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Terminological conventions

It is agreed by all, I think, that for a perfectly black surface, thermal emission as radiance is independent of direction?

I think it will be agreed that radiant intensity varies with direction, according to Lambert’s cosine law?

According to the SI, radiant intensity treats the source as a point source. Planck holds that a point source emits no power. We may be headed for trouble here?

According to the SI, I believe at present, the radiance is power per unit projected area of source per unit solid angle of detection. The projected area $\mathrm {d} A\,$ of source is the actual source area $\mathrm {d} \sigma \,$ multiplied by the Lambert factor $\mathrm {cos} \,\theta \,$ .

The traditional specific intensity is power per unit actual source area per unit solid angle of detection. An example of this is in Mandel and Wolf 1995 page 292. They actually call their quantity the (spectral) radiance but I am reasonably confident (subject to correction by the learned assembled company) that they would have in earlier days called it the specific intensity:

Start of quote from Mandel and Wolf page 292

The rate $\mathrm {d} ^{2}F_{\nu }\,$ at which energy at frequency $\nu \,$ is radiated into an element $\mathrm {d} \Omega \,$ of solid angle by an element $\mathrm {d} \sigma \,$ of a planar steady-state source $\sigma \,$ is given by the expression

$\mathrm {d} ^{2}F_{\nu }=B_{\nu }^{(0)}(\mathbf {\rho } ,\mathbf {s} )\,\mathrm {cos} \,\theta \,\mathrm {d} \sigma \,\mathrm {d} \Omega \,\,.\,\,\,\,\,\,\,(5.7-36)$

Here $\mathbf {\rho }$ is the (two-dimensional) position vector of the point $Q\,$ in the source plane at which the element $\mathrm {d} \sigma \,$ is located, $\mathbf {s}$ is the unit vector along the axis of the element $\mathrm {d} \Omega \,$ of solid angle and $\theta \,$ ($0\leq \theta \leq \pi /2$) is the angle which the unit vector $\mathbf {s}$ makes with the normal to the source plane (fig. 5.25). The function $B_{\nu }^{(0)}(\mathbf {\rho } ,\mathbf {s} )\,$ is called the (spectral) radiance or brightness.

End of quote from Mandel and Wolf page 292

As I read this it agrees with the definitions (which here I will call the traditional definitions of specific intensity) given by Mihalas and Mihalas 1984 and by Chandrasekhar 1950 and Planck 1914 for the specific intensity; they say nothing about spectral radiance or SI uints.

An obvious (but perhaps mistaken?) interpretation of the above is that Mandel and Wolf 1995 were mistaken in identifying traditional specific intensity with SI radiance. They use the traditional definition of specific intensity and label it also as the "SI" definition of radiance.

As I read it, a similar confusion, but with things reversed, arises in the case of Goody and Yung 1989. They write on page 16: "Much of the early work on radiative transfer is to be found in the astrophysical literature. SI nomenclature differs from astrophysical nomenclature and is more suited to situations involving finite sources. Equivalents are specific intensity (this book) and radiance (SI) ... This book does not use an equivalent to radiant intensity (SI)..." They use the SI definition and label it also as the "traditional" definition. Their "specific intensity" differs from the traditional specific intensity, because their "specific intensity" is a function only of line of sight, not of the orientation of the actual emitting source surface, while the traditional specific intensity is a function of the line of sight and also of the orientation of the actual emitting source surface. They go on to talk about a component of flux in a direction off the line of sight, and there the $\mathrm {cos} \,\theta \,$ comes into it, but not as part of the definition of their "specific intensity".

As I read it, the SI radiance is a property of an imaginary surface of unit area, located near the source, at right angles to the line of sight of the remote observer.

As I read it, the traditional specific intensity is a property of a real physical emitting surface, per unit area of that surface.

Rybicki and Lightman 2004 on page 3 talk about specific intensity for an imaginary surface element $\mathrm {d} A$. They do not in their definition right here say anything about the physical source surface or angle of view, but talk only of a viewing angle perpendicular to their effective or apparent source. I would attribute to their thinking a real physical surface $\mathrm {d} \sigma =\mathrm {d} A/\mathrm {cos} \,\theta \,$ . As I read it, this puts them in the Planck-Chandrasekhar-Mihalas-Mandel&Wolf camp, against the Goody and Yung camp?

Being quick and efficient by not specifying all the variables of which $L^{0}\,$ is a function, Caniou 1999 says on page 106 that

Start of quote from Caniou 1999 page 106

The radiant energy crossing $\mathrm {d} S\,$ during one second and contained within $\mathrm {d} \Omega \,$ is expressed by

$\mathrm {d} \Phi ^{0}(\theta ,T)=L^{0}(T)\,\mathrm {cos} \,\theta \,\mathrm {d} S\,\mathrm {d} \Omega \,\,,\,\,\,\,\,\,\,(4.1)$

where $\mathrm {d} \Phi ^{0}(\theta ,T)\,$ is the radiant flux and $L^{0}(T)\,$ is the radiance of the blackbody at temperature $T\,$ (section 2.2.3).

End of quote from Caniou 1999 page 106

As I read it, this puts Caniou in the majority camp listed above, though he says nothing about specific intensity, speaking only of radiance.

Paltridge and Platt 1976 issue warnings and avoid the use of the term specific intensity, speaking for themselves of radiance and of radiant intensity, and issue more detailed warnings about variations in the literature in the use of the word intensity. As I read them, they give two contradictory definitions of the radiance of an area of a source. One of their definitions is in a table on page 36 and defines the radiance as radiant flux per unit solid angle of detection per unit projected area of source surface; this is the SI definition as I understand it. Their other definition is in the text on page 37 and reads "Thus the average radiance of an area of a source in a given solid angle is the total flux radiated from that area into that solid angle, divided by the product of the area and the solid angle"; this is the traditional definition of specific intensity, as I read it.

Begone, walls of text. Strenuously avoid Latin tags. Never make a decision today that you can put off till tomorrow.Chjoaygame (talk) 04:47, 10 November 2011 (UTC)

It may be convenient here to add some other statements of definition.

Mihalas and Mihalas 1984 are virtuous in that they actually state the argument variables of which their specific intensity is a function. On page 311 they define a quantity the call specific intensity. So far as I can guess, they mean this to be identical with the quantity of the same name used by Planck and by Chandrasekhar. They write:

Start of quote from Mihalas and Mihalas 1984 page 311

We define the specific intensity $I(\mathbf {x} ,\,t\,;\,\mathbf {n} \,,\,\nu )\,$ of radiation at position $\mathbf {x} \,$ and time $t\,$ , traveling in direction $\mathbf {n} \,$ with frequency $\nu \,$ , to be such that the amount of energy transported by radiation of frequencies $(\nu \,,\,\nu +\mathrm {d} \nu )\,$ across a surface element $\mathrm {d} S\,$ in a time $\mathrm {d} t\,$ , into a solid angle $\mathrm {d} \omega \,$ around $\mathbf {n} \,$ is

$\mathrm {d} E=I(\mathbf {x} ,\,t\,;\,\mathbf {n} \,,\nu )\,\mathrm {d} S\,\,\mathrm {cos} \,\alpha \,\mathrm {d} \omega \,\mathrm {d} \nu \,\mathrm {d} t\,\,,\,\,\,\,\,\,\,(63.1)$

where $\alpha \,$ is the angle between $\mathbf {n} \,$ and the normal to $\mathrm {d} S\,$ .

End of quote from Mihalas and Mihalas 1984 page 311

This is clear. It does not refer to the SI.

Chandrasekhar also lists explicitly the geometrical argument variables of his specific intensity, and his definition is so close to that of Mihalas and Mihalas that I think it would be redundant to quote it explicitly here.

All three of Mandel and Wolf 1995, Mihalas and Mihalas 1984, and Chandrasekhar 1950 explicitly make the specific intensity a function of the same geometrical argument variables. They all multiply it by a cosine on the way to calculating a flux from it. Planck 1914 on page 13 lists the geometrical argument variables as angles in his textual definition and his definition is the same. Milne, E.A. (1930), Thermodynamics of the Stars, Handbuch der Astrophysik, volume 3, Part 1, pages 63–255, also uses this definition of specific intensity. As noted above Rybicki and Lightman and Caniou do it like this too, though they don't explicitly list the geometrical argument variables, which is naughty of them, and Caniou tells what seems like a fib (as I read it, Caniou 1999 says he is telling us the "radiance" but really he is telling us the traditional specific intensity).

I think this puts a strong case to call this definition the traditional one for specific intensity or brightness. It is a property of the actual emitting surface, not of an imaginary surface perpendicular to the line of sight such as is I think referred to by the SI term radiance.

Paltridge and Platt seem self-contradictory. That leaves only Goody and Yung 1989 consistently doing it how the SI recommends, in terms of SI radiance; and even they (mistakenly I think) say they are using the "specific intensity".

Because cavity radiation is isotropic, the geometrical angle argument variables are not critical for it. I think it likely that this is part of the reason why the above-mentioned inconsistency or mistake doesn't count too much. Thermal radiation from a perfectly black surface into a vacuum is not isotropic, in the sense that it is all coming from the perfectly black virtual surface.

What all this means for us, I am not sure. Perhaps some ideas will come.Chjoaygame (talk) 07:44, 10 November 2011 (UTC)

correction of my temporary mistake

Thanks to a helpful question from PAR, I just now reconsidered the above section "Terminological conventions" and now see that it is mistaken. My statements previous to my writing the above section now stand, and the above section is mistaken, a temporary aberration. Sorry about that.

The correct statement was and is that the concepts of specific intensity and spectral radiance are identical in every case. They both refer to the projection of the actual source area into a plane normal to the line of sight. Mandel & Wolf and Goody & Yung are not misleading when they identify spectral radiance with specific intensity. Paltridge and Platt are not mistaken on page 37 (just at one point a little loose in wording); they eventually correctly say that "the area referred to is invariably (although not necessarily) the projected area of the [source] surface on to a plane perpendicular to the direction of measurement." I think their "(although not necessarily) simply refers to the case when the source surface is already in the plane perpendicular to the direction of measurement.

To be explicit: the correct statement is: The traditional specific intensity is power per unit projected source area per unit solid angle of detection, the same as for spectral radiance.Chjoaygame (talk) 13:52, 11 November 2011 (UTC)

Crash course in radiometry

Great wall, Chjoaygame. :) Let me try to match it with an orthogonal wall concerning intensity, radiance, etc. (Note the "try to", whether I'll actually succeed is itself orthogonal.)

With my recent addition of spectral radiant exitance to Template:SI_radiometry_units on 02:08, 4 November 2011‎, item 101 in the table below, this template now has 14 entries. The template is maintained by User talk:Matthiaspaul, an administrator with rollback privileges, and a good thing too given how confusing the material is. (Were research not a higher priority for me I'd find these privileges very handy myself.)

What I'd like to do here is to separate the terms in the template.

There are only three dimensions to pay attention to in radiometry that are relevant to Planck's law, namely the transmitter area, direction of radiation (equivalent to area of the receiver as perceived by the transmitter when adjusted for separation), and frequency of radiation. (There is also whether the perspective is that of the transmitter or receiver, for which the term "irradiance" is the tip-off that the perspective is the receiver's, which is irrelevant for the Planck's law article.)

For each dimension the following transitions typify what to add to the terminology when that dimension is "activated."

Transmitter area: radiant flux → radiant exitance (aka emittance)

Direction of radiation: radiant flux → radiant intensity

Frequency of radiation: radiant flux → spectral radiant flux (far more common than radiant spectral anything).

These transitions start from the unit of watts, W, and add respectively the units m⁻², sr⁻¹, and either Hz⁻¹ or m⁻¹ (more often cm⁻¹ in spectroscopy) according to whether (in the notation of this article) we're dealing with $B_{\nu }$ or $B_{\lambda }$ respectively.

All eight subsets are common in the literature. Although nomenclature is not absolutely uniform, the following seem to be the most common. I've encoded in binary the presence of each of area, direction, and frequency, in order from least to most significant bit (so 001 is area, 010 is direction, and 100 is frequency).

000 radiant flux

001 radiant exitance/emittance

010 radiant intensity {or specific intensity; Google ratio 141K:147K. --Vaughan Pratt (talk) 22:29, 10 November 2011 (UTC)}

011 radiance

100 spectral power

101 spectral radiant exitance/emittance

110 spectral intensity

111 spectral radiance

If anyone finds a synonym for any of these that is more common according to Google, please bring it up and I'll go back and amend this table accordingly (using strikeout rather than mere deletion).

