For the moment I’m just going to ignore everything else in this debate (I have other time/energy committments...) and just focus on this particular question, since it’s one of the most fundamental questions we disagree on.
You are wrong, plain and simple. Rüdiger Mitdank is also wrong, despite his qualifications (I have equivalent qualifications, for that matter). Either that or he has failed to clearly express what he means.
If it were true that you could maintain an object colder than the background without consuming energy (just by altering surface absorption!), then you could have a free energy device. Just construct a heat engine with one end touching the object and the other end being a large black radiator.
If it were true that you could maintain an object colder than the background without consuming energy (just by altering surface absorption!),
Yea—several examples from wikipedia for the temperature of a planet indicate that albedo and emissivity can differ (it’s implied on this page, directly stated on this next page).
Here under effective temperature they have a model for a planet’s surface temperature where the emissivity is 1 but the albedo can be greater than 0.
Notice that if you plug in an albedo of 1 into that equation, you get a surface temperature of 0K!
The generalized stefan boltzmann law is thus the local equilibrium where irradiance/power absorbed equals irradiance/power emitted:
J_a = J_e
J_a = J_in*(1 - a)
J_e = eoT^4
T = (J_in (1-a) / (eo)) ^-4
J_in is the incoming irradiance from the light source, a is the material albedo, e is the material emissivity, o is SB const, T is temp.
This math comes directly from the wikpedia page, I’ve just converted from power units to irradiance. replacing the star’s irradiance term of L/(16 PI D^2) with a constant for an omni light source (CMB).
On retrospect, one way I could see this being wrong is if the albedo and emissivity are always required to be the same for a particular wavelength. In the earth example the albedo of relevance is for high energy photons from the sun whereas the relevant emissivity is lower energy infrared. Is that your explanation?
then you could have a free energy device. Just construct a heat engine with one end touching the object and the other end being a large black radiator.
Hmm perhaps, but I don’t see how that’s a ‘free’ energy device.
The ‘background’ is a virtual/hypothetical object anyway—the CMB actually is just a flux of photons. The concept of temperature for photons and the CMB is contrived—defined tautologically based on an ideal black body emitter. The actual ‘background temperature’ for a complex greybody in the CMB depends on albedo vs emissivity—as shown by the math from wikipedia.
One can construct a heat engine to extract solar energy using a reflective high albedo (low temp resevoir) object and a low albedo black object. Clearly this energy is not free, it comes from the sun. There is no fundamental difference between photons from the sun and photons from the CMB, correct?
So in theory the same principle should apply, unless there is some QM limitation at low temps like 2.7K. Another way you could be correct is if the low CMB temp is somehow ‘special’ in a QM sense. I suggested that earlier but you didn’t bite. For example, if the CMB represents some minimal lower barrier for emittable photon energy, then the math model I quoted from wikipedia then breaks down at these low temps.
But barring some QM exception like that, the CMB is just like the sun—a source of photons.
I’ve never seen someone so confused about the basic physics.
Let’s untangle these concepts.
Effective temperature is not actual temperature. It’s merely the temperature of a blackbody with the same emitted radiation power. As such, it depends on two assumptions:
The emitted power is thermal in origin,
The emission spectrum is the ideal blackbody spectrum.
Of course if these assumptions aren’t true then the temperature estimate is going to be wrong. Going back to my LED example, a glowing LED might have an ‘effective temperature’ of thousands of degrees K. This doesn’t mean anything at all.
The source of your confusion could be that emitted and received radiation sometimes have different spectra. This is indeed true. It’s true of the Earth, for instance. But at equilibrium, absorption and emission are exactly equal at all wavelengths. Please read this: https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation
Notice that if you plug in an albedo of 1 into that equation, you get a surface temperature of 0K!
