Solving the Parallel Plane Black Body Radiator Problem and Why the Consensus Science Is Wrong

I will present the Consensus, Settled Science solution to the parallel plane black body radiator problem and demonstrate that it is wrong.  I will show that it exaggerates the energy of electromagnetic radiation between the two planes by as much as a factor of two as their temperatures approach one another.  As a result, the calculations of the so-called Consensus, Settled Science dealing with thermal radiation very often result in violations of the Law of Conservation of Energy.

Their calculations of thermal radiation greatly exaggerate the density of the infrared photon radiation in the atmosphere and the extent of the absorption of infrared radiation by infrared-active molecules, commonly called greenhouse gases, such as water vapor and carbon dioxide.  Their theory of the transport of heat energy between the surface of the Earth and the atmosphere and through the atmosphere is very wrong.  It exaggerates the role of thermal radiation greatly and minimizes the role of the water evaporation and condensation cycle and the role of thermal convection.  It cannot be emphasized enough how harmful their mishandling of thermal radiation calculations is to their understanding of the critical issues pertaining to the Earth's climate and to man's role in changing the climate through the use of carbon-based (fossil) fuels. 

In the case in which the two black body parallel plane radiators have the same temperature, the volume between them becomes that of a black body radiator.  The fundamental characteristic of a black body radiator is the constant energy density in the cavity.  I will show that the so-called settled science treatment, which wrongly takes the primary characteristic of a black body radiator to be that the power of emission of radiant energy is given by the Stefan-Boltzmann Law, clearly violates the real principal characteristic of a black body cavity, namely that its constant energy density, e, is given by Stefan’s Law as

e = aT⁴,

where T is the temperature in Kelvin, a is Stefan’s Constant of 7.57 x 10^-16 J/m³K⁴ , and e is in Joules per cubic meter.

I have already demonstrated the failure of the settled science treatment of thermal radiation from black body radiators in the form of concentric spherical shells in a paper posted on 23 October 2017, entitled Thermal Radiation Basics and Their Violation by the Settled Science of the Catastrophic Man-Made Global Warming Hypothesis, but I want to post this solution with its simpler geometry so that the reason the consensus treatment is wrong will be even more apparent to thinking readers.

Numerous critics of the consensus science on catastrophic man-made global warming have argued that the Second Law of Thermodynamics claims that energy only flows from the warmer body to the colder body, but the consensus scientists have argued that thermodynamics only applies to the net flow of energy.  I have long argued that the reason that radiant energy only flows from the warmer to the cooler body is because the flow is controlled by an electromagnetic field and an energy gradient in that field.  I will offer that proof in this paper.  The Second Law of Thermodynamics is not invoked as the basis of the proof in this paper, but the minimum energy of a system consistent with the Second Law of Thermodynamics does turn out to be a consistent solution to the problem of thermal radiation, while the Consensus, Settled Science theory of thermal radiation does not minimize the system total energy, does not produce the correct energy density of a black body cavity, and is not consistent with the Conservation of Energy.  I have pointed out its failure to conserve energy in many prior posts. 

In a black body cavity, the electromagnetic radiation is in equilibrium with the walls of the cavity at a temperature T.  The energy density e is the mean value of 

½ E·D + ½ H·B,

where E is the impressed electric field, D is the displacement, which differs from E when the medium is polarized (i.e., has dipoles), H is the impressed magnetic field and B is the magnetic polarization of a medium.  If the cavity is under vacuum, then D = E and B = H in the cavity volume and |E| = |H|, so e equals |E|².  The mean value of the energy density of the electromagnetic field in the cavity depends on the temperature and is created by the oscillating dipoles and higher order electric poles in the cavity walls.  The energy density is independent of the volume of the cavity.  The radiation pressure on the cavity walls is proportional to the energy density.


This physics may be reviewed by the reader in an excellent textbook called Thermal Physics by Philip M. Morse, Professor of Physics at MIT, published in 1965 by W.A. Benjamin, Inc., New York.  Prof. Morse wrote it as a challenging text for seniors and first-year graduate students.  I was fortunate to use it in a Thermodynamics course at Brown University in my Junior year.  Alternatively, see my post The Greenhouse Gas Hypothesis and Thermal Radiation -- A Critical Review.

Inside a black body cavity radiator at a temperature T, the energy density, or the energy per unit volume of the vacuum in the cavity is constant in accordance with Stefan’s Law.  If one opens a small peephole in the wall of the black body cavity, the energy density just inside that peephole is the energy density of the black body cavity and that energy density is proportional to the square of the electric field magnitude there.  The Stefan-Boltzmann Law states that the flow rate of energy out of the peephole when the black body cavity is surrounded by vacuum and an environment at T = 0 K, is given as the power P per unit area of the peephole as

P = σT⁴

Note that P = (σ/a) e and that e in the T=0K sink is equal to zero.  A change of energy density in the vacuum volume immediately inside the peephole into the black body cavity as given by Stefan's Law to a value of zero in the T=0K outside environment causes a power of thermal radiation emission out of the peephole to the outside environment as given by the Stefan-Boltzmann Law.

