000 km altitude, the U.S. Standard Atmosphere of 1976 temperature is 1000 K and the density of the atmosphere is only 2.9 x 10^-15 times that at sea level.  Of course any surface thermal radiated heat absorbed in the upper atmosphere is generally going to be radiated out into space and not radiated back to the surface in any case.

So, there is no black body shell around the Earth at a temperature of 163.0 K.  But we can make an approximate calculation of where in the solar system we could place a black body absorber at a temperature of 163.0 K that could absorb such a large fraction of the power irradiated from the NASA Earth surface.  A black body radiator/absorber at a temperature of 163.0 K radiates 40.0 W/m² , which matches the power of the infrared radiation that NASA says escapes the Earth's atmosphere without absorption.  

So we ask where does a sphere around the Sun in radiative equilibrium with the Sun have an energy output rate of 40.0 W/m² corresponding to a temperature of 163.0 K?  This will be where the product of the power output per unit area of the photosphere of the Sun times the surface area of the sphere at the radius of the photosphere equals 40.0 W/m² times the surface area of the sphere centered on the sun with a radius r large enough that the temperature of that sphere will be 163.0 K.  That is a temperature far above that of the distant universe, so it should be found somewhere within our solar system.  We have:

Temperature of the sun on the photosphere = 5772 K
Radius, R, of the sun at the photosphere = 6.957 x 10⁵ km

We calculate the power output per unit area on the Sun's photosphere using the Stefan-Boltzmann Law to be:

Power output at the solar photosphere radius = 6.293 x 10⁷ W/m² 

For the solar photosphere radius R, the distance r at which the solar power will be reduced to 40 W/m² , will be 

r²  = [(6.293 x 10⁷ W/m² ) / ( 40 W/m² )] R²   = [(6.293 x 10⁷ W/m² ) / ( 40 W/m² )] [6.957 x 10⁵ km]²  

and so r = 8.726 x 10⁸ km.

The mean radius of the Earth's orbit is 1.496 x 10⁸ km, so according to the NASA Earth energy budget, 358.2 W/m² of power radiated as infrared radiation from the Earth's surface is dumped into the Earth's atmosphere at a distance from the Sun which is minimally 5.83 times further from the Sun than is the Earth.  Measured from the Earth, rather than the sun, the distance that portions of the surface emitted radiation is absorbed in space varies from 4.83 to 6.83 times the mean radius of the Earth's orbit.

NASA has a most interesting definition of the Earth's atmosphere.  What is more, how does that energy absorbed in the distant portions of our solar system manage to return to the Earth's atmosphere in any way that might increase the temperature of the Earth's surface?  Of course it does not.  This NASA Earth Energy Budget is a complete farce, as are the many similar Earth energy budgets used by the UN IPCC reports to justify their claims that adding carbon dioxide to the atmosphere will cause catastrophic warming problems for mankind.

One must also take note that without the absorption of this 358.2 W/m² of power from the surface, the idea that the atmosphere can back radiate 340.3 W/m² to the surface becomes impossible.  It is the absorption of the surface radiated power coupled with the back radiation of a nearly as great power that is the very basis of the claim that adding more greenhouse gases to the atmosphere will cause the Earth surface temperature to rise.  And no, there is no way that water vapor, carbon dioxide, and methane gas can absorb and emit more infrared radiation than can a black body absorber.  They fall well short of being as effective as absorbers and emitters as is a black body absorber/radiator.

According to the U.S. Standard Atmosphere of 1976 table, the coldest temperature within the Earth's reasonably dense atmosphere is the 216.65 K of the tropopause, the layer of the atmosphere between the troposphere and the stratosphere.  A black body layer placed there could at most absorb a power of infrared radiation emitted from the surface of 

P = σ (Ts)⁴ -  σ (216.65 K)⁴ ,

where σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) (216.65K)⁴ = 124.9 W/m² and  σ (Ts)⁴ =  398.2 W/m² 

so P = 273.3 W/m² 

This upper limit absorption of 273.3 W/m² is well less than the claimed absorption of 358.2 W/m² . What is more, this black body in the tropopause would radiate 124.9 W/m² to space, making this energy flux immediately unavailable in the Earth's atmosphere.

