Planck trying to convince a disbelieving Bohr that his proof of Planck’s Law is correct. On the blackboard: Maxwell’s equations.

## Backradiation and Planck’s Law

My recent debate with Roy Spencer and Fred Singer and others about the concept of **backradiation,** which has come to play a fundamental role in CO2 alarmism, illustrates the danger of misunderstanding mathematics and thus misunderstanding physics.

Central in the discussion is the interpretation of Planck’s Law of blackbody radiation expressing the energy radiated from a blackbody B as a function of temperature frequency as

- ,

where is a universal constant. The question centers around the interpretation of the physics supposedly described by Planck’s Law. There are two possibilities:

- B radiates into a surrounding S seemingly independent of the temperature of S. This allows an interpretation with R radiating into S even if S has higher temperature, which is exploited by CO2 alarmists as backradiation causing global warming.
- B and S form a system and B radiates energy into S only if B has higher temperature than S. There is no backradiation, and no ground for CO2 alarmism.

The question is now if Planck’s Law describes 1 or 2. How can we tell? By looking at the proof of Planck’s Law, which I do below. The result of the analysis is the following:

- In Planck’s proof B acts independent of S. It allows an interpretation that B emits into S even if S has higher temperature = backradiation, even if this was not the intention by Planck.
- In my proof B and S form a resonating system. B emits into S only if S has lower temperature.

My conclusion is that 2. with B and S forming an interacting system is closer to actual physics, than 1. with B independent of S which can be misinterpreted as alarming backradiation.

The analysis shows the importance of judging the proof of a mathematical result when interpreting the result. Blind interpretation can lead astray. Planck did not speak about backradiation, but his proof brings in non-physical statistics and thereby opens to a misinterpretation inventing non-physical backradiation.

## Planck’s Proof: Statistics of Quanta

The derivation of Planck’s Law (of blackbody radiation) given by Planck is based on particle statistics borrowed from thermodynamics with little connection to the real physics of radiation as interaction of matter and electromagnetic wave motion.

Planck was forced into particle statistics because a radiation law based on wave motion seemed to lead to a Rayleigh-Jeans law with an “ultraviolet catastrophe” with energy exploding to infinity like with frequency without upper bound. By a statistical assumption that highly energetic waves are rare Planck gave physics a way to avoid the catastrophe which earned almost infinite fame and a Nobel Prize.

But Planck left a question without answer: Is it possible to derive a correct radiation law without resort to statistics, using instead classical electrodynamics described by Maxwell’s wave equations?

## New Proof: Deterministic Finite Precision Computation

In Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation I show that this is indeed possible, by replacing statistics by a much more basic physical assumption of **finite precision computation. **Let me** **here presents** **the essence of my argument:

I consider mathematical wave model of the form:

where the subindices indicate differentiation with respect to space and time , and

- : material force from vibrating string with U displacement
- is Abraham-Lorentz (radiation reaction) force
- is a viscous force
- is exterior forcing,

and and are (small) positive coefficients subject to a frequency dependent switch from outgoing radiation with to absorption/internal heating defined below.

This model with is essentially the starting point also for Planck, before resorting to statistics of quanta: A system of resonators in resonance with an exterior forcing .

The finite precision computation is represented by the viscosity which sets a smallest scale in space of size by damping frequencies higher than .

The coefficients and are chosen so that only one is positive for a certain frequency in a spectral decomposition, with a switch from to at , where is temperature and is a fixed precision parameter representing atomic dimension in the string of atoms modeled by the wave equation.

The switch thus defines a temperature dependent **cut-off frequency** with the effect that only frequencies below the cut-off are emitted as radiation while frequencies above cut-off are stored as internal heat energy of the string.

The essence of the proof is the following energy balance established by a spectral analysis assuming periodicity in space and time:

with is a coefficient of emissivity (= absorptivity), which we write in condensed form as

- , or in a spectral decomposition , or in words
- Radiation + Absorption = Forcing,

where , and .

As just said, the switch effectively means that

- for
- for ,

which expresses that the forcing is remitted as outgoing radiation for , and is absorbed and stored as internal heat energy for .

The coefficient represents emissivity = absorptivity, in accordance with Kirchhoff’s Law. An essential aspect is that is independent of and , which expresses that radiation (to up to the constant ) is independent of the nature of the radiating/absorbing body.

Planck’s law in the form with only one of and non-zero for each frequency is a variant of Planck’s classical law with a sharp switch instead of a continuous transition from radiation to absorption/heating.

Planck’s law in the form thus contains the classical law for radiation as but also the further information that absorption into internal heat energy only occurs for frequencies above cut-off.

The proof that is non-trival and expresses a fundamental aspect of near-resonance in systems of oscillators with small damping. The proof shows Planck’s Law for and Kirchhoff’s Law, without resort to statistics.

The effects of Planck’s statistics in modern physics are subject to an investigation in Dr Faustus of Modern Physics: Is it true that Planck’s technique for avoiding the ultraviolet catastrophe led to a much bigger catastrophe of abandoning rationality in physics?

## Sum Up and Conclusion

Both proofs essential start from the same wave equation model but differ as concerns the mechanism for high frequency cut-off: Planck uses statistics of quanta and I use deterministic finite precision computation.

The difference comes out as a different view of the system of blackbody B and surrounding S.

With Planck’s statististics B is detached from its surrounding and B is supposed to emit whatever statistics decides B to emit irrespective of the temperature of S.

But in my derivation B and S form an interacting system and B only emits energy into S if B has higher temperature than S.

My conclusion is that my proof with B and S forming a system is closer to physics than Planck’s proof with B acting independently of S.

What is your conclusion: System or Not System? To Interact or Not To Interact?

## iceskaterfinland

I cannot see that the issue you are objecting to is something we need to call backradiation.

You are objecting to standard net radiation heat loss curves.

Ie the well estabished practical observation, that If you have a warm surface at 50C it loses heat less rapidly when exposed to a surface warmer than for example 0C or 49C.

Therefore if you place matter at 4k between an object at 298K and the coldness of space at 3k, then the object at 298K is now cooling more slowly, and if it is heated either from within the material due to a temperature gradient or externally heated it will get hotter at the surface.

There is 14.7 pounds of atmosphere above the thin surface layer occupying only one square inch. So even extremely cold temperatures in the atmosphere will reduce the cooling rate of the surface.

What actually are you arguing against? It is not clear to me.

The basic physical observations are well established