Why is it that a surface which is not a peephole into a black body cavity might act like a black body radiator?  It has to be that the energy density very, very close to that surface has the characteristic of the energy density in a black body cavity radiator, namely that

e = aT⁴.

Any flow of energy out of the surface due to its temperature T must be caused by this electromagnetic field energy density at the surface generated by the vibrational motion of electric charges in the material of the surface.  Such flow of energy from the surface only occurs to regions with an energy density that is lower.  There is no flow of energy from the inside wall of a black body radiator because the energy density everywhere inside the cavity at equilibrium is equal.  P from the interior walls is everywhere zero.  A non-zero P is the result of a non-zero Δe.

In fact, while it is commonly claimed that photons inside the cavity are being 100% absorbed on the walls and an equal amount of radiant energy is emitted from the absorbing wall, the actual case is that the radiant energy incident upon the walls can be entirely reflected from the walls.  Planck had derived the frequency spectrum of a black body cavity from an assumption of complete reflection from the walls.

Here is the problem of the parallel plane black body radiators diagrammed, where T_C is the cooler temperature and T_H is the warmer temperature:

Let us first consider the case that each plane is alone and surrounded by an environment of space at T=0.  Each plane has a power input that causes the plane to have its given temperature.  Each plane radiates electromagnetic energy at a rate per unit surface area of 

P = σT⁴.

Consequently, if neither plane were in the presence of the other and each plane has a surface area of A on each side and P_CO , P_CI , P_HO , and P_HI are all radiation powers per unit area, we have

P_C = AP_CO + AP_CI = 2AσT_C⁴

And

P_H = AP_HO + AP_HI = 2AσT_H⁴ ,

since in equilibrium the power input is equal to the power output by radiation.

In the consensus viewpoint, shared by many physicists and by almost every climate scientist, the parallel plane black body radiators above are believed to emit photons from every surface of each plane even in the presence of the other plane with a power per unit area of

P_H = σT_H⁴ and P_C = σT_C⁴,

just as they would if they were not near one another and they only cast off photons into a sink at T = 0 K.  This viewpoint takes the emitted radiation as a primary property of the surfaces rather than an electromagnetic field with a known energy density as the primary property of the surfaces.

Thus, when these planes are in one another's presence this consensus viewpoint says that

P_C = AP_CO + AP_CI - AP_HI = 2AσT_C⁴ - AσT_H⁴

P_H = AP_HO + AP_HI - AP_CI = 2AσT_H⁴ - AσT_C⁴

Note that P_C becomes zero at T_C = 0.8409 T_H and below that temperature P_C is negative or a cooling power in addition to radiative cooling.  If T_H = 288K, then T_C = 242.2K, the effective radiative temperature of the cooler plane to space if the cooler plane is thought of as the atmosphere and the warmer plane is the surface of the Earth, both the atmosphere and the surface act like black body radiators, and the atmosphere receives only radiant energy from the surface.  This is in agreement with calculations I have presented in the past and is a result which I believe to be correct under the assumptions, even though in some critical respects this consensus viewpoint is wrong.

If these planes were isolated from one another and each plane faced only that T = 0 K vacuum, then one would have

e_H = aT_H⁴ and e_C = aT_C⁴,

because these are black body radiator surfaces.  P_HI when the hotter plane is surrounded by T=0 environment provides the photon flow near the emitting surface which causes the local energy density to be

e_H = aT_H⁴ in this case.

Unlike the case of concentric spherical shells, which I considered in Thermal Radiation Basics and Their Violation by the Settled Science of the Catastrophic Man-Made Global Warming Hypothesis, there is no divergence or convergence of the photons emitted from either surface.  The relationship of the radiative power P to the energy density due to that electromagnetic radiation is always e = (a/σ) P as one traverses the distance between the planes.  

Consequently, the energy density between the planes is

e = (a/σ) P_HI + (a/σ) P_CI = aT_H⁴ + aT_C⁴

anywhere between the two planes, because photons have energy no matter which direction they are traveling and they do not annihilate one another based on their direction of travel.  The total energy between the planes is that of the electric field or it is the sum of the energy of all the photons in the space between the planes.  The energy density e would then be the total energy of all the photons divided by the volume of vacuum between the planes.

Now, let us imagine that these planes are very close together and the ends are far away and nearly closed.  Let us have T_C → T_H, then

e → 2aT_H⁴,

but this space between the planes is now a black body cavity in the limit that T_C → T_H, and we know by Stefan’s Law that

e = aT_H⁴

in this case.  In addition, we have created a black body cavity radiator here and P for the walls inside the cavity is actually zero because the interior is in a state of equilibrium and constant energy density.  P is only P = σ T⁴ just outside the peephole facing an environment at T = 0 K.  In the above consensus viewpoint case, each plane surface is emitting real photons, but these cannot annihilate those photons of the opposite plane.  There are no negative energy photons.  These respective photon streams simply add to the total energy density.

The consensus treatment of black body thermal radiation doubles the energy density in a black body cavity, in clear violation of the principal characteristic of a black body cavity upon which their treatment must be based.  Their treatment greatly increases the energy density between the planes whenever T_C is anywhere near T_H, such as is the case of the temperatures in the lower troposphere compared to the Earth’s surface temperature.  Consequently, the sum of P_HI and P_CI must be much smaller than they are thought to be in the consensus treatment of this problem or in the similar concentric spherical shell problem.

