Talk:Stefan–Boltzmann law

What dooes this mean ?
"To find the total absolute power of energy radiated for an object..." I am guessing that it should be "To find the total absolute power radiated by an object..." —Preceding unsigned comment added by 82.32.49.157 (talk) 06:32, 4 May 2011 (UTC)

Wien's displacement law
It should be included or mentioned, because it's one of the two laws that describe the blackbody radiation. 91.153.156.165 (talk) 01:12, 12 December 2008 (UTC)
 * I agree. I have mentioned it in my note below about why the Stefan Boltzmann law cannot be used to determine the temperature achieved by a combination of two or more sources of radiation. 202.172.115.20 (talk) 21:04, 16 November 2016 (UTC)

Old talk
It would be nice to give a different derivation of the intergal $$J$$ in the appendix than the one in the article Planck's law of black body radiation. I suggest to do a contour integration of the function $$\frac{\exp(i k z)}{\exp(z)-1}$$ over the contour from zero to R and then to $$R + 2\pi i$$, $$2\pi i+\epsilon$$, then a clockwise quarter circle with radius $$\epsilon$$ and center $$2\pi i$$ to $$2\pi i-i\epsilon$$ and then back to zero. Count Iblis 13:30, 4 September 2006 (UTC)

I've just finished this. Count Iblis 23:14, 4 September 2006 (UTC)

Temperature of the sun
This section should include that the energy radiated at the sun's surface, as can be calculated by the Stefan-Boltzmann law is 6.32899e+07 W/m2 —Preceding unsigned comment added by 71.111.6.64 (talk) 16:44, 4 April 2010 (UTC)

calcul Earth temperature
- The title of this section is misleading: the calcul is just plain wrong for calculating the physical Earth tempearture, it only calculates the effective temperature of Earth (otherwise it assumes than a grey bodyœ absorber (albedo 0.3) is a black body emitter (effective temperature calculated)!) To have the physical temperature, fourth root of temperature has to be extracted before integrating over the surface, not after!

- would be nice to explain that the coefficient 1/(square root(2)) is a geometrical factor coming from the earth being spherical

once I learn about Latex, I ll make the changes —Preceding unsigned comment added by Goretesque (talk • contribs) 00:25, 28 October 2007 (UTC)


 * I think what it says about albedo is wrong, as is what you say. The albedo should not affect the equilibrium temperature, since it affects absorption and emission in the same proportion (assuming it's constant with wavelength, which it really isn't). Dicklyon 00:29, 28 October 2007 (UTC)


 * Do you mean that if two objects in space have all characteristic the same (size, orbit around sun, etc), but the albedo, they should be at same temperature ? are you speaking about effective temperature or physical? Goretesque 14:03, 28 October 2007 (UTC)


 * More than that. They don't have to be the same size, just the same distance from the sun, as long as the albedo is not wavelength dependent.  In that case, effective temperature and physical temperature are not different, are they?  In reality, the picture is made much more complicated by albedo varying with wavelength, which is what greenhouse gases contribute to. Dicklyon 16:19, 28 October 2007 (UTC)


 * Ok, I begin to understand what you are telling me:
 * - I made a mistake between size and shape! you are right, size of planet has no influence on effective temperature. However, shape will have some for objects, as the effective temperature depends on the amount of energy received (cross section of the object) compare to the surface over which this energy is spread. From planet, one can say it is always a disc spread over a sphere.
 * - however, from the definition of effective temperature ($$T_{eff} = \left (\frac{L (1-A)}{16 \pi \sigma D^2} \right )^{\tfrac{1}{4}}$$), if the distance from the star of a planet with albedo of 0 and a planet with albedo 0.5, then you can easily see that effective temperatures are different. This formula is what is used in the article (by replacing L with the value for Sun, and taking the albedo into account at the end) to calculate the "temperature of Earth". In fact it is the calcul of the "effective temperature of Earth", that is the temperature of a black body irradiating as much as Earth.
 * - by definition, the effective temperature $$T_{eff}$$ and the physical temperature $$T_{phys}$$ of a grey body can never be the same. The only case this could happen, is for a perfectly homogen grey body, that is a grey body with the exact same temperature on every points of its surface (that is a planet with the same temperature on tropics and on poles!). Taking into account only radiation, then the physical temperature of the surface is lower than the effective temperature.
 * - from this you can deduce that, in case of Earth, comparing $$T_{eff}$$ 255 K and the actual measured temperature $$T_{meas}$$ 288 K, is no sense for evaluating greenhouse effect. $$T_{phys}$$ (much lower than $$T_{eff}$$) and $$T_{meas}$$ have to be compared to get it, and it will be much more than 33 K.
 * - but all that reasoning is for a non rotating, not oblique, planet. Calculation become much more complicated in case of obliquely rotating globe (I am even not sure it is possible to calculate it)
 * - and finally, up to now, one considered a average albedo for Earth, did not consider oceans/earth coverage, did not take into account the internal heat, and one consider the system in thermal equilibrium! (and I am sure, I am forgetting some other factors)
 * conclusion: it is impossible to calculate $$T_{phys}$$ for Earth; therefore impossible to evaluate a meaningful greenhouse effect, even a correct approximation Goretesque 23:07, 28 October 2007 (UTC)

