So let me see if I get this straight, if I have a diffuse heat source of area ‘A’, and a focussing mirror, I set up my experiment so that the mirror reflects the light from the source back onto the sources surface over a smaller area ‘a’ then the temperature over ‘a’ remains the same, as if there was no mirror there at all. (I’m glad I’m not a physicist any more 
)
Way back when my husband was a junior in high school, he attended a science conference at Ball State. At one point the instructors took out a huge fresnel lens and focused the sunlight on a penny.
The penny caught on fire.
So alls I gots to say is – plenty hot!
Hmmm dont quite get it. At equilibrium the enegy influx into the black sphere = energy flux out. Now for a black body emittor the energy outflux I presume is merely proportional to the temperature and the surface area. So if we dramatically increase energy influx by a lens, the energy outflux must rise, so this is can only be done by an increase in temperature.
Once we are in thermal equilibrium, adding a lens will not increase the energy influx, so no increase in temperature is needed.
If the area A can quickly maintain a uniform temperature, then the mirror will simply change the thermal coupling of the area with the environment. The mirror is basically acting like insulation.
If the area A cannot quickly maintain a uniform temperature (so that some parts could change temperatue with respect to other parts), then the mirror will heat up patch a.
scr4, i’m not sure how to meaningfully define thermal equilibrium for two objects moving with respect to each other. I’m sure it’s possible, but I bet it’s tricky once you go relativistic.
Elysian, was it really the penny on fire, or the surface it was resting on? Those big Fresnel lens can ignite asphalt, too. Was the flame green (and what color does zinc burn?) or more white? Definitely a neat trick, whatever catches fire! 
Yes, if you had a propane heater and put up a mirror in front of it, the heater would get hotter. And a hotter heater would make the butter better. That’s because the size of the mirror is very large relative to the size of the heater.
A gigantic mirror, even one covering the entire surface of the moon, is very small relative to the surface area of the sun. So yes, if you put up a mirror to the sun and reflected the light back to the sun, the sun would get hotter. By some infintesimal amount, probably not measurable.
Think this way. You have a propane heater heating up some butter, since hotter butter is better. Once the butter is as hot as the heater, it cannot get hotter, since then the butter would heat up the heater rather than the reverse. But suppose you got two heaters and pointed them both at the butter. The butter would be hotter faster. But when it got to the temperature of the heaters, it could not get hotter, since it would radiate energy back to both heaters.
This is complicated because you also have the heaters/sun and the butter radiating energy into the earth and/or the 3K universe. But a mirror or a lens works both ways. The hot butter will radiate energy. Some will go to the universe and make the universe hotter. Some will go back to the heater/sun and make the heater hotter by some marginal amount. But when we are talking about objects as small as the earth, the amount of energy an earth sized mirror will transfer back to the sun is negligible. But when the butter is the temperature of the sun, it will radiate as much energy per square centimeter as the sun. And thus the sun cannot transfer energy to the butter, no matter how much solar energy is focused on the butter.
But if the patch a will be heated up, will its temperature not then exceed the average temperature of area A. Giving us a case then if A were the sun for example, creating temperature > than that of the average temperature of the surface of the sun using a simple focusing mirror?
And if the temperature at 'a 'could be higher than A, how is this in accordance to the relivent thermodynamic laws?
If patch ‘a’ was not part of area A, then why can we now not get temperature > than the average temperature of area A in patch ‘a’.
OK, dusting of the ol’ Physics degree here…
Heat vs energy vs temperature always causes confusion.
Trying to ask what the “temperature” of a focused spot could be is a confusing red herring. One needs to ask what the energy flux is at a focused point. (i.e. what is the energy per surface area). Unfortunately, what most people think of as “temperature” is a function of many things, incident radiation, thermal equilibrium with surroundings, density and pressure if it’s a gas…
For example, the surface temperature of the sun is around 6000 Kelvin, but the corona around the sun is as high as 1 million K. The core temperature of the sun climbs to 15 million K as you get deeper. So how do we really determine the energy output of the sun? What is the maximum temp of a focused spot?
http://fusedweb.pppl.gov/CPEP/Chart_Pages/5.Plasmas/SunLayers.html
Clearly, this can be confusing then if you try and make the claim that a focused spot’s max temperature is the same temp as the surface of the sun.
