Talk:Planck's law/Archive 7

Terminology for this article
Reading various Wikipedia articles, it's apparent that Wikipedia is as about as muddled in terminology as the literature is. I am clear on a concept, which Chandrasekhar calls "specific intensity", which fully characterizes the radiation field in a medium, without physical surfaces. $$I_\nu(\mathbf{x},t,\mathbf{n},\nu)\,d\Omega\,dA\,d\nu\,dt$$ is the energy passing thru an area element dA into a small solid angle $$d\Omega$$ about the normal to the area element in a small frequency band $$d\nu$$ in a small time interval dt. It has SI units of watts/meter^2/sr/hz. Apparently many people refer to this as "spectral radiance". Can we please settle on a term for this concept, and use it consistently?

Next is the question of a physical surface. What is the term that fully characterizes the radiation coming out of a physical surface element that corresponds to "specific intensity", but is only specified for radiation out of the surface, and is undefined into the surface? It cannot have the same name, because its only like, half of the specific intensity. How do we describe the fact that this radiation is independent of direction, as long as that direction is out of the physical surface? Presently, this article uses "spectral radiance" for this concept (first sentence of article), but that is WRONG if "specific intensity" and "spectral radiance" are the same.

These are questions that need to be settled on before we can write a decent article. In my mind, I use the terms "specific intensity" or "spectral intensity" for the first paragraph above, and "spectral radiance" for the second, in which case the article's first sentence is correct, but I don't care what names are used, as long as they are precisely defined and a case can be made that it is a common definition. PAR (talk) 01:25, 19 November 2011 (UTC)


 * I don't know about specific intensity, but at least for radiance the solid angle element need not be about the normal to the area element. Radiance is in general a function of angle separately from the orientation of the area element through which the light passes. This is essential, because a common application of radiance is to take an element of area that lies on a physical surface, and a solid angle corresponding to light from that surface being received at a detector elsewhere. (Or the other way around—an area element on a detector and a solid angle subtending a source.) Radiance need not be tied to a physical surface, however; it is defined everywhere within a light field. --Srleffler (talk) 01:36, 19 November 2011 (UTC)


 * I think this is saying the same thing. Defining radiance to be the amount through the area, normal to the area is the same as saying the amount through the projection of the area in a direction off-normal. According to what you have said above, then, Chandrasekhar's "specific intensity" is identical to "radiance". But how do we characterize the emission of a physical surface element? The radiation out is what is being described, but the incident radiation is not considered, it is undefined. This cannot be called radiance since radiance is defined for all directions. The article is presently wrong when it say Planck's law describes the radiance of a black body. It defines the "out-radiance" or something. PAR (talk) 02:07, 19 November 2011 (UTC)


 * If I understand it, I think I am in precise agreement with what Sreffler says above about spectral radiance. My reading is limited but I think that specific intensity is identical in every way in physical meaning with spectral radiance. I think they both refer to both actual surfaces and imaginary or purely mathematical surfaces. The difference is only that specific intensity is used in traditional textbooks and by most of our reliable sources, while spectral radiance is a more modern term approved by various committees and official bodies.


 * My observation of the present article is that the word intensity appears once, and all other relevant term usage is of spectral radiance. I did not find a use of "specific intensity" actually present in the article as it now reads. The one use of the word intensity is not in a way that depends critically on what word is used, it being just a loose expression in more or less ordinary language. So far as I can see the present article's consistent usage of spectral radiance is in accord with the proclamations of the committees and officials and would be unobjectionable. If someone felt like putting in a small note that older and more traditional texts use the alternative term "specific intensity" where the present article refers to spectral radiance, I don't think that someone would be out of line.


 * It seems to me that there is no problem here about spectral radiance as the main term to be used in the present article.


 * As to PAR's question about terms for actual physical surfaces as contrasted with imaginary or purely mathematical surfaces, and terms that distinguish between power leaving, and power incident on, a surface. There is a term exitance (with which I am unfamiliar) that doesn't have the directional specificity of spectral radiance (= specific intensity), because it refers to all the power totalled for every direction coming from the surface. But it does distinguish leaving and arriving power. Until the recent round of edits, there was a statement of this quantity for a black body radiating in equilibrium, under a difference name, in the article on Planck's law. I do not see any reason why it should not be stated in the present article if someone feels like putting it there. I don't think that exitance should be a big player in the present article, because it can be easily calculated from the directionally informative spectral radiance. A term with which I am also unfamiliar is radiosity. I think this does refer exclusively to an actual physical surface, because it is defined to include emitted plus reflected power, and I suppose this makes sense only for power leaving actual surfaces. Again it refers to total power without directional information and I think it should not play a big part in the present article, and probably may be unnecessary. For a black body, the reflected power is zero, but for a general body in an equilibrium cavity, the reflected power does not have to be zero, while still the radiosity will equal the irradiance.


 * In summary, it seems to me that spectral radiance is nearly all we need, with perhaps some slight exceptions for future additions, in the present article, and no action is strictly necessary for it at present.


 * As for PAR's concern about the spectral radiance referring to all directions by definition. One might say that the radiation leaving a surface can be described by the spectral radiance evaluated for the senses of direction away from the surface. I think it has to be recognized also that a black body in equilibrium needs energy to supply the energy that it radiates from its surface. Strictly and exactly speaking this can be supplied only in a cavity in equilibrium. In general, a naked body will cool its surface by radiating if all the energy supply has to come from its interior, and its near-surface interior won't be at a uniform temperature unless very special and perhaps unrealistic conditions prevail in its interior. For practical purposes for this article, I think the remedy is to speak only of black bodies in thermodynamic equilibrium, with particular mention of cavity conditions as appropriate. It may be convenient in future to add some information about how closely Planck's law is or is not obeyed by bodies that are not in thermodynamic equilibrium.Chjoaygame (talk) 02:50, 19 November 2011 (UTC)


 * How do you deal with the fact that at present, the article says that Planck's law describes the radiance of a black body, when, in fact, it does not? It only describes the emitted radiance component out of the surface? The radiance at the surface of an opaque body is half-undefined. PAR (talk) 03:03, 19 November 2011 (UTC)


 * One could say that Planck's law describes the electromagnetic radiation emitted from a black body in thermodynamic equilibrium. (One would have in mind that the equilibrium practically requires incoming radiation to keep the temperature of the surface constant. Our object of interest is the radiation, not the terms by which we describe it.) Then one could say that the emitted radiation is described in terms of spectral radiance by such and such formulas. Planck thought about these problems and he stated his law for thermodynamic equilibrium radiation in a cavity with rigid opaque walls that are not perfectly reflective for any wavelength, or in a cavity that contains a speck of carbon and has walls that are perfectly reflective for all wavelengths, or for a black body in thermodynamic equilibrium. To maintain the radiation at the surface of the black body as equilibrium radiation, the radiation in the interior of the black body should in general also be black body radiation.Chjoaygame (talk) 05:07, 19 November 2011 (UTC)


 * Ok, I was hoping there was a specific term for the radiance out of a surface, with the radiance out the other side undefined. PAR (talk) 08:22, 19 November 2011 (UTC)

Hi Chjoaygame - You know I am not big on terminology, and I made a mistake calling $$E=h\nu$$ Planck's relation. I don't think we really need 6 references to correct my mistake, when maybe two good ones will do. It just looks odd. I will be away for a few days, but I will fill in the Einstein coefficient section with references, mostly Chandrasekhar and R&L when I get back. PAR (talk) 17:50, 30 November 2011 (UTC)

Absorptance vs Absorptivity
Chandrasekhar uses the phrase "absorption coefficient", it is wrong to change it to "absorptance". According to Webster


 * ABSORPTIVITY: the property of a body that determines the fraction of incident radiation absorbed by the body


 * ABSORPTANCE: the ratio of the radiant energy absorbed by a body to that incident upon it

Q Science (talk) 17:05, 20 November 2011 (UTC)


 * Exactly right, and what the article is talking about here is the second situation. PAR (talk) 17:49, 20 November 2011 (UTC)


 * I see you restored absorptance, but I don't understand why. The article is clearly talking about the property of a body. Chandrasekhar, 1950, Chapter 1, section 3, defines the "absorption coefficient", the phrase previously in the article, in the same way it was used here. You have not provided a single reference to support your change. Q Science (talk) 20:28, 21 November 2011 (UTC)


 * Chandrasekhar defines the absorption coefficent as the fractional reduction in radiance of a beam as it goes through an infinitesimal distance ds, divided by ds. It has the units of 1/length. What the article is talking about is the ratio of energy absorbed by a surface to the energy incident and it is dimensionless. (add "spectral" and its 1/length/frequency and 1/frequency respectively). I did not reference the statement, I just used the definition taken from the Wikipedia article on absorbance which states "Absorptance[5] (not absorbance) is defined as: The ratio of the radiant flux absorbed by a body to that incident upon it." So supposedly, reference [5] ([Webster's dictionary]) should affirm the statement, which, looking at it now, it does not, except as a specialty definition in the "mining domain". I Googled "absorptance", saw that the first few agreed with the second definition. Looking further, its a mess, bouncing around between the two definitions. You see the problem - the "absorption coefficient" of Chandrasekhar is not the proper descriptor of the ratio of absorbed to incident radiation from the surface of a body. What is the correct term? I'm open to suggestion.


 * We have already thrown up our hands in this article on the distinction between the radiance out of a surface (into: undefined) versus the radiance in a medium (defined in all directions). Are we going to do the same with absorptocitousness or whatever the %$#$%#! term is? Maybe we need to define terms internally in this article, without links to other Wikipedia articles and references, which are basically a conflicting mess. I hate spending half our time arguing over the terminology for a concept we understand. Anybody who thinks they know the terminology cold can come in and create some links and while they are at it, fix every Wikipedia article they link to that is in conflict with their knowledge. PAR (talk) 15:30, 22 November 2011 (UTC)


 * Well, I agree that this is a mess. I also agree that we need a standard set of definitions. However, previous experience here indicates that there will be extreme resistance because "wikipedia is not a dictionary".