- I think it should be pointed out that the 255K temperature is derived from 279K by multiplying by 0.7 ^ 1/4. But I agree with Dicklyon that this ignores Kirchhoff. This implies that an ice age would catastrophically end with the earth a snow ball with a temperature about 156K. Surely there is experimental data on the temperature of gray body as a function of its albedo. Bob Armstrong (talk) 15:22, 15 January 2008 (UTC)

- Every place in this section of the article that says "energy" should be replaced with "power". See the definition at the top. Parveson (talk) 00:08, 30 August 2012 (UTC)


 * The amount of geothermal power radiation over the whole earth is pitiful by comparison to the other difficulties, such as the sub-solar to polar gradient and day/night temperature difference, since the radiated power is of course not averagable over area, being non-linear, and a fourth power is very non-linear. A further complication is that the re-radiation is at different altitudes and different temperatures even before worrying over re-absorption. And remember that the earth's diameter is greater than it is. No, seriously. Atmospheric deflection means that rays that should (just) skim past the surface of the earth are in fact deflected onto the surface. The radiation from the surface - humm. A ray departing the surface tangentially would be refracted back down to the surface. This is the reverse direction of the incoming near-tangential ray being bent to strike the surface. One could recast the discussion into being for an airless sphere before attempting a sphere with an atmosphere. Even simpler would be to calculate the effective temperature at the radius of the earth's orbit - that is, thanks to the sun, so many watts/square metre are passing by. One could imagine a flat black body - though one side would be irradiated while both sides would radiate. At the other extreme is the example of Venus with a ferocious surface temperature and atmospheric refraction so great that there is no horizon. NickyMcLean (talk) 10:59, 19 September 2014 (UTC)

- Seriously, though? Computing the 'effective temperature' of the Earth with no greenhouse gases but with the same albedo is beyond stupid. The Earth's albedo is very much dictated by the presence of water, which is a greenhouse gas in gas form. If you remove water from Earth, you have a very different albedo. You cannot pretend to have clouds in the atmosphere but no water in it, I hope that's clear enough. Also IR emissivity of surfaces is not 1, but typically lower. It's not even a constant on the whole spectrum. Use an emissivity of 0.95 and the albedo of the Moon and you'll get different results. — Preceding unsigned comment added by 2A02:2F05:8401:7F00:812C:9531:BFC6:ECCA (talk) 22:42, 7 December 2023 (UTC)

Gray Body Temperature
I've thought about this during the day, and believe I can clarify the argument that by Kirchoff's Law the temperature of a sphere of uniform albedo does not depend on that albedo. Assuming a gray body in earth orbit, at equilibrium

where  is the Stefan-Boltzmann constant,   and   the temperatures of the sphere in earth orbit and sun ,   the constant depending on geometry derived at Temperature relation between a planet and its star and   and   are the emissivity and the absorptivity of the sphere. By Kirchoff at equilibrium,. Thus, for  ,   and   drop out and mean temperature of the sphere depends only on the surface temperature of the sun. Of course there is a singularity at , or   which means the internal temperature of a totally reflecting sphere would be disconnected from the space around it.

The idea that the temperature of an object can depend on its albedo, in particular the 4th root of its albedo , leads to absurdities such as the following : If the inner chamber of a vacuum bottle were coated with magnesium oxide with an albedo of about .9 then on the basis of a 4th root relationship , one would expect it's interior to settle at about 0.56 the temperature of the room the bottle is in , say about -100 centigrade. --Bob Armstrong (talk) 23:35, 15 January 2008 (UTC)


 * You read it wrong. It said the "effective temperature" would depend on the albedo, and that's true; review the definition.  But I fixed it to be more clear. Dicklyon (talk) 01:30, 16 January 2008 (UTC)


 * - Seems to me effective temperature doesn't mean much for a passively heated body . Seen from the outside, its albedo will restore to its radiance exactly the amount reduced by its reduced emissivity . I actually would question why the section is about that rather irrelevant number rather than the actual predicted physical mean temperature for the planet . The most fundamental point is that by Kirchoff , the temperature of a radiantly heated gray body will be no different that of a black body . Clearly the notion that its temperature is reduced by the 4th root of , or for that matter any other function of , the albedo leads to an absurdity . In fact , I believe , tho not stated in the Wikipedia entry , Kirchoff applies frequency-wise , so this should be generalizable to any colored body . In fact , it must be because anything else leads to the absurdity that you can make an energy sink or source just by painting an object.


 * The bottom line is, there is no discrepancy between the Stefan-Boltzmann temperature measured in earth orbit and the measured temperature of the planet needing to be "explained" by some greenhouse effect . ( In this regard , I find it interesting that the Wikipedia article on the greenhouse effect has no equations quantifying it , nor have I been able to find any elsewhere . ) Bob Armstrong (talk) 00:50, 17 January 2008 (UTC)


 * It's not clear what you mean by "the Stefan-Boltzmann temperature measured in earth orbit". In any case, the point seems clear, that when the albedo varies with wavelength the equilibrium temperature is affected by that; we call it the greenhouse effect.  I added another ref about it, including specific pages. Dicklyon (talk) 02:10, 17 January 2008 (UTC)