Additionally, the total amount of energy that can be focused is dependent upon many things, size of mirror and reflectivity, loss to atmosphere absorption, transparency of lens, your latitude and angle of the sun’s rays, etc.
What it boils (hee hee) down to is that we have to restrict our discussion purely to energy flux (incident energy per surface area unit, usually Joules per square-meter). The maximum amount of energy that can be delivered to a focused spot is equal to the amount that the collector receives. The flux is a function of the surface area of the collector and the surface area of the focused point.
So the real question is, how small of a point can the rays of the sun be focused at? This will determine what your ultimate “temperature” at that point will be.
I agree with scr4 and Pleonast, the maximum temperature you can reach by focusing sunlight is the surface temperature of the sun. Remember that while the coronosphere of he sun might be 16 millionº, it’s very diffuse, and radiates in x-rays and extreme ultraviolet, neither of which make it through the atmosphere to the surface anyway. Also, it’s diffuseness means that it subtents a larger angle in the sky, and can not be focused into as smmall a point.
If you could focus the sun’s light into an arbitrarily small point, you could raise the temperature of that point to an arbitrarily large value. but you cannot. If you use a larger mirror, the minumim size of the focal point gets larger. The ‘focal point’ is only a point when resolving objects that are truly points. Any non-point object has a ‘focal plane’ in reality. The sun is not a point in our sky, and when perfectly focused, it appears as a disk, a disk that gets larger the larger the mirror used.
I think this is what is tripping people up. They are assuming that you can just raise the size of the mirror however big and somehow keep the focal image size from also increasing. You simply cannot do this, so the temperature of the focal point remains the same. A mirror has a fixed aspect ratio, say 10to1 for example. That means the focal point will be 1/10 the size of a mirror with that curvature, no matter how big or small you make it.
Having said all that, I’ve heard of sunlight driven lasers that can supposedly acheive temperatures of 10,000ºF, but I’m not sure how they work, or if they are real. I’ll try to look them up after this.
For the purpose of this OP, you can treat the photosphere as a perfect blackbody and ignore everything else. The photosphere is where the sun becomes optically thick, so the temperature below this point is irrelevant. Radiation from the upper layers is extremely small compared to photospheric emission.
The OP asked for the maximum. This is attained with a 100% reflective mirror system and zero atmospheric absorption, and focusing the light onto a perfect blackbody.
Equilibrium temperature of a blackbody is directly related to energy flux, so I don’t see why you need to make the distinction.
I stand by my original statement that the theoretical maximum temperature of a solar furnace is 6000 degrees. By optimizing various parameters as stated above (mirror system design, reflectivity, etc) you can get arbitrarily close to this temperature. And you cannot exceed this temperature without violating the 2nd law of thermodynamics.
I guess that was a bit too much hand-waving. The rigorous way to treat this is to use effective temperature, which is defined as the temperature of a blackbody that would emit the same energy flux. For the sun, this is very, very close to photospheric temperature.
(By the way I said “bolometric temperature” earlier but I think that was wrong, sorry. It should be just “effective temperature.”)
Forget the single curved mirror for a moment and consider an array of small square, flat ones, each of which can be independently tilted in two axes.
Assuming parallel rays from the sun (which I know isn’t actually the case), then the ‘spot’ projected by the mirror is roughly the same area as the mirror itself.
The adjacent mirror can be angled in such a way as that the projected ‘spots’ overlap almost perfectly.
As can be done with the next and the next and so on, so a huge array of independently aimed mirrors, covering an area the sixe of, say, one side of the moon, can all be made to project their spots on the same small patch of ground on planet Earth and (assuming no complexities such as atmospheric interference , vibration, orbital movement etc (which are all there, but irrelevant to this question), the total area of the target could theoretically be not much larger than the area of a single one of the mirrors. So we’d be delivering the sun’s light from a circle the diameter of the moon (at the distance of the moon from the sun) to a spot on Earth not much larger than one of the mirrors. The focused image could be quite small.