 * By the way, I disagree with your interpretation of Chandrasekhar. On page 5, when discussing gases (not opaque objects), he defines


 * mass absorption coefficient - $$\kappa_\nu$$ - such that - $$dI_\nu = -\kappa_\nu \rho I_\nu ds$$
 * Which means that $$\kappa_\nu \rho$$ has dimension 1/length. On page 8, the emission coefficient - $$j_\nu$$ - is defined such that, at local thermodynamic equilibrium
 * $$j_\nu = \kappa_\nu B_\nu(T)_\nu$$
 * Yes, I am confused. Using these definitions, the absorption coefficient is never equal to the emission coefficient. Internet searches have not helped. And we should never trust wikipedia for these types of definitions. Based on what I've seen, it is likely that reliable sources may provide different meanings for some of the terms. Q Science (talk) 05:09, 23 November 2011 (UTC)


 * Yes, you are right about that density term being in there. I think that is referred to as the "mass absorption coefficient" to distinguish it from the "absorption coefficient". Maybe. Also, the bulk "emission coefficient" is not defined analogously to the surface "emissivity". Surface emissivity is the ratio of the emission radiance to that of a black body, while the bulk emission coefficient is just the radiance emitted (divided by density for the case of mass emission coefficient, I suppose). So we would not expect the emission coefficient to equal the absorption coefficient, but rather (emission coefficient)/(black body) = absorption coefficient. The thing about Chandrasekhar is that he is very clear and consistent in his terminology, but the terminology is "old" - not what is currently used in the literature. E.g. he uses "specific intensity" instead of "spectral radiance". Also, he does not deal with surfaces, so emissivity and absor-whatever are not discussed. I would like to eventually add a subsection in "Physics" that does a quick run-through of radiative transfer, illustrating how Planck's law is used in bulk, rather than on surfaces. Chandrasekhar and Rybicki-Lightman seem to more or less agree on terminology here. PAR (talk) 06:30, 23 November 2011 (UTC)

Chandrasekhar in Sections 3 and 4 is talking about propagation of a ray through a semi-transparent medium consisting of material particles. There is differential addition to the spectral radiance of the ray from emission and differential subtraction from it by absorption. For equilibrium, the two balance and the ray proceeds with unchanged spectral radiance.

Considering the case of local thermodynamic equilibrium, Chandrasekhar is indirectly saying that scattering adds as much to the spectral radiance as it removes from it, and can be ignored in favour of considering only true emission and true absorption by material particles. Our article so far covers only the case of actual thermodynamic equilibrium. So for safety, let us restrict our conversation from Chandrasekhar's more general case down to our case of strict thermodynamic equilibrium at temperature $T$.

In the case of thermodynamic equilibrium at temperature $T$, the (true) mass emission coefficient is governed by Planck's law. The differential volume addition to the spectral radiance by emission is $ρ j_{ν}(T) = ρ ε_{ν}(T) B_{ν}(T)$, which defines $ε_{ν}(T)$ , the "true mass emissivity at temperature $T$ " , a quantity not defined by Chandrasekhar, nor by Rybicki & Lightman, and not defined in this way by Mihalas & Mihalas, but defined thus by some textbook writers, such as Bohren & Clothiaux. The mass absorption coefficient is also determined by the fact that the spectral radiance is Planckian, because of thermodynamic equilibrium. The differential volume removal from the spectral radiance by absorption is then $ρ κ_{ν}(T) B_{ν}(T)$. In thermodynamic equilibrium, these two balance, and we have $ρ j_{ν}(T) = ρ ε_{ν}(T) B_{ν}(T) = ρ κ_{ν}(T) B_{ν}(T)$, and thence $ε_{ν}(T) = κ_{ν}(T)$.

While these statements are true, they are not a clear statement of Kirchhoff's law as it is stated in reliable sources. Kirchhoff's article introduced itself by pointing out that it had long been well-known that emissivity = absorptivity, at least for total over all wavelengths for thermal equilibrium. Kirchhoff proposed, on the basis of a proposed theoretical proof that is mostly agreed to be faulty, and without any empirical measurements of spectral radiances, that it was true also for every wavelength.

But his great invention was the concept of the perfect black body with universality of the function $B_{ν}(T)$, and that is why we speak of Kirchhoff's law of thermal radiation, and that is how reliable sources state it.

The account of Kirchhoff's law in the article does not refer to the propagation of a ray as does its cited source in Chandrasekhar, and does not make it very clear that Kirchhoff's contribution was the invention of the concept of the perfect black body with universality of the function $B_{ν}(T)$, the determination of which was finally achieved by Planck.

The account in the article is really about reflectivity and transmissivity at an interface, not absorptivity and emissivity by material particles as it seems to be. The account in the article is really about conservation of energy as expressed in terms of spectral radiance for the reflected and refracted rays, and has nothing to do with Kirchhoff's law of thermal radiation. At the surface of any body, let the reflectivity be denoted by $r(ν, T)$. Then the transmissivity $t(ν, T)$ at that surface will obey $r(ν, T) + t(ν, T) = 1$. The account in the article must rely on the Helmholtz reciprocity principle to balance the moiety of the Planckian ray that enters the body by a moiety of a Planckian ray that leaves it from the interior of the body through the same surface element. No mention of true absorptivity or true emissivity of material particles here. What stands in the article at present is more comparable with Kirchhoff's current law at a node.Chjoaygame (talk) 14:21, 24 November 2011 (UTC)

radiative transfer
The local thermodynamic equilibrium condition defined in the article allows that the local temperature be a function $T(x, t)$ of position $x$ and of time $t$. The article does not define the dependence of the spectral radiance $I_{ν}$ or of the the density $ρ$, but for the general case of space and time dependence, indicated in the definition here of local thermodynamic equilibrium, of $T$ , presumably the spectral radiance and density will also be implicit functions of space and time.

The present demonstration relies on a hypothesis of time stationarity of $T$ and hence of $I_{ν}$ and of $ρ$. The present demonstration eventually requires also a spatial homogeneity of these two variables. Can it be deduced from the time stationarity that there will also be eventual spatial homogeneity of temperature and density? The permitted time variation is not used in the present demonstration. The spatial dependence of $I_{ν}$ is used only for the mass absorption coefficient. Is it perhaps convenient simply to hypothesize eventual spatial homogeneity also of temperature and density, and to leave out dependence of temperature and density on time, making them time stationary by hypothesis? This would justify the notation $dI_{ν} / ds$, and remove a question about the deduction of eventual spatial homogeneity from time stationarity, and scattering can then be deductively set aside because of temporal stationarity and spatial homogeneity.

The subject of the present article is Planck's law. In a sense, Kirchhoff's law is an ingredient for Planck's law. A full derivation of Kirchhoff' law is perhaps not easy to fit into a single paragraph. The present article might well offer a demonstration in support of Kirchhoff's law, and the present section Radiative transfer would serve that purpose, and could be given the title Kirchhoff's law of thermal radiation and a new sequential place, replacing the present section on Kirchhoff's law. It is just a matter of saying that for thermodynamic equilibrium, one has the spectral radiance $I_{ν} (x, T)$ taking the universal value $B_{ν} (T)$, and that yields Kirchhoff's law. The final radiative transfer equation could perhaps be left to the article on radiative transfer, or stay in the Radiative transfer section of this present article in its own right?Chjoaygame (talk) 07:34, 25 November 2011 (UTC)


 * We definitely do not want to go into too much detail here, that should be left to the radiative transfer article. I think we should give an example, and show how the case for surfaces and volumes have the same idea - that equilibrium implies the equivalence of two material properties which then hold even outside of equilibrium. The radiative transfer article starts out with the full radiative transfer equation, time dependence, scattering, non-LTE, and the focus on the use of Planck's law is kind of hard to isolate, which is what we should try to do here. PAR (talk) 17:47, 25 November 2011 (UTC)


 * No one ever should go into too much detail anywhere. That is meaning of 'too much'.


 * Allowing time dependence for radiative transfer is introducing irrelevant detail, and is a distraction in the present section on radiative transfer. Allowing space dependence of density likewise.


 * The cases for surfaces and volumes are not the same. According to Goody and Yung "The treatment of Kirchhoff's laws is nowhere more explicit and readable than in Planck [1913] (our Planck 1914)." In an article on Planck's law, surely this recommendation from reliable source Goody and Yung make Planck 1914 an admirable reliable source for Wikipedia. As Planck is at pains to clarify repeatedly, a surface is only a mathematical object, and to treat it as a material object is an abuse of language, a thing I think not to be indulged in here. Kirchhoff had his perfectly black surface infinitesimally thin and still able to absorb all the heat radiation that fell on it, but Planck did not follow him in that. Planck does not permit surfaces or interfaces to absorb or emit; they only reflect and transmit, not the province of Kirchhoff's law properly understood; only the material interior of bodies emit and absorb. Proper derivation of Kirchhoff's law took a long time to be settled, partly for reason like that. Planck's derivation seems ok for the present purposes, and does not go to the greater generality of local thermodynamic equilibrium, and does refer to absorption and emission of interior material. Chandrasekhar was writing for his own context and was not concentrating on an introductory exposition of Planck's law or Kirchhoff's law as we are doing here; we do not have to follow Chandrasekhar in that respect. If the present article is to talk about Planck's law for non-equilibrium situations such as local thermodynamic equilibrium, a special section should be written about that, in advance of use of it in other sections such as this. We are not here to repair the defects of the article on Radiative transfer.


 * I think that a section on Kirchhoff's law should take precedence here over an section on the application of Planck's law to radiative transfer, because Kirchhoff's law is so essential an ingredient in Planck's law. The present Kirchhoff's law section of the present article is not satisfactory, and should be replaced.Chjoaygame (talk) 23:48, 25 November 2011 (UTC)


 * Please notice the logical development of the Physics section. Each section relies on concepts developed in the previous section. Except for the radiative transfer section which deals with bulk media, its all developed for surfaces. Surfaces are easier to understand than media. One concept that has not been developed is that of optical depth - the distance a light beam travels before being diminished by 1/e due to absorption. The optical depth for black body radiation is zero. (Rybicki & Lightman p 17). That means that the distance a beam travels when it hits a black body is zero, i.e. it is absorbed at the surface. Yes, this never happens in reality, but neither does a black body, nor does black body (or thermal) radiation.


