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Re: [Phys-l] fire starter from the sun



>>To connect this ultra-specifically to the Subject: of this
>>thread: If you could focus the sun's rays more tightly than
>>permitted by Liouville's theorem, it would be possible to
>>create a focal spot hotter than the surface of the sun.

Robert Cohen wrote:

I'm a little confused, particularly on what you mean by "hotter". Are
you saying that if Liouville's theorem did not hold then...

(a) it would be possible to create a spot through which the total energy
flux (energy per time) is greater than the total energy flux that leaves
the surface of the sun,

Of course not. The total luminosity of the sun is 4x10^26 watts. Kinda
hard to exceed that no matter what you are doing with handheld lenses.

Conventionally, "hotter" means "higher temperature".

or are you saying that...

(b) it would be possible to create a spot through which the total
irradiance (energy per time per area) is greater than the total
irradiance that leaves the surface of the sun, or

Well, that is part of the argument I was making.

Using Stefan-Boltzmann ideas, if you have high irradiance going in,
the target will heat up. It will continue heating up until it reaches
a temperature T such that the power going out is the same as going
in. There is a almost one-to-one relationship between temperature and
power ... I say "almost" because there are some solid-angle factors
involved also.

Let me explain my confusion: It seems to me that (a) would violate
conservation of energy, let alone Liouville's theorem. In addition, how
hot the spot gets depends more on the total irradiance not the total
energy flux so I'm guessing you don't mean (a).

Agreed, (a) doesn't make sense.

On the other hand, if you mean (b) then I fail to see why it is not
possible. I admit that I cannot easily follow the logic you presented
in your note. However, suppose you took an infinite plane surface of
uniform temperature T. This would produce a certain irradiance. If you
place a lens (with optical axis perpendicular to the surface), couldn't
you get the rays to focus to a point with much higher irradiance than
the irradiance at the surface of the sheet?

If I understand the question, the answer is no. Liouville's theorem forbids
it. Not coincidentally, the second law of thermodynamics also forbids it.

The key variable is not
power per unit area
but rather
power per unit area per unit solid angle.

That is: Suppose we have an opening 1cm in diameter. Putting an undirected
glowing surface behind the opening is one thing; shooting a highly collimated
laser beam through the opening is another matter entirely. The latter can be
focussed to a very tiny spot, whereas the former cannot.



P.S. I apologize in advance if I am not use the appropriate terminology.

Well, since you bring it up, irradiance is radiant power going _in_ to a
surface; it is a misnomer to use the same term for power going out.

Hopefully my concern is clear enough.

The direction of the irradiance isn't a serious source of confusion for me.