 * - By SB temperature I mean the temperature calculated as in the article from total radiant flux . That figure of 279K, incidentally , needs to be increased by 3k to 281k for the contribution from all other directions in space . This is getting mighty close to measurement error in mean solar and earth temperatures . I looked at both your references and they have the simply wrong ( by the arguments above ) assertion that the calculated mean temperature of the earth should be 255K . I see no derivation of the "greenhouse effect" in either of these references other than taking the difference between this mistaken number and the observed value which in fact agrees with notion that you can't make something cold just by painting it white . I think the idea that Venus , with the highest albedo of any inner planet can stay more than 2 times its SB temperature ( more that 3 times its "effective" temperature ) due to greenhouse heat trapping strains credulity . It violates very fundamental notions to claim you can make heat go up hill . I think it appropriate to mention that I did a "Mr Wizard" style experiment with black and white ping-pong balls at 2500M in the Colorado sun and observed very little if any difference in their asymptotic temperatures . By the logic that the "natural" temperature of the earth should be 255K , the white ball should have been freezing . The YouTube video of the experiment and a lot more on this topic is at my climate and energy page . These arguments combined with the total failure of global temperature to track CO2 over the last decade make the whole notion of anthropogenic global warming very suspect to say the least . ( Note , I clearly have more work to do to prove the case to colored balls , but it is hard to comprehend how some colored ball could chill if a white ball doesn't . ) ,  Bob Armstrong (talk) 19:33, 17 January 2008 (UTC)


 * I'm sorry, but you've totally lost me. Did you follow the links and read about where it mentioned greenhouse effect?  It has no relation to your strawman idea of a "natural" temperature; they specifically talk about a "natural" greenhouse effect.  Is your rant based on some reaction to this term? Dicklyon (talk) 19:37, 17 January 2008 (UTC)


 * - No, it's based totally on the fact that claiming the temperature of the earth should be reduced by the 4th root of 1 - its albedo is simply wrong . Both articles you cite use the difference between 279K and the reduced 255K as their total justification for a greenhouse effect . Bob Armstrong (talk) 22:37, 17 January 2008 (UTC)


 * To clarify for those reading, Mr. Armstrong is a global warming denier. He believes that the temperature of a body depends solely on the intensity of the radiation incident on the body, and that therefore changes in the composition of the earth's atmosphere cannot alter the surface temperature of the earth, so the idea that increased CO2 in the atmosphere could alter the earth's temperature must be wrong. He's attempting to get this page rejiggered to fit his ideas. (I've been having an entertaining discussion with him on a mailing list on this topic.) Pmetzger (talk) 01:41, 18 January 2008 (UTC)


 * Yes, I see. That sort of half explains his inability to read and understand what the article says, even though it is unrelated to the topic of human-caused modifications to the greenhouse effect.  I suppose it means that asking again for him to review the references is pointless. Dicklyon (talk) 03:24, 18 January 2008 (UTC)


 * May I suggest that the authors remove the two previous paragraphs? I think it would be rather difficult to classify them as a contribution to the understanding of the Stefan-Boltzmann law. I imagine that the author of the subsequent paragraph will modify his accordingly--Damorbel (talk) 13:37, 26 June 2008 (UTC)
 * - That I deny there exists such a phenomenon as the "greenhouse" effect is to state the obvious . It is the reason why I deny its existence that must be dealt with . Note : it's very clear that a "greenhouse" effect greatly moderates the variance of planetary temperature ; the issue is whether it can change the mean . Here are the points of contention :
 * Does or does not the article state that the mean temperature of the earth would be the calculated black body value 279k but that value is reduced to 255k by multiplying it by 0.91, the 4th root of earth's presumed absorptivity , 0.7 ? The article may call this an "effective" temperature using a term applied to the observed radiation versus internal temperature of gray bodies , but thereafter the article uses it as the predicted actual temperature.


 * No, it does not. Read it again. The effective temperature is not a real temperature, and is not used as such.  The real temperature requires a calculation that also involves emissivity.  If that were the same as absorptivity, over the wavelengths that matter, then the real temperature calculated would be the 279 K number, not the 255 K number which would be the "effective temperature" at that cited albedo.  The 279 K, the effective temperature in the blackbody assumption case, would be the same as the "real" temperature in any gray-body case, but that's still a hypothetical theoretical, not an actual. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - Yet you immediately use it in a calculation of greenhouse effect ! Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * Does not this same logic lead to the prediction that a ball coated with magnesium oxide with an albedo of about 0.9 will come, in a similar 279k radiant environment , to an equilibrium temperature of about 0.56 of that or about 157K ? Is this not patent nonsense ?


 * Yes, that's patent nonsense, and not what the article says or implies. The ball would come to an "effective temperature" of 157K, perhaps, under the definition that it would emit the same radiant power as a blackbody of that temperature.  If you don't acknowledge the definitions of the terms being used, you can't reason about the meaning. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - That's got the notion of "effective temperature" confused . By the equation for j* at the top of the page, the "effective temperature" of a 279K body with an emissivity of 0.1 will be 157K . That's the amount of energy it will be radiating as seen from the outside . The "effective temperature" of a 157K body with a 0.1 emissivity would be about 88K . Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * Is not the discrepancy between the measured earth temperature and this hypothesized 255K calculated temperature used as the total evidence for a "greenhouse" effect both in the article and in its references ?