Even if the theoretical limit of absorption did occur in the tropopause, one would still have the insurmountable problem of finding a way to return any significant portion of this absorbed energy to the surface from the tropopause to provide any significant warming of the surface.  Equally important, such a violation of such an easily calculated upper limit on the fraction of the surface infrared emission which can be absorbed wipes out any confidence one can reasonably have in any of the NASA science used to create this Earth energy budget.  Indeed, there are many other errors in this fanciful creation.

This upper limit as defined by the Stefan-Boltzmann Law of Electromagnetic Radiation is itself much too high an upper limit for the following reasons:

1)  The surface of the Earth does not emit the 398.2 W/m² as claimed by NASA because this surface is not emitting radiation at a vacuum interface and has competing energy loss mechanisms cooling at least local submicroscopic areas of the surface briefly.  This is a matter dictated by the Law of Conservation of Energy.  Since less energy is emitted from the surface as infrared radiation, the atmosphere cannot absorb as much either.

2)  As noted earlier, water vapor, carbon dioxide, methane, and other infrared-active molecules can each absorb only a fraction of the wavelengths of infrared radiation that a black body absorber/radiator can.  These gas molecules also have many cases of overlap of those wavelengths they do absorb. Therefore, once again, the fraction of the wide spectrum of infrared radiation that the surface emits that can be absorbed by the atmosphere is much reduced.

3)  The mean free path for the absorption of such infrared as the infrared-active molecules absorb is much too short for surface emitted infrared radiation that can be absorbed to reach the tropopause before it is absorbed.  Indeed, in most areas of the Earth, the absorption mean free path for water vapor is many, many times too short for this to happen.  Even that for carbon dioxide at present concentrations is much too short for this to happen.  This means the temperature differentials are much reduced and consequently much less energy is absorbed.

4)  The density of infrared-active absorbers in the troposphere, even if each were a black body absorber/radiator, would not be great enough to absorb all of the radiation coming from the surface even if that radiation could make it to the tropopause.

There is still another important consequence.  The black body absorber/emitter we have placed in the tropopause to maximize the absorption of radiation from the surface is, as noted above, radiating 124.9 W/m² directly into space.  Since the surface is radiating 40.1 W/m² directly into space, all but 74.9 W/m² of the total radiation from the entire Earth system being radiated into space is accounted for.  Let us add up the remaining energy flux into the atmosphere:

77.1 W/m² directly absorbed from solar insolation
273.3 W/m² deposited into the black body absorber in the troposphere
18.4 W/m² deposited in the atmosphere by thermals (conduction/convection)
86.4 W/m² deposited in the atmosphere by the condensation of water vapor

Total = 455.2 W/m²

How could an atmosphere with an additional power input of 455.2 W/m² direct 340.3 W/m² of this power toward the surface and only 74.9 W/m² toward space?  The asymmetry of radiation to the surface and into space in the NASA Earth energy budget was already a red flag.  This asymmetry is now worse.  Note also that the sum of these two radiative fluxes is now 40.0 W/m² short of the 455.2 W/m² of remaining energy flux into the atmosphere.

In short, the NASA Earth Energy Budget is based on nonsense physics.  This, we are told, is the settled science of the climate.  Is it any surprise that computer models based on nonsense physics have been making wrong predictions of climate warming for 19 years now? Garbage physics in means no reality out.

