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.
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.