This post covers some foundations which are often misunderstood.
Radiation emitted from a surface (or a gas) can go in all directions and also varies with wavelength, and so we start with a concept called spectral intensity.
This value has units of W/m².sr.μm, which in plainer language means Watts (energy per unit time) per square meter per solid angle per unit of wavelength. (“sr” in the units stands for “steradian“).
Most people are familiar with W/m² – and spectral intensity simply “narrows it down” further to the amount of energy in a direction and in a small bandwidth.
We’ll consider a planar opaque surface emitting radiation, as in the diagram below.
The total hemispherical emissive power, E, is the rate at which radiation is emitted per unit area at all possible wavelengths and in all possible directions. E has the more familiar units of W/m².
Most non-metals are “diffuse emitters” which means that the intensity doesn’t vary with the direction.
For a planar diffuse surface – if we integrate the spectral intensity over all directions we find that emissive power per μm is equal to π (pi) times the spectral intensity.
This result relies only on simple geometry, but doesn’t seem very useful until we can find out the value of spectral intensity. For that, we need Max Planck..
Most people have heard of Max Planck, Nobel prize winner in 1918. He derived the following equation (which looks a little daunting) for the spectral intensity of a blackbody:
where T = absolute temperature (K); λ = wavelength; h = Planck’s constant = 6.626 x 10-34 J.s; k = Boltzmann’s constant = 1.381 x 10-23 J/K; c0 = the speed of light in a vacuum = 2.998 x 108 m/s.
What this means is that radiation emitted is a function only of the temperature of the body and varies with wavelength. For example:
Note the rapid increase in radiation as temperature increases.
What is a blackbody?
- absorbs all incident radiation, regardless of wavelength and direction
- emits the maximum energy for any wavelength and temperature (i.e., a perfect emitter)
- emits independently of direction
Think of the blackbody as simply “the reference point” with which other emitters/absorbers can be compared.
The Stefan-Boltzmann equation (for total emissive power) is “easily” derived by integrating the Planck equation across all wavelengths and using the geometrical relationship explained at the start (E=πI). The result is quite well known:
E = σT4
where σ=5.67 x 10-8 and T is absolute temperature of the body.
The result above is for a blackbody. The material properties of a given body can be measured to calculate its emissivity, which is a value between 0 and 1, where 1 is a blackbody.
So a real body emits radiation according to the following formula:
E = εσT4
where ε is the emissivity. (See later section on emissivity and note 1).
Note that so long as the Planck equation is true, the Stefan-Boltzmann relationship inevitably follows. It is simply a calculation of the total energy radiated, as implied by the Planck equation.
The Planck law is true for radiant intensity into a vacuum and for a body in Local Thermodynamic Equilibrium (LTE).
So that means it can never be used in the real world
Or so many people who comment on blogs seem to think. Let’s take a closer look.
The speed of light in a vacuum, c0 = 2.998 x 108 m/s. This value appears in the Planck equation and so we need to cater for it when the emission of radiation is into air. The speed of light in air, cair = c0/n, where n is the refractive index of air = 1.0008.
Here’s a comparison of the Planck curves at 300K into air and a vacuum:
Not easy to separate. If we expand one part of the graph:
We can see that at the peak intensity the difference is around 0.3%.
The total emissive power into air:
E = n²σT4, where n is the refractive index of air
So the total energy radiated from a blackbody into air = 1.0016 x the total energy into a vacuum.
This is why it’s a perfectly valid assumption not to bother with this adjustment for radiation into air. In glass it’s a different proposition..
Local Thermodynamic Equilibrium
The meaning, and requirement, of LTE (local thermodynamic equilibrium) is often misunderstood.
It does not mean that a body is at the same temperature as its surroundings. Or that a body is all at the same temperature (isothermal).
An explanation which might help illuminate the subject – from Thermal Radiation Heat Transfer, by Siegel & Howell, McGraw Hill (1981):
In a gas, the redistribution of absorbed energy occurs by various types of collisions between the atoms, molecules, electrons and ions that comprise the gas. Under most engineering conditions, this redistribution occurs quite rapidly, and the energy states of the gas will be populated in equilibrium distributions at any given locality. When this is true, the Planck spectral distribution correctly describes the emission from a blackbody..
Another definition, which might help some (and be obscure to others) is from Radiation and Climate, by Vardavas and Taylor, Oxford University Press (2007):
When collisions control the populations of the energy levels in a particular part of an atmosphere we have only local thermodynamic equilibrium, LTE, as the system is open to radiation loss. When collisions become infrequent then there is a decoupling between the radiation field and the thermodynamic state of the atmosphere and emission is determined by the radiation field itself, and we have no local thermodynamic equilibrium.
And an explanation about where LTE does not apply might help illuminate the subject, from Siegel & Howell:
Cases in which the LTE assumption breaks down are occasionally encountered.
