# What can we deduce by the fact that mirrors cannot get a ray hotter than Sun's surface?

I think it is quite counter-intuitive that some lenses or mirrors focusing sunlight to a single spot cannot produce a temperature higher than Sun's Surface.

What could a scientist deduce from that peculiar behaviour? Any quantic property in example? Or something about optics?.

• agree with @DIYser : where did you get that "fact" from? Sep 9, 2016 at 13:45
• You can't make a perpetual motion machine by focusing the thermal radiation from the cold side of a heat engine? Sep 9, 2016 at 14:01
• relevant / answer: what-if.xkcd.com/145 Sep 9, 2016 at 14:20
• Presumably this question is asking about using mirrors (or lenses) to directly heat an object, as opposed to using mirrors to power a solar electric power plant and then using the generated electricity to heat an object to a much greater temperature than that of the surface of the Sun. Sep 9, 2016 at 15:49
• @JeremyHolovacs no it is true. Sep 9, 2016 at 18:10

## 7 Answers

What can we deduce by the fact that mirrors cannot get a ray hotter than sun's surface?

We can deduce that the laws of thermodynamics reign supreme, even with regard to radiative heat transfer. The second law of thermodynamics would be false were it possible to use lenses and/or mirrors to make some object get hotter than the source of the thermal radiation.

The optical reasons are quite simple. Mirrors and lenses do not focus sunlight into a point. They instead ideally focus sunlight into an image of the Sun. (I wrote "ideally" because no real mirror or lens can quite achieve perfect focus.)

There is a way to use sunlight to raise the temperature of an object a few more hundred kelvins hotter than the Sun's effective temperature of 5778 kelvins, and that is to use lots of mirrors, each focusing the middle of the Sun on the object in question, but making the focused images of the Sun from each mirror a few times larger than the object.

The effective temperature of the light coming from the middle of the Sun is considerably warmer than is that coming from the Sun's limbs due to limb darkening. The light we see coming from the middle is a mix of light from the Sun's "surface" and from a slightly below the surface (where things are a bit hotter) while the light we see from the limbs is just that slightly cooler surface light. This means one could use mirrors and/or lenses to heat an object to the effective temperature at the middle of the Sun.

However, I would argue that this still falls in the category of not being able to heat an object using mirrors and/or lenses to a temperature greater than that of the Sun itself.

• So does this also apply to solar pumped lasers?
– user56903
Sep 9, 2016 at 9:39
• @Dirk No because these lasers will generate entropy in operation and dump waste heat somewhere. Ideal lenses and mirrors don't generate entropy. Sep 9, 2016 at 12:12
• even if the size of the area being heated is smaller than the surface of the sun?
– Sam
Sep 9, 2016 at 12:34
• @DirkBruere -- It also doesn't apply to a solar powered electric generator whose generated electricity is in turn used to heat some object well beyond the temperature of the Sun. Tokamaks and the LHC would be impossible were this the case. Presumably the question instead is asking about the maximum temperature attainable by focusing sunlight directly on some object. Sep 9, 2016 at 15:13
• I suppose to stretch the same point further, one could in principle collect radiation from the corona. In practice it's a tiny fraction of the total radiation, but it is much hotter than the surface (photosphere). Still the same thermodynamic principle applies, so I agree this is just quibbling over the fact the questioner specifies "surface", which commonly means something very subtly different from "where the radiation is emitted". Sep 10, 2016 at 23:16

The colliding proton beams at LHC , page 29:

In the collisions, the temperature will exceed 100 000 times that of the centre of the Sun.

The electric currents running the LHC can easily be provided by a large series of solar panels.

Now to go to the lenses part of the question:

A series of lenses concentrating on the same spot can add up to a temperature higher than the sun without violating energy conservation. The reason is because the sun is an extended body, the rays entering each lens come from a different area, although adjacent, of the sun , and there is no limit to the number of lenses that can be added except geometrical.

