In recent posts I have exhibited a version of Stefan-Boltzmann's radiation law referred to as
False-SB which is used by climate scientists to support CO2 alarm, and I have shown that False-SB is the result of an incorrect application of Planck's radiation law. Let me here again show the incorrect argument leading to
False-SB, and the correct argument leading to
True-SB.
Stefan-Boltzmann's radiation law in its original form (SB) expressing the total radiance from a blackbody of temperature T into a background at 0 K, is obtained by integration/summation over frequencies of Planck's radiation law expressing the radiation intensity I(f,T) emitted by a blackbody as a function of frequency f and temperature T, per unit frequency, surface area, viewing solid angle and time:
- I(f,T)=yTf2theta(ν,T),y=2kc2
with the high-frequency cut-off factor
where c is the speed of light in vacuum, k is Boltzmann's constant, with theta(ν,T)≈0 for hfkT>10say and theta(f,T)≈1 for hfkT<1. Since h/k≈10−10, this effectively means that only frequencies ν≤T1011 will be emitted, which fits with the common experience that a black surface heated by the high-frequency light from the Sun, will not itself shine like the Sun, but radiate only lower frequencies.
We refer to kTh as the cut-off frequency because frequencies f>kTh will be radiated subject to strong damping. We see that the cut-off frequency scales with T, which is Wien's Displacement Law.
Normalizing and simplifying the exponential cut-off, Planck's law can be written in the form
- I(f,T)=yTf2 for f≤T
- I(f,T)=0 for f>T.
The idea is now that SB is obtained from Planck's law by summation/integration over frequencies. The basic form of SB expresses the total radiance R(T,0) of a blackbody of temperature Tradiating into a background at 0 K as
where sigma=γ3.
We next seek the radiance R(T,Tb) when the background is a blackbody of temperature Tb>0. Planck proved his law in the case Tb=0 by using an argument based on statistics of quanta and it is not clear how to extend this argument to T>Tb>0. Accordingly, a derivation of R(T,Tb) from Planck law in the case Tb>0, appears to be missing in physics literature.
To be able to compute R(T,Tb) a new proof of Planck's law was given in
Computational Blackbody Radiation in
Slaying the Skydragon, a proof which allows direct generalization to Tb>0 with (compare with
previous post):
- RTrue≡R(T,Tb)=∫TTbγTf2df+∫Tb0y(T−Tb)f2df≡I1+I2,
where the first integral I1 is the radiance from the body into the background above the cut-off of the background and the second integral the net radiance below cut-off. Notice that if T≈Tb, then I1≈3I2 and so the above cut-off contribution I1 dominates the radiance.
We see that Extra close brace or missing open brace is the sum of two integrals with positive integrands both expressing radiance from the warm body into the colder background. We should now stop here having reached an expression for RTrue=R(T,Tb) correctly derived from a correct Planck law.
However, climate scientists have introduced an incorrect version of SB named
False-SB, obtainedby rewriting I1 as follows:
- I1=∫T0yTf2df−∫Tb0yTf2df=sigmaT4−sigmaTT3b
which since I2=sigma(T−Tb)T3b gives
False-SB on the (seductively simple) form
expressing the transfer of energy from the body to the background as the difference of two gross flows in opposite directions. We see that
False-SB arises by rewriting an integral with positive integrand as the difference of two integrals with positive integrands as follows:
- ∫TTbf2df=∫T0f2df−∫Tb0f2df,
where the lower integration limit 0 could be replaced by any positive number smaller than Tb.
False-SB arises when giving this formal mathematical manipulation a physical meaning stating that one-way net flow is the difference of two-way gross flows, with the flow from the background in violating with the 2nd law of thermodynamics.
We see that
False-SB arises by writing the correct R(T,Tb) as
where C is an arbitrary positive constant, and then assigning the arbitrary constant C a definite physical meaning as the transfer of heat from the colder background to the warmer body, in violation of the 2nd law of thermodynamics. The only correct choice is C=0 which gives the correct net flow R(T,Tb) as stated above.
False-SB is thus obtained by an ad hoc generalization of SB for Tb=0 in the form R(T,0)=sigmaT4, to the incorrect form RFalse=sigmaT4−sigmaT4b in striking violation of the 2nd law.
Note that the seduction of RFalse is enhanced by the fact that RFalse gives the
correct net flow R(T,Tb), which by proponents of the correctness of RFalse is used as evidence that RFalse is correct. But this only shows that there is an aspect (one-way net flow) of RFalsewhich is correct, while the two-way gross flow suggested by RFalse is grossly incorrect.
If the only meaning of RFalse is net flow equal to R(T,Tb), then RFalse should better be eliminated from the discussion altogether by replacing it with the correct R(T,Tb). DLR should thus be eliminated as fictitious pseudo-physics.
Finally, the difference between RTrue and RFalse comes out as different stability
properties of one-way net flow and two-way gross flow, with RFalse supporting the idea of high climate sensitivity behind CO2 alarmism. If RFalse and DLR is eliminated from the discussion, then CO2 alarmism crumbles.