 * I don't think so. The actual difference in albedo between the visible wavelengths where the earth absorbs and the long wavelengths where it radiates is probably not huge, so the greenhouse effect is perhaps only a few degrees. It's still a notable difference, and still easy to reason about. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - Here you are treating your calculated "effective temperature" as real . Otherwise there is nothing to explain . This still boils down to the idea that simply changing the color of a radiantly heated ball will change its temperature . Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * Explain these points, particularly the second one , and I'll be a convert . Alternatively show me any experiment in which simply painting a sphere changes its equilibrium temperature at all , much less as radically as by the 4th root of the albedo . Show me what the flaw is in my simple application of Kirchoff's law , . I sure wish I could lay my hands on the boy's science book where I first learned back in the 1950s that white and black rocks in the desert sun end up the same temperature .


 * It's very easy to find cases where painting a sphere (or a car) will change its temperature at equilibrium in sunlight. Black cars get hotter inside than white cars, usually, because they have similar emissivities in the long wavelengths, and very different absorptivities in the solar wavelengths. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - oh ? I had a white Porsche and it got hot as the devil sitting in the sun . And, my white and black ping-pong balls didn't show any difference . And , as I mentioned , it's very hard to shake the arguments I learned from boys' science books 50 years ago . Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * Additionally, I would appreciate a pointer to any derivation of the "greenhouse effect" with the same level of rigor as the derivation of Stefan-Boltzmann rather than simply based on the claim made here that a white ball will be colder than a black ball exposed to the same radiant flux.


 * Did you try reading that ref yet? It wasn't a long drawn-out derivation, but it was certainly not the trivialization of your strawman, either. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - I saw no derivation other than taking the difference between the observed temperature and the "effective temperature" . Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * Sorry if it sounds like a rant, but answer those first 2 bullets and you'll win me over . Bob Armstrong (talk) 01:47, 19 January 2008 (UTC)


 * Good, I'll take the win. Dicklyon (talk) 02:13, 19 January 2008 (UTC)
 * - Obviously I don't cede it . In fact, the article leaves me reminded of Steven Colbert's comments on people making reality simply by posting it on Wikipedia . For the nonce , I would suggest people stick with the more conservative statement of the physics on the Black body page . Bob Armstrong (talk) 23:42, 21 January 2008 (UTC)


 * That page does also mention greenhouse effect, but in the lengthy example it only works the blackbody case, and doesn't deal with albedo even, much less wavelength-dependent albedo. It doesn't say anything that contradicts the current page.  I agree that the presentation would probably be improved if we got rid of the concept of "effective temperature," which is confusing and obviously misleading you in your attempt to apply logic to it. Dicklyon (talk) 02:01, 22 January 2008 (UTC)


 * I don't think the problem is with the explanation, which seems fine to me. I think that the problem in this case is with the particular reader having a conclusion they wish to be true in advance of reading the article. Pmetzger (talk) 12:57, 22 January 2008 (UTC)


 * Yes on the latter, but maybe also the former. The "effective temperature" is certainly a confusing concept when it's far from the actual temperature, like when the albedo is high.  Shouldn't we re-express it without that? Dicklyon (talk) 21:44, 22 January 2008 (UTC)


 * Hrm. I don't have a good textbook handy with me, but I'd recommend we stick to the terminology in common physics texts, whatever that might happen to be. --Pmetzger (talk) 19:46, 28 January 2008 (UTC)


 * Some sources do use effective temperature of the Earth; the ones we have cited now in the article do not. We should probably cite one that does, since we start that way. Dicklyon (talk) 00:30, 29 January 2008 (UTC)


 * I would think a totally reflective planet (is that a white body?) would not receive any energy from the sun, nor would it lose any energy from within. It would be fully insulated. From outside you would not know the temperature inside. A grey body is partially absorbing, partially reflective, changing with wavelength, somewhere in between black and white.

Conservation of energy prohibits a body in equilibrium from emitting more energy than it receives. With a solar constant of 1366K/m2 projected onto a sphere this limits the maximum possible mean temperature to 279K. The surface cannot possible receive more energy than top of atmosphere so someone must explain how it is possible for a surface limited to 342W/m2 incoming power can possibly emit 390W/m2 and keep it up forever. The average temperature of the global ocean is 3.9C which is about 277K. This makes more sense for the average surface temperature of the earth as it is within the limit imposed by the input power from the sun. It also implies that the greenhouse effect is within 2C of of saturation and this is the most that any possible addition of greenhouse gases can raise the average temperature. 70.123.128.84 (talk) 23:00, 24 March 2012 (UTC)


 * How much energy does the Earth release naturally from fission and fusion?

Correct value of constant ?
Any ideea why some sources give a different value for the constant? e.g.: "Stefan-Boltzmann Law This is the relationship between luminosity (L), radius (R) and temperature (T): L = (7.125 x 10-7) R2 T4 Units: L - watts, R - meters, T - degrees Kelvin" (See: http://science.howstuffworks.com/star3.htm) or "Stefan-Boltzmann Law - This is the relationship between luminosity (L), radius (R) and temperature (T): L = (7.125 x 10-7) R2 T4.            Units: L - watts, R - meters, T - degrees Kelvin" (http://www.nameastargift.com/astronomydictionary/index.html) Units look ok. Why such a difference (maybe I am missing someting?) -Paul- (talk) 01:03, 9 February 2008 (UTC)


 * The one in the article is smaller than those by a factor of exactly 4*pi, so it's probably the power per steradian instead of the total power. In other words, someone got it wrong, I bet. Here's a book with an explanation that should help straighten it out. Dicklyon (talk) 06:23, 9 February 2008 (UTC)