The Planck's law article standardizes on 111, spectral radiance. Replacing it with 101, spectral radiant exitance, merely entails multiplying by π. Independently, replacing it with 110, spectral intensity, entails multiplying by the area of the emitting surface. Lambert's cosine law applies when using the actual area, but if you use the apparent area as seen by the receiver then Lambert's law no longer applies because "apparent" cancels "Lambert." When integrating over all directions to convert x1y to x0y, no matter what x and y are, you must assume Lambert's law applies or you'll be off by a factor of 2.

A more complex notion is radiosity, which in heat transfer refers to the combination of reflection and radiation from a surface. Everything you see, whether an orange viewed from three feet away or Earth viewed from the moon, emits a bimodal distribution whose high frequency component is reflected visible light and whose low frequency component is thermal radiation governed by Planck's law for the radiating object at temperature T. Typically the two modes are a factor of 20 apart. In the case of Earth the proportions are very close to 3:7, namely 30% of the sunlight is reflected to space and the remaining 70% is absorbed by the atmosphere and the surface, with all of the absorbed radiation being radiated to space to maintain thermal equilibrium except during periods of substantial climate change up or down.

In theory the electromagnetic spectrum draws no hard line between light and heat. In practice however the bimodality of real-world radiosity draws that line at a wavelength of around 4 μm, treating near infrared (e.g. your TV remote) and beyond as light, and far infrared (what an infrared thermometer responds to) as heat. --Vaughan Pratt (talk) 05:51, 10 November 2011 (UTC)

Good, thank you. "The Planck's law article standardizes on 111, spectral radiance" - as it should. $B_{\nu }\,d\Omega \,d\nu \,dA$ is power into solid angle $d\Omega$ , in frequency band $d\nu$, from the "projected" area, or in Vaughan Pratt's terminology "apparent" area dA. That means it is independent of direction. This is in accord with the Wikipedia definition of spectral radiance. So what, now, is the problem? I still want to put the above statement into the article just to remind a reader of the definition of spectral radiance. PAR (talk) 06:35, 10 November 2011 (UTC)

As I've said, there does exist a sense in which BB radiation is direction independent. But you have to be very clear about that sense in order to avoid being swallowed by the tarpit in which the Stefan-Boltzmann constant is 11.34 instead of 5.67.

One way to look at this is to ask what is the area of the visible portion of the Sun. If you view it as a disk of unit radius then its area is π. But if you view it as a hemisphere with the same radius then its area is 2π. There's a factor of two difference. Which one is correct, and by what criteria?

The former is the apparent or projected area, the latter is the actual area. It's a factor of two, which is nontrivial. Does the article make the choice clear?

If you are using 111, spectral radiance, you have no choice: you must use Lambert or be off by a factor of two when computing the integral. If you have some other way of integrating 111 to get 101 that does not commit the factor-of-two error I'd be very interested in seeing it. --Vaughan Pratt (talk) 07:59, 10 November 2011 (UTC)

I just caught an error in my "(more often cm⁻¹ in spectroscopy)". For the spectroscopy units there are two reciprocals, and for consistency with the other two units I should have written cm rather than cm⁻¹. In the numerator of $B_{\tilde {\nu }}$ we have watts times bandwidth in cm. Not an error after all, just a moment of aberration where I confused wavelength on the y-axis with wavenumber on the x-axis (see reply to Q Science below). I was correct to begin with because I was speaking of wavelength and y-axis, not wavenumber and x-axis, which confusingly both involve the unit cm⁻¹. --Vaughan Pratt (talk) 08:20, 10 November 2011 (UTC)

No, by convention W/cm⁻¹ is never converted to W cm. It is not a bandwidth, graphs are typically labeled

frequency (cm⁻¹)

Q Science (talk) 09:38, 10 November 2011 (UTC)

Two problems:

• That's the label on the x-axis, I was talking about the y-axis, which inverts the spectral unit, turning it into bandwidth ("per unit spectrum").
• The x-axis is incorrectly labeled, it should be "wavenumber", not "frequency."

My mistake (what I just struck out) was that the cm⁻¹ on the y-axis when the spectral unit is wavelength triggered the "this must be a wavenumber" reflex (it was late at night), where the y-axis is labeled with cm, not cm⁻¹. But that was wrong because the context was wavelength, not wavenumber.

On the y-axis, we have for the per-unit-bandwidth unit the following.

• Wavelength spectrograms: cm⁻¹, or "per cm".
• Frequency spectrograms: s (seconds), aka Hz⁻¹, or "per Hz".
• Wavenumber spectrograms: cm, or "per cm⁻¹".

These are inverted on the x-axis. I believe you were referring to wavenumber spectrograms where the x-axis is labeled cm⁻¹. --Vaughan Pratt (talk) 15:08, 10 November 2011 (UTC)

On further reflection, it seems to me that all ways of looking at non-normal radiation are just different descriptions of Lambert's law. If you ask how much power is radiated from unit area into unit solid angle, Lambert's law is used to compute that power directly. If you say that the surface looks equally "bright" from all directions, Lambert's law is used twice, once in the numerator as per the previous sentence, and once in the denominator to turn actual area into projected area.

So far no one has objected to my parenthetical statement above, "(There is also whether the perspective is that of the transmitter or receiver, for which the term "irradiance" is the tip-off that the perspective is the receiver's, which is irrelevant for the Planck's law article.)" The statement "The radiation emitted is the same in all directions" if allowed to stand would be the one exception, since it is only the same from the receiver's (i.e. viewer's) perspective. Is this distinction explained in some other Wikipedia article? If not, then in order to make that statement, we would need to introduce and explain "from the viewer's perspective" and warn that this does not imply that a small flat surface irradiates a surrounding hemisphere uniformly, but according to the cosine law. If the zenith of the hemisphere is receiving unit radiation, then the horizon (zenith angle of 90°) is receiving zero radiation while at a zenith angle of 60° the hemisphere is receiving 0.5 units of radiation, despite the fact that to the viewer the surface appears equally bright from all positions on the hemisphere.

On the other hand a spherical surface placed at the center of a sphere irradiates the whole sphere uniformly. But nowhere in the article do we talk about spherical radiators, for which Lambert's does not apply (except for infinitesimal surface elements of the sphere). This idea that black body radiation is isotropic must have been obtained from some astronomy article, where the radiator is indeed spherical and the radiation is indeed isotropic. The statement is false for a flat surface element, which is what our article treats.

I suspect this is the crux of the matter. --Vaughan Pratt (talk) 17:49, 10 November 2011 (UTC)

The alternative between spectral radiance and specific intensity should be looked at carefully. The two concepts are not identical, as I have very recently learnt, and as is now stated in two sections adjacent to this present section. The two concepts refer respectively to two physically different objects. Specific intensity is a property of an arbitrarily oriented small surface element. Spectral radiance is a property of a small surface element that is constrained to be oriented at right angles to the line of sight of the observer-detector. The two coincide when the arbitarily oriented small surface element happens to be chosen to be oriented perpendicular to the line of sight of the observer-detector.

Isotropy. The only radiation that is absolutely certain to be ideally and perfectly black is the unobservable radiation in a rigid opaque imperfectly-reflecting-walled cavity in thermodynamic equilibrium. It is homogeneous, isotropic, and unpolarized. Radiation specified to be coming directly from the emission of the walls of such a cavity is not the radiation itself, but is an abstraction from it, with the immediate source specified; consequently it has no possibility of being anything like isotropic; indeed one might even say that it is tending to be strongly non-isotropic, the next extreme being a plane wave. As noted above, radiation specified to come from a flat surface element is not remotely isotropic. Unqualified thermodynamic equilibrium radiation in a suitable cavity is isotropic.Chjoaygame (talk) 23:15, 10 November 2011 (UTC)

Chjoaygame (talk) 23:15, 10 November 2011 (UTC)

Just to be clear, the reason you are saying that blackbody radiation is not isotropic from a flat surface is because it is only directed outward, not inward? In that case, a spherical radiator does not qualify either. Just because a radiating sphere radiates uniformly in all directions does not make the radiation isotropic. Radiation is isotropic at a particular point. The number of photons going in one direction are the same as those going in any other direction as determined by the number passing thru an infinitesimal directed area element at that point. Outside of a blackbody sphere, or a flat element, this is never the case. If you put a small area element at a point where the radiation is isotropic, and consider only the photons going through it in one direction, thats the same that would go through if that area element were part of the surface of an emitting body. It is in this sense that blackbody radiation is considered isotropic. PAR (talk) 23:35, 10 November 2011 (UTC)

Yes, just to be clear, I am saying saying that blackbody radiation, specified as coming from a flat surface, is not isotropic just because it is specified as only directed outward, not inward. And correspondingly I would say that a finite spherical radiator does not qualify either. I agree that radiation is isotropic or not at a particular point. I agree that a finite radiating sphere, radiating uniformly in all directions, does not make the radiation isotropic. I think that black-body radiation is considered isotropic just when it is being considered as unqualified thermodynamic equilibrium cavity radiation, located inside the cavity; not when it is qualified by a specification that it comes from a particular surface, as I think is envisaged here.Chjoaygame (talk) 23:44, 11 November 2011 (UTC)

Quoting PAR, $B_{\nu }\,d\Omega \,d\nu \,dA$ is power into solid angle dΩ , in frequency band dν, from the "projected" area, or in Vaughan Pratt's terminology "apparent" area dA. Is this in any way different from saying the same thing about $B_{\nu }\,\cos(\theta )\,d\Omega \,d\nu \,dA$ as power from the actual area? The projected area shrinks according to Lambert's law, whence a hemisphere above a flat surface element will be brightly irradiated at the top and not at all at its equator, which is the ultimate litmus test of whether the radiation is uniform. Are you claiming otherwise? --Vaughan Pratt (talk) 07:44, 12 November 2011 (UTC)

As I read you, I think we agree. Perhaps right here, keeping to the notation you cite from PAR above, we could call the actually emitting area $\mathrm {d} S\,$ and the projected area $\mathrm {d} A=\mathrm {d} S\,\cos \theta$? We have that $\mathrm {d} S\,$ is kept constant while $\mathrm {d} A\,$ shrinks like $\cos \theta \,$ as $\theta \,$ grows from $0\,$ to $\pi /2\,$. In my article entry here I used the notation of the Wikipedia article on radiance.Chjoaygame (talk) 10:19, 12 November 2011 (UTC)

surface and interior

I have, I trust with his approval, taken the liberty of making PAR's comment the lead of a new section. If he does not approve, I trust he will accepy my apology and will reverse what I have done.Chjoaygame (talk) 10:00, 12 November 2011 (UTC)

"a black body has an interior consisting of matter, and an interface with its contiguous neighbouring material medium,"

This has to be changed. It is overly verbose, and wrong. A black body does not need to have an interior consisting of matter, it could consist of pure radiation. "contiguous neighboring" is redundant. And finally the "contiguous neighboring" medium need not be "material", it may be a vacuum. PAR (talk) 03:42, 12 November 2011 (UTC)

I can see calling this "black body radiation," meaning radiation spectrally distributed according to Planck, but who has ever referred to a vacuum as a "black body?" (Not entirely a rhetorical question, some textbook might have.) The Sun is a reasonable approximation to a black body (at least by night, by day it seems more yellow) (and from this distance it also approximates an isotropic radiator with a Lambertian surface), but it is so because it consists of stuff whose surface is roughly at a uniform termperature modulo limb darkening. --Vaughan Pratt (talk)

The usual way of creating a (nearly) black body is an oven with a small hole in it. The interior of the oven and the inner walls are (nearly) in equilibrium, so the radiation exiting the hole is (nearly) Planckian. The inner walls need not be black, in fact they could be perfect mirrors. The hole behaves as the surface of a black body. I mean, how would you describe this situation? (side issue: surely you did not mean to say "The Sun is a reasonable approximation to a black body (at least by night, by day it seems more yellow)".) PAR (talk) 05:28, 13 November 2011 (UTC)

It won't work with perfect mirrors. The hole is a blackbody because any radiation that goes in does not come back out. You don't really need to say anything about the internal structure. Dicklyon (talk) 06:42, 13 November 2011 (UTC)

Quantum mechanically speaking, photons can interact with each other and exchange energy. I know, the equilibration time and required volume would be huge, but nevertheless finite. This may seem like grasping at straws, but I find it valuable because it allows you to think of the radiation field as an entity unto itself, just like a material body, which will eventually equilibrate on its own, under the right conditions. The fact that the rate is enormously faster and the minimum volume is much smaller (and photons are not conserved) when the radiation field and a material body are equilibrating together is important, but not fundamental. Actually, for small volumes, any reflection at all will screw things up, with the maximum screwup occurring for perfectly reflecting walls. I agree, you need to say nothing about the interior other than that it confines the radiation and exceeds a certain minimum dimension. That includes not needing to specify that the walls are not perfectly reflective. PAR (talk) 07:39, 13 November 2011 (UTC)