 * I am in favor of keeping the present description of Kirchoff's law for surfaces, even though it is a simple abstraction. Planckian radiation is itself an abstraction. Why drag the reader through the more conceptually difficult theory of radiative transfer right at the beginning? Do the simple stuff first, then radiative transfer, then maybe some notes on the fact that the simple stuff is not as simple as it is first made out to be. We want something that is accessible to a beginner, then get more and more complicated and exact. What changes do you propose to the Kirchoff's law section?


 * The development of the Physics section is not logical. That's why I am commenting on it.


 * Simplicity is not enough; it has to be right as well. The section on Kirchhoff's law may seem simple, but but it doesn't give any hint of what Kirchhoff's law is about. Talk of reflection and transmission as if it were the burden of Kirchhoff's law is not right, simple though it may be; and it does not properly represent the content of its cited reliable source. "Things should be made as simple as possible, but no simpler."


 * (Kirchhoff's black body absorbs all the incident radiation at its infinitely thin surface, but Planck, who had more time to think about it, didn't see it like that. For Kirchhoff, the infinitely thin surface has infinitely large optical depth. This feature of Kirchhoff's black body was one of the difficulties with his proof. I didn't find what you referred to in Rybicki and Lightman on page 17.)


 * Kirchhoff's law asserts the existence of a perfectly black body that characterizes the spectrum of radiation in a suitable cavity in thermodynamic equilibrium.


 * The present section on Kirchhoff's law makes no explicit mention of a black body, and defines the coefficient of absorption in terms of radiative transfer without reference to a black body; instead it just assumes Planck's law. As I have explained above, it then talks about reflection and transmission at an interface as if the interface were described by Kirchhoff's law, when it is only the compositions of the rays that encounter the interface that are directly relevant to Kirchhoff and Planck, being emitted and absorbed by materials as their laws say. Those rays are considered in the section on Radiative transfer (with unnecessary complication by talk about local thermodynamic equilibrium), and that section does indeed throw light on Kirchhoff's law. I think, as I wrote above, that the present section on Radiative transfer should be simplified by removing the unnecessary complication of local thermodynamic equilibrium and sticking to thermodynamic equilibrium, and then should be moved to replace the present faulty section that is called Kirchhoff's law.


 * The existence of the universal thermal radiative spectrum should be emphasized as the content of Kirchhoff's law. The reader needs to be told this. The equality of emission and absorption in equilibrium, even for wavelength-selected moieties of the radiation, having been established by others before him, is not the essence of Kirchhoff's law, especially in an article about Planck's law, which is just an exact expression of the specific character of Kirchhoff's universal function. If it were only a matter of equality of selective emission and absorption in thermodynamic equilibrium, we wouldn't talk about Kirchhoff; we would talk about others. When these things have been fixed, the section on the Black body also needs fixing. Perhaps its messages will be incorparated into the section on Kirchhoff's law, and a separate section on a black body will not be needed.Chjoaygame (talk) 07:54, 26 November 2011 (UTC)


 * First of all, I agree that the LTE stuff is not necessary, and I will remove it in favor of the idea of thermal equilibrium in some neighborhood of a point. I also agree that Kirchhoff's law implies a unique distribution of radiation energy, without specifying what it is. I will fix that as well.


 * However, Kirchhoff's law, as I understand it, and as the Wikipedia article states it, is "At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity". If we further postulate that emissivity and absorptivity are properties of the material, then this relationship holds even in the case when the body and the radiation field are not in equilibrium. It also follows that a "black body" may be defined as one whose absorptivity is unity, and whose emissivity is therefore unity, and therefore it emits the equilibrium radiation, whatever that might be. The idea that Kirchhoff's law has nothing to do with absorptivity and emissivity is far from my understanding of the law and far from the description of the law in the Wikipedia article.


 * Also, the idea that a body and its surface must be rejected as a conceptual device is wrong. If you look at the equation of radiative transfer, the optical depth in a material that is radiating as a black body is zero. That is what R&L state on page 17 when they say that "thermal radiation becomes blackbody radiation only for optically thick media". In other words, a black body radiates from its surface. Also, the conceptual device of a "grey body" that is opaque and emits only a fraction of the black body radiation at a particular temperature should not be discarded. Note that for cavity radiation, the cavity contains no contributing material and so the optical depth is not infinite and is in fact zero. When you look into the small hole of an equilibrated cavity, what you are looking at are the walls.


 * What I hear you saying is that the concept of a black or grey body along with their surfaces should be discarded and Kirchoff's law and Planck's law should be developed in the framework of the equation of radiative transfer in bulk media in one combined section. I could not disagree more. Emission and absorption from a surface is easier to understand, and should be introduced first. Radiative transfer is conceptually more difficult and should be introduced afterwards.


 * We have some work to do here.


 * A fundamental physical point needs to be considered. The clue is given by the sentence in the Kirchhoff's law section, that reads "It is generally true that the emissivity and absorptivity are properties of the material only, so that this equivalence will hold even when the radiation field is not thermal radiation." The true physics is not exactly so. Planck 1914 did not understand this, but I may yet learn that he did eventually find it out. Its understanding had to wait till Einstein's 1917 theory of A and B coefficients. That theory tells how the absortptivity depends not only on the properties of the material but also on the intensity of the field. When the field intensity is very great, the absorptivity becomes small, and under some conditions may be negative, and this is how lasers work. The condition of thermal equilibrium is the one that makes the ratio of the emission coefficient to the abosrption coefficient equal to the corresponding ratio for a unique universal black body. Chandrasekhar didn't know of the real possibility of lasing in 1950, and this point is not emphasized in his account, which refers to local thermodynamic equilibrium, far from lasing conditions.


 * Kirchhoff and Planck define the coefficient of absorption on the assumption that it is independent of the intensity of the radiation field. So far as I at present understand, although they used the equilibrium condition in their arguments, they did not know that equilibrium is necessary for that independence. The coefficient of absorption is independent of intensity in the neighbourhood of thermodynamic equilibrium. This is because thermodynamic equilibrium is a maximum entropy state, and there is a horizontal tangent there. Far from equilibrium, the coefficient of absorption depends significantly on the intensity of the radiation field. The radiation field can drive the populations of states so that stimulated emission dominates and makes the coefficient of absorption negative. This was hinted at by the Einstein A and B coefficient theory, but even so it was not fully understood in the early days of that theory, I think.


 * Stimulated emission is, like absorbed radiation, proportional to the amount of radiation. The amount of stimulated emission, or absorption, divided by the incoming radiation is more or less constant for a given material. We don't need to go into this in detail, unless we have a short section on Einstein coefficients, which might be a good idea. Again, the stimulated emission coefficient is just combined with the "true" absorptivity, and off you go. Yes, if you drive things to the outer edges of non-equilibrium, none of the theory is correct. There are non-linear effects, lasing, etc, but we don't want to burden the reader with these except perhaps to quickly and clearly state the conditions of applicability for which the results are valid. I have worked with highly non-equilibrium radiation fields, consisting of a few atomic lines, and the theory works fine, emissivity=absorptivity. PAR (talk) 23:10, 26 November 2011 (UTC)


 * Thermodynamic equilibrium is key to Kirchhoff's law. True, it is very nearly obeyed when the conditions are very nearly in thermodynamic equilibrium, namely in local thermodynamic equilibrium; but that is not the first thing that the reader needs to know. It was not understood till Milne 1928. Goody & Yung make it clear on page 31: "We may, therefore, regard Planck's and Boltzmann's laws as interchangeable; conditions leading to one lead to the other, and vice versa."


 * Also key to Kirchhoff's law is that perfect blackness is enough to uniquely define the emission spectrum of a body in thermodynamic equilibrium. That is saying essentially more than that emission is equal to absorption. In other words, there is only one kind of black radiation. This is a non-obvious, non-trivial statement. Kirchhoff (Guthrie) writes about the invisible radiation of heat: " ... it being, however, assumed that bodies only emit rays of one kind. ... in many cases in which the homogeneousness of the emitted rays could at least be so far assumed, inasmuch as they were all invisible. Whether the same law applies when bodies emit rays  of different kinds (which, strictly speaking, is always the case), has never hitherto been determined theoretically or by experiment." He was unaware of Balfour Stewart's experiments with "sifted" (= selectively filtered) rays. Stewart would have done better if he had checked other kinds of black material, as well as lamp-black, as references for his definition of absorbing power. Another way saying things is that Stewart's definition of absorbing power would give the same comparative results for every kind of black radiation, lamp-black, platinum black, etc., etc.. Kirchhoff saw the need to prove this, not just assume it.


 * Perfect blackness is a special case of Kirchhoff's law - absorptivity =1, emissivity=1. It says less than Kirchhoff's law. PAR (talk) 23:10, 26 November 2011 (UTC)


 * "Thermodynamic equilibrium in the neighbourhood of a point" is a dubious concept. Is it different from local thermodynamic equilibrium? The radiation field at a point in a semi-transparent medium of the kind that Chandrasekhar is referring to is largely determined by material sources that are more or less distant, that is to say, not in the neighbourhood of the point.


 * Material thermodynamic equilibrium in the neighbourhood of a point is the condition for LTE at that point, it is not a dubious concept. The differential form of the equation of radiative transfer is not determined by conditions that are distant, it is entirely local. The solution to the equation of radiative transfer does involve conditions along the entire length of a beam, and is not a local statement. If LTE exists along the entire beam, then the solution is greatly simplified, but we are not talking about solutions, only to the local differential statement, and so we don't need to get all entangled in the idea of LTE over an extended region. PAR (talk) 23:10, 26 November 2011 (UTC)


 * The fractionation of absorption into a positive part and a negative part (stimulated emission) was not known to Kirchhoff and not known to Planck 1914 and should be treated separately, not incorporated right here.


 * No, it is almost always incorporated into an effective absorption coefficient because it behaves like absorption - proportional to the power of the driving beam. PAR (talk) 23:10, 26 November 2011 (UTC)


 * It is commented above: " However, Kirchhoff's law, as I understand it, and as the Wikipedia article states it, is "At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity". " Wikipedia is not a reliable source. In this case I think it may in some sense be a circular reference.