Gravity, Not Just Greenhouse Gases, Provides a Warm Earth

According to the usual viewpoint, the surface of the Earth is 33K warmer than the 255K temperature of the Earth as seen from its thermal radiation temperature in space because of greenhouse gases. This is claimed on the basis of the NASA Earth Energy Budget shown below:

According to this federal government story, infrared radiation emitted by our atmosphere provides a surface warming power flux of 340.3 W/m² compared to 163.3 W/m² of radiation (UV, visible light, and infrared) directly absorbed from the sun.  In the official pronouncements of the United States of America federal government, so-called back radiation from our cooler atmosphere provides 2.084 times as much heat to the warmer surface as does the sun directly.


In this viewpoint, commonly claimed to be the consensus viewpoint of 97% of all scientists, the Earth does not have a gravitational field which acts upon air to to provide a temperature gradient in accordance with simple physics which has been well-known for a very long time. This article will explain this very simple, well-known, yet now completely ignored physics in relatively simple terms.  If one is to understand the equilibrium climate of the Earth's surface and its lower atmosphere, the troposphere, that understanding requires that we understand how gravity acts upon our atmosphere.

Yet we cannot ignore the role of radiation in developing this understanding either.  Let me reprise some considerations from my earlier article The Simple Physics Explaining the Earth's Surface Temperature by way of introduction.

NASA says that the Earth emits 239.9 W/m² of longwave infrared radiation into space.  This implies an effective Earth system radiative temperature T:

P = 239.9 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, 

so T = 255.0K, by application of the Stefan-Boltzmann equation for black body radiation.

Now we can go a step further, since NASA says that 40.1 W/m² of longwave infra-red radiation emitted from the surface passes through the atmospheric window without absorption by the atmosphere directly into space. This allows us to calculate the effective radiative temperature of the atmosphere alone.  Consequently, the effective radiative temperature of the atmosphere as a black body is found to be:

P = (239.9 - 40.1) W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴,

so T = 243.7 K, the effective radiative temperature of the atmosphere alone as seen from space.

According to the U.S. Standard Atmosphere Table of 1976, this is the temperature at mid-latitudes at an altitude of 6846 meters by interpolation of table data.  Now, the U.S. Standard Atmosphere Table of 1976 may not be a great representation of the average Earth atmosphere, but it gives us something concrete to work with and is certainly intermediate between the properties of the atmosphere over the tropics and over arctic regions.  Our effort here is to develop a physical sense of the size of the gravitational effect on the temperatures of the lower atmosphere and of the Earth's surface.  We are not trying to displace the need for future worldwide computer models to address climate issues more accurately.  But, at this time, essential physics is not going into the computer models in use. Let us investigate whether it makes sense to ignore the effect of gravity upon our atmosphere.

This effective atmospheric temperature is important because it is at an altitude at which the atmosphere is in equilibrium with radiation into it and out of it with respect to space.  This is an effectively pinned temperature in our atmosphere from the standpoint of radiation. Changes in the water vapor concentration and those of other infrared-active gases may move this point somewhat, but there is such a point given any such changes in infrared-active molecule concentrations which serves as a reference point for an equilibrium temperature of the atmosphere as seen from space.  Where this altitude is is mostly determined by water vapor in the atmosphere.

The total energy of an air molecule in the Earth's gravitational field is E = KE + PE, where KE is the kinetic energy and PE is the potential energy.  Let us take the PE at the Earth's surface to be zero and the potential energy at a height h to be mgh, where m is the mass of the average air molecule and g is a slowly decreasing value with altitude.  At the Earth's surface, g is 9.8066 m/s² in the US Standard Atmosphere Table and by interpolation of the values in that table for 6500 m and 7000 m, it is 9.7856 m/s² at h = 6848 m.

An equilibrium atmosphere will have a constant energy throughout.  The system of air molecules will equilibrate at a nearly constant total energy everywhere.  The real atmosphere will have sufficient disturbances that this will not be the case, but those disturbances will tend to operate around this equilibrium condition on average.  Consequently, the total energy of an air molecule at zero altitude will be the same as one at an altitude of 6846 m. This is actually an application of the 2nd Law of Thermodynamics.