Examples are in very rarefied gases, where the rate and/or effectiveness of interparticle collisions in redistributing absorbed radiant energy is low; when rapid transients exist so that the populations of energy states of the particles cannot adjust to new conditions during the transient; where very sharp gradients occur so that local conditions depend on particles that arrive from adjacent localities at widely different conditions and may emit before reaching equilibrium and where extremely large radiative fluxes exists, so that absorption of energy and therefore populations of higher energy states occur so strongly that collisional processes cannot repopulate the lower states to an equilibrium density.
Now these LTE explanations are far removed from most people’s perceptions of what equilibrium means.
LTE is all about, in the vernacular:
Molecules banging into each other a lot so that normal energy states apply
And once this condition is met – which is almost always in the lower atmosphere – the Planck equation holds true. In the upper atmosphere this doesn’t hold true, because the density is so low. A subject for another time..
So much for Planck and Stefan-Boltzmann. But for real world surfaces (and gases) we need to know something about emissivity and absorptivity.
Emissivity, Absorptivity and Kirchhoff
There is an important relationship which is often derived. This relationship, Kirchhoff’s law, is that emissivity is equal to absorptivity, but comes with important provisos.
First, let’s explain what these two terms mean:
- absorptivity is the proportion of incident radiation absorbed, and is a function of wavelength and direction; a blackbody has an absorptivity of 1 across all wavelengths and directions
- emissivity is the proportion of radiation emitted compared with a blackbody, and is also a function of wavelength and direction
The provisos for Kirchhoff’s law are that the emissivity and absorptivity are equal only for a given wavelength and direction. Or in the case of diffuse surfaces, are true for wavelength only.
Now Kirchhoff’s law is easy to prove under very restrictive conditions. These conditions are:
- thermodynamic equilibrium
- isothermal enclosure
That is, the “thought experiment” which demonstrates the truth of Kirchhoff’s law is only true when there is a closed system with a body in equilibrium with its surroundings. Everything is at the same temperature and there is no heat exchanged with the outside world.
That’s quite a restrictive law! After all, it corresponds to no real world problem..
Here is how to think about Kirchhoff’s law.
The simple thought experiment demonstrates completely and absolutely that (under these restrictive conditions) emissivity = absorptivity (at a given wavelength and direction).
However, from experimental evidence we know that emissivity of a body is not affected by the incident radiation, or by any conditions of imbalance that occur between the body and its environment.
From experimental evidence we know that the absorptivity of a body is not affected by the amount of incident radiation, or by any imbalance between the body and its environment.
These results have been confirmed over 150 years.
As Siegel and Howell explain:
Thus the extension of Kirchhoff’s law to non-equilibrium systems is not a result of simple thermodynamic considerations. Rather it results from the physics of materials which allows them in most instances to maintain themselves in LTE and this have their properties not depend on the surrounding radiation field.
The important point is that thermodynamics considerations allow us to see that absorptivity = emissivity (both as a function of wavelength), and experimental considerations allow us to extend the results to non-equilibrium conditions.
This is why Kirchhoff’s law is accepted in thermodynamics.
The hilarious paper by Gerlich and Tscheuschner poured fuel on the confused world of the blogosphere by pointing out just a few pieces of the puzzle (and not the rest) to the uninformed.
They explained some restrictive considerations for Planck’s law, the Stefan-Boltzmann equation, and for Kirchhoff’s law, and implied that as a result – well, who knows? Nothing is true? Not much is true?Nothing can be true? I had another look at the paper today but really can’t disentangle their various claims.
For example, they claim that because the Stefan-Boltzmann equation is the integral of the Planck equation over all wavelengths and directions:
Many pseudo-explanations in the context of global climatology are already falsified by these three fundamental observations of mathematical physics.
Except they don’t explain which ones. So no one can falsify their claim. And also, people without the necessary background who read their paper would easily reach the conclusion that the Stefan-Boltzmann equation had some serious flaws.
All part of their entertaining approach to physics.
I mention their papertainment because many claims in the blog world have probably arisen through uninformed people reading bits of their paper and reproducing them.
The fundamentals of radiation are well-known and backed up by a century and a half of experiments. There is nothing controversial about Planck’s law, Stefan-Boltzmann’s law or Kirchhoff’s law.
Everyone working in the field of atmospheric physics understands the applicability and limits of their use (e.g., the upper atmosphere).
This is not cutting edge stuff, instead it is the staple of every textbook in the field of radiation and radiant heat transfer.
Note 1 – Because emissivity is a function of wavelength, and because emission of radiation at any given wavelength varies with temperature, average emissivity is only valid for a given temperature.
For example, at 6000K most of the radiation from a blackbody has a wavelength of less than 4μm; while at 200K most of the radiation from a blackbody has a wavelength greater than 4μm.
Clearly the emissivity for 6000K will not be valid for the emissivity of the same material at a temperature of 200K.