Planck believed that the pure radiative field with perfect mirrors would never change its spectrum. That's why he demanded a speck of carbon and his theoretical oscillators in a sense played the role of the speck of carbon. Ehrenfest pointed out that the oscillators were also perfectly linear and couldn't do the job. Planck accepted that and agreed that he didn't have a physical explanation of how equilibrium would be reached, so that his oscillators were only probes, not physical objects that explained transduction between frequencies. Do you have a reliable source for the finite equilibration time in the case of perfect mirrors and no material transducer?Chjoaygame (talk) 08:08, 13 November 2011 (UTC)

Planck can be forgiven for not knowing about photon-photon interactions. They are mentioned in the Wikipedia photon article. If photons can exchange energy, then it follows that this is a mechanism for equilibration, just as collisions of material particles cause equilibration for material bodies. I do not have a source for this, however. It seems to me that googling photon-photon interactions would turn up something. PAR (talk) 08:47, 13 November 2011 (UTC)

Well, let's at least find a source that says the the walls of the cavity don't need to absorb any photons for it to reach a thermodynamic equilibrium, if you want to rely on that. Dicklyon (talk) 20:20, 13 November 2011 (UTC)

I agree, but this whole thing arose from Vaughan Pratt questioning whether a material vacuum containing only black body radiation qualifies for the term "black body" and I just gave the cavity as a counter example, where there isn't even a black body involved, and then went off the deep end saying the walls could be perfectly reflective. I am not advocating that this "deep end" be included in the article, especially without reference, even though it is reasonable to believe it is true. I just use it as a reminder that there is not that much difference in principle between a photon gas and any other gas of material particles - the need for a thermalizing piece of matter is not a fundamental physical principle. Its the massless and speed of light effects that create the biggest difference. PAR (talk) 21:03, 13 November 2011 (UTC)

What is being proposed here is not only that equilibrium can be reached with perfectly reflecting walls; Planck routinely worked with such a cavity, but the point here is that he thought that the speck of carbon was needed when so working. The source that seems to be needed must say that the speck of carbon is not needed when the walls are perfectly reflecting. It is said that perhaps gamma rays can interact as pure photons, by way of their producing virtual electrons, but that hardly seems adequate to show that low energy photons such as infrared photons can do so too.Chjoaygame (talk) 20:42, 13 November 2011 (UTC)

I'm not proposing that this be entered into the article, at least not without a good reference. Low energy photons will do the same, only at a much lower rate. There is no "cutoff energy" at which photon collisions totally cease to occur. Even though I think its true, maybe it doesn't belong in the article anyway. I just use it as a way of reminding myself that the need for a piece of thermalizing matter is not the result of any fundamental physical principle, its just a way of speeding things up. PAR (talk) 22:31, 13 November 2011 (UTC)

I have no objection to changing this. It may be overly verbose and may be wrong in some respects. But it represents something that needs to be said somehow, if Planck is to be understood. My statement in the article is intended to be pretty nearly a straight lift from Planck 1914. This distinction is fundamental to the literature at least since Helmholtz 1856, Stewart 1858, and Kirchhoff 1860, and is still valid today. All speech includes some redundancy; that is why it is more or less reliable. I put in those extra words to emphasize to what might otherwise have passed unremarked. I am not happy with the idea of making Planck think that pure radiation is a black body; I think he would resist that quite vigorously; I would even go so far as to say that I think the distinction is fundamental to his approach. In those days, both in English and in German, they distinguished between (a) ponderable matter which I have rendered as 'material' and of which bodies were constituted, and (b) etherial processes like radiation which was not ponderable matter, though it could and did permeate ponderable matter; for example Kelvin (date near there), Einstein 1905, Jeans 1905. My words mean to distinguish an interface between (a) a rigid opaque material wall of a cavity or a body of radiating material in general, and (b) the space within the cavity or around the body of radiating material in general. The space within the cavity or around the body I saw as filled with a transparent material but I have no objection to letting it be a vacuum instead (though I did not intend to exclude a transparent material medium, I didn't mean to insist on it). Perhaps you will comment further?Chjoaygame (talk) 10:00, 12 November 2011 (UTC)

We are not here to understand Planck, we are here to understand and explain how things work, using people like Planck and everybody else as guides. The fact that it is a "straight lift from Planck" does not make it true. If Planck said that, then it is out of context, or he is wrong, or I am wrong, and we have to get it right by understanding things, not by blindly quoting Planck. The point that I am trying to make is that black body radiation can exist at a point in space without there being material at that point, and just because you cannot find where Planck said this does not make it untrue. Another example of a thoroughly bad statement is:

In his law of thermal radiation in equilibrium, for radiation from any point on the surface of a perfectly black body, Gustav Kirchhoff stated that in any direction, the absorptivity and emissivity match perfectly. Consequently, the interface of a perfect black body with its contiguous medium obeys Lambert's cosine law.

No. The conclusion is not a consequence of Kirchhoff having said it. It's the truth, reference Kirchhoff. Furthermore, if we are going to introduce absorptivity and emissivity we need to quickly and cleanly explain how it follows that a black body obeys Lambert's law, using "it follows that" instead of "consequently". Furthermore "perfectly black body" is redundant. Its a black body. If its not, its a grey body, or whatever. And furthermore, why use the ponderous phrase "the interface of a perfect black body with its contiguous medium" when you could say "the surface of a black body"? Just because the person who translated Planck had a fondness for impenetrable language, doesn't mean we have to follow suit. And furthermore, it is the radiation which obeys Lambert's law, not the surface, so the statement is not only ponderous but wrong. If we leave the reader in a state of awestruck befuddlement, with vast respect for the impenetrable thoughts of the masters and for the editor's mastery of Latin, we have failed. If a reader has to have a dictionary by his side in order to wade through the article, we have failed. e.g. why use "punctate" and "contiguous" when you can say "point-like" and "neighboring"? Don't use a Latin word if there is a serviceable Anglo-Saxon equivalent, unless it's common usage. If the reader walks away laughing, saying "I didn't realize it was so simple", then we have succeeded. PAR (talk) 13:24, 12 November 2011 (UTC)

I avoided the terms absorptivity and emissivity because they did not seem quite appropriate, but another editor put them in in place of my words. Perhaps you could take that one up with him. In this game there is a difference between a Kirchhoffian perfectly black body and a Planckian black body which is not quite as perfect as the Kirchhoffian one. Contiguous means touching, while neighbouring means nearby, but not necessarily touching. The term 'interface of a black body with its contiguous medium' emphasizes that the interface is a shared thing not only a property of the black body, a point carefully emphasized by Planck. Planck is a secondary or tertiary source. He wrote many respected textbooks and handbooks and the like. Yes, good point, I agree that the radiation obeys Lambert's law, not the interface. I will fix that and "punctate".Chjoaygame (talk) 13:40, 12 November 2011 (UTC)

I wasn't blaming you, I was blaming the sentence whoever wrote it, and yes, I see your point about the difference between contiguous and neighboring. LOL - I never heard of punctate before, and if I hadn't suffered through three years of &$%#$#^& Latin, I would never have been able to guess its meaning. PAR (talk) 16:22, 12 November 2011 (UTC)

If anyone deserves to be called punctilious it would be Chjoaygame. --Vaughan Pratt (talk) 01:21, 13 November 2011 (UTC)

Perfect mirrors are good enough. What happens in this (theoretically ideal) case is that when a photon collides with a mirror and is reflected, there is an additional momentum transfer to the photon apart from the perpendicular component being reversed, due to the thermal motion of the atoms in the mirror. This will cause the photons and the mirrors to eventually achieve mutual thermal equilibrium. Count Iblis (talk) 20:53, 13 November 2011 (UTC)

That makes perfect sense, but wasn't what I took "perfect" to mean. What if the mirrors all start out at absolute zero? You have to assume that the atoms of the mirror aborb some energy and momentum from the reflecting photons to make it work. Is this still what is meant by "perfect"? If so, OK, but if you take a perfect mirror to be one where the reflected photon has the same energy as it came in with, then I think it doesn't work (or you need photon–photon interactions t make it work, but I don't know about those). It sounds to me like it gets ridiculous to pretend that it can work when the walls of the cavity don't exchange energy with the photons, which is what perfect mirror suggests. Dicklyon (talk) 21:07, 13 November 2011 (UTC)

Planck's perfectly reflecting rigid walls were Dicklyon's "perfect" mirrors which interact with the light in perfectly elastic collisions with no thermal motion of the atoms in this ideally perfect case that Planck was talking about. Planck knew that if the mirrors had thermally moving atoms they would not be perfect in his sense; that is precisely why he demanded the speck of carbon, which did have the thermally moving atoms that are needed for his theory.Chjoaygame (talk) 22:19, 13 November 2011 (UTC)

Yes, if there are no photon-photon interactions, then the inner walls or a black speck have to thermalize the photons by absorbing some radiation and re-emitting it as thermal radiation. Perfect mirrors don't do this, the incoming photon and reflected photon have the same energy and angle of incidence equals angle of reflection. Photon-photon interactions (or "collisions") thermalize photons the same way massive particle collisons thermalize the particles. You could have a box with a gas in it and if the particles collided perfectly elastically with the walls (like a perfect mirror does to photons), they would still thermalize due to their collisions. Otherwise they would not. PAR (talk) 22:31, 13 November 2011 (UTC)

an old conundrum sorted out

At 01:49, on 1 October, the article contained the following (with the heading at a higher level, here reduced to make it fit on the page, with text alignment also changed here for the same reason):

Various forms of expression of Planck's law

As a function of frequency $\nu$, Planck's law is written as:

$I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.$

This function represents the emitted power per unit area of emitting surface in the normal direction, per unit solid angle, per unit frequency. It is a specific radiative intensity.

This was correct and in agreement with the sources, which are explicitly written in terms of specific (radiant) intensity.

But it was different from a previous version.

At 23:38, on 22 May 2010 the article contained the following:

In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths emitted in the normal direction from a black body at temperature $T$. As a function of frequency $\nu$, Planck's law is written as:

$I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.$

This function represents the emitted power per unit area of emitting surface, per unit solid angle, and per unit frequency. Black bodies are Lambertian objects, which means that the radiance is proportional to the cosine of the viewing angle. Therefore, the spectral radiance of a black body surface viewed from an arbitrary angle $\theta$ is $I(\nu ,T,\theta )=I(\nu ,T)\cos(\theta )\$.

This is also correct as to its statement in the article. It is not however in agreement with its source, Rybicki and Lightman who wrote in terms of specific intensity not in terms of spectral radiance as stated in this form of the article.

One can see differences in the forms of expression. The main difference is between statement in terms of spectral radiance and in terms of specific intensity.

When the statement is in terms of spectral radiance, the qualification "emitted in the normal direction" is strictly necessary. When the statement is in terms of specific intensity, that qualification is redundant. We now know this, but I did not know it until now, and I suspect that others may not have had very clear ideas about it, otherwise they would have picked up that the second above-listed statement was not in accord with its cited source. I vaguely recall some discussion as to whether or not that qualification was necessary, but I don't recall that discussion raising the difference between the source's "specific intensity", and the article's "spectral radiance".

I recall leaving that qualification in place, not because I thought it necessary, but because I thought it provided absolute and explicit specificity, which was appropriate for a definition, but could be relaxed by subsequent reference to Lambert's cosine law and to the fact that a perfectly black surface obeys that law.

But now that difference turns out to be crucial and is not present in the current version of the article. In general, unqualified, Mandel and Wolf's and Goody and Yung's and Paltridge and Platt's identification of SI spectral radiance and traditional specific intensity is invalid.