The temperature of an ideal or perfect gas is proportional to its kinetic energy.  Air molecules are very nearly perfect or ideal gas molecules.  Consequently, the temperature of an air molecule is given by KE = C T, where C is the heat capacity (at constant pressure), or the amount of energy required to increase the temperature of the molecule by 1 Kelvin.

We can calculate the temperature difference between an air molecule at the surface (h=0) and at h = 6846 m altitude by calculating the kinetic energy difference between the molecules and using the heat capacity of the air molecule to convert the kinetic energy difference into a temperature difference.  So we have

KE (h=0) - KE (h = 6846m) = mgh = m(9.7856 m/s²  ) (6846 m) = m (66992.2   m²/s² )

According to the U.S. Standard Atmosphere Table of 1976, the average air molecule has a mass of 0.028964 kg/mol.  So the kinetic energy difference of one mole of air molecules at the surface and at an altitude of 6846 m is 

KE (h=0) - KE (h = 6846m) = (0.028964 kg/mol) (66992.2   m²/s² ) = 1940.36 J/mol

The heat capacity at constant pressure for a mole of the average air molecule is 29.07 J/K mol.  So we have 

T(h=0) - T(h=6846 m) = (1940.36 J/mol) / (29.07 J/K mol) = 66.75K

This is a temperature gradient per km of altitude of 9.75 K/km, which is a bit less than the more frequently given 9.8 K/km because we took into consideration the fact that the gravitational constant is not really quite constant and decreases slightly with altitude.

The actual temperature gradient with altitude is smaller than the 9.75 K/km lapse rate due to gravity.  This is because that measured lapse rate is reduced from this value dictated by gravity and the average mass of an air molecule by many cooling mechanisms such as the evaporation/condensation cycle of water and radiative cooling.  What I intend to extract here is the effect gravity has on the lapse rate and on the surface temperature.

Now comes the important lesson of this exercise.  We can now calculate the equilibrium surface temperature of the Earth due to the temperature gradient in the troposphere resulting from gravity acting on the atmosphere.  Recall that at 6846 m altitude, the temperature interpolated from the US Standard Atmosphere Table of 1976 is 243.7 K.  With a gravitational temperature gradient of 9.75 K/km, the surface temperature is 

T (h=0) = 243.7K + (6.846 km) ( 9.75 K/km) = 310.4 K

The air temperature at the bottom of the atmosphere, where the molecules of the atmosphere are bombarding the surface, should be about 310 K were it not for the existence of cooling effects.  This is much higher than the 288 K average temperature of the Earth's surface.

In fact, given that the surface is said in the Earth Energy Budget above to be absorbing 163.3 W/m² of direct radiation from the sun, the surface should be much hotter than 310 K. The problem is still greater if the surface is also warmed by back radiation from the atmosphere of 340.3 W/m² , despite the exaggerated surface infrared emissions of 398.2 W/m² of that energy budget.  I discussed why the surface infrared emissions are greatly exaggerated recently in Infrared Radiation from the Earth's Surface and the So-Called Scientific Consensus.  The claim of a large back radiation from the atmosphere was needed because even without an exaggerated cooling by radiation of the surface, there was no way to achieve an average temperature of 288 K at the surface without the gravitational effect being taken into account.

It is important to understand that under equilibrium conditions, the gravitational field is not transferring heat through the atmosphere or to the surface.  The gravitational field effect on the temperature of the atmosphere is trying to stabilize the temperature gradient in the atmosphere and at the surface and heat transfer only occurs when heat transferring effects such as radiation, water evaporation and condensation, and thermals upset the balance.  In this sense, the gravitational field effect does not belong in the Earth Energy Budget.  But, the fact that it does not belong there points out that the Earth Energy Budget is inadequate to understand equilibrium temperatures and is actually misleading.