Irrelevant, as I said. (The concept of albedo isn’t very useful for studying thermal equilibrium, I suggest you ignore it)
The generalized stefan boltzmann law is thus the local equilibrium where irradiance/power absorbed equals irradiance/power emitted:
Yes, this is the definition of being at the same temperature, if you didn’t know. (Assuming, of course, that the radiation is thermal in origin and radiation is the only heat transfer process at work, which it is in our example). If you disagree with this then you are simply wrong by definition and there is nothing more to say.
You seem to think that temperature is some concept that exists outside of thermal equilibrium. This is a very common mistake. Temperature is only defined for a system at thermal equilibrium, and when two objects are in thermal equilibrium with one another, they are by definition at the same temperature. It does not matter at all how fast their atoms are moving or what they are made of.
The concept of temperature for photons and the CMB is contrived—defined tautologically based on an ideal black body emitter.
No it’s not. It’s based on analysis of the spectrum, which is almost perfectly the spectrum of an ideal black body.
The ‘background’ is a virtual/hypothetical object anyway
The physics would be exactly the same if it were an actual sheet of black material at 2.7 K covering the universe.
The source of your confusion could be that emitted and received radiation sometimes have different spectra. This is indeed true. It’s true of the Earth, for instance. But at equilibrium, absorption and emission are exactly equal at all wavelengths. Please read this: https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation
That was indeed a source of initial confusion as I stated above, and I read Kirchnoff’s Law. I said:
if the albedo and emissivity are always required to be the same for a particular wavelength. In the earth example the albedo of relevance is for high energy photons from the sun whereas the relevant emissivity is lower energy infrared. Is that your explanation?
However this still doesn’t explain how passive temps lower than 2.7K are impossible. Passive albedo cooling works for the earth because snow/ice is highly reflective (inefficient absorber/emitter) at the higher frequencies where most of the sun’s energy is concentrated, and yet it is still an efficient absorber/emitter at the lower infrared frequencies. - correct?
Now—what prevents the same principle for operating at lower temps? If ice can reflect efficiently at 500 nm and emit efficiently at 10um, why can’t some hypothetical object reflect efficiently at ~cm range CMB microwave and emit efficiently at even lower frequencies?
You said: “But at equilibrium, absorption and emission are exactly equal at all wavelengths.”
But clearly, this isn’t the case for the sun earth system—and the law according to wikipedia is wavelength dependent. So I don’t really understand your sentence.
You are probably going to say … 2nd law of thermodynamics, but sorry even assuming that said empirical law is actually axiomatically fundamental, I don’t see how it automatically rules out these scenarios.
Notice that if you plug in an albedo of 1 into that equation, you get a surface temperature of 0K!
Irrelevant, as I said. (The concept of albedo isn’t very useful for studying thermal equilibrium, I suggest you ignore it)
This isn’t an explanation, you still haven’t explained what is different in my examples.
The physics would be exactly the same if it were an actual sheet of black material at 2.7 K covering the universe.
Sure—but notice that it’s infinitely far away, so the concept of equilibrium goes out the window.
Also—aren’t black holes an exception? An object using a black hole as a heat sink could presumably achieve temps lower than 2.7K.
why can’t some hypothetical object reflect efficiently at ~cm range CMB microwave and emit efficiently at even lower frequencies?
I never said it can’t. Such a material is definitely possible. It just couldn’t passively reach lower temperature than the background. Assuming it’s far out in space, as it cooled down to 2.7 K, it would eventually reach the limit where its absorption and emission at all frequencies equalled the background (due to Kirchhoff’s law), and that’s where the temperature would stay. If it started out at a lower temperature (due to being cooled beforehand) it would absorb thermal radiation until its temperature equalled the background (again, this is directly due to how we define temperature), and again that’s where it would stay.
If you have a problem with that, take it up with Kirchhoff, not me :)
The Earth system is completely different here because neither the Earth is in thermal equilibrium with the Sun, nor is the Earth in thermal equilibrium with the background, nor is the Sun in thermal equilibrium with the background. There is a net transfer of thermal energy occurring from the sun to the earth to the background (yes the sun is heating up the background—but not by much though). And ‘net transfer of energy’ means no equilibrium. Sources that use ‘thermal equilibrium’ for the relationship between the Sun and the Earth are using the term loosely and incorrectly.