But there is one specific case in which that identification is valid: when the actual surface of emission is constrained to be normal to the line of sight of the observer.Chjoaygame (talk) 21:06, 10 November 2011 (UTC)

Or when you specify, as does the Wikipedia article on radiance, that the area referred to is the projected area of the emitting element, not the area of the element itself. Then you may say that the radiance is independent of angle, and is valid for an angle to the normal. PAR (talk) 00:41, 11 November 2011 (UTC)

The radiance is independent of angle of view just when the surface is Lambertian, but not otherwise. The general terms do not require a Lambertian surface for their definition. In the present case, the perfectly black surface is Lambertian, but that is not part of the distinction between the general terms spectral radiance and specific intensity. The article at present does not explicitly say that the perfectly black surface is Lambertian.Chjoaygame (talk) 03:51, 11 November 2011 (UTC)

It doesn't really need to be said. It's enough to say that it's black, i.e. that it absorbs all the light that falls on it. Then it can't help but be Lambertian. Dicklyon (talk) 04:47, 11 November 2011 (UTC)

Its obviously a Lambertian reflector, zero at any angle, but to say it can't help but be a Lambertian emitter? That's not obviously true. Is it in fact true at all, without some additional assumptions? And to Chjoaygame, what IS the distinction between spectral radiance and specific intensity? I don't believe it has anything to do with how some observer is oriented, its a property of some point in space and time, independent of an observer. PAR (talk) 06:54, 11 November 2011 (UTC)

The obviousness comes from thinking about thermal equilibrium, in an isotropic photon cloud; the blackbody emits the same as it absorbs. Dicklyon (talk) 00:00, 12 November 2011 (UTC)

Agreed that the distinction between spectral radiance and specific intensity has nothing to do with how some observer is oriented. The distinction is as to the source. For the specific intensity, the area of relevance is that of the actual source, oriented however it may be with respect to the line of sight from source to observer/detector, while for the spectral radiance, the area of relevance is the projected area, which is normal to the line of sight.Chjoaygame (talk) 10:34, 11 November 2011 (UTC)

Source, please. Is that how you're reading Rybicki and Lightman? I could well be wrong, but I read them as using "specific intensity" as a now-outdated 1970's synonym for what we now call spectral radiance. If the distinction you're claiming existed for them, the term "spectral radiance" would appear in their book, but it doesn't, instead they use "specific intensity" indistinguishably from how we use "spectral radiance" today. Where did you get your claimed distinction from? --Vaughan Pratt (talk) 08:08, 12 November 2011 (UTC) Sorry, thought I'd deleted that. --Vaughan Pratt (talk) 01:41, 15 November 2011 (UTC) Dear Vaughan Pratt, you put this comment right in between two adjacent lines of mine, the later line correcting the earlier. If you read the correcting line, right below this one, perhaps you may like to comment again. Yes, my comment to which you are commenting here is wrong: that's why I wrote a correcting line immediately after it. Would it be easier for you if I strike it out, a statement on the next line that it is wrong not being good enough for you?Chjoaygame (talk) 09:00, 12 November 2011 (UTC)

Please see the above section "Terminological conventions" for a major correction about this.Chjoaygame (talk) 13:56, 11 November 2011 (UTC)

Ok, good. I try not to think of radiance or intensity as involving an observer out there somewhere, it only involves a particular direction, and that direction is defined at the source. That makes the radiance a completely local thing, not in any way involved with some observer who is not local to the source. Also, when it comes to intensity, you don't need an emitting area, you could be immersed in a radiation field with no actual physical emitting surface. The area is a mathematical device to define what you mean by intensity, by specifying how many photons go thru the area into a small solid angle about the normal. I like to think of a very narrow cone representing a small solid angle. Clip the cone near its vertex, perpendicular to the cone's axis. That forms a small circle and the intensity is proportional to the number of photons which pass thru that circle and remain in the cone forever. This number will change as the cone's direction is changed, unless the radiation is isotropic. It will also change as you move the little circle around in space, unless the radiation is homogeneous. PAR (talk) 19:34, 11 November 2011 (UTC)

PAR, when you say by specifying how many photons go thru the area into a small solid angle about the normal I'm guessing what you really meant was the energy going through that area. Number of photons differs from their energy in a more interesting way than you might think. The difference itself should be obvious, what's not so obvious is how that difference could be more interesting than expected. (Me being my usual cryptic self.) --Vaughan Pratt (talk) 07:58, 12 November 2011 (UTC)

Agreed. Nevertheless I like to think of an actual experiment to characterize a measurement. Somewhere, placed along a line in your "direction defined at the source", I like to think of a detector, perhaps with a telescope or some aperture arrangement, perhaps a photon counter if you like, of the transferred energy. My detector is located in the base of your very narrow cone. Perhaps you don't like to think that an observer places and reads the detector. I seem to recall I have said somewhere that the emitting surface is virtual or purely mathematical. So I think we agree apart from our differing tastes in imagining or not imagining an experimental set-up.Chjoaygame (talk) 23:25, 11 November 2011 (UTC)

Chjoaygame, this article is about radiation from a black body, not about what an observer of that radiation might see. These are two very different things, and if you don't keep that distinction clear you will draw absurd conclusions. --Vaughan Pratt (talk) 07:51, 12 November 2011 (UTC)

Dear Vaughan Pratt, the word "might" gives some very important latitude; we seen to agree that what I like to think about includes what might be. I could leave it at that, but I guess you would think I was missing your point, so I will continue, under some feeling of duress. Physics is about potentially objective facts of nature, and objectivity is established by concordance of observations between several observers. Things that are such that there is no way in which they might be observed are not potentially objectively observable and so are not potentially objective facts of nature and in my view are not in the purview of physics. While I recognize a logical distinction between "radiation from a black body" and "what an observer of that radiation might see", I do not recognize a useful physical difference there. So I do keep the distinction clear as to logic, and I cleave to what might potentially be objectively observed. This is letting you drag me unwillingly in a philosophical direction and is probably not useful to pursue far here. I will be very unwilling to be dragged further in this direction. I do not think right here it would be useful for me to say more.Chjoaygame (talk) 09:25, 12 November 2011 (UTC)

To Vaughan Pratt, yes, I was not being precise. I conceptualize radiation as photons, to get a picture in my mind, then make the conversion to energy if need be. If I make a statement about photons in which the translation to energy is problematic, please let me know. To Chjoaygame, this is classical physics, so we treat phenomena as objectively real without the need of an observer. Leave all the pondering about objective reality to the quantum physics articles. We are also dealing with a local theory, so we need to describe it using local terms. Using an observer outside the locality we are discussing might help conceptualize the local situation, but should not be used in the article to define a local concept. For example, we should try to talk about a direction at a point, not the direction of a line to some observer. Talk about an infinitesimal area element, not a finite area element. Talk about an infinitesimal solid angle, not an extended cone out to infinity, (the way I did above). PAR (talk) 12:13, 12 November 2011 (UTC)

The "interesting point" I was hinting at is that there is an explicit form of Planck's law specifically for number of photons, namely $2(\nu /c)^{2}/(\exp(h\nu /kT)-1)$, or $2\lambda ^{-2}\exp(h\nu /kT)-1)^{-1}$ when νλ is substituted for c. The units are as for spectral radiance, with the omission of Joules. It gives a fourth shape, with peak at 1.593624260040040092... kT/h Hz, even lower than the frequency peak. It's on p.117 of Caniou. --Vaughan Pratt (talk) 02:38, 15 November 2011 (UTC)

Yes, this is classical physics. But so far as I know, classical physics is based on experimental observation. You are really dragging me into philosophy here. But it seems I have to respond. Objectivity is achieved by concordance of observations between several observers. Another term for it is experimental reproducibility. Do you have some other definition of objectivity that you think I should know? Classical physics is full of pictures of eyes looking at things and suchlike. But it doesn't worry me. If you don't like how I word it, please change it how you like. Ah, yes, now I see the offensive words "from every direction of view". I have now changed them to "in every direction". If there are more offensive words, please correct them.Chjoaygame (talk) 12:40, 12 November 2011 (UTC)

Ah, yes, finite area element. That's how my source, Born and Wolf 1999 do it; but then what would they know about physics? There is no intention of letting it change during the process of interest. The point is that it doesn't change during the process. I didn't actually say that the element was finite, but if you want the word infinitesimal in there, I have no problem with that. If you want me to change it, or if you want to change it yourself, from $\delta A\,$ to $\mathrm {d} A\,$, I am happy to comply with your wish. Quantum mechanical people like to emphasize how they take the observer into account, and part of their way of doing this is to make out that classical physics doesn't do so. But an important part of the story of the discovery of Planck's law is about gradually improving observational technique.Chjoaygame (talk) 12:58, 12 November 2011 (UTC)

"Pondering about objective reality". I would have thought that you and Vaughan Pratt are dragging that in here against my will. Objective reality, however, is not under the sole governance of quantum mechanical experts; they tend think about it in their own rather dogmatic and, dare I say it, bizarre ways, which, like you, I do not wish to discuss here.Chjoaygame (talk) 13:08, 12 November 2011 (UTC)

To Vaughan Pratt you say that you conceptualize radiation as photons. In the next sentence, to me, you say we are talking about classical physics. Hmm.Chjoaygame (talk) 13:12, 12 November 2011 (UTC)

That was me who said that, and I just knew someone would catch that :) Lets just say that I prefer to conceptualize electromagnetic radiation as possessing a perfectly punctate nature without specifying that the aforesaid nature is perfectly photonic, with all of the quantum mechanical manifestations and implications attributable thereunto. PAR (talk) 13:39, 12 November 2011 (UTC)

That's what you tell Vaughan Pratt, but what about me to whom you say we are doing classical physics here?Chjoaygame (talk) 13:44, 12 November 2011 (UTC)

Ok, but this falls into the category of conceptualization, not article writing. There's nothing wrong with thinking about EM as point particles unless you start doing interference effects, etc. Its easier for me to think of photons carrying energy and passing through an area element than to think of a wave carrying that energy, or to count how many photons go into a solid angle, rather than how much properly directed wave energy goes into it. I think we should be careful about talking about photons in the article however, unless we are talking about quantum mechanics, like how the classical picture can fail when reaching a quantum limit. PAR (talk) 16:15, 12 November 2011 (UTC)

I suppose I was being a bit intrusive into your mind. But (Davisson Germer 1927) when a particle such as an electron interacts with a large thing such as a nickel crystal with a linear periodicity $L$ it can gain or lose momentum in amounts $\Delta p$ only according to the quantum rule $L{\text{ }}\Delta p=h$. If the base period of the crystal is $l$, then $L$ can take values $L=l,{\text{ }}l/2,{\text{ }}l/3,...$ (how do you force a typographic space in a situation like that without forcing the big font?). When these quantal changes are distributed statistically according to Fourier theory, particle by particle the interference pattern builds up according to Laue-Bragg rules. In the present context I hardly need say this works also for the Young two-slit experiment, and that Young was wrong to think that his experiment proved the wave nature of light. No need to think about waves at all. Of course there is no law against thinking about waves, and indeed for teaching purposes it is to be rigorously insisted upon in a random manner, now yes, now no, so that the dogma of wave-particle duality can be safely enforced, that is to say the dogma that for any one kind of quanticle you must sometimes think in terms of particles and must sometimes think in terms of waves and the students will be duly baffled (when of course in reality you always have the choice, for every kind of atomic process, because of the optico-mechanical analogy allowed by Hamiltonian mechanics). Where would we be without a good supply of choking mystification?Chjoaygame (talk) 20:21, 12 November 2011 (UTC)Chjoaygame (talk) 21:38, 15 November 2011 (UTC)

emissivity and absorptivity

The fact that "emissivity and absorptivity are both unity" does not mean that "the absorption and emission match perfectly" "from any point on the surface of a perfectly black body". In fact, in the general sense, the absorption is never equal to the emission except inside a cavity inside a blackbody. Q Science (talk) 22:39, 14 November 2011 (UTC)

I do not intend to be disputatious, and therefore wrote "with respect". This sentence is referring to Kirchhoff's perfectly black surface. For Kirchhoff (in contrast to Planck), the perfectly black surface was of an infinitely small thickness, and absorbed all; no incident radiation penetrated into the interior of the body. Kirchhoff's perfectly black surface in equilibrium emitted what it absorbed, so emitting all that came from it. For Kirchhoff's perfectly black surface, indeed the absorption and emission did match perfectly, in every direction, I think because of the Helmholtz reciprocity principle which he explicitly quoted in full. The terms "absorptivity" and "emissivity" are newcomers (e.g. Liou 2002), many orthodox texts of the twentieth century (e.g. Chandrasekhar 1950) using the terms "absorption coefficient" or "coefficient of absorption" and "emission coefficient" or "coefficient of emission". Balfour Stewart had written "The absorption of a plate equals its radiation." The point of the stage of reasoning, on the way to Kirchhoff's law, in the present article sentence, was not to compare dimensionless absorptivity with dimensionless emissivity, but to compare dimensioned absorption with dimensioned emission, a comparison that had to be established on the way to comparing present-day absorptivity with emissivity. For Kirchhoff, it was about "radiating power of the body", which was a dimensioned quantity, and about the "power of absorption of the body", which was a dimensionless ratio. I wanted to avoid that slightly complicated mode of expression at this point, so I settled for "absorption" and "emission" and used the words "match perfectly" instead of "are equal", as reflecting how I think Kirchhoff was thinking. I still think that is ok, but if you continue to find it unsatisfactory I will be happy to see another wording that conveys the meaning in context.Chjoaygame (talk) 23:44, 14 November 2011 (UTC)Chjoaygame (talk) 00:20, 15 November 2011 (UTC)

Perhaps I am not understanding you. What I am suggesting is that if blue light falls on a blackbody, 100% is absorbed. However, the energy emitted is based only on the temperature of the body and not on the frequency of the light absorbed. Thus, at thermal equilibrium, the total energy absorbed is equal to the total energy emitted even if the two spectra are not the same. As a result, the absorption and emission may, or may not, "match perfectly".

