# Absorption of Earth’s Radiated Energy

A E Banner                 September  2021

Introduction
The present climate change disasters are being explained by anthropogenic “global warming” caused by increases in the concentration of carbon dioxide in the atmosphere.  This is the classic “Enhanced Greenhouse Gas theory” which has been around for nearly forty years, but has ever been flawed by trying to cope with the whole planet on an average basis.

A new treatment is offered here which deals with individual areas of one square metre at specified latitudes.  This involves the investigation of energy balance considerations, in that the energy output per second of a body must be equal to the energy input per second.  So, in fact, we are dealing with power balance, and the units are Watts per square metre, Wm^-2.

The power output depends upon the temperature of the body and can be calculated from Planck’s Law, where the temperature is on the Absolute Scale, and so, a temperature of T degrees Celsius is (T + 273.2) Kelvin, K.

The Enhanced Greenhouse Gas theory says that the temperature will increase because the greenhouse gases in the atmosphere absorb some of the energy radiated from the surface, and some is retained, so raising the temperature, and that this will continue as the concentration of carbon dioxide is increased.  Clearly, however, no further temperature increase can occur if ALL of the absorbable power is absorbed, no matter how much CO2 is added.  Hitherto, there has been no information about the present proportion of radiated power which is absorbed, and so the GH explanation is still uncertain.

Accordingly, an attempt is made here to investigate that proportion.  It is assumed here that a proportion, n, of the absorbable power radiated from the surface is NOT absorbed.  So it is required to find n.

The problem involves detailed consideration of the power flow in the Earth’s system, as shown in Fig 1.  The power is in units of Watts per square metre.

The total power output from the land surface is shown as P, and this depends on the Absolute Temperature in Kelvin.  It can be determined from the Planck equation.

This power comprises both longwave  L, and shortwave  w, energies.
The longwave range of wavelength is from about 2000 microns to 14 microns.
The shortwave range of wavelength is from 14 microns to about 2 microns.

A proportion  f  of this shortwave range is not available for absorption, because of the weak intensities of absorbable radiation at the particular wavelengths in this region. The value of f has not been extensively determined, but it must be between zero and unity. There is some evidence that the value is about 0.493   Ref (2)

This provides the atmospheric window, fw, for which the power escapes directly to space. fw.……….SPACE

This leaves a power of  (1 – f )w  in the atmosphere.
But, a proportion n is supposed NOT to be absorbed because of insufficient GH gases, and so this power  n(1-f)w escapes directly to space.                                  n(1 – f )w ………SPACE

This now leaves a power of  (1 – n)(1 – f )w  in the atmosphere which is absorbed, and in line with the Greenhouse Gas theory, half of this is re-emitted upwards to space and half is re-emitted to Earth.
0.5(1 – n)(1 – f )w ………SPACE

And, the same amount returns to Earth. 0.5(1 – n )(1 – f )w…….Earth

The longwave power emitted has two components as shown.  If there is insufficient carbon dioxide a proportion nL cannot be absorbed, and so this power escapes directly to space. nL .……..SPACE

This leaves an amount (1 – n )L in the atmosphere.
Again, in line with the Greenhouse theory, half of this is re-emitted upwards to space.  0.5(1 – n)L………SPACE

and half is re-emitted to Earth.  0.5(1 – n)L…….Earth

Therefore, summing the several contributions,

Power to Space = fw + n(1 – f )w + 0.5(1 – n)(1 – f )w + nL + 0.5(1 – n)L

This simplifies to:-
Power to Space = 0.5[ n( L + (1 – f )w )  +  L + (1 + f) w ]

This is the general equation, but it is necessary to take account of latitude, represented by a, where this is an integer for programming reasons, more later.

Therefore, at “a-lat” a this becomes
Power to space[a]   =  0.5[ n( L[a] + (1 – f )w[a] )  +  L[a] + (1 + f )w[a] ]    . …………….(1)

Power at the Earth’s surface
Let i be the angle of incidence of the Sun’s input power to the surface, that is the angle between the Sun’s rays and the normal to the surface.  This must be a decimal number for accuracy, but for those angles considered let there be a corresponding integer variable  a for the sake of arrays in the computer program.

When dealing specifically with the maximum LST’s at local noon in mid-June, the appropriate tilt angle of the Earth’s axis in its orbit around the Sun is 23.4 degrees.  The angle if incidence is then given by ( latitude – tilt ).
That is  i  =  ( latitude – tilt ) when calculations are involved, but for simplification when writing we can use  a = latitude – tilt.  Hence, the expression “a-lat” to represent a for convenience, and the computer program.

