How?
The point is, the most you can do with absolutely perfect lenses and mirrors is to focus the sun’s rays to the point that the rock might as well be inside the sun. At that point, the rock heats up to the temperature of the sun. Then it can’t get any hotter, because if it did it would be heating up the sun rather than the sun heating up the rock.
Any system that allows energy to transfer from the sun to the rock must also allow energy to transfer from the rock to the sun. If the sun is hotter, it will heat the rock. If the rock is hotter, it will heat the sun.
The two bodies do not reach the same temperature, they reach the point where the power they give off is equal to the power they absorb. For two bodies sitting against each other that is the same thing, but for what we are talking about here it is different. Thermal conduction is dependent on the temperature difference between the bodies, whereas thermal radiation is only dependent on the temperature of the radiating body.
Radiation depends both on the temperature and on the surface area. If you have a solid block of metal it will stay warm longer than a thin plate of metal, even with the same mass and initial temperature, just from radiation. So, if you have one body with a big area it will give off energy at a faster rate than another body with a small area. You can’t have one giving off more energy than the other and have them in equilibrium. Since the big object is giving of energy faster than the small body, the small body will heat up. Eventually the small body will become hotter enough that the temperature difference makes up for the area difference and they are each giving off the energy at the same rate. Then the two objects are in equilibrium, but at different temperatures.
Lenses, mirrors, just like in my previous posts.
This is the problem right here. That is only a true statement if the heat is being conducted. Radiant heat does not depend on temperature difference. It depends upon power balance. Energy in must equal energy out. In conduction, temperature difference is the driving force. In radiation it is temperature and surface area.
Well, crap. Now I think I must be wrong, because if what I was saying was true, you would be able to drive a heat engine off of the temperature difference, and that would be a source of infinite energy. The math still seems to support what I am saying though. Anyone able to find a hole in the math?
OK, agreed that an object with a large surface to volume ratio will cool off faster via radiation. So a metal plate radiates heat into the environment faster than sphere of the same mass, it cools off faster. But the object with the large surface to volume ratio also warms up faster via radiation, so a plate will warm faster than a sphere of the same mass. The plate will heat up and cool off faster than the sphere.
Except in our case, the sun is the sphere and the rock is the plate. You’ve confused your terms. The “big” object in your example is an object with a “big area”, which gives off energy faster than the “small body”. But “big” in your example refers to an object that has a large surface to volume ratio. That is, an object with low mass compared to it’s surface area. If both objects are solid spheres of the same material, then the “big” object that gives off energy faster must actually be a small sphere and the “small” object that gives off energy more slowly must actually be a larger sphere.
Exactly. You can’t sidestep thermodynamics simply because the two objects are radiating at each other instead of being directly in contact. (At the lowest level, there is no difference: charged particles exchanging photons.) Thermal coupling is thermal coupling.
I think the problem is there’s probably some unstated assumptions in the black-body radiation equation that’s being violated. All I know is that if you pit a derived equation vs some fundamental principle, I’ll go with the fundamental principle every time.
I think there are some semantic and mechanical/chemical issues that need to be addressed. The photosphere, as previously noted, is about 6000°K. It appears to be the coolest region of the sun, the core and corona both are orders of magnitude warmer than that. What is the temperature of the sun? If we use the 6000°K value, that is a tick warmer than the point at which tungsten boils. This is where we run into the mechanical problem. The sun remains in one piece because it is big enough to hold itself together. But if you focus solar-scale heat into a small ingot of tungsten, it will ultimately not hold itself together, which will make it quite difficult to add more heat to. And if you put it in a glass ball so that you can keep heating it, I am going to guess that the hot tungsten gas will destroy the container.
If you possibly can heat a thing hotter than the sun, either it better be as big as the sun or you need to have some really amazing way to keep the target tightly confined in the focus of the light beam.
It is the photosphere value (6000°K, as you note). The reason is that is where the photons we are using are coming from. Matter deeper in the sun emits photons as well, but those photons are absorbed by higher-up layers and aren’t accessible directly.
In this world of perfect mirrors and lenses the hypothetical inhabits, it is probably fine to also posit an infinitely strong, perfectly transparent box to hold your sample in place while we do the experiment. 
As always, xkcd has something on point for the OP: Sunbeam
In this case we just beat Randall to the punch by a year. Nice of him to finally join the party on this one. 
Hmm, makes me wonder if he trawls through GQ looking for interesting topics? Probably not; his fan base is pretty good at asking more destructive questions than he could ever answer.
Somebody has seen SW:TFA.
But Munroe talks about the Earth’s atmosphere being heated to millions of degrees in a fraction of a second. According to half of the posters here, it shouldn’t be possible to heat it to more than ~6000º, right? Just 6000º really, really fast. So, white-hot like a carbon arc electrode, with lots of UV emitted, but none of this X-ray business.
At the risk of stirring up old arguments, I imagine the following scenario, where the Sun has a surface area of 1,000,000 m^2 and our test object has a surface area of 1 m^2. They start at the same temperature, such that for an arbitrary wavelength of light each will emit 1 photon/m^2s. So in the first second the Sun emits 1,000,000 photons and the test object emits 1. If we funnel those Sun photons into the object, and the object photons into the Sun, have we not transferred more energy from the Sun into the object than vice versa? If we want them at thermal equilibrium, where they are exchanging equal numbers of photons per unit time, doesn’t the smaller object have to be emitting more photons per unit area per second than the Sun? Meaning it is hotter?
