Energy Accounting and Carbon Dioxide


A E Banner, June 2020

A great amount of work has been done over many years by brilliant climate scientists to establish the details of the Greenhouse Gas Theory, whereby increases in the concentration of carbon dioxide in the atmosphere over the last 50 years or so are claimed to have caused increases in the surface temperature of the Earth

The explanation seems to come from the well-known energy diagrams of Trenberth et al, followed by others, including NASA.  Unfortunately, the diagrams are puzzling, principally because of the large amounts of energy shown to be circulating in the atmosphere and being sent to the Earth.  The energy escaping to space is correctly shown equal to the energy received from the Sun, but it is not clear from where these extra amounts derive.  Moreover, the figure given for the power of the atmospheric window, a highly significant feature, although changed, is still not in agreement with the figure used in this work, which is based upon the reference and the theory given below. 

However, the Greenhouse Gas Theory is, in fact, correctly based upon the fundamental property of carbon dioxide and some other gases, water vapour in particular, to absorb radiative photons of certain energies, and subsequently to re-emit this energy equally in all directions.  Nevertheless, the computer models are still said to be “running hot”, and with the difficulties mentioned above, it seemed to be worth a re-evaluation of the problem.

When considering the possibility of global warming, there is a fundamental principle which applies.  This is the requirement for energy balance: the power emitted to space by radiative emission from the Earth’s surface must be equal to the power received by the Earth’s planetary system from the Sun.  This is generally taken to be 239 Wm^-2.

The Earth’s surface emits electromagnetic energy as photons, very small quanta of energy, with a wide range of wavelengths from 400 microns down to 2 microns, in accordance with Planck’s Law.  It should be noted that the Planck program gives values in milliWatts per square metre per cm^-1.  This applies for emission per steradian, but for our purposes, this needs to be converted to units of Watts per square metre per cm^-1, in line with the Law of Stefan-Boltzmann which covers the full wavenumber spectrum.  This involves multiplying the Planck numbers by “pi”.

This paper is essentially about energy accounting.  

All of the energy, and therefore power, emitted from the Earth’s surface must be accounted for, and its destination determined. 

All the radiative power emitted from the Earth’s surface is shown in column A of Table 1 as the Planck distribution for the full range of wavelengths for a surface temperature 288 K, duly converted into units of Wm^-2.  The emission between wavenumbers 25 and 714 cm^-1 is absorbed by the greenhouse gases, and is entered into column C.  

The power between wavenumbers 715 and 4500 cm^-1, flows through the atmospheric window and some of it escapes directly to space, but allowance must be made because the proportion transmitted is well short of 100%, depending on the actual wavenumber.  The successful, transmitted proportions are shown, together with the other window data, in green italics in the table. 

Insert Fig 1 Planck

Ref (1) has enabled initial, approximate visual estimates to be made of the proportions of power transmitted in each micron section within the window.  Later calculation from theory, see below, suggests that the overall power of the window is 96.26 Wm^-2, and the successfully transmitted powers are shown in column B.  The differences between columns A and B represent the powers which are not successfully transmitted, because of absorption, and are entered into column C.

To summarize, photons which do not escape directly to space are subject to absorption by the greenhouse gases in the atmosphere.  These are within the wavenumber range from 25 to 714 cm^-1, and comprise the power in column C, which is the value in column A minus the value in column B, ie (A – B), so for these wavenumbers the entry in Column C is equal to that in column A.  However, the energy which is not transmitted directly to space through the window, which is value in A minus value in B, must also be entered in column C. 

In accordance with the Greenhouse Gas Theory, half of this power in column C escapes to space, and is given in column D.  So, the total power to space is the sum of columns B + D, that is column E. Note, also, that the value in column D is also equal to the power returning to the Earth’s surface, where it is added to the power received by Earth’s planetary system, 239 Wm^-2, to make the required value of 381.74 Wm^-2, the amount emitted by the surface.

It is seen that for the value of window used, 96.26 Wm^-2, and 288K, the total power to space is equal to 239 Wm^-2, as required.  This means  that all the power which is not directly transmitted to space has been absorbed by the greenhouse gases and then re-emitted; and the energy flowing directly through the window has also been taken into account.  Therefore, the total energy radiated by the Earth’s surface has been accounted for.  Therefore, no further increase in carbon dioxide can cause any further increase in temperature.

The 32 deg C Rise

Consider the atmosphere without any greenhouse gases.  Planck results show that the total radiative emission from the Earth’s surface at about 256 K is 239 Wm^-2. 
Suppose greenhouse gases are added so that some energy absorption can occur.  Let the amount be x, and in a new table this would be added into column C, and the corresponding increase in column D would be 0.5x. 
But, absorption of x in C means a decrease of x in B. 
So E increases by (0.5x – x), which is a decrease of 0.5x, and this causes the total output to space to fall below 239 Wm^-2.  This continuing process is corrected by increase in temperature until stability is achieved at 288 K with all the power absorbed, apart from the window power.