While it is wrong to think of the gravitational effect as a power input in the system, it is still instructive for the sake of comparison to those factors that do put power into the Earth system to calculate its effective power input.  Using the Stefan-Boltzmann equation this would be given as

P = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) (310.4 K)⁴ =  526.3 W/m² 

If one adds to this the 163.3 W/m² of direct solar radiation and subtracts the 18.4 W/m² of cooling thermals and the 86.4 W/m² of cooling water evaporation power, the Earth's surface temperature would be given by

P = 584.8 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, 

so T = 318.7 K.

The net radiation from the surface is another cooling mechanism, which according to NASA  is (398.2 - 340.3)W/m² = 57.9 W/m² .  If we subtract this power from the 584.8 W/m² , we get 526.9 W/m² for the effective surface power emission given these inputs and outputs.  So, 

P = 526.9 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, 

and T = 310.5 K.

This temperature is almost exactly the same as the temperature calculated from the effect of gravity alone, which is to be expected because the heat flows into and out of the surface in the NASA scheme were equal.

Clearly, the main question we should be addressing as scientists is not why is the Earth's surface as warm as 288 K, but why is it as cool as 288 K.  The mechanisms cooling the Earth's surface must be much more robust than those of the NASA Earth Energy Budget.  The sum of thermal, evaporative, and radiative cooling mechanisms has to be greater than those given in the NASA scheme.  A surface temperature of 288.0 K requires an effective heat input of 390.1 W/m² , so we need to find another 136.8 W/m² of cooling for the surface.

I do not trust the NASA value for the net cooling by radiation of the surface, which is given as 57.9 W/m² in their scheme at all.  I do believe that they can measure the 40.1 W/m² of surface radiation through the atmospheric window into space.  I also believe that this implies an actual surface emission about three times that amount.  Furthermore, there is no back radiation in the equilibrium condition, though given that the atmosphere is often not in equilibrium, there are times when the air is warmer than the surface, so there is an average low value of back radiation.  I do not have a decent number for the back radiation, but it is much smaller than the heat loss by thermals, given by NASA as 18.4 W/m² .   Therefore, I will ignore back radiation as being in the noise.  Taking net surface radiation as about 120 W/m² then instead of 57.9W/m² , we get a surface temperature of

P = 464.8 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴, 

and T = 300.9 K.

Consequently, with a more realistic surface radiation cooling, the surface temperature is still about 12.9 K too warm.  Another 74.7 W/m² of cooling is still missing.  Most of this is certainly due to underestimating the cooling effects of thermals and water evaporation.

Looking at the NASA Earth Energy Budget, the infrared-active gases emit all but 40.1 W/m² of the 239.9 W/m² of outgoing infrared radiation into space, making this a cooling effect of 199.8 W/m² for the entire Earth system.  But if we look at the effects on the surface temperature, then the 77.1 W/m² of solar radiation absorbed by the atmosphere is a cooling mechanism with respect to the surface.  Much of the 77.0 W/m² due to cloud and atmospheric reflection owes to the so-called greenhouse gas water vapor, which again is a cooling mechanism.  Another important cooling mechanism for the surface is the evaporation of water to produce water vapor which removes 86.4 W/m² of heat.  Adding these three surface cooling effects together gives us a surface cooling effect due to infrared-active or greenhouse gases of 240.5 W/m² , which is much more than the 163.3 W/m² of direct solar radiation absorbed by the surface.

Already, one should be able to see that the only way to support the idea that infrared-active gases are alone warming the surface is to believe in the reality of the 340.3 W/m² of back radiation shown in the diagram.  This back radiation requires that a black body radiator be at a temperature of 278.34 K, which in the U.S. Standard Atmosphere Table of 1976 occurs at an altitude of 1510 m.  The infrared radiation from such a black body would have to travel 1510 m to reach the surface without any absorption by the infrared-active molecules in that path. In reality, because water vapor and carbon dioxide only emit a portion of the spectrum of a black body radiator, to emit so much radiation they would have to be at a much cooler temperature than 278.34 K, which means that the radiation they emitted exclusively at the wavelengths that they also want to absorb would be emitted at a very much higher altitude than 1510 m.  This would require a much longer mean free path for their emitted radiation than 1510 m.  In reality, the mean free path is very much shorter than 1510 m.  This makes this large back radiation from the atmosphere a fiction.

