måndag 17 oktober 2011

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

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.

1 kommentar:

  1. Claes, I feel that the most important aspect of your work on all this lies in the conclusion that a body will not convert to thermal energy any radiation with frequency below its cut-off. Assuming we are only talking about spontaneous emission (not induced emission, as with lasers) then we have one way transfer of thermal energy from warmer to cooler bodies only.

    I have some trouble in imagining why a molecule of carbon dioxide in the atmosphere would only emit in the direction of cooler air. And what happens if, as in the stratosphere, there is in fact cooler air below it? Does it then emit in the direction of Earth and, if so, does such radiation continue through lower-down warmer air until it reaches the surface?

    Personally, I don't think it matters all that much, so long as we show that no additional warming of the surface will occur. When such radiation reaches the surface it is as good as reflected, the effect being the same. Hold a mirror just above the surface at night and see if it warms it!

    I understand that a gas surrounding an emitter (which is being warmed) will only start to absorb radiation when the emitter becomes warmer than the gas. Clearly this confirms your conclusions, at least for gases, and you may wish to repeat such experiments. Obviously it is easy to detect absorption by gases using spectroscopy. But surely a similar experiment could be carried out with accurate temperature measurements on metal plates. This should be all it takes to demolish the greenhouse effect completely with physical evidence supporting your computations.

    In general, keep it simple! They will always be able to show some radiation from the atmosphere at night - perhaps from warm air pockets as Nasif suggests in his September 2011 paper. If you push this too much they will dismiss everything you say, as some do because of your discussion of the particle nature of photons. Photons can still exist as wave packets, I believe.

    It seems to me these issues may eclipse the main one regarding no heat generation below the cut-off frequency, so I suggest concentrating on that and backing it up with empirical results.

    They may argue that carbon dioxide slows the cooling process, as indeed so does the rate at which warm air rises by convection, but so what? Who cares if the atmosphere from a hundred metres upwards is a little less cold, or if the warmth of the day extends a few minutes longer into the night? None of this is going to effect the local maximum temperatures each day - maybe just the mean of hourly temperatures over each 24 hour cycle.

    One further point I make on my site http://climate-change-theory.com is that it would take far, far more thermal energy to be "trapped" in order to cause any given temperature rise, because of the stabilising effect of underground temperatures. (See my "Explanation" page.) If the surface temperature were to rise and stay higher for hundreds of years or more, then the whole temperature plot from the core to the surface would have to be raised at the surface end. The additional area under this near linear plot represents far more energy than would be required just to warm the land and oceans by the equivalent temperature difference.