Take this solar furnace :

A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light (Insolation) onto a focal point. The temperature at the focal point may reach 3,500 °C

There is no conservation rule that forbids using more than one furnace to fall on the same focal point but there exists an entropy argument I find convincing given by @BebopButUnsteady

However to ensure that entropy increases we must perform additionally perform an irreversible process. Otherwise we are simply taking heat out of a cold body and moving it to a hot body. Optics are a reversible process, and therefore you cannot use them alone to make a heat pump.

The reversibility manifests itself in the fact every optical path from the Sun to the object can be traversed backwards. Recall as well that the probability to absorb and the probability to emit must be related by thermodynamics. Therefore once you object reaches the same temperature as the Sun your object must be emitting as much radiation at the Sun as the Sun is emitting at you object. So its temperature cannot increase past the temperature of the Sun.

bold mine.

Note that the Sun is not a perfect blackbody. Therefore, one may be able to heat a furnace slightly beyond the usual stated "surface temperature of the sun". Although this would all be practically indistinguishable, the correct quantity to use is the entropy of the radiation.

Edit after comments.

It is entropy that is the problem in reaching temperatures higher than the sun which is generated plentifully at the LHC. With two heat furnaces concentrated on different small volumes within a larger volume, the paths are randomized/disordered and entropy has to increase, and higher temperatures arrived thermodynamically.

• You seem to be covering "is the claim true" not "what can we derive from the truth of this claim". Or am I missing something?
– Yakk
Sep 9, 2016 at 14:44
• The first part about the LHC is a bit off-topic. Presumably the question asks about how hot an object can be made by focusing sunlight on the object in question as opposed to focusing the sunlight on a solar power generator and using the generated electricity to heat the object. The second part about a solar furnace is also misleading; the effective temperature of the Sun is about 5505 °C (much hotter than the cited 3500 °C). Sep 9, 2016 at 15:00
• The final part about heating an object slightly beyond the "surface temperature of the Sun" is also a bit misleading; the oft-cited surface temperature of 5505 °C (5778 kelvins) is in fact the Sun's effective temperature. This accounts for the non-thermal radiation in the UV and beyond. Sep 9, 2016 at 15:06
• "...once you object reaches the same temperature as the Sun your object must be emitting as much radiation at the Sun as the Sun is emitting at you object." That seems to be obviously false to me. I imagine some sort of ideal lens concentrating all of the sunlight from one hemisphere of the sun onto the Earth. Suppose the Earth heats up until it reaches the temperature of the sun. Now the Earth is a black body at sun surface temperature - why would all of that radiation head back to the sun and not radiate in every direction like any other black body? Sep 9, 2016 at 20:30
• @AmericanUmlaut It's impossible for a lens to concentrate all the sunlight from one hemisphere of the sun onto the earth. You can't "squeeze light together" like that. Sep 10, 2016 at 2:31

I think it is quite counter-intuitive that some lenses or mirrors focusing sunlight to a single spot cannot produce a temperature higher than Sun's Surface.

It is indeed counter-intuitive because it is not possible to focus sunlight to a single spot.

As a simplification, let us consider the sun to be a perfect black body. Then if we know its surface temperature (5772K according to wikipedia), we can calculate its heat flux (63000kW/m2) according to the stefan boltzmann equation. If we know its thermal power output, we can then derive its diameter (which is quite a bit more than zero.)

The sun is not seen in the sky as a single point. From Earth, it appears to have a diameter of about 0.5 degrees (that is to say, if you point a telescope at one side of the sun, you have to turn it through 0.5 degrees to point to the other side.) This angular diameter of 0.5 degrees (or about 1/100 of a radian)is a function of Earth's distance from the sun, and the sun's diameter, which is related to its heat flux as described above.

Imagine I have a parabolic dish reflector on Earth that is perfectly manufactured. With this I can concentrate the sun's rays about 10000 times (no more, because of the 0.5 degree angular diameter of the sun.) I will not give a geometric proof here but if you use a ray tracing program you will find that a parabolic trough can concentrate a light source of 0.5 degree angular diameter about 100 times, and a parabolic dish will be the square of that.

The normal radiant flux on Earth is about 1kW/m2, so I can get about 10000kW/m2 heat flux and a temperature of about 3644K.

As you can see, this is comparable to but slightly lower than the temperature of the sun.

