From Correct Planck Law to True-SB and False-SB

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:

• $I(f,T)=\gamma Tf^2\theta (\nu, T),\quad \gamma =\frac{2k}{c^2}$

with the high-frequency cut-off factor

• $\theta (f, T)=\frac{\frac{hf}{kT}}{e^\frac{hf}{kT}-1}$,

where $c$ is the speed of light in vacuum, $k$ is Boltzmann's constant, with $\theta (\nu ,T)\approx 0$ for $\frac{hf}{kT}>10$ say and $\theta (f ,T)\approx 1$ for $\frac{hf}{kT}<1$. Since $h/k\approx 10^{-10}$, this effectively means that only frequencies $\nu\le T 10^{11} $ 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 $\frac{kT}{h}$ as the cut-off frequency because frequencies $f>\frac{kT}{h}$ 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

• $I(f ,T) =\gamma Tf^2$ for $f\le T$
• $I(f,T) = 0$ 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 $T$ radiating into a background at 0 K as

• $R(T,0)=\gamma T\int_0^{T}f^2df = \sigma T^4$

where $\sigma =\frac{\gamma}{3}$.

We next seek the radiance $R(T,T_b)$ when the background is a blackbody of temperature $T_b > 0$. Planck proved his law in the case $T_b=0$ by using an argument based on statistics of quanta and it is not clear how to extend this argument to $T > T_b > 0$. Accordingly, a derivation of $R(T,T_b)$ from Planck law in the case $T_b > 0$, appears to be missing in physics literature.

To be able to compute $R(T,T_b)$ a new proof of Planck's law was given in Computational Blackbody Radiation in Slaying the Skydragon, a proof which allows direct generalization to $T_b>0$ with (compare with previous post):

• $R_{True}\equiv R(T,T_b) =\int_{T_b}^{T}\gamma T f^2 df + \int_0^{T_b}\gamma (T - T_b) f^2df \equiv I_1 + I_2$,

where the first integral $I_1$ 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 $T\approx T_b$, then $I_1\approx 3 I_2$ and so the above cut-off contribution $I_1$ dominates the radiance.

We see that $R(T,T_b}$ 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 $R_{True}=R(T,T_b)$ correctly derived from a correct Planck law.

However, climate scientists have introduced an incorrect version of SB named False-SB, obtained by rewriting $I_1$ as follows:

• $I_1 = \int_0^{T}\gamma Tf^2df - \int_0^{T_b}\gamma Tf^2df =\sigma T^4 - \sigma TT_b^3$

which since $I_2 = \sigma (T-T_b)T_b^3$ gives False-SB on the (seductively simple) form

• $R_{False} = \sigma T^4 - \sigma T_b^4$

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:

• $\int_{T_b}^{T} f^2 df = \int_0^{T}f^2df - \int_0^{T_b}f^2df$,

where the lower integration limit $0$ could be replaced by any positive number smaller than $T_b$. 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 $R(T,T_b)$ as

• $R_{False} = (R(T,T_b) + C) - C$,

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 $R(T,T_b)$ as stated above.

False-SB is thus obtained by an ad hoc generalization of SB for $T_b=0$ in the form $R(T,0)=\sigma T^4$, to the incorrect form $R_{False} = \sigma T^4 - \sigma T_b^4$ in striking violation of the 2nd law.

Note that the seduction of $R_{False}$ is enhanced by the fact that $R_{False}$ gives the

correct net flow $R(T,T_b)$, which by proponents of the correctness of $R_{False}$ is used as evidence that $R_{False}$ is correct. But this only shows that there is an aspect (one-way net flow) of $R_{False}$ which is correct, while the two-way gross flow suggested by $R_{False}$ is grossly incorrect.

If the only meaning of $R_{False}$ is net flow equal to $R(T,T_b)$, then $R_{False}$ should better be eliminated from the discussion altogether by replacing it with the correct $R(T,T_b)$. DLR should thus be eliminated as fictitious pseudo-physics.

Finally, the difference between $R_{True}$ and $R_{False}$ comes out as different stability

properties of one-way net flow and two-way gross flow, with $R_{False}$ supporting the idea of high climate sensitivity behind CO2 alarmism. If $R_{False}$ and DLR is eliminated from the discussion, then CO2 alarmism crumbles.