When is Colder Thermal Radiation Incident Upon a Warmer Body?

In my recent articles on thermal radiation physics (here and here), I showed that the consensus physics on thermal radiation is wrong in its belief that black body radiators near one another and in a state of equilibrium emit the same radiation as they would when isolated from one another and surrounded by an environment at T = 0K.  That is, they do not each emit power per unit area as given by the Stefan-Boltzmann Law when they are in the vicinity of one another in a state of equilibrium.

How then is it that one can detect the Cosmic Microwave Background Radiation with a detector at 4K when the Cosmic Microwave Background Radiation is that of a black body radiator at 2.73K?  This might at first blush seem to be an argument against my claims.  It may not be a very sound argument, however, due to the very unusual circumstances of detecting this radiation from gigantic distances and long ago when there could not be an equilibrium condition between the detection location on the Earth and the matter which emitted the radiation.

The Cosmic Microwave Background Radiation is believed to be radiation left over from the Big Bang.  The remaining radiation has had little in the universe that can absorb it since the universe cooled to the temperature at which electrons and protons could form hydrogen atoms.  The really long wave radiation left now is not absorbed by hydrogen atoms, which are about all that fill most of the space in the universe.  This microwave background radiation is cooling with time as the universe expands.

This microwave background radiation is now commonly detected by transition edge detectors using the rapid increase in resistance with temperature increase as a superconductor warms through the transition to the normal state at temperatures such as 0.1K.  A map of this background radiation is shown below:

Why does this microwave radiation background not violate my explanation of thermal radiation when it was originally detected by Arno Penzias and Robert Wilson using a detector warmer than 2.73K?  One reason is that this radiation was not emitted from a cooler body to a warmer body.  It is in fact the radiation from much hotter matter and it has been in the form of photons flying about the universe for eons of time.  The Earth did not even exist when this thermal radiation was emitted, so the relative electromagnetic field energy density at the surface of a detector on the Earth to that adjacent to some matter from which this radiation was emitted is not at all in play.  No equilibrium condition between them has ever existed.

Let us take a case of thermal radiation emitted from an actual body in the universe today far from Earth.  Any photons emitted because that body has a temperature higher than that somewhere else in the universe at a particular instant in time could be in flight for a long time.  Let us suppose that the warmer Earth moves into the trajectory of that long ago emitted photon and as a result the photon strikes a warmer detector on Earth.  There is every reason to believe this could happen.

These are two cases where one expects to detect radiation with a lower characteristic black body radiation temperature than a sensor might have on Earth.  Indeed, such radiation can and does bombard our planet.  This does not invalidate the theory on thermal radiation between two bodies in equilibrium with one another that I have presented.  In the context of the Earth's surface emission of infrared radiation and its absorption of infrared radiation emitted by greenhouse gases, colder greenhouse gases do not emit radiation which is absorbed by the Earth's warmer surface.  There is no such back radiation contributing to catastrophic man-made global warming.

Updated 25 March 2018 to emphasize that a cooler body in equilibrium with a warmer body does not emit radiation which is incident upon the warmer body.

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.

Thermal Radiation Basics and Their Violation by the Settled Science of the Catastrophic Man-Made Global Warming Hypothesis

The Catastrophic Man-Made Global Warming Hypothesis rests upon a false thermal radiation foundation.  It is riddled with further errors, which I have written many papers about.  This foundational thermal radiation error needs more examination.

Basically this error assumes that the Earth’s surface and the infrared-active gases in the atmosphere, commonly called the greenhouse gases, act like black body radiators and absorbers with respect to longwave infrared radiation.  This infrared longwave radiation is the important thermal radiation at temperatures in the range of those of the Earth’s surface and the infrared-active gases in the atmosphere.  It further assumes that the thermal radiation emitted by these bodies is the same as it would be if that body is surrounded by vacuum at a temperature of absolute zero, or 0 Kelvin.