Note that adding still more CO2 to the atmosphere will actually further reduce the mean free path for infrared radiation and reduce the temperature differentials that are required to transport much heat between areas at different temperatures by radiation.  Because of the T to the fourth power dependence of radiative emissions, smaller temperature differentials yield rapidly decreasing radiative heat transport back from the atmosphere to the surface. Most of the time, when an infrared-active molecule absorbs a photon, it transfers almost all of that energy to other infrared-inactive molecules such as nitrogen, oxygen, or argon during many collisions with them.

The large back radiation of the NASA Earth Energy Budget is a fiction inserted because they have failed to acknowledge the critical role of gravity in providing us with a warm surface.

There is much more that is very odd about the NASA notion of back radiation here.  Let us do a little accounting.  Note that the atmosphere absorbs energy at the following rates:

77.1 W/m² from incoming solar radiation
358.2 W/m² of radiation from the surface
18.1 W/m² from thermals rising from the surface
86.4 W/m² from the condensation of water evaporated at the surface
Total atmospheric energy flux input is 540.1 W/m² .

Note that the atmosphere radiates the following energy fluxes:

(239.9 - 40.1) W/m² = 199.8 W/m² into space
340.3 W/m² to the surface
Total atmospheric energy flux output is 540.1 W/m² .

These sums are in balance.  However, one has to note that of the total atmospheric infrared radiation, only 37% is radiated to space, while 63% is somehow radiated back to the surface.  As each infra-red active molecule absorbs an infrared photon, it is equally likely to subsequently emit an infrared photon in any direction according to the ideas of photon emission to which the scientists who created this vision of the Earth's energy budget ascribe.  If so, because the atmosphere becomes less dense with altitude and because beyond the altitude at which water vapor condenses there is much less absorbing water vapor, more infrared radiation should escape the atmosphere to space than should be returned to the surface.  The reality is that in the lower troposphere, the transport of energy or heat as radiation is small compared to the transport by convection.  The transport by convection is much greater because the rate of molecular collisions is much greater than is the rate of photon emission by the infrared-active molecules in the lower, very dense atmosphere.  But even if radiative transport of energy were the dominant means of energy transport in the lower troposphere, one could not have more than half of the energy transported downward.

Here is still another unphysical oddity:  According to NASA and the many similar energy diagrams adopted by many other governmental organizations such as the UN and the European Union, all but about 10% of the infrared radiation of the surface is absorbed by the atmosphere.  Here, the surface emits 398.2 W/m² and the atmosphere absorbs 358.2 W/m² of that.  If the surface temperature is Ts and the effective atmospheric temperature is Ta, then one has

σ (Ts)⁴ -  σ (Ta)⁴ = 358.2 W/m² 

and we know that for the NASA Earth energy budget the first term on the left is 398.2 W/m² , so we have 

Ta = 163.0 K

Now this is a very interestingly low temperature.  There is no temperature this low in the U.S. Standard Atmosphere Table of 1976.  The temperature with altitude drops in the troposphere, stabilizes at 216.65 K in the tropopause and then increases with altitude until there is essentially no atmosphere left to do any absorbing of infrared photons.