BTW. in Chandrasekhar 1950, the "coefficient of absorption" is NOT equal to the "coefficient of emission". In fact, the ratio is defined to be the Planck distribution. (I am not sure I understand that either.) At any rate, if we are having this much trouble understanding each other, then someone reading this article has no hope. Q Science (talk) 00:51, 15 November 2011 (UTC)

Yes, absorption is the rate of energy absorption, emission is the rate of energy emission. When the radiation field is in equilibrium with the black body, then yes, of course they are equal. When the radiation field is not in equilibrium, it is not true. Q Science's blue light scenario above, for example. For another example, if you have a black body in a vacuum, with no EM radiation other than what the black body emits, then absorption will be zero, emission will not be zero, it will be black body emission. The Wikipedia articles on absorptivity, absorbance, absorptance, absorption coefficient, are a mess and need to be straightened out so that we can settle on some terminology. $/alpha$ (the absorptocity or whatever) is the ratio of the amount absorbed to the amount incident (spectral). The emissivity is the ratio of the amount emitted to the amount that would be emitted if it were a black body at the same temperature. They are equal, no equilibrium needed. The whole subject of radiation transfer is being ignored in this article, and it should not be. Absorption coefficient is the infinitesimal change in a beam of light as it moves an infinitesimal distance in an optical medium. Maybe per unit mass, I think. It is for describing things in an optical medium, not something applicable to a surface, which is what we are restricting things to. PAR (talk) 02:29, 15 November 2011 (UTC)

The sentence was originally in a particular context, intending to provide some answer to the question why radiation from a Kirchhoff-perfect black body obeyed Lambert's law. Now the context is different, and the sentence may now be considered as orphaned. The presence and purpose of the sentence is in need of reconsideration. Perhaps the whole sentence is now irrelevant or redundant. I do not wish to try to remedy this. Anyone is of course free to delete or otherwise edit it.Chjoaygame (talk) 03:58, 15 November 2011 (UTC)

material body

"There are two ways that radiation may leave such an area element - reflection and emission." This assumes, without it being stated, that the "material body" is perfectly opaque.Chjoaygame (talk) 08:20, 13 November 2011 (UTC)

Good point - I will fix that. That falls under "absorption", but just not absorption by the surface. Also - on Stefan-Boltzmann, do you have a reference? Rybicki and Lightman give the constant as 5.67 not 5.6700400 times ten to the something. PAR (talk) 15:19, 14 November 2011 (UTC)

Kirchhoff's black surface was of "infinitely small thickness" (Gurthrie's translation) [bei unendlich kleiner Dicke] and absorbed all. Planck's interface was a two-dimensional mathematical fiction that neither absorbed nor emitted, because it was not a body; it only reflected and transmitted with refraction.Chjoaygame (talk) 21:38, 14 November 2011 (UTC)

The correct source for all constants is CODATA. Stefan-Boltzmann = 5.670 373(21) × 10−8. Q Science (talk) 22:53, 14 November 2011 (UTC)

If you express temperature in hectokelvins (so essentially all natural terrestrial temperatures except lava and lightning are between 1 and 9.999, and 288 K becomes 2.88 hK), the constant is just 5.67... with no "ten to the something." The radiation from one sq m (whether flat or spherical) at 300 K is then simply 5.670373*81*π = 1442.94 watts. --Vaughan Pratt (talk) 02:54, 15 November 2011 (UTC)

I'm confused. Where does the π come from? Q Science (talk) 06:30, 15 November 2011 (UTC)

Sorry, "radiation" was too vague, I meant radiant flux, not radiant intensity. The π is explained in Lambert's law in the section about the relationship between intensity and flux (for their "luminous" read "radiant"). --Vaughan Pratt (talk) 17:23, 15 November 2011 (UTC)

Stefan's equation provides the total energy (watts/m²) emitted by one square meter into 2π steradians. To get the radiant flux, I think you need to divide by π and somehow convert to photons/(s·cm²·sr). However, I still find this all very confusing. Q Science (talk) 20:57, 15 November 2011 (UTC)

Emitted?

The first line refers to "Planck's law describes the amount of energy emitted by a black body". Surely not, doesn't Planck's law describe the spectrum of cavity radiation? The important feature of a cavity is that the radiation is confined to the cavity, it isn't emitted. --Damorbel (talk) 09:10, 22 November 2011 (UTC)

It describes both. Poke a small hole in the cavity, it emits Planckian radiation. There is also the concept of a black body. Suppose you had a star like the Sun, but no limb darkening, emitting Planckian radiation. It would be a spherical "black body". PAR (talk) 15:34, 22 November 2011 (UTC)

A "small hole" is not very scientific! Surely a small hole is just a device invented to allow for the concept of gaining access to the cavity. The point being that the "small hole" allows energy to leak out of (or into) the cavity. This is not really a trivial objection because, if energy is allowed to exit (or enter) the cavity through the small hole, then equilibrium is destroyed and equilibrium is a fundamental requirement for Planck's Law to be consistent. I have long wondered about this and I'm sure I'm not alone. --Damorbel (talk) 16:32, 22 November 2011 (UTC)

Darmorbel is quite right to wonder about this, and he is right that he is not alone in that. His point is not a trivial objection. Every physical problem needs to be considered with this kind of objection very much in mind. The task is to find a model that makes the effect of the objection negligibly small. Finding such a model is an important part of examining a problem in theoretical physics. It is part of understanding what is important for the problem.

In accord with this, Planck thought about these problems for thermal radiation and he stated his law for thermodynamic equilibrium radiation in the interior space of 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 in a cavity. 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. For the maintenance of its thermodynamic equilibrium, the black body must be supplied from its exterior with uniformly incident heat radiation at its own temperature. The walls of the cavity must be maintained at the given temperature by a heat bath so large that no events in the cavity can overload the capacity of the bath to supply heat at the given temperature. Planck pointed out that to observe the radiation in the cavity, one needs to imagine a small hole through which to make observations. The hole must be small in comparison with the size of the cavity. Some textbooks (e.g. Hottel and Sarofim 1967) and some early research publications give explicit calculations of how small the hole has to be to yield a prescribed closeness of approximation. Such calculations are necessary for a thorough understanding of the physics.

The idea of a large heat bath, or of a small hole, is not unphysical; it is an idealization of a kind that is always involved or implied in physical theory. One thinks in terms of limiting cases, with the hole being progressively smaller in comparison with the size of the cavity, until, to the accuracy that one desires, the hole makes negligible difference to the values that one finds. All physics is about approximations of this kind. For example, conceptually, Newton's laws require an ideally inertial frame, which must be supplied by a reference body so heavy that it entirely dwarfs the various forces and momenta of the bodies of interest, and that they cannot affect its state of rest or uniform motion in a straight line.Chjoaygame (talk) 17:16, 22 November 2011 (UTC)Chjoaygame (talk) 17:19, 22 November 2011 (UTC)

A "small hole" is code for something very scientific. It's a limiting process. To paraphrase what Chjoaygame said, the smaller the hole, the closer you get to Planck's law. Pick how close you want to get, and there will be a hole that provides that closeness and any smaller hole will do even better. That is the definition of a "small" hole. PAR (talk) 18:16, 22 November 2011 (UTC)

Yes PAR, you are quite correct, I have raised a false dilemma. It is quite reasonable (but I still think it is distracting) to consider a "small hole" since the internal distribution of energy will converge to the equilibrium case as the "small hole" approaches zero. Does anybody think this point is worth making? --Damorbel (talk) 06:16, 23 November 2011 (UTC)

I think its worth making, but not as precisely as I stated it above. Something about the hole must be small enough so as not to appreciably disturb the equilibrium. The hole is the thing that allows one to access the blackbody radiation, its the best laboratory realization of a black body emitter, but its not the only way it can happen. PAR (talk) 06:36, 23 November 2011 (UTC)

Next step

An answer to my suggestion that we might be able to work things out together here for ourselves has been given, in answer PAR's question "How does this help?", by Headbomb's writing "It does not."

I think mediation would demand more time from editors than is appropriate and might be impracticable. I think Vaughan Pratt's suggestion, that his own complaints define our problem, would not make a compelling cae and is not the best approach.

I think the most practical option may be to ask for Headbomb to be blocked from editing this article and the article on History of Planck's law for one year. Does anyone know, is it likely that such a request would be granted? Does anyone know how to do this, in specific detail? How much support is there for this option?Chjoaygame (talk) 15:14, 8 November 2011 (UTC)

I don't think that there's any basis for seeking to block headbomb. I'd rather block you, so we wouldn't have you stirring things up so. Dicklyon (talk) 15:19, 8 November 2011 (UTC)

Thank you Dicklyon for letting us know how you feel. I can agree with you this far, that we have a tedious problem here.Chjoaygame (talk) 17:19, 8 November 2011 (UTC)

Agree with Dicklyon here. You're not interested in improving the article, only interested in "winning". Headbomb {talk / contribs / physics / books} 16:46, 8 November 2011 (UTC)

Headbomb, I tried in the section issues above to start dealing with our problem by working things out together here, and would still prefer to do so, but your answer above, "It does not", made me think you did not want to to do so. If you do want to do so, I would still prefer that.Chjoaygame (talk) 17:19, 8 November 2011 (UTC)

I'd prefer to be rid of people who cause trouble on article pages and don't follow content policies than one who complain on the talk page. Dmcq (talk) 16:58, 8 November 2011 (UTC)

Policy guidelines are being followed. Stop lying. Headbomb {talk / contribs / physics / books} 17:47, 8 November 2011 (UTC)

->WP:DUE and you sticking in what you think should be in rather than being guided by neutral point of view. ->WP:CIVILITY as well. Can we just have people stopping attacking each other please? I provided a counterpoint argument to Dicklyon not that I actually want rid of anyone. It would really really help getting somewhere towards a constructive environment if you toned down your language. This is the sort of thing I was suggesting mediation instead of. Dmcq (talk) 17:53, 8 November 2011 (UTC)

and p.s for Chjoaygame, do you think you could try and just avoid referring to Headbomb and just refer to what has been stuck in instead please? If you find you've written 'Headbomb' down have a good try at rewriting. Dmcq (talk) 18:01, 8 November 2011 (UTC)

I agree with Dmcq. Being an editor on Wikipedia is not easy if you want to do it right. Every time you focus on the motives of someone who disagrees with you, you are doing it wrong. Every time you characterize the behavior of someone who disagrees with you, you are doing it wrong. The edits and the reversions speak for themselves. When you focus on the article and how to improve it, you are doing it right. If you think the fact that I (regretfully) sometimes break these rules myself proves that these rules are wrong, then not only am I at fault, but so are you.

What I want is that everybody, in their edits, try to "do it right" no matter how painful and unfulfilling it is. No personal attacks, no moaning about how many times one has been reverted, no multiple reversions. In my opinion, this article is in better shape than it was a few weeks ago, and its as a result of all this s**t, rather than in spite of it. How cool would it be if we focused on getting this article good article status?. Then we could attack the real enemy, the people who would deny us this. Those who are more interested in drama will drift away, and the people who are really interested in this article will have some work to do.

If the antagonisms are irreparable, and people cannot resist moaning and attacking, then people who continue to do it wrong should be blocked for a week at a time until they do it right. Nobody gets punished for the past, only the future. PAR (talk) 19:10, 8 November 2011 (UTC)

@PAR In my opinion, this article is in better shape than it was a few weeks ago If you mean the tables and bibliography are more neatly arranged, then yes. If you mean it is factually more accurate, then a resounding no. Currently it contains a lot of rubbish, which I see has been continuing to accumulate during the past three weeks.

With regard to accuracy if not neatness, the article was in far better shape in September. Accuracy-wise it's being ripped to shreds. I've spelled out why in detail, with many repetitions. At this point I'm not in a good position to repeat myself yet again, which will have to wait until mid-December. Meanwhile I see no one has attempted my three-part true-or-false question, from which I infer that no one feels they understand this material adequately.