Tyndall's measurements
I've been doing some checkings how Stefan experimentally derived the law. Here it is just briefly mentioned that he deduced it on the basis of experimental measurements made by John Tyndall. Stefan in fact never knew for Tyndall's measurements directly, but he learned of them from Wülner's textbook on thermodynamics, where Wülner in his 2nd and 3rd editions had included Tyndall's data. He also arranged temperatures to Tyndall's colours of radiated light, which were a little bit arbitrary. As Tyndall he refered to Draper's measurements. Stefan read about Tyndall's data in Wülner's textbook and fortunately correctly changed two temperatures into absolute scale. The questions is - to which Draper do both, Tyndall and Wülner, refer? My source says that Henry Draper in 1847 tried to establish experimentally at which temperature warmed body starts to radiate. He didn't succeed but he found out that energy density is increasing rapidly (exponently) and not strait proportional to temperature. Stefan read about Draper's measurement from his article in 1878. I guess Henry Draper can't be the right person, since in 1847 he was 10 years old, and I assume that it was his father John Draper. Tyndall also refers to Draper all over simply as Dr. Draper, and John Draper was also a physician. --xJaM (talk) 01:31, 5 March 2008 (UTC)


 * Yes, John Crepeau from the University of Idaho in his article Josef Stefan: His life and legacy in the thermal sciences cites John Draper's article from 1878 Scientific Memoirs: Being Experimental Contributions to a Knowledge of Radiant Energy, so this is the right source and John Draper the right person. --xJaM (talk) 17:49, 5 March 2008 (UTC)

Diagram request
A graph plotting power vs. temperature would be helpful to visualize the implications of the law. -- Beland (talk) 00:39, 6 March 2008 (UTC)

GW v AGW
" Global warming is an increase in this equilibrium temperature due to human-caused additions to the quantity of greenhouse gases in the atmosphere." is not true. At the very minimm, the phenomenon described is AGW. How about "Anthropogenic Global warming is an increase in this equilibrium temperature which is widely accepted to be due to human-caused additions to the quantity of greenhouse gases in the atmosphere. These differentially affect the earth's albedo at different wavelengths, ceteris paribus raising the equilibrium temperature."


 * AGW is most often referred to simply as global warming. Adding the unfamiliar word anthropogenic there mostly distracts the reader from understanding what we're referring to.  It would be better to put it in parens if you feel a clarificatin is needed, but then you'll want it not at the beginning of a sentence.  Alternatively, just leave out human-causeed, perhaps? Dicklyon (talk) 15:20, 27 June 2008 (UTC)


 * GW could be caused by natural or artificial causes. AWG is the human caused version. The debate is whether AWG is significant or not. — Preceding unsigned comment added by 101.170.255.231 (talk) 13:42, 25 June 2012 (UTC)

Stefan's constant
I just wonder if this realy is a wide used name for the boltzman's constant. I have never heard of it befor —Preceding unsigned comment added by 213.153.113.18 (talk) 14:44, 15 September 2008 (UTC)


 * Stefan's constant and Boltzmann's constant are two different constants entirely. CrispMuncher (talk) 20:22, 16 May 2009 (UTC)

OP is correct here. e = a T^4 is Stefan's Law, the equation for the radiation energy density. 'a' is Stefan's Constant. The Stefan-Boltzmann constant σ = (a * c)/4.

We must use different forms of the S-B equation, one form for idealized blackbodies, one for graybodies.

https://i.imgur.com/QErszYW.gif

Idealized Blackbody Object (assumes emission to 0 K and ε = 1 by definition):

q_bb = ε σ (T_h^4 - T_c^4) A_h = 1 σ (T_h^4 - 0 K) 1 m^2 =   σ T^4

Graybody Object (assumes emission to > 0 K and ε < 1):

q_gb = ε σ (T_h^4 - T_c^4) A_h

The 'A_h' term is merely a multiplier, used if one is calculating for an area larger than unity [for instance: >1 m^2], which converts the result from radiant exitance (W m-2, radiant flux per unit area) to radiant flux (W).

Temperature is equal to the fourth root of radiation energy density divided by Stefan's Constant (ie: the radiation constant).

e = T^4 a

a = 4σ/c

e = T^4 4σ/c

T^4 = e/(4σ/c)

T = 4^√(e/(4σ/c))

T = 4^√(e/a)

q = ε_h σ (T_h^4 – T_c^4)

[1] ∴ q = ε_h σ ((e_h / (4σ / c)) – (e_c / (4σ / c)))

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).

W m-2 = W m-2 K-4 * (Δ(J m-3 / (W m-2 K-4 / m sec-1)))

[2] ∴ q = (ε_h c (e_h - e_c)) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).

W m-2 = (m sec-1 (ΔJ m-3)) / 4

One can see from the immediately-above equation that the Stefan-Boltzmann (S-B) equation is all about subtracting the radiation energy density of the cooler object from the radiation energy density of the warmer object.

[3] ∴ q = (ε_h * (σ / a) * Δe) / 4

Canceling units, we get W m-2.

W m-2 = ((W m-2 K-4 / J m-3 K-4) * ΔJ m-3) / 4

You will note that σ = (a * c)/4… the Stefan-Boltzmann Constant equals Stefan’s Constant multiplied by the speed of light in vacua, all divided by 4.

[4] ∴ q = (ε_h * ((a * c) / a) * Δe) / 4 = (ε_h * c * Δe) / 4

Canceling units, we get J sec-1 m-2, which is W m-2 (1 J sec-1 = 1 W).