 * That statement is not an adequate statement of Kirchhoff's law of thermal radiation, as may be checked by reference to reliable sources. I did not write that 'Kirchhoff's law has nothing to do with absorptivity and emissivity.' I wrote that: "The equality of emission and absorption in equilibrium, even for wavelength-selected moieties of the radiation, having been established by others before him, is not the essence of Kirchhoff's law." Kirchhoff took it as long known that in equilibrium, emission and absorption, in total over all wavelengths, were equal. Stewart showed experimentally that they were equal for "sifted" qualities of thermal radiation in equilibrium. Kirchhoff reasoned that for equilibrium thermal radiation they must be equal for every wavelength and that this was associated with the uniqueness of the black-body spectrum; the latter was his special contribution, and this is reflected in the statements of his law in reliable sources, though not always made quite clear.


 * I really disagree - How would you state Kirchhoff's law? PAR (talk) 23:10, 26 November 2011 (UTC)


 * It took Planck 1914 to make it quite clear. Planck 1914 remains the best reliable source, according to Goody & Yung page 64, themselves a reliable source. Kirchhoff's original arguments were much criticized, and Planck's 1914 argument for Kirchhoff's law is a valuable resource. Planck 1914 himself says that Pringsheim (1901) "greatly simplified" Kirchhoff's proof. Argument went on after Pringsheim, especially from Hilbert.


 * I did not write that 'the concept of a black or grey body along with their surfaces should be discarded'. For cavity radiation I think it necessary to think of bodies of colour, including grey. I see no special place for talk of a specifically grey body here.


 * I mean grey at a particular wavelength. Yes, we should consider varying shades of grey at various wavelengths, in other words "colored". PAR (talk) 23:10, 26 November 2011 (UTC)


 * A black body is defined as one that absorbs all the radiation that falls on it. This is not identical with the statement that its absorption coefficient is unity. Argument is needed to show their equivalence.


 * Are you referring to the complication of stimulated emission? If not, then the two statements are in fact identical. How do you see them as different? PAR (talk) 23:10, 26 November 2011 (UTC)


 * On Kirchhoff's law, Planck 1914 supported by Goody & Yung is a preferred reliable source, and outweighs the presentation by Rybicki & Lightman. Planck does not allow that a 'surface' can emit or absorb, although Kirchhoff nearly did so; but his 'surface' was 'infinitely thin', not a true mathematical surface. Planck's 1914 treatment is preferable and is not hard to understand, and has the advantage of being physically realistic and useful for theoretical reasoning and for applications. The optical depth of Kirchhoff's infinitely thin black 'surface' is infinitely great.Chjoaygame (talk) 21:28, 26 November 2011 (UTC)
 * Chjoaygame, would you by any chance have a one-sentence summary of your point? And also a one-sentence summary of the point of the Kirchhoff section, to fit into the appropriate part of the article without creating a whole section?
 * What percentage of those coming to this article do you feel are likely to care about your personal, deep, philosophical, and unsourced insights into the importance of Kirchhoff's work for the much later Wien-Rayleigh-Jeans-Planck-Einstein work? And what percentage will judge you by them?
 * Apologies for any harshness there, but between you and Headbomb I'm really getting fed up with this article, which is unlike any other I've had to deal with in my past 5 years of Wikipedia editing, averaging around three edits a day. If and when I get the time I will be completely agreeable to any proposal to take both Headbomb and Chjoaygame to mediation together.  Although HB is more disruptive it seems to me that Chj is a close second.  --Vaughan Pratt (talk) 05:12, 28 November 2011 (UTC)
 * An emitting layer can only contribute to the emission at its surface from a depth of a few times the optical depth. The emitting layer of a black body has infinite optical depth, so the contribution comes from zero thickness. What is the problem? PAR (talk) 23:10, 26 November 2011 (UTC)
 * I wish you were editing this article, PAR, and not Chjoaygame and Headbomb. --Vaughan Pratt (talk) 05:12, 28 November 2011 (UTC)

Why not just mention the principle of detailed balance? Count Iblis (talk) 20:08, 28 November 2011 (UTC)
 * I agree, a SHORT section on Einstein coefficients would be useful. PAR (talk) 20:28, 28 November 2011 (UTC)
 * I added a section on Einstein coefficients, hopefully enough to show the use of Planck's law in their derivation without going into non-Planckian detail. PAR (talk) 05:20, 30 November 2011 (UTC)
 * I think you used a copy and paste, but then changed the wrong values for absorption. Q Science (talk) 06:45, 30 November 2011 (UTC)
 * I think its fixed, could you check to be sure? PAR (talk) 07:14, 30 November 2011 (UTC)


 * Thanks. I wasn't confident making that edit myself since this now "disagrees" with Atomic spectral line unless you assume that
 * $$\left(\frac{dn_1}{dt}\right)_{B_{12}}=-\left(\frac{dn_2}{dt}\right)_{B_{12}}$$


 * (which seems logical). Also, it should be made clear that all 3 constants depend not only of the atom (or molecule), but also on the frequency. Q Science (talk) 00:44, 1 December 2011 (UTC)

approach to thermodynamic equilibrium
Dear Waleswatch, I was wondering how long it would take for this to spread to the Planck's law article. I am not going to enter into an editing contest with you, of course. But I may point out that for thermodynamic equilibrium of electromagnetic radiation to be reached in the time frames of ordinary laboratory experiments, matter is necessary as a transducer between wavelengths. True, as PAR lovingly points out, one can think of photon-photon interactions being another mechanism for approach to thermodynamic equilibrium. But even he admits that we are then talking about a time frame on the order of the age of the universe, utterly irrelevant to ordinary laboratory experiments. PAR came up with some data showing that photon-photon interactions can be detected when gamma rays are collided. But for laboratory experiments, the thermal production of gamma rays is fantastically small, and the mechanism would be irrelevant. Moreover, these photon-photon interactions result in the production of electrons, which are ponderable matter, so that the ponderable matter thing hasn't really been expunged. And if we are considering letting electrons be produced, presumably protons and other particles will come into the picture. It gets more and more fantastic. And the story of gravitational thermalization, without matter is fantastic too; the gravitational attraction of photons to the sun is small enough, but gravitational attraction of photons to each other?

I have to say I think this business of expunging the material connection as you have just done is more a matter of you and PAR proving how clever you are than of considering the needs of an average Wikipedia reader. The reality is that you haven't even read Planck's account of his own law, which is the right account for laboratory conditions, and is basic physics that even an expert in quantum field theory ought to have as background knowledge superseded by his quantum field theory, and here you are forcing your quantum field theory or whatever on the article about Planck's law, when it is admitted that it will only be relevant on a time scale fantastically far from laboratory conditions.

I would like to ask you to reconsider what you have just done.Chjoaygame (talk) 00:13, 24 January 2012 (UTC)


 * The time scale for thermalization depends on all sorts of factors - for example, the initial energy distribution. If the energy scale (defined by the equilibrium temperature, say) of the photons in our perfectly reflecting cavity without any other matter is of order MeV or more, the thermalization time will be very short.  If the energy scale is small, it will be significantly larger.  At what point it gets to the age of the universe, I'm not sure - have you done that calculation?


 * Regardless of the time, if the photons are in thermal equilibrium, then they constitute a Planck spectrum. That's all the article says, and it's quite correct.  You seem to be concerned over how the photons got that way - but that's really a separate issue, particularly when we're discussing idealizations like perfectly reflective cavities.  It's true that in a laboratory experiment the fastest source of thermalization will be from either interactions with the walls of the cavity or perhaps atoms or molecules inside it.  But it's not true that such interactions are necessary, and inserting language about specks of "black carbon" is both unnecessary and quite confusing to the reader.  Waleswatcher  ( talk ) 00:33, 24 January 2012 (UTC)


 * Chjoaygame - The point is not how small or large the interaction is. The point is that equilibrium requires a thermalizing process in order to occur, and that there is no fundamental physical principle which restricts this thermalizing process to photon-matter interactions alone. To say or imply that material MUST be present as a matter of principle is simply wrong, and we should not leave the reader with that impression.


 * What do you think is the best way to assure that the reader is not left with this erroneous impression?PAR (talk) 03:32, 24 January 2012 (UTC)

Citation needed for Planck's views on the quantization of radiation
The article states: "He plainly explains[74] in his book "Theory of Heat Radiation" that his constant refers to hypothetical material Hertzian oscillators. The idea of quantization of the free electromagnetic radiative field was developed by others and was eventually incorporated into what we now know as quantum mechanics."

As far as I can see, the first sentence quoted above is highly misleading, and the second requires a citation, and I think overstates the facts. Planck states in his book that his oscillators can have any energy, but that they always emit radiation in quantized bunches. He even includes a plot, Fig. 7 on p.161 of "Theory of Heat", that shows a continuous increase in energy of his oscillators, that then emit in quantized bunches. In words, he says "we have...the following law: The oscillator emits in irregular intervals, subject to the laws of chance; it emits, however, only at a moment when its energy of vibration is just equal to an integral multiple n of the elementary quantum e =hv, and then it always emits its whole energy of vibration ne." Looking for example at p.186, you can see he is one small step away from proposing that the modes of the electromagnetic field themselves (as opposed to just the emission of them) are quantized, but he doesn't quite take it.  Waleswatcher  ( talk ) 13:46, 15 February 2012 (UTC)

Kuhn citation for Planck's views on the UV catastrophe
In The Theory of Heat, Planck comments that the Rayleigh-Jeans law would lead to an infinite amount of radiation. Granted that was written in 1914, but if Planck's views on that changed over time that should probably be noted. Moreover, I read pages 151-152 of Kuhn, and I don't really see how they support the language in the article they are attached to. Chjoaygame, can you comment?  Waleswatcher  ( talk ) 00:09, 20 February 2012 (UTC)


 * The colourful term "ultraviolet catastrophe" was invented by Paul Ehrenfest in 1911. This invention is mentioned in the section of the article headed 'Subsequent events'. This heading means to indicate that there was a time lapse from (a) the discovery and initial explanatory efforts mentioned in the just previous section headed 'Trying to find a physical explanation of the law' to (b) the subsequent events. The section on subsequent events mentions a time lapse of five years, and then a development of Planck's thinking by 1908 and another by 1912 and another by 1919. There is in Kangro's rather detailed history of events no mention of any kind of catastrophe, and Kuhn writes on page 151: "Planck's model, in contrast, seemed free of such difficulties. On page 152 Kuhn writes: "The Rayleigh-Jeans law and what came to be called the "ultraviolet catastrophe" did not yet pose problems for more than two or three physicists." It seems reasonable to read this as meaning that Planck was not one of those two or three in those early days. The present Wikipedia article uses the past perfect tense: "But this had not been part of Planck's thinking, because he had not tried to apply the doctrine of equipartition: he had not noticed any sort of "catastrophe"." This indicates that the thinking of Planck that is referred to was prior in time to the invention of the colourful term by Ehrenfest, and was not in terms of a "catastrophe", specfically marked with quotation marks in the article. The word "catastrophe" does not appear in Planck 1914.


