__Introduction__

The Greenhouse Gas Theory is 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. However, after 30 years with little progress in satisfactorily establishing the theory, it seemed to be worth a complete re-evaluation of the problem, from a different starting point.

Hitherto, the general approach has been to consider the workings within the atmosphere, whereas the present paper deals instead with the simple Physics and Maths of the energy from the Sun which reaches the Earth’s surface, and the energy emitted to space by the Earth.

There is a fundamental requirement for energy balance, which means that the power emitted to space by radiative emission from the Earth must be equal to the power received by the Earth’s surface from the Sun. This is generally taken to be 239.7 Wm^-2.

It is clearly shown that at current temperatures all the energy is totally accounted for, so that an increase in concentration of carbon dioxide cannot produce an increase in temperature because there is no more energy left to be absorbed.

The graph on the website of the GWPF (1) shows the Earth surface temperatures from 2001 to 2019. The values of the ordinates have been manually measured from the base temperature of 14.0 deg C, and have been found to be in excellent agreement with the temperature **anomalies **on the HadCRUT4/Met Office data (2),(3). This means that the ordinate values are themselves the temperature anomalies referred to a base temperature of 14.0 deg C. This temperature is also, therefore, the base temperature for the anomalies in the Met Office data.

The average value of the Met Office anomalies from 2001 to 2014 is 0.49 deg C, and so this gives 14.49 deg C as the average surface temperature for this period, in line with the GWPF graph. __This is 287.6 K.__

The surface temperature is seen to have been substantially constant at this value

**in spite of the fact that the atmospheric concentration of carbon dioxide has been increasing steadily throughout this period.**Since extra CO2 has had no effect on temperature, this means that ALL of the emission power from the Earth’s surface was already being absorbed even in 2001, and has been continually totally absorbed since then, and must proceed so in the future.

__Therefore, no further temperature increase above 287.6 K can occur as a result of increased carbon dioxide, or other greenhouse gases.__ **Earth’s Surface Emission**

The Earth’s surface emits electromagnetic energy as photons, very small quanta of energy, with a wide range of wavelengths, in accordance with Planck’s Law, in two broad regions; 400 to 14, and 14 to 2 microns.

The emitted Planck power for long wave region 400 to 14 microns is denoted as

**u**, and the power for the shorter wavelengths 14 to 2 microns, which is the atmospheric window region, as

**w.**

For a surface temperature of 287.6 K, the

**Planck Law**provides the values

u = 195.52 Wm^-2

w = 184.10 Wm^-2

The

**complete power emission**is, therefore, 379.62 Wm^-2. It is this total which is used in the following calculations.

**Long wavelength**

**400 to 14 microns**

Power emitted from surface = u

Power absorbed in atmosphere = u

In accordance with natural Greenhouse Gas theory, half of this absorbed power is re-emitted upwards to space, and half is re-emitted downwards to the Earth’s surface.

So, power to space = 0.5u ———————–space

and power to Earth = 0.5u ———————-Earth return

**Short wavelength 14 to 2 microns**

For this wavelength range, the power emitted from the surface into the atmosphere is **w.**A fraction

**f**of this

**w**is transmitted directly to space, and so the term “atmospheric window” has come to be applied to the amount of power,

**, escaping without absorption.**

__fw__The remaining power,

**(1 – f )w,**is absorbed and re-emitted, 50% to space and 50% to Earth.

Power to space = fw ——————————space

Power to space = 0.5(1 – f)w ———————space

Power to Earth = 0.5(1 – f)w ———————Earth return

The power returning to the Earth’s surface, the “Earth Return”, is of vital importance because it is needed, together with the power received by the surface from the Sun, to supply the power emitted from the surface in accordance with the Planck Law for the given temperature.

Summing the terms above:-

**Total power to space**= 0.5u + fw + 0.5(1 – f)w ———————–(1)

For energy balance, this must be 239.7

Therefore 0.5u + fw + 0.5(1 – f)w = 239.7

0.5u + fw + 0.5w – 0.5fw = 239.7

0.5(u + w) + 0.5fw = 239.7

0.5fw = 239.7 – 0.5(195.52 + 184.1)

= 49.89

fw = 99.78

**This is the Atmospheric Window**

f = 0.541988

__f = 0.542__**Total Earth return**= 0.5u + 0.5(1 – f)w ———————————(2)

For re-supply, this must be equal to total emission from surface, less the energy from the Sun at the surface, ie u +w – 239.7 ie 139.92 Wm^-2

Therefore Earth return = 0.5u + 0.5(1 – f)w

= 0.5(195.52) + 0.5(0.45801)(184.1)

= 97.76 + 42.16

= 139.92 Wm^-2, as required.