Let us now examine the correct solution to this parallel plane black body radiators problem.  It is the electromagnetic field between the two planes that governs the flow of electromagnetic energy between the planes.  Or one can say it is the energy density at each plane surface that drives the exchange of energy between the planes due to the energy density gradient between the two planes.  The critical and driving parameter here is

Δe = e_H - e_C = aT_H⁴ – aT_C⁴ ,

where each black body radiator surface maintains its black body radiator requirement that the energy density at the surface is given by Stefan’s Law.

Electromagnetic energy flows from the high energy surface to the low energy surface, as is the case in energy flows generally.

P_HI = (σ/a) Δe = (σ/a) (aT_H⁴ – aT_C⁴ ) = σ T_H⁴ – σ T_C⁴

P_CI = 0,

which is consistent with experimental measurements of the rate of radiant heat flow between two black body radiators.  Note that as T_C approaches T_H, P_HI approaches zero as should be the case inside a black body cavity in thermal equilibrium.  There is no thermal emission from either of the black body cavity walls then.  Note also that when T_C = 0 K, P_HI is given by the Stefan-Boltzmann Law

P_HI = σ T_H⁴.

Let us recalculate P_H and P_C in this correct formulation of the problem:

P_C = AP_CO + AP_CI - AP_HI = AσT_C⁴ + 0 – [AσT_H⁴ - AσT_C⁴] = 2AσT_C⁴ - AσT_H⁴

P_H = AP_HO + AP_HI - AP_CI = AσT_H⁴ + [AσT_H⁴ - AσT_C⁴] - 0 = 2AσT_H⁴ - AσT_C⁴

And we see that the power inputs to each plane needed to maintain their respective temperatures as they cool themselves by thermal radiation are unchanged in this correct energy density or electromagnetic field centered viewpoint from the consensus viewpoint. Experimentally, the relationship between the power inputs to the thermal radiation emitting planes at given temperatures are exactly the same.  This fact causes the proponents of the consensus viewpoint to believe they are right, but they nonetheless violate the energy density requirements of electromagnetic fields and of black body radiation itself.

Because P_CI = 0, back radiation from a cooler atmosphere to the surface is also zero and not 100% of the top of the atmosphere solar insolation as in the current NASA Earth Energy Budget.  Because P_HI = σ T_H⁴ – σ T_C⁴ , the Earth's surface does not radiate 117% of the top of the atmosphere insolation either.  These radiation flows are hugely exaggerated by NASA and in similar Earth Energy Budgets presented in the UN IPCC reports.  See the NASA Earth Energy Budget below:

This is very important because reducing these two radiant energy flows of infrared photons reduces the effect of infrared-active gases, the so-called greenhouse gases, drastically.  Many fewer photons are actually available to be absorbed or emitted by greenhouse gases than they imagine.  This is a principal error that should cause the global climate computer models to greatly exaggerate the effects of the greenhouse gases, just as they have.

As I have pointed out in the past, one fatal consequence of the exaggeration of thermal radiation from the surface of the Earth is readily calculated from the fact that even if the atmosphere acted as a black body absorber, which it does not, it can only absorb the thermal radiation said to be emitted from the surface at an absurdly low atmospheric temperature.  Observe that the power of infrared radiation from the surface which is absorbed by the atmosphere P_SA is

P_SA = σ ( T_S⁴ – T_A⁴)

P_SA = (1.17 – 0.12) (340 W/m²)

The first equation is from the discussion above with T_S the surface temperature and T_A the temperature of the atmosphere.  The second is according to the NASA Earth Energy Budget above.  If one takes T_S to be 288K, then the temperature of the atmosphere required to absorb as much infrared energy as NASA claims is absorbed is 155.4K.  There is no such low temperature in the Earth’s atmosphere.  To find so low a temperature, one has to go far out into the solar system many times the radius of Earth’s orbit.  That being the case, any such thermal radiation absorbed by matter at such a low temperature is as much lost in the Earth Energy Budget as is the power equal to 12% of the solar insolation emitted from the Earth’s surface which NASA allows passes through the atmospheric window into space.  It should be apparent to the reader that the NASA Earth Energy Budget is nonsense.

It is not at all surprising that physics adheres to a minimum total energy in the system and does not generate a superfluous stream of photons from the colder body to the warmer body and does not have any more photons flowing to the colder body from the warmer body than necessary.  By means of the electromagnetic field between these two planes, the photon emission of the planes is coupled and affected by the presence of the other plane.  This is in no way surprising for an electromagnetic field problem.  One needs to remember that photons are creatures of electromagnetic fields.  Opposing streams of photons do not annihilate one another to cancel out energy, they simply add their energies.  Treating them as though one stream has a negative energy and the other a positive energy is just a means to throw the use of the Conservation of Energy out the window.  That is too critical a principle of physics to be tossed out the window.