Is there a radiation physicist in the house? If not this is one Wikipedia article that is doomed to mediocrity. --Vaughan Pratt (talk) 04:34, 28 November 2011 (UTC)

Three problems in the different-forms section

I've been unable to participate lately as I have an upcoming deadline that will keep me away from all but very brief interactions with Wikipedia for the next three or four weeks. I'm not sure if any of the following problems are in the process of being resolved above, but since they still exist in the article I'll bring them up anyway. I don't mind being contacted by email in the interim if it will clarify anything.

Problem 1

The unnecessarily belabored and questionably sourced treatment of the various possible constant-factor scalings of $B_{\nu }$ and $B_{\lambda }$ in the table. This has been discussed to death, and, like the climate debate, has come to a standstill, with people simply talking past each other. 'Nuff said on that. If Chjoaygame and Dmcq and anyone else want to replace the table with the original two functions and the observation that scaling the input of either of them by m requires scaling the output by m⁴, illustrated say with the examples of ν̃ = cν and ω = ν/2π, I will strongly support that. These should be more than sufficient to convey the idea.

Problem 2

The identification of isotropic and Lambertian radiation. These are not the same thing, contrary to what the article claims. A simple test of whether a radiator is radiating isotropically or according to Lambert's law is to enclose it in a sphere large relative to the radiator. If the interior of the sphere is uniformly illuminated, it is an isotropic radiator. If there is a brightest point on the sphere (call it the "north pole"), and the equator is in darkness, and the 60th parallel (the border between British Columbia and the Yukon) is receiving half the intensity received by the north pole, the radiation obeys Lambert's law. More completely, it obeys it when the intensity is proportional to the cosine of the colatitude (angular distance from the north pole). Isotropic radiation is easier to calculate with than radiation that follows the cosine law, so if they were the same thing one would have to ask, why bother with Lambert's law?

Up to a constant factor, Planck's law as a function of both frequency and temperature holds equally well for both kinds of radiation. However this does not make them the same, in fact far from it, as common sense should indicate. How could the 60th parallel be receiving half the illumination as the north pole and all of it at the same time? Only if 1 + 1 = 1. Abstract arguments about rotating imaginary surfaces that supposedly "prove" the two kinds of radiation are the same only serve to show how easy it is to slip fallacies into that kind of argument.

Suppose you are in a region of space where there are no physical surfaces. Suppose the radiation field is isotropic. That means that no matter which direction you choose, the rate of energy into some small fixed solid angle about that direction will be the same. Now create an imaginary surface element, and pick a side of that element. The radiation coming from the other side and out that side (or vice versa) will be Lambertian and it won't matter which way you orient the element.

The above definition is the definition of isotropic radiation, not the one you gave. But to continue, suppose you pass a plane thru your area element. If you blocked off all the radiation thru the plane except that passing thru this area element and projected it onto a distant hemisphere, yes it would have a cosine dependence, dropping to zero at the horizon. But its this "blocking off" to a constant area that causes the problem. You cannot have it both ways. If you pick a direction, and a small fixed solid angle about it, the energy into the solid angle will be independent of angle, but the area that it collects from will vary, going to infinity when the direction is towards the horizon. The projected radiation will be uniform. If you fix the emitting area, then the solid angle that collects from it will vary with direction, going to zero when the direction is towards the horizon. As the solid angle varies, so does the amount of energy collected. The projected radiation will have a cosine dependence, dropping to zero at the horizon.

The above definition is the definition of isotropic radiation, not the one you gave. Guess you'd better fix the article Isotropic radiator then. It claims "[An isotropic radiator] radiates uniformly in all directions over a sphere centred on the source" (a theoretical point source, e.g. the center of the Sun). It also says "At a distance, the sun is an isotropic radiator of electromagnetic radiation." I said essentially the same thing when I wrote "a sphere large relative to the radiator." The advantage of my wording is that the same sphere can be used to distinguish isotropic radiation from radiation obeying the cosine law.

The one puzzling thing about the article is the last sentence of the lead, "Isotropic radiators obey Lambert's law." I can't figure out what that even means for an isotropic radiator, it makes no sense. If there is no preferred direction, what is Θ in cos(Θ)? If I had time I'd work out who wrote that and dig deeper. --Vaughan Pratt (talk) 04:36, 15 November 2011 (UTC)

Isotropic radiator is not the same as isotropic radiation. Isotropic radiation is as defined above. I don't know, but I am guessing from the article that an isotropic radiator is something that if you put a very large sphere around it with the radiator at the center, it will illuminate the sphere evenly. Very large means large with respect to the maximum dimension of the radiator. In that case, not all black bodies are isotropic radiators. If you had a black body in the shape of a pancake, it would radiate more in the direction of its short dimension than in any other direction onto that distant sphere. (Roughly Lambert's cosine law.) Nevertheless, if you picked a point a bit underneath the surface anywhere on the pancake, constructed a thin cone (i.e. solid angle) pointing in some direction with that point as a vertex, the energy radiated into that cone by the area element formed by the intersection of that cone with the surface of the pancake would always be the same, no matter what direction the cone was pointing. That's because blackbody radiation is isotropic. Visually, if you look at any black body, it could have hills and valleys in it, but you could not see them, because your eye collects from a solid angle, not from a fixed surface area on the black body. And that solid angle always delivers the same energy, because blackbody radiation is isotropic. I agree, an isotropic radiator need not be Lambertian, the article is wrong. The sun is not Lambertian, it has limb darkening, but if you put a large sphere around the sun with the sun at its center, it would illuminate the sphere very evenly. PAR (talk) 06:35, 15 November 2011 (UTC)

I am familiar with the idea of a radiative field being isotropic at a point, but I am not familiar with an 'isotropic radiator', and I think it is not a customary locution, and should not be used in our article. Thermal radiation in a cavity in equilibrium is isotropic. I am very uncomfortable with the locution of 'radiation from a specified surface being isotropic', even from a specified perfectly black surface; the radiation from a flat surface is all going away from the surface; that's not my idea of isotropy.Chjoaygame (talk) 07:44, 15 November 2011 (UTC)

The article Isotropy bears both of you out, with plenty examples of what you're calling isotropic radiation. A cavity is one example, a cubic meter of the Venusian atmosphere at an altitude of 1 km (at least at wavelengths on the order of 15-20 μm) would be another. However I disagree with Chjoaygame that there is no notion of isotropic radiator: consider the field from a star, it's not isotropic over a volume but over the center of the star. Being a black body radiator this does not violate the Helmholtz equation, which is for a single frequency. The article Isotropy links to Isotropic antenna which redirects to Isotropic radiator. --Vaughan Pratt (talk) 17:36, 15 November 2011 (UTC)

Yes, "isotropic radiator" has nothing to do with isotropic radiation. It's an unfortunate clash of terminology. But isotropic radiation has everything to do with isotropic radiation from a surface. If you imagine a point and its neighborhood, where radiation is isotropic but there are no physical surfaces, and you construct a small flat area element there, then the radiation coming out of one side of that element obeys Lambert's law. PAR (talk) 17:28, 15 November 2011 (UTC)

I started writing an explanation of why Vauhan is wrong, but in the process convinced myself that he is at least mostly correct: a flat Lambertian radiator will more brightly illuminate the part of the surrounding sphere where the surface normal of the radiator meets it, and will not illuminate the "equator" at all. This is an ideal diffuse emitter—obviously the intensity on the sphere has to go to zero as one approaches the plane parallel to the emitting surface, because there is no line of sight from the sphere to the emitting surface in that plane. What is uniform about a Lambertian emitter is that its surface will appear equally bright when viewed from any angle, because the falloff in the total flux captured by the eye is compensated for by the fact that the apparent (projected) size of the emitter is smaller when viewed from an angle.

Vaughan's claim about the intensity on the inside of a sphere falling to zero at the equator of course presumes that the Lambertian emitter is flat. --Srleffler (talk) 06:13, 17 November 2011 (UTC)

Ok, you've convinced me that I'm underestimating the difficulty of explaining this stuff. Come mid-December I'll have another go at this. --Vaughan Pratt (talk) 23:48, 27 November 2011 (UTC)

Problem 3

The claim $B_{i}(T)=B_{j}(T)\left|{\frac {dj}{di}}\right|$ whenever i and j are in bijection on the positive reals. I pointed out that this was wrong days ago, but seemingly I'm alone on that and everything else thinks the formula is just hunky-dory.

Since T plays no role in the truth of this, let's simplify it to $f(x)=f(y)\left|{\frac {dy}{dx}}\right|$ (following the usual convention that x and y are real variables, unlike i and j which are usually integer variables.) For a simple counterexample, take f to be the identity function, satisfying f(x) = x, and take y = 2x, which is certainly a bijection. Then $\left|{\frac {dy}{dx}}\right|=2$. The statement then asserts $f(x)=2f(y)=2f(2x)$. Since $f(x)=x$ while $f(2x)=2x$, and since this holds for all positive x, we can set x to 1 and obtain 1 = 4. --Vaughan Pratt (talk) 00:38, 15 November 2011 (UTC)

B_i(T) and B_j(T) do not stand for the function B, expressed in terms of i and j, they stand for two different functions B_i and B_j [which we should, in all mathematical rigourousness, write B_i(i, T) and B_j(j, T)]. Or if you want, f(i) and g(j), and not as your "example" says, f(i) and f(j). This should be pretty damn apparent, since one features [spectral variable]³, and the other [spectral variable]⁻⁵. Your example compares B_i(i, T) with B_i(j, T) for some weird reason. You indeed are alone in thinking $B_{i}(T)=B_{j}(T)\left|{\frac {dj}{di}}\right|$ is nonsense because you stand alone in thinking B_i and B_j somehow represents the same function. Which BTW, is exactly what the section is warning against. Headbomb {talk / contribs / physics / books} 05:28, 17 November 2011 (UTC)

No, the reason it's nonsense is because you neglected to specify what the relationship was between the "two different functions" B_i and B_j.

If you're claiming that the result is true no matter what the relationship is, then in particular it will hold when they're the same function, which I showed was impossible.

If you're not claiming that, then you need to specify a relationship between the two functions before what you wrote makes any sense. If you're saying it is sufficient for the two functions to be different then I'll be happy to provide another counterexample. But as it currently stands it is not even wrong, contrary to what I originally wrote, it is just meaningless because of undefined terms. --Vaughan Pratt (talk) 23:48, 27 November 2011 (UTC)

Problem 3

You have missed two important points. First, it is more like

$f(x)=g(y)\left|{\frac {dy}{dx}}\right|$
$f(x)=x\implies g(y)=y/2$
$f(x)=x=g(y)\left|{\frac {dy}{dx}}\right|=y/2*2=y$

which obviously still fails. The other important point is that both blackbody equations are equal to zero at both zero and infinity. As a result, they have a finite integral over that range. No one is arguing that the equations have the same values at the same points, they don't. Thus, the correct equations are more like

$|\int \limits _{0}^{\infty }{f(x){dx}}|=|\int \limits _{0}^{\infty }{g(y){dy}}|$

Q Science (talk) 02:44, 15 November 2011 (UTC)

I mainly wanted to point out that the formula wasn't at all correct, even when f = g. An earlier version pointed out that f != g but I decided that was Too Much Information and deleted it.

Note that you need to strengthen the bijection condition, which isn't enough. The specific bijection (assuming f and g are the two standard Planck functions) has to be precisely y = c/x.

The integral equation is right. The abs functions can be removed by reversing the limits of the second integral. The article used to point this out (I made that edit back around 2008 IIRC) until HB deleted it in the past couple of weeks. Reversing them actually makes sense when you consider that integrating frequency from 0 to infinity should have the same effect signwise as integrating wavelength from infinity to 0. But there is another, simpler, equation that's just as correct: just drop the integral sign and limits and the two d's to give $f(x)x=g(y)y$ where xy = c. One nice thing is that this is true regardless of which way round you assign the two Planck functions to f and g. --Vaughan Pratt (talk) 03:14, 15 November 2011 (UTC)

Suppose you have $y=x^{5}$, a bijection for $x\geq 0$. Then $f(x)=g(y)|dy/dx|$ gives $f(x)=5x^{4}\,g(x^{5})$. The restriction that y=c/x is not necessary. It is necessary that $\int _{0}^{\infty }f(x)dx$ and $\int _{0}^{\infty }g(y)dy$ be finite and positive (they are ultimately energies).