My discussion in this paper will be centered on this last assertion by the catastrophists that thermal radiation emitted by a body at temperature T is at a rate per unit area of P = σT⁴, which is called the Stefan-Boltzmann Law of Thermal Radiation, even when that body is surrounded by, or itself surrounds a body, which is not at 0 K.  This wrongheaded belief is one widely held by physicists as well as by climate scientists.  I will show that the application of this idea of radiative emission by black bodies violates the most fundamental property of a black body radiator.

I am going to address this issue by means of the most fundamental characteristic of the thermal properties of black body radiators, namely that the energy density e or the energy per unit volume of the electromagnetic field inside the black body cavity is given by Stefan’s Law:

e = aT⁴ ,

where a is Stefan’s constant and

a = 7.57 x 10^-16 J/m³K⁴

where J is the unit of energy Joules, m is the unit of meters, and K is the unit of the absolute Kelvin temperature for which 273.15 K = 0.00⁰C.  A 1 K temperature change is a 1⁰C temperature change.  I showed the derivation of this energy density for a black body cavity in an earlier paper, The Greenhouse Gas Hypothesis and Thermal Radiation – A Critical Review.  I am interested in the energy density because it is easier to correctly tie to the Conservation of Energy and to a rational appreciation of the properties of a black body radiator.  This will allow me to prove that the Catastrophic Man-Made Global Warming Hypothesis is based upon a false introduction into the Earth climate system of massive amounts of radiation energy which do not really exist.

Why does a surface, which does not really have the characteristics of the model used to derive the properties attributed to a black body cavity, nonetheless emit thermal radiation as a black body cavity with a tiny peephole in the wall of the cavity would?  When one peers into that tiny peephole in the wall of the black body cavity, one sees an interior whose electromagnetic field has a constant energy density e = aT⁴ as noted above.  Consequently, a more realistic and common kind of surface exhibiting the thermal radiation characteristics of a black body radiator will do so only if it has an electromagetic field energy density infinitesimally close to the surface which is equal to e = aT⁴.  When e = aT⁴ at a surface, P = σT⁴ for that surface if the surface is emitting radiation to an environment at T=0 K.  Solving for e in terms of P we find that e = (a/σ)P.

Let us now consider a black body behaving sphere of radius R_H with a temperature T_H which is surrounded by vacuum at 0 K.

The immediate space against the sphere’s surface has a boundary condition in which the energy density e = a T_H⁴.  The power output per unit area of the sphere surface is P = σ T_H⁴   in accordance with the Stefan-Boltzmann Law of Thermal Radiation and σ is the Stefan-Boltzmann constant.  The power at a distance r, measured from the center of the sphere and greater than R_H, will be given by

P_r = (4πR_H²/ 4πr²) σ T_H⁴ = (R_H²/r²) σ T_H⁴

This means that the energy density at this distance r on a sphere centered on the center of our thermal radiation emitting sphere surface will be

e_r = (a/σ) P_r = (a/σ) (R_H²/r²) σ T_H⁴ = (R_H²/r²) a T_H⁴

Now let us add a second black body surface spherical shell concentric with our first sphere having a radius R_C and with vacuum between the inner sphere and the outer spherical shell.  Outside this larger radius spherical shell is nothing but vacuum whose distant temperature is 0 K.  Let the temperature of this spherical shell be T_C.

In the case the T_H = T_C, the entire volume between the sphere and the outer spherical shell will have the energy density of a black body cavity, namely e = a T⁴, with T = T_H = T_C.  Logically, if T_c is less than T_H, the total energy E between these two spherical boundaries must be lower than that contained volume times aT_H⁴ or

E_H = (4/3)π (R_c³ – R_H³) aT_H⁴.

But the Catastrophic Man-Made Global Warming Hypothesis claims that the inner sphere still emits thermal radiation as though the spherical shell were not present and the spherical shell emits it as though the inner sphere was at temperature zero.  They claim that the energies emitted cancel each other in part, yet the vessels of the energy are photons with non-zero energies and they do not actually come in the form of positive energy photons and negative energy photons.  There are only positive energy photons.  We will follow their belief now and examine the consequences.  I am going to color this section with their viewpoint and its consequences in red so the reader will readily recognize that I do not believe this to be correct theory.  I am working out the consequences of an incorrect theory here.