In order to get enough energy flux to return a large back radiation to the surface to replace the role of gravity, NASA has to find a way to absorb almost all of the infrared radiation emitted from the surface in the atmosphere and to do that they have to make the atmosphere very cold.  They have to make it colder than it is.  Even if it were cold enough in its upper reaches and dense enough to absorb the radiation, how then could it actually return that radiation to the surface and also appear to be a black body radiator with a temperature of 378.3 K, which we calculated above?  And note that this problem only becomes worse when we recognize that none of the infrared-active gases are black body absorbers or emitters.  They are much less efficient both as absorbers and emitters of radiation.

Having more infra-red active molecules in the upper portion of the troposphere provides more emitters of radiation into space, which will lower the temperature of the altitude whose temperature is in radiative equilibrium with space, but will do so without moving that equilibrium altitude very much at all.  This is because water vapor already has a very low concentration in the upper troposphere, after decreasing in concentration rapidly at altitudes colder than the condensation temperature of water.  CO2 even doubled does little to shift the equilibrium altitude given water vapor dominance in the emission spectrum and given the temperature uniformity in the tropopause.  The net effect is simply more heat emission to space and little change in the distance to the surface over which the gravity-induced temperature gradient works.  The net effect of adding carbon dioxide is a cooling effect in terms of the atmospheric radiation of infrared to space.

The infrared active molecules speed the upward transport of solar insolation which has warmed the surface by short hop radiation from a warmer layer of air to a cooler layer just above it. The mean free path for the reabsorption of emitted infrared is short, so the amount of energy transported this way is small.  But, this effect is a cooling effect in that the energy transport at the speed of light is much faster than the velocity of air molecules traveling upward in convection currents.

The primary infrared active molecule, water vapor, has a mass to heat capacity ratio less than that of the average air molecule, so it reduces the temperature gradient created by gravity in accordance with the calculation performed above.  Because the water molecule is lighter than the average air molecule, it also has a tendency to increase updrafts of air at higher humidity and thus speed up heat dissipation from the surface by carrying heat to the altitude at which water condenses.  The Earth is also cooled by plant growth and the absorption of water and carbon dioxide by minerals, which increases with more carbon dioxide and water vapor in the atmosphere. 

All of these factors are candidates for being underestimated by the NASA Earth Energy Budget and provide mechanisms in which infrared-active molecules actually cool the surface of the Earth and, ultimately, the atmosphere.

The fact that the greater part of the surface temperature of the Earth is due to our gravitational field acting on air molecules goes a long way to explain why our day to night temperature differences are relatively moderate.  If the electromagnetic spectrum of radiation played as big a role as that implied in the usual greenhouse gas warming hypothesis, this relative moderation of day and night temperatures is difficult to explain. Very difficult. The fact that the direct solar absorption at the surface is only 27.9% of the effective net energy in and out the surface helps to explain this beneficial stability of the surface temperature.  The surface and lower atmosphere radiative emissions over the daily cycle act to increase the day to night temperature differences and also the seasonal differences.

We are very fortunate to have a deep atmosphere in a substantial gravitational field composed mostly of infrared inactive molecules with enough water to dissipate heat adequately by evaporation at the surface and enough infrared active molecules to dissipate heat from the upper troposphere.  It is also critically important that the mean free path length for the absorption of infrared radiation emitted by our infrared-active molecules is short so that heat near the surface is not immediately and directly dissipated to space, but must instead rise slowly through the troposphere to the top portion of the troposphere before it is lost to space.  The infrared-active gases play a critical role, but they do so in conjunction with the powerful effect of our Earth's gravitational field acting on our air molecules.

Summary:  The surface temperature of the Earth would be much higher than 288K due to the action of the Earth's gravitational field on our atmosphere, were it not the case that the non-radiative cooling mechanisms in the troposphere are much stronger than NASA and the UN IPCC claim them to be.  It is often said that a large back radiation from the atmosphere is necessary to provide a surface temperature of 288K.  This is a false claim because, as shown above, gravity acting on the atmosphere easily serves the purpose claimed for back radiation.

Additions made on 21 October 2017 are in green.
Additions made on 17 March 2020 are in burnt orange.