The situation here is far from equilibrium because of the massive amounts of energy that the sun is putting out. This is the very opposite of ‘passive’ operation.
why can’t some hypothetical object reflect efficiently at ~cm range CMB microwave and emit efficiently at even lower frequencies?
I never said it can’t. Such a material is definitely possible. It just couldn’t passively reach lower temperature than the background.
The ‘background ‘is just an incoming flux of photons. If you insist that this photon flux has a temperature, then it is obviously true that an object can have a lower temperature than this background flux, because an icy earth can have a lower temp than it’s background flux. As you yourself said earlier, temperature is only defined in terms of some equilibrium condition, and based on the equations that define temperature (below), the grey body ‘temperature’ for an irradiance distribution can differ from the black body temperature for the same distribution.
The math allows objects to shield against irradiance and achieve lower temps.
The general multispectral thermal emission of a grey body is just the black body emission function (planck’s law) scaled by the wavelength/frequency dependent emissivity function for the material (as this is how emissivity is defined).
G_{lambda,T}=epsilon_{lambda}B_{lambda,T}.
The outgoing thermal emission power for a grey body is thus:
}).
The object’s temperature is stable when the net energy emitted equals the net energy absorbed (per unit time), where the net absorption is just the irradiance scaled by the emissivity function.
Where E_{lambda}. is the incoming irradiance distribution as a function of wavelength, and the rest should be self-explanatory.
I don’t have a math package handy to solve this for T. Nonetheless, given the two inputs : the material’s emissivity and the incoming irradiance (both functions of wavelength), a simple numerical integration and optimization can find T solutions.
The only inputs to this math are the incoming irradiance distribution and the material’s emissivity/absorptivity distribution. There is no input labeled ‘background temperature’.
Assuming I got it right, this should model/predict the temps for earth with different ice/albedo/greenhouse situations (ignoring internal heating sources) - and thus obviously also should allow shielding against the background! All it takes is a material function which falls off heavily at frequencies before the mean of the input irradiance distribution.
This math shows that a grey body in space can have an equilibrium temperature less than a black body.
A hypothetical ideal low temp whitebody would have an e function that is close to zero across the CMB frequency range, but is close to 1 for frequencies below the CMB frequency range. For current shielding materials the temp would at best only a little lower than CMB black body temp due to the 4th root, but still—for some hypothetical material the temp could theoretically approach zero as emissivity across CMB range approaches zero. Put another way, the CMB doesn’t truly have a temperature of 2.7k—that is just the black body approx temp of the CMB.
If your position is correct, something must be wrong with this math—some extra correction is required—what is it?
Assuming it’s far out in space, as it cooled down to 2.7 K, it would eventually reach the limit where its absorption and emission at all frequencies equalled the background (due to Kirchhoff’s law), and that’s where the temperature would stay.
No—that isn’t what the math says—according to the functions above which define temperature for this situation. As I pointed out earlier 2.7K is just the black body approx temp of the CMB (defined as the temp of a black body at equilibrium with the CMB!), and the grey body temp can be lower. Kirchhoff’s law just says that the material absorptivity at each wavelength equals the emissivity at that wavelength. The wikipedia page even has an example with white paint analogous to the icy earth example.
There is a net transfer of thermal energy occurring from the sun to the earth to the background (yes the sun is heating up the background—but not by much though). And ‘net transfer of energy’ means no equilibrium. Sources that use ‘thermal equilibrium’ for the relationship between the Sun and the Earth are using the term loosely and incorrectly.
