Theoretical maximum temperature with solar heating

Question

I've heard that since sun is around 5800K hot at its surface, the hottest temperature one can get with solar heating is that temperature. From a thermodynamics standpoint, this makes very much sense. An object cannot heat another object to a higher temperature than itself.

However, I am not exactly satisfied with that answer. The sun emits around $3.8\cdot10^{26} W$ of power. If one gathers even a fraction of that and then concentrates that power to a small point, shouldn't it not heat up to millions of degrees or more to balance out the heat coming in with heat radiated out via $\frac{P}{A}=\sigma T^4$?

One explanation I've heard that the albedo of the object itself becomes higher and higher as the temperature approaches the incoming temperature, 5800K in this case. To me, this sounds a bit ridiculous, though. How can be recieving object know the temperature it should not go over, and then act appropriately? It's like the object itself performs some fourier analysis of the incoming radiation, deduces that the radiation is thermal, then calculates the absolute temperature and finally changes its own albedo accordinly. I didn't know rocks or tables could think and alter their own properties.

Even if the scenario about the albedo that I presented above would make some sense, I still refuse to believe that if you gather the entire solar output for one second and shoot it at a pencil, that the pencil gets heated up just to 5800K.

The final thought I had for the example given above, is the realization that the light gets concentrated. If one concentrates the solar light, then the 5800K constraint should be removed right? Well, from a thermodynamics standpoint this still does not make that much sense to me at least. In order for an object (sun) to heat up another object (target) to a higher temperature than itself, additional energy should be supplied. This is how a heat pump works. However, if one uses only mirrors, there is no additional energy anywhere, so this cannot be the full explanation.

Any thoughts on this?

Answer 1

This is answered in Randall Munroe’s What If, question 145.

The question asks “Can I start a fire with moonlight?” to which the answer is “no”, and it turns out the answer is applicable here too. All the images I’ve used here belong to him, from that linked page.

You can’t make heat flow from something at $T_1$ to something at $T_2>T_1$ without expending energy

Lenses and mirrors work for free (ideally): they do not take energy to work. An ideal mirror will just be an ideal mirror; you don’t have to plug it in. As such, they have zero energy to work with.

If you could heat something with only the Sun and lenses and mirrors to a greater temperature than the Sun, you’re expending zero energy to make temperature flow from a cold(er) place to a hotter place, which violates thermodynamics.

The optical system would be irreversible

If you want to focus all the Sun’s energy to a point, you will have a lens looking like this,

which is not reversible and thus not possible to construct. Light from both B and A will go to C, heating it up beyond the temperature of the Sun. Yay! But optical systems should be reversible, i.e. if light goes from C through the lens it should end up where it was emitted. Unfortunately, if the lens focuses all the light onto a point, how will the light emitted at C know whether to go to A or B?

This implies that the optical system cannot exist. Optical systems focus things onto areas, not points. As such, you will find difficulty with…

Conservation of étendue

There is such thing as conservation of étendue, which means that for light coming into an optic spread by angle $\theta_i$ on an area $A_i$ and emitted with angle $\theta_o$ and area $A_o$, you have $\theta_iA_i=\theta_oA_o$. Munroe explains it graphically in that What If question:

Nope, you can’t do this.

The best you can do is equivalent to being inside the Sun, which won’t make you hotter than the Sun

The absolute best an optical system can do is make all lines of sight end at the light source, like so:

And, of course, if you’re just surrounding yourself with the Sun, you cannot get hotter than your surroundings without violating thermodynamics, and as such you will quickly reach the temperature of your surroundings, i.e. the temperature of the Sun and no hotter.