According to their viewpoint, the outer spherical shell emits a power per unit area toward the inner sphere of

P_C (r = R_C) = σ T_C⁴

The inner sphere emits a power per unit area toward the outer sphere of

P_H (r = R_H) = σ T_H⁴

The power per unit area incident upon the inner sphere from the outer spherical shell is

P_C (r = R_H) = (R_C²/R_H²) σT_C⁴.

The power per unit area incident upon the outer spherical shell from the inner spherical shell is

P_H (r = R_C) = (R_H²/R_C²) σT_H⁴ .

If each of these surfaces emits thermal radiation as though they were in vacuum and surrounded only by an environment in the distance at T = 0 K, then the energy density corresponding to these respective powers approaching each surface arbitrarily closely is

e_H = (a/σ) [P_H (r = R_H) + P_C (r = R_H)] =  a T_H⁴ + (R_C²/R_H²) aT_C⁴

and

e_C = (a/σ) [P_C (r = R_C) + P_H (r = R_C)] =  a T_C⁴ + (R_H²/R_C²) aT_H⁴.

In the limit that T_C approaches T_H and R_C approaches R_H, the energy density just inside the outer spherical shell and that just outside the inner sphere both approach 2aT⁴, or twice that of a black body cavity at a temperature of T.  Clearly, they must instead approach an energy density of aT⁴.  Therefore, it is impossible for two such nested thermal radiation emitters to continue each to emit thermal radiation as though they were each only surrounded by a distant radiation sink at T = 0 K.  The actual radiation that each emits is highly influenced by the fact that the other black body emitter is nearby.  Photons do not have positive and negative energies as I noted above.  Photons that would violate the Conservation of Energy are simply not generated in the first place.

In reality, photons are a manifestation of an electromagnetic field.  Thermal radiation is emitted from a material or a molecule due to dipole vibrations and the vibration effect of higher order poles, though the higher order poles have much shorter electromagnetic ranges than do the dipoles in vibration.  The acceleration and deceleration of charges in dipoles is the primary source of the electromagnetic field that generates photons.  An energy density e_H = a T_H⁴ in the vacuum immediately outside the surface of the inner sphere and an energy density of e_C = a T_C⁴ immediately inside the surface of the outer spherical shell cause a gradient in the electromagnetic field (the energy density of an electromagnetic field in vacuum is proportional to the magnitude of the electric field squared) from the inner sphere surface to the surface of the outer spherical shell.  The total energy gradient between the two surfaces is given by 

ΔE = 4πR_H²e_H - 4πR_C²e_C

and

4πR_H² P_H = (σ/a) ΔE, where P_H is the power emitted per unit area from the inner sphere surface, so

P_H = σT_H⁴ – (R_C²/R_H²) σT_C⁴

And

4πR_H²P_H = 4πR_C²P_C

Where P_C is the power per unit area incident upon the inner wall of the spherical shell at the lower temperature, so

P_C = (R_H²/R_C²) σT_H⁴ – σT_C⁴ 

It is the energy gradient that is fundamental here and it determines the flow of energy and hence the incidence of photons upon the outer spherical shell.  Because the energy gradient is much affected by nearby objects with temperatures well above absolute zero, most objects in our life experience are not emitting photons as they would if they were isolated in vacuum with a distant photon sink at T=0 K.  They are much more parsimonious in emitting photons because their surfaces radiate to other bodies with not too dissimilar temperatures commonly and the electromagnetic field gradients or the energy density gradients between them are modest.

Despite this, the power transferred from one body to another is the same as though one imagines them isolated from one another and throwing out photons as though they were in vacuum with a photon sink at T = 0K.  That simple-minded approach to many calculations of temperatures works and this convinces most scientists that photons are so flung about without discrimination to energy density gradients and electric fields.  As we have seen though, that viewpoint leads to incorrect energy and therefore to incorrect electromagnetic field conclusions.  It also leads to serious problems when the imagined large fluxes of photons are imagined to be absorbed by such things as infrared-active gases, the so-called greenhouse gases.