Extending the Solution to Gray Body Thermal Radiators and Other Real Materials:

Many real materials do not behave like black body radiators of thermal radiation.  Those that do not radiate as black body materials would, radiate less than the black body radiator would.  Why would they radiate less?  This is because they do not create as high an electromagnetic field energy density at their surfaces as does a black body radiator.  From Stefan’s Law for a black body radiator, the energy density at the surface is

e = aT⁴

but for a gray body radiator the energy density at each wavelength λ is

e(λ) = εaT⁴.

This means the energy density at any given frequency is a constant fraction ɛ of that of a black body radiator, with 0 ≤ ɛ ≤ 1.

In general, a material may not behave like either a black body or a gray body radiator at a particular wavelength.  In that case,

e(λ) = ε(λ)aT⁴,

where the fraction of the black body output at wavelength λ is variable.

An isolated material surrounded by vacuum and a T=0 K environment then has a power per unit area output of

P(λ) = εσT⁴ for a gray body and ε is seen to be the emissivity, and

P(λ) = ε(λ)σT⁴, for a general material, such as carbon dioxide or water vapor, where the absorption and emission become variable fractions of that of a black body as a function of wavelength.

For our two parallel plates above, if both are gray bodies, then between the plates

Δe = e_H – e_C = ε_H a T_H⁴ – ε_C a T_C⁴

P_HI = (σ/a) Δe = ε_H σ T_H⁴ – ε_C σ T_C⁴

P_CI = 0.

Here we see that the emissivity which determines the electromagnetic field energy density at the surface is also playing the role of the absorptivity at the absorbing colder surface.  So of course, Kirchhoff’s Law of thermal radiation that the emissivity equals the absorptivity of a material in a steady state process applies.  There is really nothing at all to prove if one starts with the primary fact and boundary condition that the energy density is the fundamental driver of the thermal radiation of materials.

Update:  Considerable additions were made on 8 November 2017.

Update:  Additions made on 12 November 2017.
Update:  The extension to gray bodies and other bodies was made on 14 November 2017.
Update:  Explanatory additions on 27 November 2017.
Update:  Further explanatory text added on 29 November 2017.
Update:  Further efforts to help the reader to understand this post and to provide the reader a reason to read it in the introductory paragraphs on 24 March 2018.  I am amazed how difficult many readers find it to alter their viewpoint of thermal radiation even when they have no counter argument to my argument presented here.

The Settled Science of Catastrophic Man-Made Global Warming Violates the Laws of Physics

President Trump has just recently announced that the U.S. will not participate in the Paris Climate Agreement.  He implied that he regarded it as a bad deal for the U.S. economy, while not necessarily disputing its premise that man’s use of fossil fuels was going to cause a disastrous deterioration of the Earth’s climate.  Only a minority of Americans seriously claim that the science behind the catastrophic man-made global warming hypothesis is wrong.  Most of mankind has been seriously misinformed on this critically important topic.

Almost all of the debate about the science is about whether the number of hurricanes of late; the world area under drought conditions; the area of melting ice somewhere in the arctic, Antarctica, or at the site of some glacier in the mountains is greater or reduced recently; the temperature of the ocean surface or at land stations near population centers has increased; or whether some area of the ocean has an increasing acidity or not.   Given the complexity of all the factors that might be linked to increased carbon dioxide in the atmosphere and their cycles over decades and more, this debate could easily go on for decades with no certain conclusion.  The possible symptoms of man-made global warming are too much affected by weather, cycles in solar energy output, ocean heat cycles, and high altitude wind currents to clearly see the underlying long-term climate equilibrium trends.

Meanwhile, the alarmists will go on claiming that the Precautionary Principle requires us to make radical cut-backs in our use of carbon-based fuels, no matter the consequences in the cost of energy and its availability for the poor and most everyone else, on jobs, and on economic growth.  Preventing an imagined disaster decades into the future takes precedence over the well-being of billions of humans in the meantime in the eyes of the many advocates of the catastrophic man-made global warming hypothesis.

This long, drawn out debate is unnecessary.  Worrying about a future climate catastrophe has no basis in the physics of the climate.  It turns out that the so-called settled science has internal flaws in the physics that constitutes its very most fundamental basis.  The advocates of the global warming by carbon dioxide hypothesis have told us that the science is settled. They make some very specific claims about how so-called greenhouse gases cause the present temperatures of the Earth’s surface and atmosphere to be warm as a part of what they claim to be the settled science.  Consequently, we can challenge the physics they claim to be behind the present conditions that control the Earth’s basic climate.  We do not even need to wait to see if their catastrophic man-made (anthropogenic) global warming hypothesis makes correct predictions for the future.  In challenging the physics of their theory, we may well learn more about what the effect of additional carbon dioxide on the climate in the future will be.  In any case, if their reasoning for the present temperatures of the surface and atmosphere is substantially wrong and in violation of fundamental laws of physics, then any further predictions they may make have no basis in science.  If they do not understand the present, they certainly do not understand the future.

Let me first lay out what their basic argument for how infrared-active (so-called greenhouse) gases cause the Earth to be much warmer than it would be without them.  Having done so, I will proceed to demolish their argument by showing the many ways in which it defies well-known physics principles, none of which the scientific advocates of the strong greenhouse effect will actually deny are true principles of science.  In fact, they make a pretense of using these same principles of science themselves.

