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23 October 2017

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 = 0.00C.  A 1 K temperature change is a 1C 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 RH with a temperature TH 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 TH4.  The power output per unit area of the sphere surface is P = σ TH4   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 RH, will be given by

Pr = (4πRH2/ 4πr2) σ TH4 = (RH2/r2) σ TH4

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

er = (a/σ) Pr = (a/σ) (RH2/r2) σ TH4 = (RH2/r2) a TH4

Now let us add a second black body surface spherical shell concentric with our first sphere having a radius RC 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 TC.

In the case the TH = TC, 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 = TH = TC.  Logically, if Tc is less than TH, the total energy E between these two spherical boundaries must be lower than that contained volume times aTH4 or

EH = (4/3)π (Rc3 – RH3) aTH4.

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

PC (r = RC) = σ TC4

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

PH (r = RH) = σ TH4

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

PC (r = RH) = (RC2/RH2) σTC4.

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

PH (r = RC) = (RH2/RC2) σTH4 .

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

eH = (a/σ) [PH (r = RH) + PC (r = RH)] =  a TH4 + (RC2/RH2) aTC4


eC = (a/σ) [PC (r = RC) + PH (r = RC)] =  a TC4 + (RH2/RC2) aTH4.

In the limit that TC approaches TH and RC approaches RH, 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 eH = a TH4 in the vacuum immediately outside the surface of the inner sphere and an energy density of eC = a TC4 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 = 4πRH2eH - 4πRC2eC


4πRH2 PH = (σ/a) ΔE, where PH is the power emitted per unit area from the inner sphere surface, so

PH = σTH4 – (RC2/RH2) σTC4


4πRH2PH = 4πRC2PC

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

PC = (RH2/RC2) σTH4 – σTC4 

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

RC2/RH2 = (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 TH = 288 K and

TC / TH = 0.9663, so

eH = (a/σ) [PH (r = RH) + PC (r = RH)] =  a TH4 + (RC2/RH2) aTC4 

eH = a TH4 + (1.00047) a (0.9663)4 TH4

eH = 1.8723 a TH4

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 TH.  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.


Anonymous said...

I thought I would offer a comment of general interest. In case you missed it, Anthony Watts's blog has started posting physics papers on gas pressurization (I'm not a scientist) as a cause of planetary temperature gradient. The first reference, is to three interesting papers by two British physicists, Michael and Ronan Connolly suggesting a faster and more significant heat transfer mechanism than convection.

Here are the URLs:

Charles R. Anderson, Ph.D. said...

Thank you for the links to this interesting work, Tom.

Steve Titcombe said...


Brilliant piece of work. Thank you. Will you be posting this over at the "Science of Doom" site? They're a smug bunch over there, and they need to know this stuff.

By the way, In the first section where you use red font, you appear to have a typo for your equation for: The inner sphere emits power per unit area toward the outer sphere of;

PH (r = RH) = C

I think that you meant to type;

PH (r = RH) = σ TH4

Please tag up with Joseph Postma (over at the Climate of Sophistry site) if you've not already done so - he's fighting on the same side as you are.

Best wishes
Steve Titcombe

Charles R. Anderson, Ph.D. said...

Thank you for your comment Steve and thanks for the correction. I have fixed the problem which I expect occurred when I tried to do a Ctrl C to copy as an easy way to minimize the problem of subscripts and superscripts and I likely hit Shift C instead.

Joe Postma and I have had many discussions in the past, though not many lately. Between running my lab and writing on this blog, my time is commonly very limited, especially now that American companies are again eager to innovate in their materials usage and do a greater due diligence in quality controls and failure analyses. It apparently helps when you think the government is not determined to keep your business efforts from succeeding.

Steve Titcombe said...


Thanks to your article, and a trilogy of articles by Joe Postma, I really do get it. Back-radiation (also inferred by the use of the word “net” in any false heat transfer equation) does not exist.

The S-B law of radiant exitance = σ T^4 applies only for a blackbody when radiating into a 0K environment. In all other circumstances, radiant exitance = ɛ σ (T WARM^4 – T COOL^4). When used in conjunction with the Stefan Law for Radiant energy density = a T^4 , it can be shown that only the warmer object can emit photons to the cooler object across the radiation energy density gradient that will exist between the two objects. It can also be shown that an object with an internal power source will, when the steady-state has been established between another other passive object in a vacuum, radiate all of the power generated by that object according to ɛ σ (T WARM^4 – T COOL^4) . However, prior to the steady-state being established, the cooler object (when constructed of an insulating material) will need to establish a kinetic energy gradient across it’s mass before it is, itself able to radiate (from a different surface) all of the radiant energy that the warmer object wishes to dissipate. The other object will thus require a higher temperature on it’s warmer (receiving) side to radiate on it’s cooler (emitting) side. Although those photons emitted from the warmer object are thermalized on the surface of the cooler object, the kinetic energy that now exists on this thermalized surface is not readily conducted away from this surface so, by the equation ɛ σ (T WARM^4 – T COOL^4), the radiant exitance from the warmer object will be restricted, such that the ‘excess power’ within the power-generating object will be converted to kinetic energy inside the warmer object (and thus results in increased temperature) until the sufficient radiant energy density gradient can be established between the two objects. I have written an essay on this topic and submitted it to PSI, but I suspect that that last revelation is causing some consternation at PSI – it is a concept that us RGHE-deniers have always denied – but I can show that it is a real thing. I have emailed the essay to you – perhaps you and John O’Sullivan can discuss it. I believe that my essay really does blow the whole “back-radiation” argument away.