Let us examine the Earth energy budget currently propagated by NASA on one of their websites:

We see that the back radiation claimed to be incident and absorbed by the Earth’s surface is 100% of 340 W/m².  What is the temperature of a black body sphere that would emit this power of energy?

P = 340 W/m² = σ T⁴ = (5.6697 x 10^-8 W/m²K⁴) T⁴,

so T = 278.3K.

The effective temperature of the atmospheric outer spherical shell is then 278.3K.

This temperature in the U.S. Standard Atmosphere Table of 1976 is at an altitude of about 1500 m or 1.500 km.  The radius of the Earth is about 6,371 km.  So

R_C²/R_H² = (6371 + 1.5)²/(6371)² = 1.00047,

so this ratio for the Earth and the atmosphere is for all intents and purposes 1.

If one applies these values in the Catastrophic Man-Made Global Warming version of black body thermal radiation to the calculation of the energy density at the surface of the Earth, one has T_H = 288 K and

T_C / T_H = 0.9663, so

e_H = (a/σ) [P_H (r = R_H) + P_C (r = R_H)] =  a T_H⁴ + (R_C²/R_H²) aT_C⁴ 

e_H = a T_H⁴ + (1.00047) a (0.9663)⁴ T_H⁴

e_H = 1.8723 a T_H⁴

This energy density at the surface of the Earth, by the false settled science of the Catastrophic Man-Made Global Warming Hypothesis, is 1.87 times the energy density in a black body cavity with a temperature of T_H.  This is a fundamental violation of black body cavity physics.  It is also a violation of the Conservation of Energy.

It is this exaggeration by 87% of the energy that a black body atmosphere can produce at the surface of the Earth through an imagined back radiation that allows the Catastrophic Man-Made Global Warming Hypothesis to ignore the role of the temperature gradient in the atmosphere.  See my paper mgh, Not Just Greenhouse Gases, Provides a Warm Earth.

Of course there is a further major error in their assumptions.  There is no black body absorber or emitter in the atmosphere.  There are only infrared-active molecules such as water vapor and carbon dioxide that absorb and emit radiation at wavelengths corresponding to only a fraction of the range of a black body absorber and emitter.  According to the following NASA diagram,

the atmosphere only absorbs 29% of the thermal radiation emitted by the Earth’s surface, not the 100% that a black body absorber would absorb.  Treating the atmosphere as a black body radiator and absorber is another way the hypothesis inflates the radiation energy incident upon the surface of the Earth.  I discussed major problems with the Catastrophic Man-Made Global Warming Hypothesis on this issue in Climate Change Settled Science: The Atmosphere Absorbs 90% or 29% of Surface Radiation.  The proponents of that hypothesis do not worry about such inconsistencies.

The Catastrophic Man-Made Global Warming Hypothesis is based on wrong physics inconsistently applied.  It is most deplorable that so many scientists actually believe this nonsense.  It is beyond belief that this massively wrong hypothesis, which has been around so long and upon which so much scientific funding has been directed, has not been laughed out of the universe by rational and intelligent scientists.  Apparently, intelligence and rationality have been wanting, while a religious sort of fervor and faith in this hypothesis has taken over.  Unfortunately, many scientists are susceptible to the bandwagon effect and will not argue against any crackpot theory believed in by those who control funding for their R&D work.

Horrifically, this extremely wrongheaded hypothesis has been accepted by many as good theory and is used to justify the destruction of whole industries, with the loss of many jobs and much capital investment.  It is used to justify imposing much higher energy costs on everyone, which most hurts the poor.  The higher energy costs imposed on other industries also cause the loss of jobs and a decrease in the rate of growth of our economy.  This erroneous hypothesis justifies subsidies and mandates for the use of energy on the grid which is highly unreliable and which matches peak needs for electricity very poorly.

Errors have consequences.  Denying reality has consequences.  Those consequences are very harmful for mankind.  They are also very harmful to the reputation of science.

The segment in this color was changed on 30 October 2017 to properly take into account the spherical geometries.  It was initially written properly for two parallel planes.