Let us begin with an examination of the Earth Energy Budget NASA currently has on its website to explain the effect of infrared-active gases in warming the Earth.  Because this and many similar Earth energy budgets have been a key part of the claimed science of anthropogenic global warming by infrared-active gases for many years now and because they are adamant in claiming that the science is settled, then any violations of scientific principles in this NASA Earth Energy Budget invalidate the complete argument for catastrophic man-made global warming.

When faced with a complex scientific problem, a good scientist should have the wisdom to first establish whatever limits he can on what the range of physically possible solutions might be by seeing if he can establish lower or upper bounds for the values of a property he wishes to measure or calculate.  He will learn something about the essence of the complex problem in the process and he will protect himself from becoming an advocate for a physically impossible solution to the problem.  This has the real benefit of protecting him from making ridiculous conclusions.

According to this energy budget, the total photon energy flux absorbed by the entire Earth system is the incoming solar radiation absorbed by the atmosphere plus that absorbed by the surface, which is

P (solar absorbed) = (77.1 + 163.3) W/m² = 240.4 W/m²

Watts are Joules/second, or the rate at which energy flows.  This 240.4 W/m² is the largest possible photon energy supply that the surface might ever be able to absorb.  It is an upper limit, since nothing on the Earth can add to the energy introduced into the Earth system from the solar radiation, save a possible very minor contribution of heat leakage from the hot interior of the Earth, which they do not consider in their energy budget.  This upper limit is actually further reduced by the amount of any energy absorbed by the atmosphere which is directly emitted as radiation into space, which is the case for sure of much of the 77.1 W/m² absorbed from sunlight in the atmosphere.  This upper limit is a simple matter of conservation of energy.

Now we can immediately see that this energy budget claims that there is a miraculous supply of energy-bearing photons absorbed by the surface with a power of

P (surface absorbed) = (163.3 + 340.3) W/m² = 503.6 W/m²

Where does this addition energy in the form of photons come from?  According to the settled science, this is not additional energy because the total energy flowing out of the Earth system at the top of the atmosphere is the maximum power of the first equation minus the net absorbed heat (0.6 W/m² ) in the surface of a slowly warming Earth.  According to them, one can add as many photons as one wishes with as much energy as one wishes within the Earth system of atmosphere and surface, so long as enough of those photons have negative energy to cancel out the positive energy of other photons.

Of course, they never say it this way because there is no such thing as positive and negative energy photons.  The reality is that there are only positive energy photons and this settled theory has created a huge supply of energy carried by photons in violation of the Law of Conservation of Energy.  Somehow, greenhouse gases are supposed to have created a supply of photons with an additional energy flow rate of (503.6 – 240.4) W/m² = 263.2 W/m²  compared the maximum possible energy in the form of photons, which is even more energy than was absorbed in the entire Earth system from the sun!

If the total rate of energy into the Earth system is 240.4 W/m², this is the highest rate at which energy might leave the surface.  However, since the energy budget provides for a net absorption, thermals losses, and water evaporation, the largest possible flux of energy carried by photons leaving the surface on invoking the Law of Conservation of Energy is

P (photon energy flux leaving surface) = (240.4 -0.6 – 18.4 – 86.4) W/m² = 135.0 W/m²

But having no regard for the Conservation of Energy, the settled science insists that photons carrying energy at the rate of 398.2 W/m² are leaving the surface.  Somehow photons carrying energy at a rate of (398.2 – 135.0) W/m² = 293.2 W/m² have materialized out of nowhere through the miracle of greenhouse gases.  The photon flux leaving the surface is 398.2 / 135.0 = 2.95 times the theoretical upper limit.  Well, never let mere physics stop you from making whatever argument you might think will be good for the maintenance of the environment or the justification to propagate more governmental controls and taxes.

The ridiculous physics does not stop with these violations of the Conservation of Energy.  When once one has deviated so far from physical reality, it is necessary to violate still more well-known physics.  I will now apply a similar examination of physical limits to the issues of the amount of infrared radiation from the surface that is absorbed by the atmosphere and the amount of radiation from the atmosphere which is absorbed by the surface.

There is a well-known law of thermal radiation called the Stefan-Boltzmann Law which says that the power per unit area of thermal radiation from a black body radiator is

P (W/m²) = σ T⁴,

which is the maximum power that can be radiated by a black body when it has no competing power loss mechanisms to radiative cooling and is surrounded by vacuum at absolute zero temperature, or 0 K.  σ = 5.6697 x 10^-8 W/m² K⁴ and T is the absolute temperature in Kelvin, represented by K.  Of course, real materials usually radiate less power than this and do so to an environment which is not at 0 K, though deep space is at least close to that temperature.

The power that a cooler black body at T_c absorbs from a warmer black body radiator at T_w is

P (absorbed, W/m²) = σ ( T_w⁴ – T_c⁴ )

This power is greater than that which can be gained by a real material which is not a black body material from warmer real material which is more like a black body material.  Consequently, we will be using this to establish the upper limits of the transfer of heat by thermal radiation from the warmer Earth surface to the cooler atmosphere.