So, for example, with the tilt angle of 23.4 degrees, a=0 at latitude 23.4 degrees, then a true latitude of 35.0 degrees  gives an angle of incidence  i = 11.6 deg.

Let the input power from the Sun arriving at the Earth’s surface at normal incidence where a = 0 be G Wm^-2.  In the calculations, the cosine is involved, so it is necessary to include the expression cos(i).

So, the power arriving at the surface at angle of incidence 0.0  is  G.cos(0) Wm^-2
and the power arriving at angle of incidence is G.cos(i) Wm^-2.

Not all of this power ENTERS the surface.  Some is reflected away.  The proportion which enters the surface and is absorbed is explained by the work of Fresnel (1) and it is seen that this proportion reduces to zero at 90 deg latitude.

Therefore, a new term is introduced, the energy absorption factor, r, which is changes with a, so the term r[a] is used.  Therefore, at a=0, the power from the Sun entering the surface and being absorbed, is G.r.cos(0),  and at “a-lat” = a  it is G.r[a].cos(i)

Power balance in the Earth system
Equation (1) gives the output power to space:-
Power to space[a]  =   0.5[ n( L[a] + (1 – f)w[a] ))  +  L[a] + (1 + f )w[a] ]

The input power from the Sun is G.r[a].cos(i)

For power balance
G.r[a].cos(i)  =  0.5[ n( L[a] + (1 – f)w[a] ))  +  L[a] + (1 + f )w[a] ]   . …………………..(2)

Now, there is a problem because accurate values for G and r[a] are not available.  This can be overcome as follows.

Consider a chosen location at any particular latitude represented by a.  Equation (2) can be simplified for this case.

Therefore,
G.r.cos(i)  =  0.5[ n( L + (1 – f)w ))  +  L + (1 + f )w ]  ………………………………(3)

Now the land surface temperature (LST) for the chosen location can be obtained from Copernicus or from NASA’s Earth Observatoty for local noon (temp1) and for local, say, 4pm (temp2) on the same day.  Let the relevant Planck powers for these temperatures be denoted by L1, L2, w1, w2, and Planck is run to obtain their values.  So two equations are obtained from (3) for the same location; G and r have their values common to both equations.

G.r.cos(i)   =   0.5[ n( L1 + (1 – f)w1 ))  +  L1 + (1 + f )w1 ]

G.r.cos(i).z  =  0.5[ n( L2 + (1 – f)w2 ))  +  L2 + (1 + f )w2 ]

Combining these two equations,
0.5[ n( L1 + (1 – f)w1)) +L1+(1 + f )w1 ]
=  (0.5/z)[ n( L2 + (1 – f)w2)) + L2+(1 + f )w2

Simplification can be obtained as follows.
Let   A = L1 + (1 – f)w1
B = L2 + (1 – f)w2
C = L2 + (1 + f )w2
D = L1 + (1 + f )w1

Hence,  n  =  (C/z)  –  D
A  –  (B/z)

Results
The chosen location was at a=20, latitude 43.44 degN, longitude 93.04 degW.
Copernicus gave the LST at local noon on 21/06/2020 as 303.12 K, temp1,
and at local 4pm as 299.18 K, temp2.

Planck’s Law provided the corresponding values for the longwave powers L1 and L2, and the shortwave powers w1 and w2, in Wm^-2 as follows.
L1 = 223.6     w1 = 244.9                   L2 = 216.4     w2 = 228.2

The value of the atmospheric window factor f was taken to be 0.493 and that of z to be 0.50, but moderate differences do not affect the overall outcome.

The value of  n  was found to be   – 1.672

For comparison, n  was determined for a wide range of temperatures, and therefore for latitudes, and the results are shown in the table.  In the case of the chosen location just used, the 4pm temperature was a factor of 0.987 of the noon temperature, and this factor was applied throughout the range.

It is clear that n is not positive throughout the range, and so all of the absorbable energy radiated from the surface is, therefore, totally absorbed.

Conclusion
All of the absorbable energy radiated by the Earth is fully absorbed, and so further addition of carbon dioxide, or any other greenhouse gas, to the atmosphere cannot cause any further temperature increase.

This is not to deny the effect of greenhouse gases.  The Earth’s “comfortable” temperature before anthropogenic activities was, of course, caused by the Natural Greenhouse effect.  The overall stability of this temperature until the 20th century indicates that Earth’s radiant emission was already fully absorbed at this stage.

Any further increase in atmospheric temperature, as reported by NASA and claimed to be due to anthropogenic activities emitting more carbon dioxide, must have a different explanation.

References