The calculated value for

**f**has been confirmed within about 10% by visual estimations of the window data available in the link below.

https://en.wikipedia.org/wiki/Infrared_window#/media/File:Atmosfaerisk_spredning.png

**Conclusion**

At a surface temperature of 287.6 K, the total power emission from the Earth’s surface has been shown to be completely absorbed / directly transmitted to space through the atmospheric window.

**because there is no more energy to absorb.**

__Therefore, increased concentration of carbon dioxide cannot cause an increase in surface temperature__(1)https://www.thegwpf.org/

(2)https://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/download.html

(3)https://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/time_series/HadCRUT.4.6.0.0.annual_ns_avg.txt

(4)https://en.wikipedia.org/wiki/Infrared_window#/media/File:Atmosfaerisk_spredning.png

__Appendix A__

Suppose, contrary to this hypothesis, that not all the power emitted from the Earth’s surface is absorbed. What would the surface temperature be if this were the case?

Let the power NOT absorbed be **h**.

Therefore, this power **h** is transmitted directly to space.**Long wavelength region**

Power from surface = u

Therefore, power absorbed in atmosphere = u – h

Then power to space = h ——————————space

power to space = 0.5(u – h ) ——————–space

power to Earth = 0.5(u – h ) ——————–Earth**Short wavelength region**

Power from surface = w

Then power to space = fw ———————space

power to space = 0.5(1 – f)w ——————space

power to Earth = 0.5(1 – f)w ——————Earth

Summing,

Total power to space = h + 0.5(u – h ) + fw + 0.5(1 – f)w ———(3)

Total Earth return = 0.5(u – h) + 0.5(1 – f)w ——————-(4)

In order to re-supply the (u + w) Planck emission from the surface,

Earth return + 239.7 = u + w

Therefore, 0.5(u – h) + 0.5(1 – f)w + 239.7 = u + w

Whence, 0.5u + 0.5h + [1 – 0.5(1 – f)]w = 239.7

Substituting for f = 0.542,

0.5u + 0.5h + 0.771w = 239.7

u + h + 1.542w = 479.4

** h = 479.4 – u – 1.542w** ————————-(5)

So we need to know the relationship between the power which is not absorbed, h, and the stable surface temperature supplying the powers

**u**and

**w.**From the Planck Law we find:-

Temp K u w h Wm^-2

287.6 195.52 184.1 0.00

287.0 194.46 182.0 4.30

286.0 192.71 178.5 11.38

285.0 190.96 175.1 16.65

This clearly shows that if not all the Earth’s emitted power is absorbed, the surface temperature would be significantly less than the present 287.6 K.

__Appendix B__** Suppose an increase in surface temperature due to El Nino**The Met Office data show a very small temperature increase around 2016. This was only about 0.2 K, and could well have been caused by El Nino effects. But for the sake of argument, let us consider a value of 1 K greater than that used above, more in line with NASA’s claims. So, take a figure of 288.6 K.

For 288.6 K, the Planck data are:-

u = 197.29 Wm^-2

w = 187.64 Wm^-2

Therefore, total power emission from surface = (u + w) = **384.93 Wm^-2**

Substituting these figures into equations (1) and (2), and with f = 0.542, we find

Total Power to space = 243.32 Wm^-2

Total Earth Return = 141.62 Wm^-2

Therefore, the available power for emission from the surface is qual to the 239.7 from the Sun PLUS the Earth Return of 141.62, which makes **381.32 Wm^-2.** So, the available power is short by 3.61 Wm^-2, and this means that the 1 K rise considered is NOT sustainable.

Therefore, any temperature increase such as might be caused by El Nino effects would return to the stable 287.6 K as shown.

__Appendix C__

__Variation of surface temperature with transmission factor f __

The power emitted from the Earth’s surface is (u + w).

For energy balance conditions, this must be re-supplied at the surface. The Sun supplies a power of 239.7 Wm^-2.

More power is returned to the surface by downward emission from the absorbing greenhouse gases in the atmosphere. This total Earth return is shown in equation 2.

Therefore, 0.5u + 0.5(1 – f )w + 239.7 = u + w

(1 – f)w = u + 2w – 479.4

Whence ** f = [(479.4 – u) / w] – 1.0** ———————(6)

The surface temperature determines the powers **u** and **w**, and so for energy balance there is clearly a relationship given by this equation between temperature and factor f. This is shown in the graph below, where the limits of **f** are 0 and 1.0

The importance of the transmission factor is clearly demonstrated.

The presence of HFCs and HCFCs in the atmosphere is a great potential threat because they have very large absorption coefficients. Also, sulfur hexafluoride should be included. Although this does not affect absorption of energy at wavelengths outside the window as already explained, these compounds do, nevertheless, have absorbing wavelengths __within__**the window.** Therefore, less energy can be directly transmitted to space. That is, the value of the transmission factor f is reduced, and so the surface temperature increases as shown in the graph

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