Returning to the NASA Earth Energy Budget, the Earth’s surface is shown to deposit 358.2 W/m² of energy flux into the atmosphere out of the 398.2 W/m² that it emits.  This surface emission of thermal infrared radiation from a black body behaving as though it were surrounded by 0 K vacuum implies a surface temperature of

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

Which is the highest average surface temperature for the Earth that I have ever seen claimed.  The usual range of temperatures claimed in the settled science is 287.5 to 288.5 K.

As I have done in some of the recent articles I have written, we will now calculate the temperature of a black body spherical shell surrounding the Earth which would be capable of absorbing the 358.2 W/m² said to be absorbed from the surface thermal radiation by the atmosphere.  This is

P (absorbed, W/m²) = 358.2 W/m² = σ ( T_w⁴ – T_c⁴ ) =

(5.6697 x 10^-8 W/m²K⁴) [(289.5 K)⁴ – T_c⁴], so T_c = 163.0 K

As I have previously pointed out, there is no part of the Earth’s atmosphere with a temperature as low as 163.0 K.  Insofar as infra-red absorbing gases in the atmosphere fall short of absorbing all of the infrared radiation from the Earth’s surface as efficiently as a black body absorber would, the required temperature of the absorbing gas molecules would have to be lower yet.  In fact, since they simply cannot absorb some of the infrared radiation frequencies or wavelengths from the surface at all, even if the so-called greenhouse gas were at zero Kelvin, absolute zero temperature, those greenhouse gases could not absorb as much energy flux as a black body absorber can at 163K.

Of course water vapor tends to condense in the atmosphere when it reaches just 273.15 K (0⁰C), so there is a rapid decrease in the water vapor concentration even in the upper troposphere.  However, at pressures below 5.1 atmospheres carbon dioxide does not liquefy, let alone solidify.  In fact, the coldest part of the Earth’s atmosphere, the mesopause, at 86 km to 90 km has a minimum temperature of about 187 K because carbon dioxide radiates heat to space so efficiently from that super low density layer of the atmosphere.  Carbon dioxide is generally an effective coolant in the atmosphere above the altitude of water vapor condensation.  This is just one of many ways that CO₂ acts to cool the Earth.

So let us examine the spectrum of energies of the photons that make up the thermal radiation from the Earth’s surface modeled as a black body radiator in vacuum and surrounded by nothing but space at absolute zero temperature.  Because we are interested in the flux of energy carried away from this hypothetical surface, which is exactly the surface assumed real in the settled science of the NASA Earth Energy Budget, I will provide the spectrum of radiated energy flux versus the radiated photon energy E = h ν, where h is Planck’s constant equal to 6.625 x 10^-34 Joule-sec and ν is the frequency per second.

Because the energy of each radiated photon is very small, I will plot the energy scale in eV, a unit of energy equal to 1.602 x 10^-19 Joules.  One Joule/second is a Watt.  This plot of power in W/m² is for a more commonly accepted Earth average surface temperature of 288 K.  The area under the power output curve is the total power output due to the radiation from a black body radiator at 288 K, which is 390.06 W/m².

Below the power output of the black body model of the Earth’s surface is a plot showing the infrared absorption spectra of water vapor and carbon dioxide taken in my laboratory using my FTIR infrared spectrometer.  This spectrometer does not produce an infrared beam with adequate intensity to make absorption measurements below an energy of 0.05 eV, so the absorption spectra start there and extend to about 0.9 eV for generally useful data.  The carbon dioxide concentration was several times that of the normal atmosphere and the water vapor was above 50% RH measured against an instrumental background spectrum made at about 25% RH. 

The important point here is not one of the relative degree of absorption of water vapor compared to carbon dioxide.  The critical point here is that from about 0.09 to 0.155 eV there is no absorption by either water vapor or carbon dioxide.  A good fraction of the infrared radiation emitted by the black body modeled surface of the Earth lies in this range, which is called the atmospheric window.  This portion of the power spectrum in which no absorption takes place is about 35.5% of the power radiated from the black body modeled Earth surface.  This is about 138.5 W/m² which should be emitted to space if the Earth’s surface really radiated as a black body radiator as the settled science says it does.  Any incomplete absorption of the surface radiation at other energies would have to be added to this.

So how is it that the NASA Earth Energy Budget says that only 40.1 W/m² is emitted from the surface into space?  If a minimum of what they say is emitted from the surface and will not be absorbed by water vapor or carbon dioxide is 138.5 W/m², then the power they claim is emitted to space is only 29% of that minimum.  NASA says only 10.1% of the surface emitted radiation is emitted to space through the atmospheric window.  This sure would make the use of infrared sensors a lot more difficult than it is.  Fortunately, our military infrared sensor designers know that the NASA story on the atmospheric window is wrong.  In fact, many scientists in NASA surely know this is wrong, except for those steeped in Climate Change funding for their work there.  About half of the NASA budget recently has been directed at Climate Change investigations, so NASA has a huge incentive to hype climate alarmism.