None of this word level logic actually shows up in the math. The CMB is just an arbitrary set of photons. Translating the actual math to your word logic, the CMB set can be split as desired into subsets based on frequency such that a material could shield against the higher frequencies and emit lower frequencies—as indicated by the math. There is thus a transfer between one subset of the CMB, the object, and another CMB subset.
Another possible solution in your twisted word logic: there is always some hypothetical surface at zero that is infinitely far away. Objects can radiate heat towards this surface—and since physics is purely local, the code/math for local photon interactions can’t possibly ‘know’ whether or not said surface actually exists. Or replace surface with vaccuum.
For your position to be correct, there must be something extra in the math not yet considered—such as some QM limitation on low emission energies.
You’re abusing the math here. You’ve written down the expanded form of the Stefan-Boltzmann equation, which assumes a very specific relationship between temperature and emitted spectrum (which you say yourself). Then you write the temperature in terms of everything else in the equation, and assume a completely different emitted spectrum that invalidates the original equation that you derived T from in the first place.
What you’re doing isn’t math, it’s just meaningless symbolic manipulation, and it has no relationship to the actual physics.
If you insist that this photon flux has a temperature,
Of course it does—this is a very very common and useful concept in physics—and when you say it doesn’t this betrays lack of familiarity with physics. Photon gas most certainly has temperature, in exactly the same way as a gas of anything else has temperature. Not only that, but it also has pressure and entropy.
In fact, in some situations, like an exploding hydrogen bomb, a photon gas has considerable temperature and pressure, far in excess of say the temperature in the center of the sun or the pressure in the center of the Earth.
You say yourself that you assume ‘local equilibrium with it’s incoming irradiance’. Firstly, you’re using the word ‘local’ incorrectly. I assume you mean ‘equilibrium at each wavelength’. If so, this is the very definition of being at the same temperature as the incoming irradiance (assuming everything here is thermal in nature) and, again, there is nothing more to say. http://physics.info/temperature/
You’re abusing the math here. You’ve written down the expanded form of the Stefan-Boltzmann equation, ..
EDIT: I fixed the equations above, replaced with the correct emission function for a grey body.
Yes I see—the oT^4 term on the left needs to be replaced with the black body emission function of wavelength and temperature—which I gather is just Planck’s Law. Still, I don’t (yet) see how that could change the general conclusion.
Where E_{lambda}. is the incoming irradiance distribution as a function of wavelength, and the rest should be self-explanatory. For any incoming irradiance spectrum, the steady state temperature will depend on the grey body emissivity function and in general will differ from that of a black body.
Photon gas most certainly has temperature, in exactly the same way as a gas of anything else has temperature.
I’m aware that the concept of temperature is applied to photons—but given that they do not interact this is something very different than temperature for colliding particles. The temperature and pressure in the examples such as the hydrogen bomb require interactions through intermediaries.
The definition of temperature for photon gas in the very page you linked involves a black body model due to the lack of photon-photon interactions—supporting my point about photon temps such as the CMB being defined in relation to the black body approx.
You say yourself that you assume ‘local equilibrium with it’s incoming irradiance’. Firstly, you’re using the word ‘local’ incorrectly
I meant local in the physical geometric sense—as in we are modelling only a local object and the space around it over small smallish timescale. The meaning of local equilibrium should thus be clear—the situation where net energy emitted equals net energy absorbed. This is the same setup as the examples from wikipedia.
I’m sorry that you’re so insistent on your incorrect viewpoint that you’re not even willing to listen to the obvious facts, which are really very simple facts.
Actually I’ve updated numerous times during this conversation—mostly from reading the relevant physics. I’ve also updated slightly on answers from physicists which reach my same conclusion.
I’ve provided the radiosity equations for a grey body in outer space where the temperature is driven only by the balance between thermal emission and incoming irradiance. There are no feedback effects between the emission and the irradiance, as the latter is fixed—and thus there is no thermodynamic equilibrium in the Kirchoff sense. If you still believe that you are correct, you should be able to show how that math is wrong and what the correct math is.
