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With enough flux can low energy photons warm a black body to any temperature? eg a solar furnace can approach the surface temperature of the sun, but could it in theory go to any temperature? If all the photons are absorbed then in equilibrium the radiated power and the temperature can be raised to any level. But maybe there is a limit on how much can be absorbed?

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  • $\begingroup$ It would not be an equilibrium if the emitter and absorber had different temperatures. There could be a steady state based on the zero net radiative power at the BB receiver side. $\endgroup$
    – Poutnik
    Nov 17, 2022 at 12:37

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With enough flux can low energy photons warm a black body to any temperature?

Yes. A black body absorber can absorb all of the incident radiation. If that amount of energy exceeds the amount that it can radiate at its current temperature then its temperature will increase. However, this does not imply the answer you expect in the next part of your question.

a solar furnace can approach the surface temperature of the sun, but could it in theory go to any temperature?

No, a solar furnace can only reach the temperature of the sun, not exceed it.

This seems like a contradiction, but it is not. The issue is the conservation of etendue, which is an important but little known concept in geometric optics.

The conservation of etendue states that in an ideal passive optical system (ideal mirrors and lenses, no scattering) the product of the area and the solid angle is constant. This is just a geometrical fact of ray optics. The solid angle is limited to $2\pi$ so the maximum etendue of a system is $2\pi A$ where $A$ is the surface area of the smaller object. That means that it is fundamentally impossible to capture all of the thermal radiation from the larger object and direct it to the smaller object. The light from the larger object simply cannot be focused that small.

Planck’s law gives the thermal power radiated at each wavelength per unit area and unit solid angle. In other words, the power is proportional to the etendue and therefore is also fundamentally limited. At a given temperature there is a limit on how much radiative power can be exchanged between two objects. This makes it so that the thermal radiation power will always flow from the colder object to the hotter regardless of how you arrange the mirrors and lenses and regardless of the sizes of the objects.

So a solar furnace cannot exceed the temperature of the sun, but how is that consistent with the earlier “yes” answer? For thermal radiation and passive optics like a solar furnace, the power will always flow from the hotter to the colder object, due to Planck’s law and the conservation of etendue.

But it is possible to use non-thermal radiation, like a laser, and non-passive optics, like a solar panel. You could take solar energy, collect it with a photovoltaic array, then use it to power a laser, which could heat a target to a higher temperature than the sun (and the laser), despite using relatively low energy photons.

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  • $\begingroup$ I understand the last part, but not the limit to the solar furnace - For the furnace there is no limit in principle to the area of mirror that can be focused at the furnace (just go further out). So there is no limit to the power going in, and hence equilibrium temperature? We don't have to capture all the energy from the sun, just enough to raise the temperature enough of the small furnace. $\endgroup$ Mar 9, 2023 at 13:14
  • $\begingroup$ @DylanMenzies that is covered in the section on the conservation of etendue. If you make your mirror bigger then the region that you can focus down to also becomes bigger. So increasing the size of the mirror will increase the size of the furnace, but will not increase its temperature. $\endgroup$
    – Dale
    Mar 9, 2023 at 15:24
  • $\begingroup$ Sorry still not past your initial point - "Yes. A black body absorber can absorb all of the incident radiation" If that is the case then once a steady temperature is reached, the radiated power equals the incident power, which determines the surface temperature by Stefans law. Hence the temp can be made arbitarily high by increasing the input flux. $\endgroup$ Mar 10, 2023 at 12:19
  • $\begingroup$ In the case of the solar furnace we need to exceed the flux at the surface of the sun, which is possible using focused mirrors for which the total surface area to furnace area ratio exceeds (Reo/Rs)^2, square of ratio of earth orbit radius to sun radius, roughly ~10^7. If you combine n identical mirrors each with flux f on the furnace the total is nf, you don't need to make a bigger mirror that focuses differently. $\endgroup$ Mar 10, 2023 at 12:19
  • $\begingroup$ @DylanMenzies the mirror you are imagining does not exist and cannot exist because of the conservation of etendue. $\endgroup$
    – Dale
    Mar 10, 2023 at 13:07

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