The bottom line is that for the 358.2 W/m² to be absorbed in the atmosphere out of the 398.2 W/m² that NASA says is emitted from the surface, the absorber needs to be a black body absorber located about 6 Earth orbit radii from the sun.  That is how far one has to go out into our solar system to find sufficiently cold space to serve as the absorbing black body sphere to absorb so much radiation.  This is in no way a part of the Earth’s atmosphere and energy dumped into space cannot become significant back radiation to constitute a good part of the 340.3 W/m² that NASA is claiming is back radiation from the atmosphere to the surface.  Of course we already know that 340.3 W/m² of back radiation is impossible because there is only 240.4 W/m² of power entering the entire Earth system.  But if you are willing to violate the Conservation of Energy, it is hardly surprising that you are driven to violate the Stefan-Boltzmann Law of thermal radiation.  One violation of reality inevitably leads to numerous other violations of reality.

We can also ask what the temperature of the black body radiator would have to be to generate the 340.3 W/m² of back radiation that NASA is claiming in the settled science occurs.  We have

P = 340.3 W/m² = (5.6697 x 10^-8 W/m²K⁴) T⁴, so T= 278.3 K,

which in the U.S. Standard Atmosphere of 1976 means this radiation could originate from a black body sphere at an altitude of about 1500 meters.  Now we have learned that the 398.2 W/m² emitted from the surface could not contribute 358.2 W/m² to the atmosphere at an altitude of 1500 m.  Let us calculate about how much it could deposit into the atmosphere if it were a black body absorber.  This number will of course be a generous upper limit.  In fact, we could say the upper limit will be only 64.5% of this taking into account our measurement of the minimum power lost through the atmospheric window.

The maximum black body absorbed power at 278.3 K in the atmosphere from the surface would then be:

P = (5.6697 x 10^-8 W/m²K⁴) [(289.5K)⁴ - (278.3K)⁴] = 58.1 W/m²,

where I have used the NASA Earth average surface temperature of 289.5 K.  If we then subtract from this the minimal fraction lost through the atmospheric window, which is 35.5%, the power at about 1500 m altitude absorbed from the surface is only 37.5 W/m².

Now let us determine an upper limit for the amount of power available in the atmosphere at 1500 m altitude.  We clearly have less than

77.1 W/m² total atmospheric absorption from the sun, a large over-estimate

18.4 W/m² in thermals

86.4 W/m² in latent heat due to water condensation (much of which happens well above 1500 m altitude

37.5 W/m² absorbed from the surface

Total Upper Limit Power Available in Atmosphere near 1500 m altitude = 219.4 W/m²

Now converting this upper limit power into radiation properly would mean that it would flow from warmer air to cooler air and therefore would almost all be transported upward through the atmosphere.  But in the NASA mindset, one fallaciously a part of the settled science viewpoint, thermal radiation is isotropic.  If one accepts that viewpoint, then only half of this upper limit of 219.4 W/m² could become back radiation to the surface.  That value of half is then 109.7 W/m².  Any claim of back radiation in excess of this amount is clearly absurd.  So, the NASA claim of 340.3 W/m² of back radiation to the surface is at least (a lower bound) 3.1 times absurd.  In fact, any real back radiation is many times smaller than 109.7 W/m².

Having discovered so many ways in which the NASA Earth Energy Budget of the settled science is ridiculous, one has to wonder if they got anything right.  I find myself very suspicious of the surface power loss to thermals and to latent heat through water evaporation.  On the other hand, one would think that they might be able to make good measurements of the reflection of solar radiation at least in total if not separately for that from the atmosphere and the surface.  I would also think that they could measure the radiation from the surface through the atmospheric window quite well also.

Let us examine some implications of this atmospheric window measurement.  If it is minimally 35.5% of the surface emission, then the surface radiative emission is at least 113.0 W/m².  Let us now calculate the maximum power that the surface can emit in the form of radiation now that we have a very high upper bound for the back radiation under the assumption that NASA has the other absorbed and loss energy fluxes correct:

163.3 W/m² absorbed from the sun

-18.4 W/m² lost to thermals

-86.4 W/m² lost to water evaporation

-0.6 W/m² residual absorbed heat causing long term warming of the surface

109.7 W/m² in greatly exaggerated back radiation

Maximum total radiation emitted from the surface = 167.6 W/m²

This generous maximum is only 42% of what NASA claims the thermal radiative emission to be.  Apparently, the surface of the Earth cannot radiate thermal energy as a black body radiator in vacuum surrounded by absolute zero would.  This should be no surprise.  This is once again a consequence of the Conservation of Energy.  Consequently, we know that within the framework of some of the measurements claimed as a part of the settled science that the thermal radiation from the Earth’s surface has be between 113.0 and 167.6 W/m².

Why do materials emit thermal radiation?  The temperature of a material is determined by its kinetic energy.  Materials with internal modes of vibration share the kinetic energy they have equally with all the modes of motion available to the material.  The internal vibrations of atoms bound to one another necessarily cause electric charges to accelerate and decelerate due to oscillatory vibrations in the material.  The emission of electromagnetic radiation then depends upon the kinetic energy of bound atoms and their resonance states.  Solid materials tend to have so many modes of vibration that they emit radiation across a very wide spectrum of photon energies.  Liquids tend to be similar in this regard, though the range of energies tends to be a bit less.  Gas molecules, however, consist of very small numbers of atoms and consequently have few modes of vibration.  There emission spectra tend to have more discrete and narrow emission lines.  Contrary to popular belief, nitrogen and oxygen molecules do have electromagnetic emission spectra, though they lie outside of the infrared portion of the spectrum.  Oxygen actually has a small absorption and emission line near the border of the red visible light to near infrared spectrum that allows it to absorb some of the incoming solar insolation.