 * The Wikipedia article, of which I was unaware until I looked it up just now, on the Ultraviolet catastrophe has a section that reads as follows.


 * Historical inaccuracies 


 * Many popular histories of physics, as well as a number of physics textbooks, present an incorrect version of the history of the ultraviolet catastrophe. In this version, the "catastrophe" was first noticed by Planck, who developed his formula in response. In fact Planck never concerned himself with this aspect of the problem, because he did not believe that the equipartition theorem was fundamental – his motivation for introducing "quanta" was entirely different. That Planck's proposal happened to provide a solution for it was realized much later, as stated above.
 * Though this has been known by historians for many decades, the historically incorrect version persists, in part because Planck's actual motivations for the proposal of the quantum are complicated and difficult to summarize to a lay audience.


 * It is common enough to read in unreliable sources implications that Planck's discovery was motivated by a concern to remedy the ultraviolet catastrophe.


 * For example, at http://physics.about.com/od/quantumphysics/a/blackbody_2.htm, we read: "In 1900, the German physicist Max Planck proposed a bold and innovative resolution to the ultraviolet catastrophe" and "Planck’s solution to the ultraviolet catastrophe is considered ..."


 * And at http://spiff.rit.edu/classes/phys314/lectures/planck/planck.html, we read about "Planck's solution to the Ultraviolet Catastrophe" and that "Max Planck was aware of the "Ultraviolet Catastrophe"."


 * And at http://www.egglescliffe.org.uk/physics/astronomy/blackbody/bbody.html, we read: "The failure of these formulae to account for the decrease in energy emitted at short wavelengths (the ultraviolet wavelengths) became known as the ultraviolet catastrophe. A major breakthrough was made by Max Planck who made a formula that agreed with experimental data, ...", as if Planck was working in an environment that spoke of an "ultraviolet catastrophe".


 * So it seems to me reasonable, in context, that our present Wikipedia article explicitly deny the suggestion that Planck at the time of his discovery was concerned about a "catastrophe".Chjoaygame (talk) 08:17, 20 February 2012 (UTC)
 * I think you're probably right, but I don't think the Kuhn reference establishes it at all clearly. In any case, by 1914 Planck seems to have known of this (it's not called "catastrophe", but he comments on it).  So I think we need to make especially clear that the article is discussing his thinking around the time of 1901.  Would you object to adding something like "at the time of XX paper" or "circa 1901" or something like that?  Waleswatcher  ( talk ) 11:49, 20 February 2012 (UTC)


 * No objection to that. Done.Chjoaygame (talk) 14:19, 20 February 2012 (UTC)

Is this correct for angular frequency?
Starting with:


 * $$B_\nu(T) =\frac{ 2 h\nu^{3}}{c^2} \frac{1}{e^{h\nu/(k_\mathrm{B}T)} - 1}$$


 * $$B_\omega(T) =\frac{ 2 (2 \pi \hbar) \left(\frac{\omega}{2 \pi}\right)^{3}}{c^2} \frac{1}{e^{\hbar \omega/(k_\mathrm{B}T)} - 1}$$

Doesn't that leave a 2 and a π2 in the denominator as in:


 * $$B_\omega(T) =\frac{ \hbar\omega^{3}}{2 \pi^2 c^2} \frac{1}{ e^{\hbar \omega/(k_\mathrm{B}T)} - 1 }$$

not


 * $$B_\omega(T) =\frac{ \hbar\omega^{3}}{4 \pi^3 c^2} \frac{1}{ e^{\hbar \omega/(k_\mathrm{B}T)} - 1 }$$

? I'm tempted to fix this, but there are two other equations with ħ instead of h. Can someone check this and fix it please? I will take it to the expression at Planck units if it gets fixed here. 71.169.186.145 (talk) 22:12, 29 February 2012 (UTC)


 * See the remark just under the table, about converting Planck's law between different spectral units. Headbomb {talk / contribs / physics / books} 22:18, 29 February 2012 (UTC)


 * Okay, so we mean intensity per rad/sec vs. Hz, right? Looks like I might have jumped the gun a little at Planck units. 71.169.186.145 (talk) 22:47, 29 February 2012 (UTC)

Correct symbol for spectral radiance is L, not B
Most of the references I can find, as well as the IUPAC Gold Book and Wikipedia itself, use the symbol L for spectral radiance. Robert Hiller (talk) 02:20, 3 March 2012 (UTC)


 * Yes, except for the spectral radiance of a black body. PAR (talk) 04:19, 3 March 2012 (UTC)


 * I think that the Wikipedia is for reporting things, not arbitrating or deciding what is "correct". It is nowadays conventional to use the letter $L$ for spectral radiance, but I don't think it is appropriate for Wikipedia editors to say that it is "correct" to do so, even in a talk page header. This is without prejudice as to whether the article should use $L$ or $B$.
 * It so happens that I agree with PAR that the better symbol for this article is $B$.Chjoaygame (talk) 06:10, 3 March 2012 (UTC)


 * B is the usual symbol for Planck's law, just as Γ is the usual symbol for the gamma function. L is the usual symbol for spectral radiance as the value of B at any given point of its domain, just as x,y are the usual symbols for a real as the value of Γ at any point of its real domain, and z as the usual symbol for a complex number as the value of Γ at any point of its complex domain.
 * My impression was that B originally stood for brightness, but that these days brightness as a radiation physics term is deprecated as per FS-1037C, so today it should just be considered the usual symbol for Planck's law. It may or may not be a coincidence that B also stands for both black and body.  --Vaughan Pratt (talk) 18:44, 3 March 2012 (UTC)

reason for undoing good faith edit
The notation is not perfect, and is unlikely to become so. The edit that I undid was well intended and in some respects reasonable, but was not enough to make an actual improvement in the article. It was an attempt to prescribe for the article a particular notational convention as if it were somehow universal.Chjoaygame (talk) 16:27, 27 April 2012 (UTC)

Spectral Radiance as function of wavelength
Isn't B(lambda) wrong? hc^2/lambda^5 units are J/(m^3 s). Shouldn't it be hc/lambda^3 instead? — Preceding unsigned comment added by 194.65.138.120 (talk) 22:31, 30 April 2012 (UTC)

No, those are the correct. B(lambda) has units of energy/(time*area*wavelength).  Waleswatcher  ( talk ) 02:34, 1 May 2012 (UTC)


 * This apparent strangeness (c^2/lambda^5 where one might expect c/lambda^3) arises from directly substituting wavelength for frequency in a derivative, namely spectral intensity, that has been taken with respect to frequency. One fix (not necessarily the best but it works) is to first undo the derivative by integrating $$\nu^3/c^2$$ to give the radiant intensity $$\nu^4/4c^2$$.  Substituting $$c/\lambda$$ for $$\nu$$ then gives $$c^2/(4\lambda^4)$$, still radiant intensity, whose derivative, now with respect to $$\lambda$$, is $$-c^2/\lambda^5$$ (back to spectral intensity, but with the sign changed reflecting the fact that $$\lambda$$ decreases with increasing $$\nu$$).


 * Incidentally the article used to draw these distinctions about spectral and radiant intensity. Did some editor find them too weird and confusing for this article? My bad, for "intensity" read "radiance" in the preceding paragraph.  Intensity refers to the radiation from the entire radiator, radiance is area density of intensity (quotient of intensity by area of radiator).  --Vaughan Pratt (talk) 09:10, 2 May 2012 (UTC)

Excessive complexity, nonstandard notation
Planck's law takes two distinct forms $$B_\nu$$ and $$B_\lambda$$, depending on whether it is expressed as a function of frequency or wavelength. Note that wavelength is not simply a different unit for frequency based on a change of scale but varies with the reciprocal of frequency, resulting in two qualitatively different laws having distinctly different shapes.

Six months ago Wikipedia invented the symbols $$B_{\tilde\nu}$$, $$B_\omega$$, $$B_y$$, and $$B_k$$, apparently by analogy with the notations $$B_\nu$$ and $$B_\lambda$$, in order to have separate laws for each of certain scales for each of frequency and wavelength, some in common use (e.g. cm&minus;1 instead of Hz for frequency) and others much less so.

The counterpart of this for Newton's law F = ma would be to have various subscripts for F and m according to the units used for force (newtons, poundals, etc) and mass (kilograms, grams, pounds, ounces, etc.). This is unprecedented: physics is never organized that way in standard physics texts used in accredited physics courses.

That these notations are nonstandard can readily be seen from the article's sources supporting the annotation "Planck's law expressed in terms of different variables" attached to the article's tabulation of these four trivial variants of the two basic forms of Planck's Law. The three sources are (i) Caniou (the English translation of a French engineering monograph about principles of, materials for, and applications of passive infrared detection devices, Amazon rank 2,832,875, price $381, i.e. ultraobscure); (ii) Kramm and Moelder (a paper by two climate scientists originally written for an Indian math journal that no longer appears on its authors' respective publication lists -- only the unpublished arXiv version seems to exist); and (iii) Sharkov (a monograph on passive remote microwave sensing of the Earth which Amazon ranks 3,433,164 at $345, i.e. at least as obscure as Caniou).
 * None of these sources can be considered representative of how Planck's law is customarily taught in accredited physics courses.
 * With the exception of Caniou, none of them tabulate a wide range of formulas for different units.
 * None of them use these Wikipedia-invented notations.
 * The fact that these were the best available sources highlights the inappropriateness of these invented notations.