This grey body radiosity function should be able to model the temp of say an icy earth, and can show how that temp changes as the object is moved away from the sun such that the irradiance shifts from the sun’s BB spectrum to that of the CMB.
We know for a fact that real grey body objects can have local temps lower than the black body temp for the irradiance of an object near earth. The burden of proof is now on you to show how just changing the irradiance spectrum can somehow lead to a situation where all possible grey body materials have the same steady state temperature.
You presumably believe such math exists and that it will show the temp has a floor near 2.7K for any possible material emissivity function, but I don’t see how that could possibly work.
I assume you mean ‘equilibrium at each wavelength’
No. In retrospect I should not have used the word ‘equilibrium’.
As you yourself said, the earth is not in thermodynamic equilibrium with the sun, and this is your explanation as to why shielding works for the earth.
Replace the sun with a distant but focused light source, such as a large ongoing explosion. The situation is the same. The earth is never in equilibrium with the explosion that generated the photons.
The CMB is just the remnant of a long gone explosion. The conditions of thermodynamic equilibrium do not apply.
If you are correct then you should be able to show the math. Provide an equation which predicts the temp of an object in space only as a function of the incoming (spectral) irradiance and that object’s (spectral) emissivity.
For the moment I’m just going to ignore everything else in this debate (I have other time/energy committments...) and just focus on this particular question, since it’s one of the most fundamental questions we disagree on.
You are wrong, plain and simple. Rüdiger Mitdank is also wrong, despite his qualifications (I have equivalent qualifications, for that matter). Either that or he has failed to clearly express what he means.
If it were true that you could maintain an object colder than the background without consuming energy (just by altering surface absorption!), then you could have a free energy device. Just construct a heat engine with one end touching the object and the other end being a large black radiator.
Yea—several examples from wikipedia for the temperature of a planet indicate that albedo and emissivity can differ (it’s implied on this page, directly stated on this next page).
Here under effective temperature they have a model for a planet’s surface temperature where the emissivity is 1 but the albedo can be greater than 0.
Notice that if you plug in an albedo of 1 into that equation, you get a surface temperature of 0K!
The generalized stefan boltzmann law is thus the local equilibrium where irradiance/power absorbed equals irradiance/power emitted:
J_a = J_e
J_a = J_in*(1 - a)
J_e = eoT^4
T = (J_in (1-a) / (eo)) ^-4
J_in is the incoming irradiance from the light source, a is the material albedo, e is the material emissivity, o is SB const, T is temp.
This math comes directly from the wikpedia page, I’ve just converted from power units to irradiance. replacing the star’s irradiance term of L/(16 PI D^2) with a constant for an omni light source (CMB).
On retrospect, one way I could see this being wrong is if the albedo and emissivity are always required to be the same for a particular wavelength. In the earth example the albedo of relevance is for high energy photons from the sun whereas the relevant emissivity is lower energy infrared. Is that your explanation?
Hmm perhaps, but I don’t see how that’s a ‘free’ energy device.
The ‘background’ is a virtual/hypothetical object anyway—the CMB actually is just a flux of photons. The concept of temperature for photons and the CMB is contrived—defined tautologically based on an ideal black body emitter. The actual ‘background temperature’ for a complex greybody in the CMB depends on albedo vs emissivity—as shown by the math from wikipedia.
One can construct a heat engine to extract solar energy using a reflective high albedo (low temp resevoir) object and a low albedo black object. Clearly this energy is not free, it comes from the sun. There is no fundamental difference between photons from the sun and photons from the CMB, correct?
So in theory the same principle should apply, unless there is some QM limitation at low temps like 2.7K. Another way you could be correct is if the low CMB temp is somehow ‘special’ in a QM sense. I suggested that earlier but you didn’t bite. For example, if the CMB represents some minimal lower barrier for emittable photon energy, then the math model I quoted from wikipedia then breaks down at these low temps.
But barring some QM exception like that, the CMB is just like the sun—a source of photons.