Table 1
Micron range  400  14
Planck energy           196.23 Wm^-2     A
Transmitted power       0                        B
Absorbed power       196.23                   C  =  (A – B)
50% Absorbed pow.   98.12                   D = 0.5*C
Power to space            98.12                  E  =  (B + D)

Micron range
   14   2
Planck energy           185.51 Wm^-2     A
Transmitted power      96.22                  B
Absorbed power          89.29                  C  =  (A – B)
50% Absorbed pow.    44.65                  D = 0.5*C
Power to space           140.87                 E  =  (B + D)
Total Power to space    239 Wm^-2

The Atmospheric Window
The power transmitted directly through the window without absorption by greenhouse gases is of critical importance in determining surface temperature.  The values in Table 1 were calculated for the theoretical window power of 96.26 Wm^-2, derived below.

Let P be the power emitted from the Earth’s surface into the atmosphere.
      w be the power transmitted directly through the window to space.
Then the power in the atmosphere equals ( P – w )
But, by greenhouse theory, half of this is emitted upwards, and half is emitted downwards.
So the power emitted to space is 0.5( P – w ). This adds to the power through the window, w.

Therefore, the total power to space = 0.5(P – w ) + w
                                                         = 0.5P – 0.5w + w
                                                         = 0.5( P + w )

This must be equal to 239 Wm^-2.
So 0.5( P + w ) = 239
             P + w = 478
                    w = 478 – P
At 288 K, P = 381.74 Wm^-2
So          w = 478 – 381.74
              w = 96.26 Wm^-2     This is the value used in Table 1 and in the figure.

The figure below shows the total emission of power to space for a range of surface temperatures, and for selected values of window power, confirming the window of 96.26 Wm^-2 is consistent with the required emission of 239 Wm^-2 at 288 K. 

A counter-opinion will doubtless be expressed by the confirmed believers in the greenhouse gas theory that this work is nevertheless wrong because not all the Earth’s emitted energy has been absorbed and that there is still insufficient carbon dioxide to accomplish this absorption.  This is clearly a false idea.  If some “excess” energy were not absorbed, it would have to be removed from columns C and D in the table, and put instead into column B.  But a decrease of 0.5x in D means in increase of x in B, so the total B increases, and so does the total emission to space, column E. 

An increase in B can be considered as an increase in window power as in Fig 2, which shows that the required 239 Wm^-2 would have already been attained at a lower temperature, and so the counter-opinion is groundless. 

Although it has been shown that the usual GHGs cannot cause any further temp increase, there remains nevertheless a potential problem with hydrochlorofluorocarbons HCFCs an chlorofluorocarbons CFCs being admitted into the atmosphere.  Also, sulphur hexafluoride.  These compounds have very high absorption cross sections at wavelengths within the atmospheric window, and so could reduce the window power, thereby causing an increase in temperature to be needed in order to achieve the required 239 Wm^-2 to space, as in the figure. 


With the present conditions of 288 K Earth surface temperature and the required 239 Wm^-2 power outflow to space, all of the power emitted by the surface is transferred to space, apart from that returned to Earth. Therefore, no increase in greenhouse gas concentration can cause a further rise in surface temperature.

A warning: fluorocarbon compounds can cause serious temperature rises. 



Revised Greenhouse Gas Theory

 A E Banner March 2020


The Greenhouse Gas Theory is intended to explain the increase in the temperature of the surface of the Earth in the last few decades, due to the effects of the actions of human-kind.  This is known as “anthropogenic global warming”.  The theory depends upon the property of the so-called “greenhouse gases” in the atmosphere to absorb electromagnetic energy of certain wavelengths and then to re-emit this energy into the atmosphere, when the process can then be quickly repeated.  The energy is absorbed/emitted in quantum amounts called “photons”, and is specific to the particular gas concerned.  A fundamental fact is that the energy is emitted equally in all directions, and so energy emitted upwards is equal to that emitted downwards.  The energy emitted downwards warms the Earth’s surface.

The most important gases are water vapour and carbon dioxide, and it follows that more carbon dioxide in the atmosphere will cause more warming. 

The theory also requires that the Earth should be in energy balance, and so the power emitted to space must be maintained equal to the power received from the Sun.  This is achieved by changes in the surface temperature, in line with the Stefan-Boltzmann law for “black body” radiation.   

These two tenets are not in dispute, and are generally accepted by the scientific community. 
They are included in the following revised treatment.    

Calculation of the numerical relation between the surface temperature increase and the concentration of atmospheric carbon dioxide is very complicated because there are many factors involved.  Although HITRAN provides the absorption cross section for carbon dioxide, this may be modified by the pressure, and there may be cross contribution from other gases at particular wavelengths.   