W m-2 = (m sec-1 * ΔJ m-3) / 4

Note that [2] and [4] are identical, arrived at via two different avenues.

So radiant exitance at its most simple (and thus the S-B equation at its most simple) is just the emissivity of the warmer object (because emissivity only applies to objects which are emitting, and only the warmer object will be emitting… the colder object will be unable to emit in the direction of the warmer object because energy cannot spontaneously flow up an energy density gradient) multiplied by the speed of light in vacua, multiplied by the energy density differential, all divided by 4.

For graybody objects, it is the radiation energy density differential between warmer object and cooler object which determines warmer object radiant exitance. Warmer objects don't absorb radiation from cooler objects (a violation of 2LoT in the Clausius Statement sense and Stefan's Law); the lower radiation energy density gradient between warmer and cooler objects (as compared to between warmer object and 0 K) lowers radiant exitance of the warmer object (as compared to its radiant exitance if it were emitting to 0 K). The radiation energy density differential between objects manifests a radiation energy density gradient, each surface's radiation energy density manifesting a proportional radiation pressure.

76.30.103.137 (talk) 08:17, 28 February 2024 (UTC)

Issues with graph
The graph is misleading. It talks of total energy emitted rather than power per unit area. I did a spot check of the figures and they seem about right - it is only the caption that is wrong. CrispMuncher (talk) 20:29, 16 May 2009 (UTC)

Planck's Law
I think it should be $$I(\nu,T)d\nu=\frac{2h\nu^3}{c^2}\frac{1}{e^\frac{h\nu}{kT}-1}$$. In the article, is $$d\nu$$ missing or there's something I don't know? ("Integration of intensity derivation" part) —Preceding unsigned comment added by 79.114.188.53 (talk) 18:35, 13 June 2009 (UTC)

Climate Change
It would be nice to have a short explanation of how Stefan-Boltzmann relates to climate models.DaveCrane (talk) 14:30, 17 December 2009 (UTC)
 * I agree this should be included, because people will ask it. I know that Earth isn't a black body because of albedo and the insulating layer of the atmosphere, and its insulation changes with composition, and the albedo changes with what is on the surface (plants, rock, water, ice, etc.). But I don't know nearly enough to write an accurate explanation. 2601:441:5000:ADF0:35DA:C87E:75C2:A251 (talk) 14:22, 26 July 2023 (UTC)

"Black body radius"
Someone added The black body radius is defined by Stefan–Boltzmann law. It is the radius of an object given its temperature and its total absolute power of Energy radiated:


 * $$ R = \sqrt{\frac{P}{\pi\varepsilon\sigma T^{4}}}.$$

--Netheril96 (talk) 17:07, 6 July 2011 (UTC)
 * 1) What is ε here? Emissivity? Then why is it called black body radius, not grey body radius?
 * 2) Is this the radius of a black body disk radiating the same amount of power with the same temperature as the object? If that is the case, then it should be $$ R = \sqrt{\frac{P}{2\pi\sigma T^{4}}}$$ since every disk has two sides that both emit radiation.
 * 3) It is claimed that "the black body radius is a concept that is excessively encountered in astrophysical science". Give a reference.

Radiant flux
I removed this: "irradiance, energy flux density, radiant flux, or the emissive power)"

The page for radiant flux gives the units of W, rather than W/m^2 as it is here, and "energy flux density" is a cryptic term. Especially since flux defines "energy flux" to mean what "energy flux density" means here, meaning they are using incompatible definitions for the word "flux".

I do not believe that the terms should be added back -- the intro should be clear and concise, and giving every synonym is confusing. If they are added back (preferably not in the intro), we need to make sure that we are using terms consistently. Kjsharke (talk) 01:53, 26 September 2011 (UTC)


 * Hi, u have given reason for removing the flux related terms, but why did u remove emissive power term?? I agree that need not be in intro, but is there any other reason too? Like any other inconsistency? Yashpalgoyal1304 (talk) 17:13, 28 March 2021 (UTC)