I prefer to think of the defining equation as $f(x)\,dx=\pm g(y)\,dy$ where it is assumed that $f(x)dx$ is finite positive to begin with and the plus or minus sign is chosen to assure that $g(y)dy$ is positive.

Looking at it this way is intuitive - The bottom line is that the energy in an infinitesimal x-band dx must equal the energy in the corresponding infinitesimal y-band dy. Those energies are f(x)dx and g(y)dy and they must be positive.

I agree with all that except for the ±. Its drawback is that it discards information unnecessarily, in this case the antimonotonicity of reciprocation c/x, which differentially shows up as negation. By keeping the endpoints of the infinitesimal strips paired up, you can sum algebraically (i.e. sign-sensitively), which is almost always preferable. The signs cancel nicely in the equation f(x)x = g(y)y since dy/dx = - y/x. Note that the c vanishes; it returns as an arbitrary constant of integration when going the other way by solving the differential equation f(x)dx = -g(y)dy to express f in terms of g or vice versa. Fixing both f and g determines c. --Vaughan Pratt (talk) 04:51, 15 November 2011 (UTC)

The restriction that y=c/x is not necessary. That's only true if you're allowed to pick f based on g and the bijection between x and y, as in your example. In this case when we take f and g to be the two Planck functions it forces the bijection to satisfy xy = c.

But you're right about matching up areas in infinitesimal strips, and the funny thing was we were both typing that essentially simultaneously, except that once again I decided "TMI" (or "walls of text") and deleted it. --Vaughan Pratt (talk) 04:17, 15 November 2011 (UTC)

Perhaps we should be discussing six (6) Planck functions, not two (2). In that case, it will be wrong to assume that y=c/x. Q Science (talk) 06:05, 15 November 2011 (UTC)

For those advocating retaining the table, this is quite right. For each of the 15 pairs of distinct functions in HB's table (it does not matter which function of the pair we call f and which g or it would be 30 pairs), y is either mx (when they're the same shape) or m/x (when they're not), where m is a constant formed by multiplying or dividing by some combination of c and π (both to get HB's highly prized k). For cm⁻¹, include 100.

Every pair uniquely determines m, which is the multiplier or divisor for the input, while the corresponding multiplier or divisor for the output is always m⁴.

The table is far from exhaustive, but would be easy to grow yet further: why stop at a mere 6? On the other hand once you've seen a couple of example of the uniform rule m⁴, why go as far as 6? It makes the article look conceptually clumsy.

This misses the point. Why try to come up with an equation that covers only what we have in the table and nothing more? Look at the Maxwell distribution article. They are using bijections that cover energy, wavelength, momentum, velocity, speed, which steps out of bounds of the table. All are based on the bottom line - a fundamental physical principle - the energy in the infinitesimal band of one variable must equal the energy in the band of another variable, energy is positive and total energy is finite. This is taken from probability theory - the probability in one infinitisimal "band" of one variable must equal the probability in the band of another variable and probability is positive, and total probability is one. And in probability theory even the bijection requirement can be dropped. Every distribution, Maxwell, Maxwell-Boltzmann, Bose-Einstein, Planck, Fermi-Dirac, is essentially a probability distribution. Why not give the fundamental principle here, instead of trying to come up with a neat equation that covers frequency and wavelength only, and has no fundamental physical meaning? Then saying, oh, no, we have to come up with another neat equation that covers some table in some article, and which again has no fundamental physical meaning. What if we want to express Planck's law as a distribution of vector momenta? Then we wouldn't have to add the proviso that the radiation is isotropic - isotropicosity (or whatever the term is) would be implicit in the equation. Why lock ourselves in an artificial box for the sake of some nice-looking equation with no physical significance? PAR (talk) 12:39, 15 November 2011 (UTC)

Apart from your second last sentence, I couldn't agree more. There is only one physically meaningful energy distribution, having a basis-dependent representation as a function, where the basis is the choice of function of frequency. (There is also a physically meaningful photon count distribution, obtained by dividing energy by hν, which would double the number of shapes to six except that four of them are duplicate pairs so there's only four. However I wouldn't put that in unless it's used a lot.) Word it suitably (I can't do so until mid-December) and I'll give it my full support.

Regarding Lambertian vs isotropic, why talk about either Lambertian or isotropic at all when we can pass that buck to the black body article? Black bodies are Lambertian independent of shape, but whether they are isotropic strongly depends on shape. Any uniform sphere, regardless of its emissivity, radiates isotropically, whereas a flat surface does not (there is no radiation in the plane of the surface but lots normal to it). A finite object with no other radiating source nearby can't generate an external "isotropic field" in your sense, but a spherically symmetric one can be an isotropic radiator by virtue of generating a spherically symmetric field. These issues should be dealt with in the relevant articles, to which we can merely refer instead of confusing things with our own prejudices about how black bodies and different shapes of radiator work. We might however want to point out that Planck's law holds not only for spectral radiance but also for spectral radiant exitance as its integral over all directions (which introduces a factor of π while removing the unit sr⁻¹), and moreover when scaled down in proportion to emissivity. --Vaughan Pratt (talk) 19:52, 15 November 2011 (UTC)

Problem 2

Isotropy of thermal radiation is usually referring to a three-dimensional radiative field. Lambert's law refers to a two-dimensional surface; as I see it, the word isotropy is not quite right for it. Planck 1914 does not name Lambert but does repeatedly speak of "diffuse refraction" and of "diffuse reflection". According to Born and Wolf "when [Lambert's (cosine) law] is satisfied, one speaks of diffuse emission or diffuse reflection ....". Hapke also talks of a "diffuse surface" and of "diffusive reflectance"; he says that "no natural surface obeys Lambert's law exactly."

The word for the surface phenomena seems to be diffuse, not isotropic.

For Kirchhoff, the perfectly black surface could absorb and emit and would be Lambertian. For Planck, a perfectly black body had an interface with zero reflectivity. The interface transmitted all from the (semi-)transparent external medium that fell onto it, to the interior of the black medium, and vice versa. The Planckian perfect black body, material-and-interface, would emit diffusely, but of course not reflect at all. Planck's black surface would derive its Lambertian character from the isotropy of the body's material interior radiative field combined with the non-effect of the mathematical immaterial interface.Chjoaygame (talk) 04:48, 15 November 2011 (UTC)

While I agree with most of that, I would add that the cosine law doesn't make much sense when the normal can change direction at different points of the radiator. Which normal do you measure Θ with respect to in cos(Θ)? For the normal to maintain a fixed direction, the radiator must be a flat surface. --Vaughan Pratt (talk) 04:56, 15 November 2011 (UTC)

To solve problems involving non-flat surfaces, you typically have to integrate over the whole surface. If you're not familiar with how that goes, imagine breaking the surface up into a large number of flat or almost-flat sections, and summing up the contribution of each. Each of these "elements" of the surface has its own normal.--Srleffler (talk) 06:19, 17 November 2011 (UTC)

I said "Most of that" because I didn't understand your sentence "Planck's black surface would derive its Lambertian character from the isotropy of the body's material interior radiative field." Sufficiently far into the interior, radiation to the exterior is extinguished. At some sufficiently small distance d from the surface, the Beer–Lambert–Bouguer law kicks in and radiation will be emitted in the normal direction, falling off not as 1/sec(Θ) (Lambert's cosine law in disguise) however but as 1/exp(sec(Θ)) on account of the BLB law. This approaches Lambert's law as Θ approaches 0 because in the neighborhood of 1, exp(x)/e = x, i.e. exp(x) behaves linearly in that (and any other) small neighborhood. I don't know what experimental data supports this, but in any event Lambert's law is derived as an approximate geometric idealization of radiation purely from the surface and not from the interior that you speak of, where Beer's law applies nontrivially when the radiation is not fully extinguished.

I had even less luck with your "combined with the non-effect of the mathematical immaterial interface." What exactly is the result of combining an effect with a non-effect? --Vaughan Pratt (talk) 05:21, 15 November 2011 (UTC)

I am talking about how Kirchhoff and Planck deal with the matter. To get a better idea of what they say, perhaps it would be easier to read them directly than to get them second-hand from me. Kirchhoff's perfect black body works purely from the surface. Planck's doesn't.Chjoaygame (talk) 05:29, 15 November 2011 (UTC)

Meanwhile section 7 of the late Jim Palmer's notes explains the difference between lambertian and isotropic in terms of respectively flat and spherical radiators, exactly as I've been saying (but see below for a caveat).

Incidentally you can read about Palmer's 30 years of teaching at the University of Arizona's College of Optical Sciences here. He is the author of The Art of Radiometry.

Another source of the same information is here, where it says "Lambertian refers to a flat radiating surface."

Another way of expressing the distinction between an isotropic and Lambertian radiator is that the radiator's brightness at distance r is independent of the viewer's position when brightness is defined (in photometric terminology) as respectively luminous intensity (total power received by the viewer from the radiator) and luminance (ditto divided by apparent area). Caveat: by this definition a Lambertian need not be flat. However the utk.edu reference adds "(The flat surface can be an elemental area of a curved surface)", which I would phrase more succinctly as "locally flat." This amounts to requiring that every point of the surface have a well-defined normal which varies suitably smoothly with the point's position on the surface.

To quote a fragment of the introduction to Palmer's FAQ, "There is so much misinformation and conceptual confusion regarding photometry and radiometry, particularly on the WWW by a host of authorities." Amen to that. --Vaughan Pratt (talk) 07:42, 15 November 2011 (UTC)

I have read much of what Palmer wrote, including buying his article. I am not impressed that he is necessarily a reliable source on every question. He is bursting with self-confidence and dogmatism, and he obviously knows a whole lot about it. But his zeal to impose his own personal one right approach makes me feel it isn't always the one right approach for the Wikipedia. I think his dogmatic talk of an "isotropic radiator" is a bit idiosyncratic; it would need a better source than Palmer. I think it is not right for us to write about an 'isotropic radiator' when talking about radiation specified to be coming from a surface. The article on isotropic radiator is specific that it refers to a point radiator. That is not good language. What is isotropic is the radiation field at a point. That is not the same thing as the point being a radiator. No radiation comes truly from a point; it comes only from a finite volume. The field is defined at a point, but that does not make the point a radiator. I suppose I am saying that the article on an isotropic radiator is wrong, but I don't feel like trying to fix it. Just let's not repeat their mistake here.Chjoaygame (talk) 07:55, 15 November 2011 (UTC)Chjoaygame (talk) 08:04, 15 November 2011 (UTC)Chjoaygame (talk) 15:01, 15 November 2011 (UTC)

The Isotropic radiator article was indeed wrong, claiming that every isotropic radiator is Lambertian when the Sun is an obvious counterexample, being isotropic by symmetry but obeying (more or less) the 1/exp(sec(Θ)) limb darkening law I mentioned above rather than Lambert's 1/sec(Θ) law. (This incidentally answers my question there about "experimental evidence.") I took a shot at fixing the lead just now, let me know if you still have concerns with it (but perhaps after reading the following). Incidentally I take back my earlier characterization of "Lambertian" in terms of the total radiation obeying the cosine law, which only needs to hold for each elemental surface of the radiator, allowing an isotropic radiator to also be Lambertian. --Vaughan Pratt (talk) 17:42, 15 November 2011 (UTC)

Chjoaygame, now that you've questioned the authority of both Palmer and Nobel laureate Cohen-Tannoudji, which would you say was less qualified?

Your objection to the notion that a point can radiate is a close parallel to the objection to calculus that the real number field contains no infinitesimals and therefore dy/dx at a point x0 must denote 0/0, which could have any value since every number is a solution in x to x*0 = 0.

This inconsistency is standardly handled by considering dy to be a function of dx as the latter goes to zero about x0, taking dx at x0 to be something like the interval [x0, x0 + dx), or [x0 - dx/2, x0 + dx/2) for faster (linear vs. quadratic) convergence. When this limit exists we say that dy/dx is defined at x0, despite being 0/0.

The counterpart of this for the Sun is to consider a growing sphere centered on it (or equivalently a shrinking Sun centered in a fixed sphere) and see what distribution the radiation falling on the sphere converges to. The limiting case of this is what is meant by a "point source." Stars are millions of times further away than the Sun and therefore nicely illustrate the concept of a point source.

This resolution of the point source paradox reveals in the process that point sources can have different shapes. The limiting distribution of radiation for a shrinking flat source obeys the cosine law, whereas that for a shrinking sphere is uniform. Yet at that limit both shapes have converged to a point.