Again, the power received by the planet depends upon the surface reflectivity, the albedo, which in turn might be affected by deforestation.  Yet another problem is the effect of aerosols emitted into the atmosphere.  So, altogether, the determination of the total “radiative forcing” is an extremely difficult problem.  A huge amount of work has been done over many years by brilliant climate scientists to produce models to emulate these processes, but there are still claims that the models are “running too hot”.  That is, calculating temperatures noticeably greater than observed figures.

Therefore, the following revision of the GHG theory of recent years is approached in a different way.  Rather than trying to improve on the radiative forcing calculations, and so to produce a quantitative theory, this revised method starts instead with the known effects of the greenhouse gases in the atmosphere, and then proceeds to include the requirements of energy balance. 

It is generally accepted that in the absence of anthropogenic effects the power emitted to space is 239 Watts per square metre, and so this must also be the power received by the planet from the Sun.  This power emitted to space comprises the energy of the upwards flowing photons having gone through absorption/emission by the ghgs, together with the power of the Atmospheric Window.  This is found to be a critical feature of the global warming problem and seems not to have been adequately addressed in previous explanations of the GHG theory. Trenberth et al have previously suggested 40, and again, 22 Watts per square meter for the power transmitted to space through the window, but these figures are shown here to be serious under estimates, and this has important implications.

The following revision enables correct determination of the required values for the temperature of the Earth’s surface both for an atmosphere with, and also without, greenhouse gases and water vapour, (or indeed, no atmosphere), and hence quantitatively explains the “32 deg C rise”.  It also shows that with the recent value of window power determined below, increased greenhouse gas concentrations can have only a limited further effect of about 1 degC on surface temperature. This, in turn, offers a credible explanation for the “temperature hiatus”, starting around 1998.

Any significant reduction in the available window power can have serious results.

The Atmospheric Window
The Atmospheric Window has a vital role in governing the temperature of the Earth’s surface.  Without it, the temperature would be too great for life.  Radiative energy of the appropriate wavelengths emitted from the surface can ass through the window to space with only moderate absorption because there are few greenhouse gas wavelengths within the wavelength range of the window.  This range is generally taken to be from 8 to 14 microns.                                                       

Reference (1) shows the observed transmission of radiative energy emitted from the surface of the Earth through the window.  The transmittance is between 80% and 70%, but reducing to zero through the 13 and 14 micron sections.  A computer program for the Planck distribution at 288 K has enabled

The Atmospheric Window has a vital role in governing the temperature of the Earth’s surface.  Without it, the temperature would be too great for life.  Radiative energy of the appropriate wavelengths emitted from the surface can pass through the window to space with only moderate absorption because there are few greenhouse gas wavelengths within the wavelength range of the window.  This range is generally taken to be from 8 to 14 microns.                                                       

Reference (1) shows the observed transmission of radiative energy emitted from the surface of the Earth through the window.  The transmittance is between 80% and 70%, but reducing to zero through the 13 and 14 micron sections.  A computer program for the Planck distribution at 288 K has enabled the transmitted power to be calculated for each micron segment within the range, and allowance has been made for the amount absorbed.  It provides a total value of 90.2 Wm^-2 for this range of microns.

The Planck curve for a surface temperature of 288K, and this window power of 90.2 Wm^-2 is given in Fig 1.  It must be pointed out that this data is the total energy in Joules per square metre per second, radiated across all wavelengths.  This is in line with the Stefan-Boltzmann equation. 
The Planck figures, however, apply in terms of the steradian, and must be multipied by ‘pi’ to achieve agreement with the S-B figures.

A revised approach
In the following treatment, the term “Greenhouse Gases” includes water vapour and clouds, in addition to carbon dioxide, methane and all the other radiative energy absorbing molecule
The greenhouse gases have very little effect within the window and so photons with wavelengths within the window pass through to space with only a little absorption. 
This power emitted to space is represented by w.
Radiation from the Earth’s surface is absorbed and re-emitted by the greenhouse gases with wavelengths outside the window.

Fig 2 shows schematically the basis of this revised approach.

Fig 2 The energy in the atmosphere

It starts with the emission of radiant energy from the Earth’s surface in line with the equation of Stefan-Boltzmann for a “black body”.  This is acceptable for Earth with an emissivity taken to be 0.98

Let   P = output power from the surface in Wm^-2
e = emissivity of the surface
s = Stefan-Boltzmann constant, 5.67*10^-8 Wm^-2K^-1
T = surface temperature in K
w = power emitted through the window to space, Wm^–2
Take T = 288 K and e = 0.98
P = e.s.(T^4)  ………………………………………….(1)
= 0.98*5.67*(10^–8)*(T^4)
= 382.28 Wm^–2
This is the power emitted as photons from the surface of the Earth into the atmosphere.
Some of this power, w, escapes directly into space through the window, because there is little greenhouse absorption in the window.
Therefore, the power remaining in the atmosphere is (P – w).
But there are greenhouse gases effective in the wavelengths outside the window, and so absorption and emission occurs here.
Now, it may be that not all of the energy (P – w) is absorbed/emitted.  This might be due to insufficient greenhouse gases in the atmosphere, or too small a molecular cross section for absorption.
So let  f  be the energy absorption factor combining these effects.