Proposed deletion of a phrase
I propose to delete the phrase "So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation," from the article on the ground that it does not impact the Stefan-Boltzmann constant. If you disagree, please state whether any such geometry would over- or under-estimate the constant. --Vaughan Pratt (talk) 08:41, 8 November 2011 (UTC)
 * It seems to be a necessary qualifier in the statement "So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface." What would you say instead?  Dicklyon (talk) 15:34, 8 November 2011 (UTC)
 * Excellent question: easy to ask, hard to answer. With more time I could shorten this answer.
 * There are two things "total energy radiated" could refer to. The one I was assuming was "total energy radiated from the surfaces," in which case the question of whether the energy is subsequently reabsorbed by some other region of the surface or sent elsewhere is irrelevant and the statement is almost a tautology (or at least an analytic truth) even without the qualifier.
 * If however it refers to that portion of the total energy radiated away from the surface never to return to any of the surfaces in question, which may be how you're interpreting it, then I would agree that the statement needs the qualifier.
 * But the latter is an obscure notion of "total energy radiated," since "geometry of the surface" is by no means the only way radiation can return. For example it may return via reflection off a nearby surface reflective at the relevant wavelengths, even if the reflector is far colder than the radiator.
 * An extreme case is a black body constructed as a cavity with a small hole to "peer into," which has for a long time been a reliable method of approximating ideal black bodies. In this situation every surface element within the cavity is radiating roughly according to SB, and the entrance to the cavity when treated as a surface in its own right, even if merely air or a vacuum, is radiating almost exactly according to SB, which is the whole point of the cavity method of manufacturing ideal black bodies for calibration purposes.
 * This case and the case of convex bodies without such cavities can be treated uniformly by defining the surface of the radiating body to be its convex hull and treating surfaces inside cavities as merely part of the interior of the body not to be considered "surface." This works equally well whether the cavity is a large interior volume with a small exit hole, or a slight depression in an otherwise convex surface, or anything in between.
 * Since either of these two definitions of "total energy radiated" can be made geometry-independent by suitable choice of definition of "surface," respectively as the neighborhoods of the surfaces and as their convex hull, it seems to me that the qualifier creates confusion where none should exist.
 * It seems to me that the subtleties here need to be clarified to prevent this sort of confusion. Would a short section on the role of geometry in the law help, touching on the all the above points, clear up these confusions or make them worse?
 * Given the lack of definitions of the relevant terms, something surely needs to be improved.
 * Switching to a marginally related thread, this potential for confusion may be why Gerlich and Tscheuschner claimed the SB constant was not universal (though Kramm and Molder went off on a tangent with a completely different argument for the same proposition). Headbomb (as the prime instigator of that thread) seems happy with both reasons, treating the qualifier in the SB law article as "saying the exact same thing," though whether he means "same as G&T" or believes that K&M have merely fleshed out G&T's cryptic argument is Too Much Information for me.  What does interest me is HB's dismissal of $&nu;B_&nu;$ as "a silly multiplication of  --Vaughan Pratt (talk) 02:04, 11 November 2011 (UTC)

A question about the derivation given
I am a little puzzled by this statement in the derivation, which appears to be vital to obtaining the result: 'Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume.' I know the pressure depends only on the temperature, but I'm not convinced that the reason given for this is correct. For example, in an ideal gas, the pressure is also proportional to the internal energy density: P = (3/2) (U/V), yet the pressure of an ideal gas certainly depends on the volume as well as the temperature. I think the key difference is that in this case, the energy density does not depend on the volume, but only on the temperature, and maybe there should be some explanation of why this is the case? Alternatively, I may have missed something simple in which case I would appreciate it if someone could point that out to me. — Preceding unsigned comment added by Tobycrisford (talk • contribs) 12:45, 18 December 2011 (UTC)

Do we need the Appendix?
Do we need an appendix that shows how to calculate the integral that appears in the derivation? This is standard maths stuff and tangential to the subject matter of the article. I propose to remove it. --Lambiam 11:37, 28 July 2012 (UTC)


 * We need it because it is related to the subject matter in the way most sources treat it. I.e. many books on this subject matter explain how to compute this integral, because the symbolic expression of sigma in terms of the other constants is notable, including its derivation. The derivation is not entirely "standard math stuff", that's e.g. not the judgment F. Reif had when he wrote his book on statistical mechanics. The issue is that if you ask a class of 30 second year physics students to integrate the Planck distribution, chances are that no one will be able to do so, not even those who have happened to take the math course on complex analysis. But this is the judgement that is made by the sources that are chosen by the profs at university. Count Iblis (talk) 15:32, 28 July 2012 (UTC)


 * I've the same opinion as Count Iblis's. In the case of Planck's law I add some sentences as an Appendix because the special value of Riemann's zeta function plays an important role to open the door to quantum theory. But we need the new article about the relation between zeta functions and some physical phenomena, for example Boltzmann's law, Planck's law, Casimir effect and quantum Hall effect. Are they related to physical phenomena via renormalization, analytic continuation and something another important ways? Enyokoyama (talk) 14:30, 29 July 2012 (UTC)

I agree with Lambian, that the appendix is inappropriate for this article, and I propose moving it elsewhere.

Basically, a physics article should describe how the equations are set up and what the results are, but should not be deeply concerned with the intermediate method of calculation. Much better to refer the interested reader to an article on the relevant mathematical methods. The content might be more appropriate there, where it could constitute an excellently-motivated example (rather than an off-topic appendix). Another advantage is that in such a context the premises won't seem so arbitrary (e.g., the choice to expose this particular method, instead of another alternative, would no longer need justifying - certainly not by the absurd postulate that special functions like zeta are "not available"). At the very least, the derivation should be collapsed.

Maybe creating a "list of applications of contour integration" article would be helpful to more readers. In this age of computer-algebra, I think Iblis would discover a class of physics 2nd years is perfectly capable of finding the solution to this integral. Many (text-)books on this subject matter accompany it with certain exercises, but that does not imply our treatment should include exercises for the reader. MOS(math) askes whether a derivation illuminates a topic or is intended merely to establish correctness of a result? Cesiumfrog (talk) 04:30, 21 May 2013 (UTC)

Name: Stefan's Law not Stefan-Boltzmann Law
In all my years, I've always seen this referred to as Stefan's Law rather than the Stefan-Boltzmann Law. A quick search of the literature will confirm this. The constant $\sigma$ is referred to as the Stefan-Boltzmann constant. Would the Wikipedia administrators mind updating this site? An example reference would be {\it Thermal Physics} by Daniel V. Schroeder (2000). — Preceding unsigned comment added by 192.80.95.242 (talk) 21:25, 9 May 2014 (UTC)
 * My many years of experience tell me that different authors variously refer to it as either Stefan's Law or the Stefan-Boltzmann Law. I find this variation within my own library of physics texts. It's just an example of some saying 'to-may-to' and others saying 'to-mah-to'. — Preceding unsigned comment added by 24.223.130.32 (talk) 23:10, 21 May 2017 (UTC)