I’ve never seen someone so confused about the basic physics.
Let’s untangle these concepts.
Effective temperature is not actual temperature. It’s merely the temperature of a blackbody with the same emitted radiation power. As such, it depends on two assumptions:
The emitted power is thermal in origin,
The emission spectrum is the ideal blackbody spectrum.
Of course if these assumptions aren’t true then the temperature estimate is going to be wrong. Going back to my LED example, a glowing LED might have an ‘effective temperature’ of thousands of degrees K. This doesn’t mean anything at all.
The source of your confusion could be that emitted and received radiation sometimes have different spectra. This is indeed true. It’s true of the Earth, for instance. But at equilibrium, absorption and emission are exactly equal at all wavelengths. Please read this: https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation
Irrelevant, as I said. (The concept of albedo isn’t very useful for studying thermal equilibrium, I suggest you ignore it)
Yes, this is the definition of being at the same temperature, if you didn’t know. (Assuming, of course, that the radiation is thermal in origin and radiation is the only heat transfer process at work, which it is in our example). If you disagree with this then you are simply wrong by definition and there is nothing more to say.
You seem to think that temperature is some concept that exists outside of thermal equilibrium. This is a very common mistake. Temperature is only defined for a system at thermal equilibrium, and when two objects are in thermal equilibrium with one another, they are by definition at the same temperature. It does not matter at all how fast their atoms are moving or what they are made of.
No it’s not. It’s based on analysis of the spectrum, which is almost perfectly the spectrum of an ideal black body.
The physics would be exactly the same if it were an actual sheet of black material at 2.7 K covering the universe.
That was indeed a source of initial confusion as I stated above, and I read Kirchnoff’s Law. I said:
However this still doesn’t explain how passive temps lower than 2.7K are impossible. Passive albedo cooling works for the earth because snow/ice is highly reflective (inefficient absorber/emitter) at the higher frequencies where most of the sun’s energy is concentrated, and yet it is still an efficient absorber/emitter at the lower infrared frequencies. - correct?
Now—what prevents the same principle for operating at lower temps? If ice can reflect efficiently at 500 nm and emit efficiently at 10um, why can’t some hypothetical object reflect efficiently at ~cm range CMB microwave and emit efficiently at even lower frequencies?
You said: “But at equilibrium, absorption and emission are exactly equal at all wavelengths.”
But clearly, this isn’t the case for the sun earth system—and the law according to wikipedia is wavelength dependent. So I don’t really understand your sentence.
You are probably going to say … 2nd law of thermodynamics, but sorry even assuming that said empirical law is actually axiomatically fundamental, I don’t see how it automatically rules out these scenarios.
This isn’t an explanation, you still haven’t explained what is different in my examples.
Sure—but notice that it’s infinitely far away, so the concept of equilibrium goes out the window.
Also—aren’t black holes an exception? An object using a black hole as a heat sink could presumably achieve temps lower than 2.7K.
I never said it can’t. Such a material is definitely possible. It just couldn’t passively reach lower temperature than the background. Assuming it’s far out in space, as it cooled down to 2.7 K, it would eventually reach the limit where its absorption and emission at all frequencies equalled the background (due to Kirchhoff’s law), and that’s where the temperature would stay. If it started out at a lower temperature (due to being cooled beforehand) it would absorb thermal radiation until its temperature equalled the background (again, this is directly due to how we define temperature), and again that’s where it would stay.
If you have a problem with that, take it up with Kirchhoff, not me :)
The Earth system is completely different here because neither the Earth is in thermal equilibrium with the Sun, nor is the Earth in thermal equilibrium with the background, nor is the Sun in thermal equilibrium with the background. There is a net transfer of thermal energy occurring from the sun to the earth to the background (yes the sun is heating up the background—but not by much though). And ‘net transfer of energy’ means no equilibrium. Sources that use ‘thermal equilibrium’ for the relationship between the Sun and the Earth are using the term loosely and incorrectly.