The only versions of Planck's law needed for this article are those for the two forms of Planck's law, which is how the article was organized prior to the introduction of the six forms of Planck's law. Following universal practice in physics for formulas, these two forms should be given independently of the choice of units, just as F = ma is invariably given as a law that is independent of units. (It should however be noted that the differentials $$d\nu$$ and $$d\lambda$$ must be considered part of the respective formulas for the purpose of changing units, since the differentials themselves have dimensions of respectively frequency and length and therefore participate in that scale change.)

What support is there for continuing to introduce Wikipedia readers to Planck's law in the form of a table of six formulas, as done currently? --Vaughan Pratt (talk) 23:17, 3 May 2012 (UTC)


 * The table was introduced by a particular editor who does not right now need to be named. It was a trojan horse by which he bluffed or bullied people into allowing his fantastic inventions. The table was used because it has undesirable elements of one-size-fits-all inflexibility which suited the trojan horse purpose. Your criticisms of the referencing of the table are justified. I would like to see the table go.


 * On the other hand I do not accept your rejection of the wavenumber formulation. It was not invented by Wikipedia. One editor wrote that it was common in his line of work. True, it is notationally changed from its sources to conform with the present Wikipedia notation, but that is not "invention". The wavenumber formulation is explicitly stated in what I think is accepted as a reliable source, Paltridge and Platt 1976, and in Caniou 1999, which I think would qualify as reliable (there is no requirement for non-ultraobscurity, and the reference is directly linked to a valid internet source), though, as you say, in different notations.


 * The literature presentations of the formulas are not in some universal unique notation. It is up to Wikipedia to use what best suits it. I do not agree with the idea that the Wikipedia should limit itself to what is customarily taught in accredited physics courses.


 * Before the aforementioned unnamed editor's efforts, the article was not limited to two forms of the law. I think it reasonable to allow other forms, in particular forms for energy density, which occur commonly in the literature. And perhaps others.Chjoaygame (talk) 00:46, 4 May 2012 (UTC)
 * Excellent, 2 votes (counting mine) for eliminating the table, so far 0 for keeping it. Let's see how this progresses.
 * The possibility Chjoaygame raises of Planck's Law having more than two forms surely depends on the definition of "form." I would say that the unit sphere has the same form in MKS and CGS units, though obviously not the same size, but I'm open to alternative views on this question (though I notice that the sphere article does not list these separately).
 * There is no requirement for non-ultraobscurity. Let me recommend WP:RS as an excellent read on this topic.  Without that guideline Wikipedia editors would be able to cite the most ridiculous sources!  --Vaughan Pratt (talk) 03:58, 4 May 2012 (UTC)

I'm not sure what the problem with the table is; at first glance, it appears to be in agreement with the cited source. What is it a Trojan horse for? Dicklyon (talk) 04:29, 4 May 2012 (UTC)
 * There's a considerable body of literature on Planck's law in the mainstream physics literature, including all the material standardly used in teaching Planck's law. How come Wikipedia has to go to a very obscure engineering monograph on passive infrared detection as its sole support for the idea that Planck's law needs a long table of trivial variants of the two standard formulas?  The mainstream texts that treat Planck's law don't do this; for one thing it forces readers to plow through tons of formulas only to discover at the end that they're all describing the same thing at different scales.  How can that be considered a clear exposition of Planck's law?
 * Regarding the "Trojan horse" Chjoaygame was referring to, he may have had in mind that HB had embarked on a major edit war in order to prevent the reversion (by at least two editors) of his replacement of k by kB.  HB came up with the idea of introducing the angular wavenumber k in order to create a notational conflict within the Planck's law article with Boltzmann's constant k so as to justify his kB replacement.  (Personally I'd be fine with kB in place of k if that were standard in Wikipedia, but it's not, see e.g. Boltzmann's constant.)  HB devised his massive rewrite of the article on October 13 for no other purpose than to justify his preferred notation for Boltzmann's constant.  This can all be verified (by those with copious free time) by following the long history of the article step by step during that period, along with this supplementary discussion.   --Vaughan Pratt (talk) 05:25, 4 May 2012 (UTC)


 * One cannot read others' minds. Maybe Vaughan Pratt is right, but I am not sure why the unnamed editor wanted to introduce the form of Planck's law that takes reduced wavenumber $k$ as independent variable. But I think it's pretty clear that that's why he introduced the table.


 * I think there is a good case for prudent selection of reliable sources, not just any source will do; that's why the sources are required to be "reliable".Chjoaygame (talk) 06:26, 4 May 2012 (UTC)


 * It is very clear - the person wanted to force us to change the Boltzmann constant from k to kB. Everything in his arguments was driven by that single goal. I support any change that fixes that.


 * As for $$\nu$$, in most references this represents frequency, therefore $$\tilde\nu$$ is used for wavenumber. When writing programs, it took a long time to get the equations correct because various references used the symbols differently. Therefore, I think it is useful to keep the 3 forms - wavelength, frequency, wavenumber. Q Science (talk) 13:27, 4 May 2012 (UTC)
 * The accusation that I wanted to force the use of kB over k for Boltzmann constant is one of the most ridiculous things I've head on Wikipedia. As for which variable to use for the spectral radiance, it matters little. I prefer I, but the article uses B for some reason. There's no point in changing variable names when changing the spectral variable. That would just confuse the reader for no reason to read e.g. B&nu; vs. Ik vs B(&omega;) instead of being consistent in notation and use B&nu;, Bk and B&omega;. Consistency is key, not to mention doing things like B(&lambda;) vs B(&nu;) would be mathematically misleading, as B would not be the same function. Headbomb {talk / contribs / physics / books} 15:11, 4 May 2012 (UTC)


 * The nu-tilde is for wavenumber in cycles, and the k for wavenumber in radians, usually. Some sources do use the subscript B on Boltzmann's constant to distinguish it from wavenumber (see ).  And I think the table is actually quite useful, to all a comparison of all the various ways of writing the same law.  Some sources have only one, two, or three of the variants, but a few have a table like that.  In articles that only have one or two forms, we constantly see editors changing the exponents on pi or lambda or something because they're looking at a source with a different form.  I don't see the problem with the present arrangement.  Dicklyon (talk) 15:27, 4 May 2012 (UTC)


 * Dicklyon writes: "I don't see the problem with the present arrangement." That is why it is good when judged as a trojan horse. It is a trojan horse for the purpose of making it hard to remove the version with reduced wavenumber $k$ as independent variable.Chjoaygame (talk) 21:52, 4 May 2012 (UTC)Chjoaygame (talk) 22:06, 4 May 2012 (UTC)


 * I agree with Headbomb that the notation $B_{ν}$ and $B_{λ}$ has merit. I expect that some will find $B_{ν} (ν, T )$ cumbersome and will refuse to use it. I find it hard to believe that that "accusation" is one of the most ridiculous things in Wikipedia: there is plenty more ridiculous in Wikipedia than that!! Many texts use $B$ to point out that they are talking about a particular case of $L$ for spectral radiance. $I$, dare I say it, is not the latest.Chjoaygame (talk) 21:00, 4 May 2012 (UTC)


 * I believe though that the table should be replaced with a list with separate commentary and citations. The table has been a long standing problem and separating the entires out will make this area easier to edit and improve. Dmcq (talk) 21:46, 4 May 2012 (UTC)
 * Turning the table into a list would be lose the clarity and conciseness of this approach, as one could no longer easily compare the forms of Plank's law with h to those with hbar. Headbomb {talk / contribs / physics / books} 22:22, 4 May 2012 (UTC)
 * People can compare them easily enough when they are separate and having them separate would enable a better description of each of them. Comparing is just one use and should not override describing them properly. Anyway I'd be interested in what Dicklyon thinks as I believe the other two above have made their feelings known, or perhaps they don't agree? Dmcq (talk) 22:34, 4 May 2012 (UTC)
 * Here I agree with Dmcq, not with Headbomb.Chjoaygame (talk) 22:59, 4 May 2012 (UTC)
 * Having a list of forms does not enable better descriptions of the various form in the least. If you want to talk about B&nu;, you say "B&nu; is ...". If you want to talk about B&omega;, you say "B&omega; is ...". Table vs list makes zero difference when it comes to that; all using a list achieves is that information is presented in a much less reader-friendly way, and makes the comparison of the h vs hbar forms much harder. Headbomb {talk / contribs / physics / books} 01:59, 5 May 2012 (UTC)

two forms
One thing that seems to have gotten lost in both this discussion and the article itself is that there are two essentially different forms of Planck's law, namely $$B_\nu$$ and $$B_\lambda$$. The table has the following problems. There is nothing wrong with having a separate section explaining how to deal with change of scale for each of frequency and wavelength. This would supplement the two basic forms of Planck's law while meeting the need of Chjoaygame's contact who asked for the scaling of the frequency form appropriate for frequency in units of cm&minus;1 instead of Hz as very commonly used in spectroscopy.
 * It creates the impression that there are lots of different forms when there aren't. The second row of the table contains two copies of the wavelength form and the rest of the table contains four copies of the frequency form, with the copies of each form differing only in the trivial matter of a units-dependent constant factor.  One would not give a table of different multiples of the sine function, for example.  The table is therefore both duplicative and confusing.
 * It overwhelms the reader with many formulas right at the beginning. This is bad pedagogy.
 * The three cited sources (Caniou, the unpublished Kramm-Moelder paper, and Sharkov) do not support the claim that this is how Planck's law is usually presented. It is customary to give it in just two forms which is how the article was organized prior to October 13 last (more on this below).  While it is true that the first of these three sources has chosen to package Planck's law in this complicated way, that does not of itself make it the best source for justifying any given presentation.  WP:RS says "Each source must be carefully weighed to judge whether it is reliable for the statement being made and is the best such source for that context."  Given that Caniou is a highly specialized engineering monograph about passive infrared detection, it is hard to see how it can be considered "the best source" other than for the purpose of defending a pedagogically bad idea.