The Law cannot be used to determine the temperature achieved by two or more sources
Because it is based on the integral of a single Planck function, the calculated temperature for the sum of fluxes from separate sources will not be what is observed. This can also be understood by the fact that Wien's Displacement Law tells us that the temperature is inversely proportional to the peak wavelength. Clearly adding separate fluxes will not achieve the same peak wavelength as would be the case for a single Planck function having the same total flux. For example: if one electric bar radiator at a given distance warms your cheek to 42°C (315K) then you would calculate (using the law) that 16 such radiators would roast you at double the temperature (630K) namely 357°C. 202.172.115.20 (talk) 20:55, 16 November 2016 (UTC)


 * I cannot get your argument about Wien's Law (How does it prevent you from adding energy fluxes? Energy is additive.) But I do see that your example about the radiators is completely flawed. The result in the section Effective Temperature of the Earth can be rewritten as
 * $$\frac{T_\oplus^4}{T_\odot^4} = \frac{\Omega}{4\pi},$$
 * where Ω is the solid angle subtended by the Sun, as seen from the Earth. If you try to apply this reasoning to your radiator collection you will see that, since Ω can be no larger than $4π$, the radiators cannot make you hotter than their own temperature. The limiting case ($Ω = 4π$) is called an “oven” and, if you stay inside long enough, you will eventually reach the temperature of its inner walls.
 * — Edgar.bonet (talk) 09:05, 17 November 2016 (UTC)
 * No, my argument is not flawed. You will never find an experiment confirming your conjecture that the Stefan-Boltzmann Law can be used to determine the temperature based on the sum of two or more sources. Not all the energy in radiation is thermalized (ie converted to molecular kinetic energy) in a target.  Read my 2012 peer-reviewed paper "Radiated Energy and the Second Law of Thermodynamics" Douglas Cotton 2001:8003:2683:D300:C060:91A5:26AE:42E (talk) 11:01, 12 October 2023 (UTC)

Effective temperature of the Earth
I am moving here a contribution made to the main article which seems to belong to this talk page instead. — Edgar.bonet (talk) 21:26, 24 January 2017 (UTC)

The temperature of 255 K corresponds to an S-B power flux of 239 W/m^2 or the ISR/OLR radiative balance at ToA, 100 km, and not at the surface. Comparing this ToA S-B calculated temperature to an S-B calculated surface temperature of 288 K (396 W/m^2, more than the ISR of 342 W/m^2 violating conservation) for a difference of 33 C is erroneous and inappropriate. The earth's surface is warm because of Q = U*A*dT not S-B. Btw this calculation models the earth as a ball suspended in a hot fluid which is not even close to how the heat balance really works. (Ref: Trenberth et al 2011jcli24, https://www.acs.org/content/acs/en/climatescience.html) — NickReality (talk) 11:59, 24 January 2017 (UTC)


 * The figure of 396W/m^2 would, if it came from a sun several times stronger than ours, raise a small black object orbiting the Sun (such as a metal marble) to about 288K because it would be uniform flux and the object would have high conductivity. But, because of T^4 in Stefan-Boltzmann calculations and because the flux to any location on Earth's surface varies, it would only achieve a somewhat lower global mean temperature. But our Sun is not that sun. Climatology energy diagrams clearly indicate that they get to about 396 by adding to solar radiation about twice as much back radiation and deducting non-radiative surface cooling. This is a totally incorrect application of the Stefan-Boltzmann Law. The back radiation, no matter its value (which they calculated to get the right answer) would never make the surface hotter than the region in the troposphere from whence it originated. No experiment anywhere confirms such a use of that law. Their diagrams show more energy coming out of the base of the atmosphere than enters at the top. What really happens is a totally different paradigm as explained in my 2013 paper "Planetary Core and Surface Temperatures" that is based on correct physics and which nobody has ever correctly refuted. You only have to think of Venus to realize how wrong is climatology science. If that doesn't convince you then find another explanation than mine as to what energy keeps the base of the Uranus troposphere hotter than Earth's surface even though it is on average more than 19 times the distance from the Sun than we are. Douglas Cotton, Centre for the Refutation of False Science. 2001:8003:2683:D300:C060:91A5:26AE:42E (talk) 11:18, 12 October 2023 (UTC)

Confusion in symbol
The symbol used for radiant emittance in this page (Stefan–Boltzmann_law) is $$j^{\star}$$ , while on other pages, it is $$M_\mathrm{e}$$. This was soo confusing. Yashpalgoyal1304 (talk) 17:42, 28 March 2021 (UTC) Ref: Other pages: the page linked for the definition of radiant emittance i.e. Radiant_exitance, and also Emissivity


 * Why is this so??
 * Can this be resolved?

Scope of expansion
update: I've moved/copied this suggestion over to Radiant_exitance talk page. Is radiant emittance same as emissive power? If yes, then can it be mentioned somewhere in this page? Ref: Emissivity In Kjsharke mentioned the removal of emissive power term, but did not give any reason for it. Yashpalgoyal1304 (talk) 17:42, 28 March 2021 (UTC)