The situation here is far from equilibrium because of the massive amounts of energy that the sun is putting out. This is the very opposite of ‘passive’ operation.
EDIT: fixed equation
The ‘background ‘is just an incoming flux of photons. If you insist that this photon flux has a temperature, then it is obviously true that an object can have a lower temperature than this background flux, because an icy earth can have a lower temp than it’s background flux. As you yourself said earlier, temperature is only defined in terms of some equilibrium condition, and based on the equations that define temperature (below), the grey body ‘temperature’ for an irradiance distribution can differ from the black body temperature for the same distribution.
The math allows objects to shield against irradiance and achieve lower temps.
The general multispectral thermal emission of a grey body is just the black body emission function (planck’s law) scaled by the wavelength/frequency dependent emissivity function for the material (as this is how emissivity is defined).
G_{lambda,T}=epsilon_{lambda}B_{lambda,T}.
The outgoing thermal emission power for a grey body is thus:
The object’s temperature is stable when the net energy emitted equals the net energy absorbed (per unit time), where the net absorption is just the irradiance scaled by the emissivity function.
So we have:
Where E_{lambda}. is the incoming irradiance distribution as a function of wavelength, and the rest should be self-explanatory.
I don’t have a math package handy to solve this for T. Nonetheless, given the two inputs : the material’s emissivity and the incoming irradiance (both functions of wavelength), a simple numerical integration and optimization can find T solutions.
The only inputs to this math are the incoming irradiance distribution and the material’s emissivity/absorptivity distribution. There is no input labeled ‘background temperature’.
Assuming I got it right, this should model/predict the temps for earth with different ice/albedo/greenhouse situations (ignoring internal heating sources) - and thus obviously also should allow shielding against the background! All it takes is a material function which falls off heavily at frequencies before the mean of the input irradiance distribution.
This math shows that a grey body in space can have an equilibrium temperature less than a black body.
A hypothetical ideal low temp whitebody would have an e function that is close to zero across the CMB frequency range, but is close to 1 for frequencies below the CMB frequency range. For current shielding materials the temp would at best only a little lower than CMB black body temp due to the 4th root, but still—for some hypothetical material the temp could theoretically approach zero as emissivity across CMB range approaches zero. Put another way, the CMB doesn’t truly have a temperature of 2.7k—that is just the black body approx temp of the CMB.
If your position is correct, something must be wrong with this math—some extra correction is required—what is it?
No—that isn’t what the math says—according to the functions above which define temperature for this situation. As I pointed out earlier 2.7K is just the black body approx temp of the CMB (defined as the temp of a black body at equilibrium with the CMB!), and the grey body temp can be lower. Kirchhoff’s law just says that the material absorptivity at each wavelength equals the emissivity at that wavelength. The wikipedia page even has an example with white paint analogous to the icy earth example.
None of this word level logic actually shows up in the math. The CMB is just an arbitrary set of photons. Translating the actual math to your word logic, the CMB set can be split as desired into subsets based on frequency such that a material could shield against the higher frequencies and emit lower frequencies—as indicated by the math. There is thus a transfer between one subset of the CMB, the object, and another CMB subset.
Another possible solution in your twisted word logic: there is always some hypothetical surface at zero that is infinitely far away. Objects can radiate heat towards this surface—and since physics is purely local, the code/math for local photon interactions can’t possibly ‘know’ whether or not said surface actually exists. Or replace surface with vaccuum.
For your position to be correct, there must be something extra in the math not yet considered—such as some QM limitation on low emission energies.
You’re abusing the math here. You’ve written down the expanded form of the Stefan-Boltzmann equation, which assumes a very specific relationship between temperature and emitted spectrum (which you say yourself). Then you write the temperature in terms of everything else in the equation, and assume a completely different emitted spectrum that invalidates the original equation that you derived T from in the first place.