All these changes we're debating were introduced starting on October 13 last. Prior to that date the article had at most the two basic forms. When the article was first written in March 2003 it gave just the frequency form (expressed as $$I(\nu)$$), and remained that way for two years until the wavelength form was added by User:Unc.hbar on 07:23 on 21 May 2005.

My preference would be for the article to present the law in the two forms it gave during the intervening six years between 2005 and 2011, and to supplement this with a suitable section on how to deal with change of scale (as distinct from change of form) so as to accommodate other units than Hz for frequency and meters for wavelength. I would expect readers to find this less confusing, especially once they realize that only two essentially different forms are involved, which is not at all clear from the current version of the article. --Vaughan Pratt (talk) 06:15, 5 May 2012 (UTC)


 * With respect, I agree that there are two basic forms for the two basic independent variable, the spectral variable, but there is also room for various forms of the dependent variable, such as the spectral radiance, the spectral energy density, and some others. I am not in favour of trying to put them in a table. I don't have a special "contact": my "customer" was just Q Science.Chjoaygame (talk) 07:26, 5 May 2012 (UTC)


 * Which conveniently ignores Kramm & Molders, Caniou, Loudon, etc..., as well as show blatant disregard for both people who prefer to work with hbar over h and students who may be taught in a different manner than what you personally prefer. Headbomb {talk / contribs / physics / books} 07:44, 5 May 2012 (UTC)


 * People who prefer to work with hbar are smarter than the average bear, as we all know. Changing from one form of the law to another is trivial, so you have been consistently maintaining. So people who prefer to work with hbar don't need to have the law specially written for them; they will find it trivial to do it for themselves.Chjoaygame (talk) 13:19, 5 May 2012 (UTC)


 * And if it's so trivial, how about we remove your favourite forms instead? After all, if you're working in physics, you just as easily be able to convert from hbar to h. Headbomb {talk / contribs / physics / books} 14:46, 5 May 2012 (UTC)

rolling into a table
Chjoaygame, I agree that the other scales you mention (besides Hz and meters, e.g. cm&minus;1) can be usefully treated in the article, and said as much above. What I'm proposing is the original form of the article, which treated the two basic forms, and a separate treatment of scale-related matters based on the various units. I see no benefit to rolling all that information into a single table consisting of many forms at the start of the article, and several disadvantages as listed above. --Vaughan Pratt (talk) 08:20, 5 May 2012 (UTC)
 * Yes that's my basic feeling. If it was split up then it could be developed more logically dealing with the different options. The topic is supposed to be Planck's law whereas the table gives various representations of the law. That should come later complete with reasons and explanations. It is as though on started the article Derivative with a list of differentiation rules as in Table of derivatives. Dmcq (talk) 08:42, 5 May 2012 (UTC) .Dmcq (talk) 08:42, 5 May 2012 (UTC)
 * Yes, Dmcq, let's get rid of the table format.Chjoaygame (talk) 13:24, 5 May 2012 (UTC)


 * Vaughan Pratt, I was not referring to different scales, but to different physical quantities. Spectral radiance is of a different nature from spectral energy density. I am as keen as you are to see the table format removed.Chjoaygame (talk) 13:24, 5 May 2012 (UTC)
 * No, let's not. Removing the table would do a great disservice to readers (see e.g. Dicklyon's take on it just above), and obsfucates things greatly. We are an encyclopedia, we should aim for completeness, not restrain ourself of a small subset of knowledge (similar to how Maxwell's equations contain both differential and integral forms, both the E/B formulation and the D/H formulations, standard and gaussian units, etc...). Headbomb {talk / contribs / physics / books} 14:46, 5 May 2012 (UTC)
 * Very good example. The Maxwell's equations article nicely illustrates what's being proposed here.  The counterpart for Planck's law of the differential and integral forms of Maxwell's equations would be the frequency and wavelength forms of Planck's law, which is what is being proposed.  The counterpart for Planck's law of the separate section on Gaussian units would be a separate section on treatment of non-SI units for Planck's law, which again is what is being proposed.  --Vaughan Pratt (talk) 16:55, 5 May 2012 (UTC)

structure of the article
There's a couple of other good lessons from that article as well. This article doesn't have a readable description of the law except in the lead. The lead should summarize the article and should then be followed by an good introduction to what the law is about and says. One could then put in the multiple forms. Note that the citation used to back up the table is after a description of the law and a deerivation which is a far better order for someone reading about it. Dmcq (talk) 17:16, 5 May 2012 (UTC)

hbar
Some fonts can provide a character for lambdabar, but so far as I know, not yet the Wikipedia? Presumably someone will fix this deficiency?Chjoaygame (talk) 21:58, 5 May 2012 (UTC)
 * You mean the ƛ character? --Vaughan Pratt (talk) 18:45, 6 May 2012 (UTC)


 * Yes, that's what I meant. It does not appear in our article but is represented by the Roman letter $y$ and I assumed that this was due to the absence of the character, but as you note, I was mistaken. I find I can read that character better with a magnified view on my monitor, 200% is good enough. Where is the place to find such things in general, please? I was not intending to push for the inclusion of this form, and still am not intending to do so.Chjoaygame (talk) 22:11, 6 May 2012 (UTC)
 * Like most characters it's in unicode. I found it by googling for "unicode for lambda-bar" which took me here. --Vaughan Pratt (talk) 23:14, 6 May 2012 (UTC)
 * Thank you.Chjoaygame (talk) 00:15, 7 May 2012 (UTC)
 * So that's ƛ and there I was thinking the Chinese character 大 for big or eldest looked good ;-) Dmcq (talk) 01:10, 7 May 2012 (UTC)
 * Ah, the dreaded radical 37 for "big" or "very." ;-)
 * If Wikipedia has an entire article on that one radical alone (!), maybe it's time to spin off this article's history section into a separate article. At 35 kB it's already bigger than many other Wikipedia articles, including radical 37. --Vaughan Pratt (talk) 04:10, 8 May 2012 (UTC)

Some candidate sections for a new start
Given that there seems to be at least some interest in replacing the current start of the body with something less overwhelming than a large table, I've written some candidate sections for consideration at User talk:Vaughan Pratt/Planck's Law. Comments welcome. Obviously these aren't intended to replace everything since there's plenty of sound material currently in the article. --Vaughan Pratt (talk) 06:50, 6 May 2012 (UTC)


 * With respect, I think some of your suggestions are a bit on the chatty side.Chjoaygame (talk) 22:13, 6 May 2012 (UTC)
 * True enough, it's an ongoing fault with my style. Happy to entertain suggestions for improvement starting with the most egregious instances first.  --Vaughan Pratt (talk) 22:30, 6 May 2012 (UTC)
 * See your talk page.Chjoaygame (talk) 00:16, 7 May 2012 (UTC)

(I moved your version/comments to the bottom of User talk:Vaughan Pratt/Planck's Law so as not to break up the flow of what I'm proposing.) Ok, I think I see what you mean by "chatty." Here's the introductory sentence of our respective "non-chatty" and "chatty" accounts.

Non-chatty (Chj): "Planck's law describes the spectral power distribution of the electromagnetic radiation in thermodynamic equilibrium in a cavity with rigid walls that are opaque to all wavelengths."

Chatty (VP): "Planck's law expresses the quantity of radiation emitted by a black body, or ideal radiator, as a function of the absolute temperature T of the radiator and the frequency $&nu;$ of the portion of radiation being so expressed."

It sounds from this as though what you mean by "chatty" is "insufficiently technical and precise." --Vaughan Pratt (talk) 16:49, 7 May 2012 (UTC)
 * No I don't think so. By chatty I believe is meant more like for instance 'We have thus arrived at Planck's law via a little inspired guessing, as promised.' Dmcq (talk) 18:16, 7 May 2012 (UTC)
 * Yes, that section (on unifying Wien and Rayleigh-Jeans) was what I'd assumed he was referring to at first, which I agree could be tightened up (or deleted altogether if no one thinks the section is helpful). I'd intended to go back and "un-chatify" it as soon as he confirmed it, but based on his suggested changes it seems like he was complaining about the first section. --Vaughan Pratt (talk) 20:44, 7 May 2012 (UTC)
 * No.Chjoaygame (talk) 20:58, 7 May 2012 (UTC)
 * For a talk page that's a tad cryptic. Feel free to clarify.  --Vaughan Pratt (talk) 03:58, 9 May 2012 (UTC)

I revised the unification section. Is it less chatty now? Any other excessive chattiness? --Vaughan Pratt (talk) 21:45, 7 May 2012 (UTC)

I tightened up the derivation of $B_{&lambda;}$ to be less chatty -- lot shorter now. Hopefully that was the last of the seriously chatty bits. Also eliminated the limb darkening bit at Chjoaygame's suggestion, speak up if you want it back. --Vaughan Pratt (talk) 23:17, 7 May 2012 (UTC)


 * It seems you are building another trojan horse.Chjoaygame (talk) 01:35, 8 May 2012 (UTC)
 * Whether I'm building that (whatever it means) or an invisible pink unicorn is moot unless and until there's some sort of consensus that this will do as a replacement for the table-organized version, which not everyone seems to like (though User:Dicklyon does, so perhaps the tabular version should stay). I'm a fan of solving one problem at a time. --Vaughan Pratt (talk) 02:33, 8 May 2012 (UTC)