I’m still not getting the equlibrium thing in this context.
“then the ‘spot’ projected by the mirror is roughly the same area as the mirror itself.”
Well…no, the beam from each mirror will get larger as it moves towards the focal point. This spreading cannot be ignored, since it is not trivial. This configuration would actually be worse than a parabolic mirror, since in the parabolic case, the rays converge to the maximum that thay can.
In a moon sized accumulation of tiny flat mirrors, each of the rays from the mirrors would be very spread out by the time it made it to the center. This would more than make up for the increase in reflective area. ( In a smaller configuration, the spreading would be smaller, but the area collected would be smaller as well, the two effects cancel out, roughly)
Your total heat energy deleivered would be tremendous, but your heat energy per unit area, the temperature, would not rise above the maximum. The ‘focal point’ would be huge, probably many miles across. It doesn’t matter that all the mirrors are pointing on the same tiny mirror shaped peice of space.
By assmuing paralellel rays from the sun, you totaly change the reality of the thought experiment. If the sun was a giant laser pointed at the mirrors, it would probably work, but then you wouldn’t even need the mirrors. 
if you concentrate a beam AND stop or minimize heat escaping from the area you are concentrating the beam on, would’nt the overall energy/heat continue to rise?
I also am curious if the fact that the concentrated beam from a magnifying glass can never exceed a certain point (based on both the surface temperature of the sun, and the quality of the light gathering/concentrating implement), does it also hold true that given equivalent quality and focusing abillity, a small magnifying glass will generate the same temperature as a larger one but over a smaller area?
OK; how about if each of the tiny mirrors were a very shallow parabolic reflector of the correct curvature to focus the smallest possible dot on an target the distance of Earth…
I was already assuming that the small mirrors were aranged in a parabolic configuration. But since the surface is not truly parabolic, you get less concentration than a parabolic surface, which cannot focus things beyond a certain point as already discussed. If i could draw in this box I’d make you a nice diagram. Remember, the sun is almost a million kiometers across, and it’s focused image will always be a minumim of a ratio of this size to the size of the parabolic mirror used. A small mirror can give you a small image , but at a cost of less heat.
Quint Essense- Yes, the small magnifying glass can acheive the same temperature, in a much smaller area in theory. But in practice the amount of total heat being concentrated is so small that the heat will leak into the surrounding area fast enough to keep the spot from getting as hot as a larger glass.
There is no way to make a body with a non-zero temperature stop radiating. Suppose you enclosed the focal object with a one-way mirror, that lets light in but doesn’t let it out. The mirror will reach the same temperature as the object, and radiate the excess heat into space.
Yes, the mirrors would be arranged in a parabolic configuration (would this be called a fresnel mirror?), but (conveniently ignoring engineering limitations, as we are talking about laws and principles, rather than stuff we can actually build), the surface of each tiny mirror could be slightly dished so as to focus the beam as tightly as possible for each mirror. Unless there’s something else I’ve missed, this should allow multiple mirrors that are spread across a wide area to superimpose their targets into a very small area.
Put simply, I’m talking about an array of small parabolic mirrors.
Hmm…tough one.
The way I see it, each small parabolic mirror would have a focal point that is some small distance in front of it. By the time the rays made it out to the center of focus for the whole array, the beams would be even more distended.
If you tried to extend the focus of each mirror to the focus of the array, you’d find that they are nearly flat anyway. ( They would all be slightly curved such that they are all surfaces of a parabola of the size of the array. In other words, we are back where we started.)
If you use a parabolic mirror to focus light at a distancd D from the mirror, then the size of the focused spot is D*A[sub]sun[/sub], where A[sub]sun[/sub] is the angular size of the sun. You can’t create a smaller spot than that, it’s simple geometry.
You might want to look up the Brightness Theorem in a book on radiometry and photometry. It basically says that you can’t increase the flux of light per unit area per unit solid angle, regardless of the optical system you use. (Actually, the refractive inde is in there, too – but the sun’s in vacuum, so we can ignore that). A good book will give you a proof. If the flux of photons per unit area and per unit solid angle an’t be increased over that of the source, I doubt if you can incease the temperature, either.