To get higher temperatures from our solar concentrator we would need the sun to maintain its output but be smaller to give us a better focus, but then of course the sun would have a higher surface heat flux and surface temperature!

What could a scientist deduce from that peculiar behaviour? Any quantic property in example? Or something about optics?.

As explained above, the behaviour is not peculiar. What can be deduced about optics is that an optical system cannot produce an infinitesimally small point focus from a spherical black body light source of finite diameter. This conclusion can also be derived geometrically.

If it were possible for a temperature source to raise the temperature of another body above its own temperature in a closed system, there would be interesting consequences for the validity of the zeroth and 2nd laws of thermodynamics.

https://en.wikipedia.org/wiki/Laws_of_thermodynamics

From the fact that the "image" formed by a mirror (of the sun) cannot be hotter than the sun's temperature, we can safely deduce that the law of conservation of energy is valid in this process. It also implies that the second law of thermodynamics is not violated in this process.

Just for fun, if we assume that the image formed by the sun has a higher temparture than the sun itself, then it will contradict our original assumption that of the image and the object, i.e. the sun will have to be the image of the so called image formed by the mirror, which does not make sense.

To expand a bit on the reply of Chetan Pandey above, the image of the sun from a (perfect) lens or mirror is always the same color as the original source. Since color is a function of temperature, you get the same temperature, you do not get a blue-shifted image.

• This one is pretty interesting. So a Red Laser cannot heat something to 7000K ? I though there was no limit to amplitude in a wave given a frequency Sep 9, 2016 at 13:22
• Color of an ideal black body is a function of tempurature.
– Yakk
Sep 9, 2016 at 14:45
• @Yakk Yes - I was assuming the sun as a BB, which is a good approximation. But not perfect. I wonder whether that imperfection offers some scope here?
– user56903
Sep 9, 2016 at 15:11
• @DirkBruere You'd have to show that the image of the sun under optics remains an approximation of a black body for this to give you any interesting conclusions, no?
– Yakk
Sep 9, 2016 at 15:16
• @Yakk What the optics would not do is upshift the wavelengths of light, although an imperfect lens might filter out some wavelengths. If the lens could (say) frequency double like a crystal in common green lasers we are not longer talking about "ordinary" lenses en.wikipedia.org/wiki/Laser_pointer#Green
– user56903
Sep 10, 2016 at 17:35

The fact that we cannot use any number of lenses to get a point hotter than the surface of the distant heat source sounds utterly unintuitive, but is true.

XKCD explains this far better than I ever could: https://what-if.xkcd.com/145/ - this is a really great explanation of the second law of thermodynamics, conservation of étendue, and that the very best that any set of lenses can do is make it so that every single line of sight ends at the surface of the sun.

Which is exactly what you get when you're embedded in the surface of the sun anyway.

What we can deduce from this is that we will never light anything on fire from the light of the moon; that we can never take "multiple solar furnaces" and add them together to sum their temperatures; and that we cannot create an infinite heat pump by making a point near a heating coil hotter than the coil itself with lenses.

Think of it as a thermal circuit.

At one end you have the sun radiating infrared photons through your lens (or mirror array) onto the target object. You also have the target object radiating photons out back through the same optics back to the sun.

As the Suns radiation starts to warm up your target object it will radiate photons more intensely back at the sun.

For now, if we assume your target object is suspended in a vacuum which would prevent other heat transfer mechanisms then:

The sun would keep warming up the target object up to the point where the energy radiated from the target was the same as the energy received from the sun, at which point the target will no longer get any hotter.

Having more mirrors or a bigger lens just means more routes for the photons to travel, and so the equilibrium point is reached quicker.

A white reflective surface on the target will emit less photons, but also absorb less sent from the sun. A black matt surface will absorb more photons from the sun, but will also emit more itself for the same temperature. Having a black matt surface again helps the equilibrium point get reached quicker, but does not raise the equilibrium temperature.

The limit to how hot the target object can get, is the temperature where it is emitting photons at the same rate it is absorbing them from the sun...

So assuming that the original statement is true, the conclusion is that for a given temperature the sun is either as good or more efficient emitter of photons than any material on earth.