If all the radiative power is being absorbed, then  f = 1.0
If none of the radiative power is being absorbed, then  f = 0.0
Therefore, the power absorbed and then re-emitted is (P – w)f.
Since greenhouse gas molecules emit photons equally in all directions, the power radiated upwards is 0.5(P – w)f , and this is equal to the power radiated downwards, 0.5(P – w)f.
However, if the energy absorption factor f is less than 1.0, there is energy still left unaccounted for in the atmosphere.  Let this remainder be R.
Therefore, it follows that  R = (P – w)(1.0 – f)
So the total power into space  = w + R + 0.5(P – w)f
And for Earth’s energy balance this must equal 239 Wm^-2.
So           w + R + 0.5(P – w)f  =  239
Hence,   P(1.0 – 0.5f) + 0.5wf  =  239
So           P = (239 0.5wf ) / (1.0 – 0.5f )
Substituting for P from eqn (1), this gives
T^4 = (239.0 –­ 0.5wf ) / (e.s.(1.0 – 0.5f ))
T^4 = 0.179966*(239 – 0.5wf )*10^8 / (1.0 – 0.5f )

The value of  T  has been determined for a range of energy absorption factors f,  and for specified values of window power w; the results are given in Fig 3.

For the current window power  w = 90.2 Wm^-2 , it shows that the surface temperature of 288K is obtained with an energy absorption factor  f = 0.981

If there were no greenhouse gases in the atmosphere, or indeed, no atmosphere, this would be equivalent to zero energy absorption factor, f = 0.0.  This gives a temperature of 256 K as shown, which is correct for an emissivity of 0.98
This provides the temperature rise of 32 deg C.

The energy flux returning to the surface from the atmosphere is 0.5(P ‒ w)f.
For T = 288 K, P = 382.28 Wm^-2, and the value of  w = 90.2 Wm^-2, the downward power to the surface is 143.27 Wm^-2.  In addition, there is 239 from the Sun, making a total of 382.27 Wm^-2, as required for energy balance.

Fig 4 shows the critical role of the window.

For any value of w, the temperature cannot exceed that given by the curve for f = 1.0, because with f  = 1.0 all of the radiant energy in the atmosphere is already being absorbed and emitted by the greenhouse gases.  Further increases in greenhouse gas concentrations will, therefore, have no effect. 
With the current window of 90.2 Wm^-2, the temperature of 288 K is obtained with f = 0.981
It is clear that an increase of 1 K to 289K could occur, or has already occurred, if greenhouse gas concentrations increased f to 1.0   But no further rise in temperature would happen.
This may be an explanation for the “temperature hiatus”.

However, if the window power were to be reduced, the results would be serious. 
Fig 5 shows the temperature increases for convenience. Even without any more carbon dioxide, the temperature rise with w = 0 could be 15 deg C.

The potential problem is due to the increasing use of compounds of fluorine; particularly the CFCs and the HCFCs.  Also, sulphur hexafluoride.  These substances have very significant wavelengths within the window, and so are very dangerous.   Fortunately, these are man-made substances, and so in principle it should be possible to exert some control on their use, in accordance with the Montreal Protocol.
However, these ozone destroying substances are being superseded by HFCs which also have high radiative absorption wavelengths within the window. And so the problem could continue.

The Revised Greenhouse Gas Theory, is much simpler than previous versions.  It certainly depends upon the effects of the several greenhouse gases, and Stefan-Boltzmann radiative emission from the Earth’s surface, but it avoids the complicated problems of a multi-layered atmosphere. 
It gives the correct values for the two “fixed points” of Earth’s surface temperature, namely for the conditions of “no atmosphere”, 256 K, and for the atmosphere with pre-anthropogenic concentrations, 288 K.   And hence the “32 deg C rise”.

It shows the importance of the greenhouse gases, and clearly demonstrates that the effect of carbon dioxide is almost exhausted, and so more CO2 would not cause any further rise.  Indeed, this condition may already have been reached around 1998, with the ensuing temperature “hiatus”.

Further temperature increases since then might be attributable to increased atmospheric concentrations of fluoride gases, HFCs, because of their absorbing wavelengths within the window.  This would reduce the available window power, and so cause surface temperature increases as explained above.  Unfortunately, information on these concentrations is not immediately available. 

Ref (1)