Let us suppose we are examining a few atoms on the surface of wet soil.  Each time a water molecule is evaporated from that wet soil, that molecule carries away an energy equal to its latent heat of evaporation.  This cools those atoms immediately around the site from which the water molecule evaporated from the surface.  Now heat will flow into this microscopic region from the warmer surrounding atoms, but for an instant in time this fractional area of the surface is cooler than the nominal average temperature of the surface.  As a result, any thermal emission of radiation from this microscopic area will be that from a slightly cooler surface.  This means that fractional area will be emitting slightly less power than will areas of the surface at the average temperature of the material.  The mean emitted photon energy will be slightly less also.  Similarly, if a cooler air molecule strikes a warmer surface and carries away a bit of energy as a result of that collision, then the surface will locally be cooler for a moment and that area will emit less thermal radiation.

Materials with water in them or in contact with air do not act exactly like black body radiators in space for this reason.  Energy is conserved.  Energy used to evaporate water cannot also be used to emit photons.  Energy transferred to the air molecules by collisions to increase their kinetic energy cannot be emitted as a photon.  This is not to say that there are not fractions of a macroscopic surface which are radiating on the microscopic level as one expects the same material to do in vacuum with no energy loss mechanisms except thermal radiation.  A square meter of the Earth’s surface might have 60% of the area emitting thermal radiation as though it were in vacuum, while the rest of the surface is radiating slightly less power and with a slight shift in the distribution of emitted photons to lower photon energies.

The emission is also limited by the temperature of the materials surroundings.  As we noted above, the radiation emitted is proportional to the difference of the fourth power of the temperature of the emitter and the absorber of the radiation.  Photon emission occurs because of electromagnetic fields and those photons travel along an electric field dependent upon the gradients of the electromagnetic field.  Photons are not just flung about as a primary entity.  They are creatures of electromagnetic fields.  The emitting higher temperature source is connected to the absorbing lower temperature photon sink by this electromagnetic field.  When you assume that every material is emitting radiation as a black body would when surrounded by vacuum at 0K, you will necessarily generate a host of non-existent photons which will upset the energy balance.

The act of ignoring the proper transfers of radiative power between objects has played a great role in the failure to recognize that the Earth’s gravitational field causes a kinetic energy gradient in the atmosphere.  Because the temperature of perfect or ideal gas is proportional to its kinetic energy, this means that gravity produces a temperature gradient in the atmosphere.  This long known gravitational temperature gradient does not imply a heat transfer or flow along the temperature gradient, though most scientists are under the misconception that any temperature gradient implies a flow of heat energy.  That belief is wrong.  Because the settled science ignores this gravitational temperature gradient and posits an exaggerated surface thermal radiative cooling compensated with a gigantically exaggerated back radiation, it could hardly be more in error.

As I have shown, the settle science represented by the NASA Earth Energy Budget is nonsense science.  It is self-contradictory in numerous ways.  It violates the Conservation of Energy.  It violates the transport of energy between objects of different temperatures by thermal radiation.  These are simple problems clearly and easily observed.  Given the huge expenditures of federal funds, United Nations funds, and the funds of many governments of developed nations around the world on issues relating to the catastrophic man-made global warming hypothesis, it is impossible to believe that there are not many scientists who are aware of these violations of physics and reality.

Yet there are too few scientists who are trying to bring these colossal mistakes to light. There are many pressures on scientists to conform to the prescribed theory.  Most scientific research performed at universities is funded by the federal government and it has been the policy of the government to only fund those who accept or at least do not deny the truth of the catastrophic man-made global warming hypothesis.  There are also many government employee scientists under great pressure to conform.  Publishing papers opposed to the hypothesis in scientific publications has been made very difficult by what can only be termed a conspiracy.  There are many scientists who have told me that catastrophic man-made global warming theory is wrong, but they will not tell the public that out of fear for their careers.  This is why many of the open critics of this nonsense theory are retired scientists.

Nonetheless, the claims that 97% of scientists believe in catastrophic man-made global warming are completely bogus.  A more realistic assessment would be that about one-third do believe in it, about one-third are not sure, and about one-third think the hypothesis is unlikely true.

I am not retired.  I operate a materials analysis laboratory which uses electromagnetic radiation extensively to characterize material properties.  Relatively little of our funding comes directly from government agencies.  We have performed analyses for carbon fuel companies, but have actually done more work for so-called green energy companies.  I really do not have a horse in this race from a financial standpoint.  I am only concerned about the integrity of science, the freedom to think and act independently, and the well-being of my fellow man.  There is much evil being done in the name of the catastrophic man-made global warming hypothesis and the alarmist attempts to promote it at the expense of rational thought and action.