Why wouldn't a mirror next to a blackbody heat parts of it higher than the initial temperature

Question

It's generally accepted that you can't use passive optics to focus light from one black body onto another back body and create a spot hotter than the source. Otherwise you could put a heat engine between the hot and cold spot and generate work. https://what-if.xkcd.com/145/

But imagine taking your bathroom mirror out into space somewhere near the sun, and some of the light gets radiated back, would not the patch of sun directly bellow the mirror get a bit hotter due to incoming light from the mirror? Is this a different problem because normally it is more than one body? I could still imagine some kind of machine connecting two patches of the single body and generating work, I think it is problematic.

While thinking about this I was reminded about sunspots, and I think I've read somewhere that from the outside they appear a little colder than the rest of the surface of the sun, because the magnetic fields somehow trap that heat, and in theory there is a hotter region bellow, which if you think about it is pretty similar to putting a mirror out there.

Answer 1

In such problems, when one is using thermodynamics, one should keep in mind that the statements about black body radiation are at thermodynamic equilibrium, at a given temperature and pressure.

Take a burning wood stove, it has a given temperature reached when coming into equilibrium with the room temperature. This equilibrium is reached with the convection currents removing heat and the radiation balance with walls furniture etc, which are all giving black body radiation at the temperature of the room. If you put a mirror in front of the stove, the black body temperature of the stove will go up because of the reflected heat . It will not be cooling fast enough and the temperature of the room will be lower at equilibrium. If you completely surround it with mirrors, the black body temperature of the stove in equilibrium (same amount of wood) will be really high, and the room really cold. (It is called insulation).

Is this a different problem because normally it is more than one body?

It is different boundary conditions reaching a different equilibrium temperature.

When there is an extended hot source, like the sun, energy can always be extracted from it, and once extracted there is no memory of the source. That is how electricity is generated with solar panels and there is no limit to the temperatures that can be reached by using it.

The thermodynamic equilibrium argument with sun temperature and focal spot of a lens on earth holds for one optical setup, between the patch of the sun from which the rays came to the lens. Suppose the focal point gets as hot as 2/3 of the sun temperature , if one uses a second lens system , it will be coming in equilibrium with a different patch of the sun and if the two focal points are made to coincide, in principle there would be no problem in getting a higher temperature than the sun surface. ( although I am sure this cannot be done for technical reasons, space and sizes of lens systems). This link is an interesting direction in solar power use .