To make the article Isotropic radiator more precise I would be inclined to phrase the definition as "A radiator is isotropic when its radiation falling on a growing sphere centered on it becomes uniformly distributed in the limit." This is completely consistent with what it currently says, while giving the notion of "point source" a more rigorous meaning. If someone brings this up at the article it may be worth clarifying the notion of "point source." --Vaughan Pratt (talk) 17:42, 15 November 2011 (UTC)

Nobel laureate Planck is my source for saying that a point can't radiate. I didn't question the authority of Cohen-Tannoudji. As I noted in the past, I wrote that you were misquoting him: "I did not contend that Cohen-Tannoudji is not a reliable source. I wrote that he was not accurately cited for this purpose by Vaughan Pratt. I think a careful read of what I wrote should clarify things.Chjoaygame (talk) 09:10, 17 October 2011 (UTC)" But, yes, I reject Palmer as a reliable source for our using the word isotropic to make it refer to radiation specified to come from a surface, and I am very unhappy to see the word isotropic used to refer to the field of radiation from a uniformly radiating sphere or to see it used as in 'the sphere is an isotropic radiator'. I do not deny that one can bend language to support such a usage, but I think it undesirable to do so. Unlike some mathematical language, ordinary language does not in general obey a rule of compositionality, and dictionary definitions cannot safely be bent under the assumption that it does.Chjoaygame (talk) 21:22, 15 November 2011 (UTC)Chjoaygame (talk) 21:25, 15 November 2011 (UTC)

"Point source" is like dx

A point source does not literally mean a source with exactly zero volume, it means a source that is arbitrarily small compared with the environment it is irradiating. Taking the concept literally is like arguing that calculus is impossible on the ground that dy/dx means 0/0. For all practical purposes a star is a point source, so tiny in fact that Heisenberg uncertainty makes photons from it hundreds of meters wide. Even though Venus looks to us like a bright star, it subtends an angle many thousands of times larger than any star, and the photons we receive from it when it's 200 gigameters from us are on the order of a millimeter wide.

Isotropic radiation vs. isotropic field

The final sentence of the lead of the Isotropy article says "Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented." The link for the first half of that sentence says that "[isotropic radiation] radiates uniformly in all directions from a point source sometimes called an isotropic radiator." Again point source should be understood as meaning "arbitrarily small," not "zero volume" which as you say is nonsensical, just as interpreting dy/dx as 0/0 is nonsensical. --Vaughan Pratt (talk) 22:33, 15 November 2011 (UTC)

@Chjoaygame: I reject Palmer as a reliable source for our using the word isotropic to make it refer to radiation specified to come from a surface See the two very favorable reviews of his (posthumously published) textbook on radiometry. He taught this material for decades at the College of Optical Sciences in the University of Arizona. You seem to have extraordinarily high standards for what is admissible as a source, when you reject an authority of long standing because (a) he's too "dogmatic" for you, whatever that means, and (b) you apparently have never encountered the notion of isotropic radiation, only isotropic fields. To me he seems remarkably accurate, non-dogmatic, and consistent with Wikipedia articles other than this one, which makes blatantly false statements.

I don't have any strong preference either way as to whether this article mentions isotropy at all. But if it does, it should not say "Since the radiance is isotropic" which is clearly false for radiation for a flat surface, as radiance from a surface decreases with increasing angle from the normal. I don't understand how you can possibly agree with that false statement, unless you have a different notion of "radiance" than the one in the article radiance. --Vaughan Pratt (talk) 01:19, 16 November 2011 (UTC)

No, radiance from a flat surface element of a black body IS isotropic. That same flat surface element is NOT an isotropic radiator. The radiance of that surface element does not decrease with increasing angle from the normal, it is constant.

You can see this with your eye. A flat surface element on the surface of a black body has the same brightness, no matter which angle you look at it from. That brightness is a measure of the radiance. However, the area covered on your retina by the retinal image of that element, decreases to zero as you go to 90 degrees from its normal. On your retina, the number of photons per unit retinal area inside that retinal image will remain constant, while the total number of photons collected will decrease as you go away from the normal because the size of the retinal image decreases. Radiance is proportional to the number of photons per unit area inside the retinal image. Total number of photons collected is equal to the photons per unit area times retinal area. What illuminates a sphere at infinity is the total number of photons. Your eye collects by a fixed solid angle, its a focused system, it responds to radiance. The sphere does not, it is not focused, it responds to radiance times the projected area of the emitting surface, which is goes to zero as you go to 90 degrees from the normal. Please, this is the third time I have said this, if you don't understand this, don't just ignore it, tell me I am being unclear.

More important facts - The radiance of the element does not change as you move further away from it. If the Sun were a black body, its brightness would be the same over the whole image of the sun, if you looked at it from Earth. If you then moved half that distance to the sun, it would appear just as bright, but it would appear four times as large, area-wise. If you put a sphere around the sun whose radius was that of the Earth, you would collect a certain number of photons per unit area on that sphere. If you reduced the radius of the sphere by half, the number of photons collected per unit area on that sphere would quadruple. PAR (talk) 04:11, 16 November 2011 (UTC)

Dear Vaughan Pratt, there are competing sources and opinions about the use of words. I carefully worded what I said about sourcing in this case, but it seems not carefully enough to make myself clear.Chjoaygame (talk) 04:51, 16 November 2011 (UTC) I think the problem is only about use of words, and not about physics. I am happy to say of the equilibrium cavity radiation that it is isotropic, and that its radiative field is isotropic, but very uncomfortable with the locution that 'its radiance is isotropic'; its just a sensitivity to language. I am not willing to apply the word isotropic to radiation specified to come from a particular surface or from a body like a uniformly radiating sphere, because such radiation is defined to be only in directions away from the surface or body; I am happy to call such radiation diffuse, and to say that the surface is a diffuse radiator; I am not willing to say that the uniformly radiating sphere is an isotropic radiator. This is just about sensitivity to language. I think it very unlikely that any amount of lecturing, or any physics teacher, no matter how much he knows, will alter my sensitivity to language in this; it's not a matter of physics: it's a matter of language. As I said above, ordinary language does not in general obey rules of compositionality.Chjoaygame (talk) 06:39, 16 November 2011 (UTC)

I think you may be right The radiation out of the surface of a black body is certainly not isotropic at that surface, since there is no requirement that the incoming radiation be Planckian at the same temperature. Nevertheless, it IS the radiation that would be passing out of that surface if the radiation were isotropic. I think there is a better term, excitance, or something. I disagree about a uniformly radiating sphere, it IS an isotropic radiator, because a sphere at infinity is equally radiated. I think you are not sensitive enough to language on this point. PAR (talk) 10:19, 16 November 2011 (UTC)

I think I found the proper terminology - radiance refers to the radiation from a surface, according to the Wikipedia article on radiance. Spectral intensity, or specific intensity (Chandrasekhar) or spectral radiant intensity (Wikipedia) refers to the radiation field without surfaces. Radiant intensity is isotropic if it is the same in all directions. Radiance is isotropic if it is the same in all directions out of the surface. If a radiation field has isotropic radiant intensity and you create an imaginary surface element in that field, then the radiance for that area element will be isotropic independent of which side of the element you choose as "out" and independent of how the area element is oriented, and will obey Lambert's law. PAR (talk) 17:58, 16 November 2011 (UTC)

Wikipedia articles are in general not reliable sources. For example, they can be circular.

The official term spectral radiance refers to radiation from a surface either actual or imaginary. The traditional term specific intensity has an identical meaning. They mean spectral radiant flux density per unit projected area of source per unit solid angle of detection per spectral unit. Above in this talk page are citations for this. Paltridge & Platt on page 35 and Hapke on page 64 concur in noting that this definition is suitable for uncollimated beams but not suitable for dealing with parallel or collimated beams.

For the other extreme case, Born & Wolf write on page 195: "Often it is permissible to assume that, to a good approximation, B [the spectral radiance] is independent of the direction. The radiation is then said to be isotropic." Paltridge & Platt write on page 37: "Thus in the special and common case where [spectral radiance] is independent of direction — that is, the case of isotropic radiation: ...". Chandrasekhar writes on page 1: "A radiation field is said to be isotropic at a point, if the [specific] intensity is independent of direction at that point." Bohren & Clothiaux on pages 6 and 207, 208 write of an isotropic radiation field, and on pages 13 and 14 of isotropic radiation, and on page 21 of an isotropic source, and on page 208 of an isotropic radiance, and on page 210 of downward radiance above the water being isotropic, and on page 209 of "an isotropic diffuse reflector", and on page 211 of "(isotropic) incident irradiance". Mihalas & Mihalas write on page 24 of an isotropic material, and on page 24 of an isotropic distribution function; I like the isotropic material but not the isotropic function. In a different usage, one which I do not like, Rybicki & Lightman write on page 2: "A source of radiation is called isotropic if it emits equally in all directions. An example would be an isolated star." They also write on page 4 of "an isotropic radiation field (not a function of angle)". On page 8 they talk of an extended "sphere of uniform brightness" as an "isotropic source", again a usage of isotropic that I do not like. They also on page 15 talk of the "absorption coefficient [being] isotropic", again a usage that I do not like. I prefer the usage of Born & Wolf, of Paltridge & Platt, and of Chandrasekhar, but I cannot say that the other usages are eccentric, and perhaps I would use them at times.

The official term spectral radiant intensity is fundamentally different from spectral radiance (or specific intensity). Official spectral radiant intensity means spectral radiant flux density from a hypothetical or virtual point source per spectral unit per unit solid angle of detection. Paltridge & Platt explain more about this. They write on page 35: "Radiant intensity refers only to a source and furthermore only to a point source." I will not fully quote them at length because I am not sure about copyright rules. They continue further on: "Thus WMO is careful to refer to the radiant intensity of an extended source as the ratio of radiant flux received at an elementary surface to the solid angle which this surface subtends at any point of the source, when this ratio is taken to the limit as the distance between the surface and the point is increased." For Born & Wolf, the radiant intensity refers to the whole surface of a remote body (not to a ratio as detailed by the WMO). It is the integral (not normalized) over the whole remote surface of the projected area of source body per unit solid angle of detection.

The ever-charming and much respected James Palmer's 1993 paper refers to official sources and ignores the traditional literature. He writes in his 1993 paper: "The current misuse of the term intensity in phyics and optics is deplored. Means of remedying the situation are suggested." He goes on to tell editors of journals to summarily reject papers that do not agree with him. He tells his reader: "Consider yourself informed."

Born & Wolf and Paltridge & Platt and Goody & Yung also give careful warnings about the word intensity, but in less charming language. Liou gives practically no warnings and might be confusing to some readers. Bohren & Clothiaux say they do not use the term intensity in their book, and read Palmer as being "humorous but biting"; they say he "shouts" that "intensity is an SI base quantity"; they will use only radiance and irradiance for radiometry; their actual usage is very predominantly of irradiance as against radiance.

Hapke on page 64 explains about irradiance, which is suitable for collimated beams. It is collimated flux per unit projected area of collimated beam. Hapke uses the term irradiance in the course of his exposition. Goody and Yung note that they do not use it (they use spectral radiance). Paltridge & Platt use both spectral radiance and spectral irradiance for relevant purposes, but do not make a point about collimation for spectral irradiance. Born & Wolf do not use the term irradiance, but on page 10 they carefully define their term light intensity to mean power per unit projected area of detection, and they identify it with the time-averaged Poynting vector, without specifying that the beam must be collimated.

Official exitance and radiosity refer to a distinction between the power emitted by the surface itself, and the power reflected from it.Chjoaygame (talk) 22:10, 16 November 2011 (UTC)

LOL Chjoaygame, thank you for a helpful "wall of text" summarizing the lunacy of the terminology on this subject. What we need is a terminology that is crystal clear and standard across Wikipedia articles, and this will help. PAR (talk) 00:32, 17 November 2011 (UTC)

The only article I've run across so far where the definitions aren't clear is this one. But I may well have overlooked some.

Nice list of definitions, Chjoaygame. Couple of questions, starting with three true-or-false ones.

A. The radiant flux equals 2π times the maximum radiant intensity for a uniform radiator that is

1. isotropic;
2. spherical;
3. ellipsoidal.

B. Do your answers depend on the definitions? --Vaughan Pratt (talk) 00:50, 17 November 2011 (UTC)

A slight mystery solved. Bohren & Clothiaux state Planck's law in terms of irradiance. Why do they do that? Should we do the same? No. Now I have found out why they did it. They state of page 208 "In Section 1.2 the Planck function P_e [...] was taken as that for irradiance to avoid a digression on radiance, ..." They just hadn't defined radiance at the stage of their exposition that presented Planck's law.Chjoaygame (talk) 20:10, 17 November 2011 (UTC)