What you’re doing isn’t math, it’s just meaningless symbolic manipulation, and it has no relationship to the actual physics.
Of course it does—this is a very very common and useful concept in physics—and when you say it doesn’t this betrays lack of familiarity with physics. Photon gas most certainly has temperature, in exactly the same way as a gas of anything else has temperature. Not only that, but it also has pressure and entropy.
In fact, in some situations, like an exploding hydrogen bomb, a photon gas has considerable temperature and pressure, far in excess of say the temperature in the center of the sun or the pressure in the center of the Earth.
https://en.wikipedia.org/wiki/Photon_gas
You say yourself that you assume ‘local equilibrium with it’s incoming irradiance’. Firstly, you’re using the word ‘local’ incorrectly. I assume you mean ‘equilibrium at each wavelength’. If so, this is the very definition of being at the same temperature as the incoming irradiance (assuming everything here is thermal in nature) and, again, there is nothing more to say. http://physics.info/temperature/
I’m sorry that you’re so insistent on your incorrect viewpoint that you’re not even willing to listen to the obvious facts, which are really very simple facts.
EDIT: I fixed the equations above, replaced with the correct emission function for a grey body.
Yes I see—the oT^4 term on the left needs to be replaced with the black body emission function of wavelength and temperature—which I gather is just Planck’s Law. Still, I don’t (yet) see how that could change the general conclusion.
Here is the corrected equation:
Where E_{lambda}. is the incoming irradiance distribution as a function of wavelength, and the rest should be self-explanatory. For any incoming irradiance spectrum, the steady state temperature will depend on the grey body emissivity function and in general will differ from that of a black body.
I’m aware that the concept of temperature is applied to photons—but given that they do not interact this is something very different than temperature for colliding particles. The temperature and pressure in the examples such as the hydrogen bomb require interactions through intermediaries.
The definition of temperature for photon gas in the very page you linked involves a black body model due to the lack of photon-photon interactions—supporting my point about photon temps such as the CMB being defined in relation to the black body approx.
I meant local in the physical geometric sense—as in we are modelling only a local object and the space around it over small smallish timescale. The meaning of local equilibrium should thus be clear—the situation where net energy emitted equals net energy absorbed. This is the same setup as the examples from wikipedia.
Actually I’ve updated numerous times during this conversation—mostly from reading the relevant physics. I’ve also updated slightly on answers from physicists which reach my same conclusion.
I’ve provided the radiosity equations for a grey body in outer space where the temperature is driven only by the balance between thermal emission and incoming irradiance. There are no feedback effects between the emission and the irradiance, as the latter is fixed—and thus there is no thermodynamic equilibrium in the Kirchoff sense. If you still believe that you are correct, you should be able to show how that math is wrong and what the correct math is.
This grey body radiosity function should be able to model the temp of say an icy earth, and can show how that temp changes as the object is moved away from the sun such that the irradiance shifts from the sun’s BB spectrum to that of the CMB.
We know for a fact that real grey body objects can have local temps lower than the black body temp for the irradiance of an object near earth. The burden of proof is now on you to show how just changing the irradiance spectrum can somehow lead to a situation where all possible grey body materials have the same steady state temperature.
You presumably believe such math exists and that it will show the temp has a floor near 2.7K for any possible material emissivity function, but I don’t see how that could possibly work.
No. In retrospect I should not have used the word ‘equilibrium’.
As you yourself said, the earth is not in thermodynamic equilibrium with the sun, and this is your explanation as to why shielding works for the earth.
Replace the sun with a distant but focused light source, such as a large ongoing explosion. The situation is the same. The earth is never in equilibrium with the explosion that generated the photons.
The CMB is just the remnant of a long gone explosion. The conditions of thermodynamic equilibrium do not apply.
If you are correct then you should be able to show the math. Provide an equation which predicts the temp of an object in space only as a function of the incoming (spectral) irradiance and that object’s (spectral) emissivity.