 * A trojan horse is a metaphor for a vehicle that introduces something surreptitiously. It seems that the partial agreement to get rid of the table is tempting you use such agreement as a way to try to introduce a bundle of Uncle Vaughan's Fireside Chats as set out on your talk page. The table was a trojan horse to introduce the hbar (ħ, $$\hbar$$) versions en masse, especially the reduced wavenumber $k$ ; the aim was to make it hard to remove specific items, because some people feel more comfortable with tables.
 * I have been trying gently to hint to you that agreement to get rid of the table does not mean agreement to the introduction of another trojan horse containing Uncle Vaughan's Fireside Chats as set out on your talk page. It seems I have to be more explicit to make myself clear. In agreeing that the table should go I am not thereby agreeing to the putting in of the Fireside Chats, or to anything else, for that matter.Chjoaygame (talk) 17:00, 8 May 2012 (UTC)
 * Hmm. Ok, I've removed the k conflict so that the article can go back to using k for Boltzmann's constant as done in the Boltzmann's constant article, which should address that particular trojan horse.  I've also removed anything I could identify as "chatty," let me know if I overlooked anything.  --Vaughan Pratt (talk) 17:58, 8 May 2012 (UTC)
 * I was not concerned about the conflict between different meanings of $k$. I was saying that in agreeing to get rid of the table, I am not thereby agreeing to put in a raft of Uncle Vaughan's Fireside Chats, that is to say nearly all of what appears on your talk page.Chjoaygame (talk) 18:20, 8 May 2012 (UTC)
 * You still haven't answered my question. Have I overlooked some sentence that you still find chatty?  Better yet, do you have a general criterion for when a sentence is veering into "chattiness."  Dmcq's example was very helpful and I fixed that one along with a bunch of others at a similar level of "chattiness" but now I don't see any further such examples.  Also note that the sections I wrote were not intended as an all-or-nothing "raft" as you put it but as (more or less) separate candidate sections for possible inclusion or at the very least a source of ideas for material.  Any section that people consider to add nothing to the article can simply be ignored.  (Incidentally I've been trying to respect the talk page guidelines, in particular "Comment on content, not on the contributor," such as not belittling you with terms like "Uncle Chjoaygame.")  --Vaughan Pratt (talk) 19:01, 8 May 2012 (UTC)


 * To answer your question, as you peremptorily demand. My non-consent to what you propose is not limited to possible chattiness. I am saying that I am not agreeing to nearly all of what appears on your talk page. I am commenting on your proposed content, not on you. Your proposed content is avuncular, chatty, and has a personalized style. That is not to belittle it, but to try to indicate why I think it is not suitable for the Wikipedia; it might well be suitable for a blog. There is nothing wrong with avuncular personalized fireside chats, but they are not my idea of Wikipedia material. I do not wish to be lured into further chatter about it. Now that you start cross-examining me ("You still haven't answered my question"), I see that I made a mistake to comment at all; I should have just said nothing, as did other wiser editors. I was trying to get you to try "solving one problem at a time": getting rid of the table, for which there seems to be some support.Chjoaygame (talk) 20:42, 8 May 2012 (UTC)
 * I interpret your "I was trying to get you to..." as indicating we have the common goal of improving the style of my content. Thus far I only have your examples to go on in order for me to be able to do so.  I've understood that my first sentence,
 * "Planck's law expresses the quantity of radiation emitted by a black body, or ideal radiator, as a function of the absolute temperature T of the radiator and the frequency $&nu;$ of the portion of radiation being so expressed,"
 * can be made less "avuncular, chatty, and personal" by being rephrased as,
 * "Planck's law describes the spectral power distribution of the electromagnetic radiation in thermodynamic equilibrium in a cavity with rigid walls that are opaque to all wavelengths."
 * So far so good, but a few more examples from other sources would clarify this further. Should I be aiming for something more like the style used in the section on Percentiles, for example?   --Vaughan Pratt (talk) 02:56, 9 May 2012 (UTC)


 * When I wrote "I was trying to get you to try “solving one problem at a time”: getting rid of the table, for which there seems to be some support" I was indicating that I would like to restrict attention to getting rid of the table and its immediate consequences. I cannot offer a style manual. In a suggestion on your talk page I have mixed $k$ and $k_{B}$ because I don't want to get involved in the debate about that.Chjoaygame (talk) 10:57, 9 May 2012 (UTC)
 * I'd go for a step at a time rather than big bang. I'd confine each step to one of - introduce or delete a section. Rearrange sections with only the absolute minimum necessary change to content, or develop one section. That way you can get immediate feedback as the stuff is done and people don't have to check the whole business all at once. Dmcq (talk) 17:55, 9 May 2012 (UTC)
 * Thanks for stepping in, Dmcq. As I've already said to Chjoaygame, "one section at a time" is what I meant by "candidate sections" plural (as distinct from "wholesale replacement" by one big "raft" of text).
 * My first step would be to replace the current "Different forms" section with my "Main forms" section, which does only four things:
 * States the law.
 * Explains what it means. (I still haven't understood Chjoaygame's objections to my explanation.)
 * Treats the non-normal case. (Important to have this up front because almost all black body radiation is non-normal).
 * Gives the wavelength form and its differences from the frequency form. (Those two forms are the only ones the article had between 2003 and last October.  Furthermore no one saw a need even for the wavelength form for the first two years of the article.  I'd be perfectly happy to use that as a rationale to simplify the first section by moving it the "other forms" section, I could go either way on that.)
 * The "other units" section is only to meet a need perceived by Chjoaygame and Headbomb, and can be included when someone requests it. I'm not going to promote it myself since it's not something covered by any standard physics text treating Planck's law, though given the thousands of engineering texts it's no surprise to find one or two with a table of half a dozen forms. Applied engineering books tend to be full of tables.
 * Likewise the integrals and unification sections are only there to give people ideas; if there's a strong interest in either one they can be included at a later date.
 * Sound like a plan? --Vaughan Pratt (talk) 18:41, 9 May 2012 (UTC)

still looks like a plan for a trojan horse
Still looks like plan for a trojan horse. If you mean only a section, you should make just that how you set it out on your talk page; just remove the distractions. You persist in misreading my wish, that the format allow for possible future different kinds of dependent variable, as if it were about different units, what you call "other units". It is about different physical quantities, as I have mentioned already. For example, spectral energy per unit volume is not the same quantity as spectral radiance, even if you like to think they differ only in "units", as it seems you do. I think there should be room in the format for future insertions of such extra quantities as spectral energy per unit volume, even if you don't, and even though they are not necessarily stated right now. I am not, repeat not, as I have repeatedly indicated above, wanting to retain the reduced wavenumber $k$ form. I do not like your plan to use a format that would in effect censor what may go into the article in future; that is one reason why I speak of a trojan horse. I find your statement or explanation of the law messy and incomplete and not the best way to state the physics; I have offered one that I prefer. I think the directional story is a story of its own and should have a section or subsection of its own, and not be merged in the basic statement.Chjoaygame (talk) 19:24, 9 May 2012 (UTC)
 * Excellent, you've raised some specific objections. Since I'm proposing only the first section for now I've added remarks on the talk page to make this clear, which hopefully addresses your "trojan horse" claim (which still makes no sense to me).
 * Your only comments that need to be addressed for now are the ones bearing on the first section, namely that the two paragraphs with the explanation are "messy and incomplete," and the proper location of the Lambertian aspect.
 * That some editors of this article have a passion for completeness is why this article is now over 100 kilobytes and has over 130 references, making it very unwieldy as Wikipedia articles go. May I suggest that some of the material needed to "complete" the explanation (if such a thing is even possible!), such as consideration of rigid cavities with opaque sides, be put in a later section rather than packing it all into the first two paragraphs following the law?  There is a limit to how long a first explanation can be before it gets out of hand.
 * As for "messy," I don't see the mess, please point out where it occurs. Every word in my explanation was chosen with great care to be as precise as possible without overburdening the reader with vast amounts of technicality, and is exactly correct.  I see no room to improve even a single word.  How can "correct" be "messy"?
 * Your point about the correct location of the Lambertian aspect is worth discussing. Since it's important and fits neatly in a single paragraph it seemed appropriate to bring it up early on, but you evidently disagree.  What do others think?  --Vaughan Pratt (talk) 19:56, 9 May 2012 (UTC)


 * You haven't bothered to get your head around the trojan horse idea, though I have repeated it several times. You don't see why I think your account is messy, imcomplete, and does the physics very poorly indeed. This is an article about physics, not a list of formulas. But you "see no room to improve even a single word." Why would I bother to make any more suggestions?Chjoaygame (talk) 22:52, 9 May 2012 (UTC)
 * Evidently I've misunderstood you, please pardon my denseness. I thought you were saying that your "trojan horse idea" referred to the later sections and not the "Main forms" section.  If I was wrong about that then what we have here is a failure to communicate.  How could the main forms section possibly constitute a "trojan horse?"  It's just a section, for heaven's sake, how can a section be a "trojan horse" as you call it?  This seems like a non-objection.
 * What you haven't "bothered to get your head around" as you put it is that repeating the insults "avuncular," "chatty," "personal," "messy," "incomplete," and "poor" n+1 times without explaining them has exactly the same effect as repeating them n times. When n is zero I trust you will agree that they have zero effect.  It follows that when n is 100 they will still have no effect.  n isn't quite that large now, but I would have thought you would be starting to wonder about whether you're going about this the right way.  You're like someone who blows on a wall to see if it will fall down, and when it doesn't, blows again and again each time muttering "why would I bother to blow any more times?"  You did the same thing in October on a small scale, but it seems to have gotten much worse in the intervening six months.   --Vaughan Pratt (talk) 02:37, 10 May 2012 (UTC)
 * Regarding This is an article about physics, not a list of formulas. But you "see no room to improve even a single word," you may have overlooked that my "no room to improve" was in reference to my two-paragraph explanation of the frequency formula. Since I don't see any "list of formulas" in either of those paragraphs, what is the reasoning behind your "but" here?  --Vaughan Pratt (talk) 02:51, 10 May 2012 (UTC)
 * Regarding I think there should be room in the format for future insertions of such extra quantities as spectral energy per unit volume, even if you don't, while your completely-out-of-the-blue "even if you don't" might explain your strange "trojan horse" theory, wherever did you get the idea that I don't? I think such future insertions would be terrific, that is, assuming going well over 100 kilobytes is perfectly reasonable for a single Wikipedia article!
 * In light of the unwieldy size of the present article, I have an alternative constructive suggestion. Since you felt it would be beneficial to expand on the Radiance article with a separate article on Specific radiative intensity, why not do the same with this article?  Expand on this article in a separate article with those aspects of Planck's law that you personally are interested in such as spectral energy per unit volume.  This would be a net benefit to both this article and yours by shrinking the former down to be a counterpart of the Radiance article while the material you'd like to see could be a counterpart to the Specific radiative intensity article.
 * There's a limit to how big a Wikipedia article can get before the time comes to split it into different topics meeting the needs of editors with divergent interests. --Vaughan Pratt (talk)