Maybe it’s not hotter, it’s just brighter? But how do you make the test object brighter and brighter at a constant temperature without changing its radiating area?
I’m just glad to see another answer. September to January is a long time to wait for the next installment.
Also, if you use your Dyson sphere of mirrors and lenses to collimate the Sun’s entire output into a pencil-thin beam, with the final lens arbitrarily close to the surface of the test mass, won’t the majority of the radiation from the test mass be sent off in directions that don’t feed back through the apparatus to reciprocally heat the Sun? It would seem like you could put a second sphere of mirrors and lenses around the target so that an arbitrarily large percentage of its radiation output was sent out into deep space.
The huge difference is this:
In the OP’s scenario we have a lens on or near the Earth’s surface. That lens can only capture the rays that flow from each point on the face of the Sun to the surface area of the lens. Considering all the rays leaving all the surface of the Sun going every which way you can see that’s a very small fraction of the Sun’s total output.
OTOH, Randall’s scenario captures all the Sun’s rays going out in all 360 spherical degrees from every bit of the Sun’s surface. That’s a HUGE (astronomical in fact) difference.
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As you sort-of say, hotter & brighter are connected but *not *the same. Brighter is a measure of the energy flux passing through a cross-section of space. Hotter is a measure of how much energy flux over time has been absorbed by how much mass of what type net of how much energy flux has been re-radiated. Those are two very different things.
Not I.
It may be helpful to think of this thought experiment:
Start with a single flat mirror directing the sun’s light directly at you. You look around, and see & feel the equivalent of two identical suns worth of light.
Now start adding mirrors one at a time. 3, then 4, then 10, then 20.
At some point, you’re completely covered, everywhere you see is a flat mirror with a sun on it. But maybe the mirrors are rectangular, so you take all the mirrors down and change them to circles so that when you look at each mirror from your central position, they are all exactly the apparent size of the sun with no extra mirror space wasted. Then in-between all the gaps you put more flat mirrors. Now everywhere you look is sun.
But at that point, you can’t do anything more. Ok, you could probably tweak your design a little bit by taking all your flat mirrors down again and break them up into infinitely small mirrors individually directed at you (or realize you’ve just basically reinvented a giant parabolic reflector, and build that instead). Or you could move your design to space to avoid the pesky atmospheric scattering of the Earth. But the point remains - Everywhere you look is sun. There’s no other places you could fit more mirrors, as they would just end up being shaded by your existing mirrors. You’ve reached the upper limit for how much you can focus sunlight on a single spot (yourself).
At this point, with a complete blanket of sun around you, you are functionally equivalent to being inside the surface of the sun. And therefore you’d equalize at the exact same temperature as the surface of the sun is. (Maybe a little lower due to atmospheric scattering of the suns light, but no higher).
But the xkcd scenario isn’t taking a giant mirror or set of lenses and focusing the Sun’s rays on the Earth. It’s concentrating all the Sun’s output via some arbitrary method, and using all that power to create a laser 1 meter wide aimed at the Earth.
If your method was just to surround the sun with a spherical shell of perfect mirrors, with a 1 meter tube open pointing at the Earth, that wouldn’t create a million degree laser. In real life putting a shell around the sun, even if it was a really good mirror, would just mean that eventually the shell would heat up to whatever the temperature would be if the sun had the same energy output but with the radius of the shell.
To add to the point above, suppose you really had a perfect mirror around the sun, that reflects all types of photons from radio to gamma rays. Put your hand on the outside, and it’s 3K, the background temperature of the universe.
Now, what’s happening inside? The inside gets hotter and hotter and hotter, since the sun cannot radiate energy away into space, it all gets reflected back by the “perfect” mirror. Pretty soon the temperature inside is millions of degrees. Eventually physics breaks inside the sphere, at some point the temperature and pressure goes crazy and you can’t get nuclear fusion anymore. Beats the hell out of me what sort of quark soup you end up with, or if you create a supernova or black hole or God knows what, when you’re allowing a perfect mirror you’re going to break physics somewhere.
So the reason you can get something hotter than the sun if you put a bunch of perfect mirrors around the sun and then poke a hole and let some energy out, is that the region of space inside the perfect mirrors is going to become much much hotter than the sun is, since the sun in real life can radiate energy into space.
The ‘perfect mirror around the sun’ scenario has another problem, because as the Sun heats up inside the sphere it expands and becomes less dense, slowing the rate of fusion. Eventually the Sun becomes a hot gas or plasma inside a silver balloon. Given a perfect mirror (cold on the outside!) fusion wouldn’t stop until all the elements inside have fused into iron. The balloon would be very hot inside at that point. Note that the perfect mirror would have to reflect neutrinos as well.
Something similar would happen with a real world imperfect mirror around the Sun as well, except that the mirror would heat up and evaporate in due course, allowing the Sun to shine out once more. But that’s a different question.