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



Bob Sciamanda wrote:

Michael's analysis is correct.

Yes.

Let me restate in my own words a subset of the key points made so far:

1) It is not always possible to focus the sun's rays to a single point.
A single example suffices to prove this point. In particular, either
of the following items (2) and/or (3) suffices.

2) For a lens of moderately long focal length, the size of the image
of the sun is easy to observe experimentally.

3) For a thin lens, the size of the image is easy to calculate.

==========

Now I would like to supplement those points with the following:

4) It is not *ever* possible to foxus the sun's rays to a single point.

This statement is harder to prove. Examples won't suffice. It can,
however, be seen as an immediate consequence of Liouville's theorem:

5) Phase space is conserved.

There is an important physics lesson here. The fundamental importance of
Liouville's theorem beggars description.
-- It is key to understanding the second law of thermodynamics; any system
that violated Liouville's theorem would immediately be usable as a perpetual
motion machine.
-- It is key to understanding the uncertainty principle; any system
that violated Liouville's theorem would immediately be usable as a Heisenberg
microscope.
-- It is useful for many other things.


In this case: Let our Z axis be the line between the middle of the sun and the
middle of the lens. The aperture of the lens extends in the X direction from
+D/2 to -D/2 for a total aperture of D. For simplicity we suppress the Y
dependence for now; it is trivial to add it back in.

We are concerned with a certain parcel of phase space. The parcel in question
can be understood by reference to:
http://www.bluffton.edu/~edmistonm/sun_lens_image.pdf

One axis of our phase space is X, and the other axis is dX/dZ. At the plane of
the lens, the relevant parcel extends in the X direction from +D/2 to -D/2.
You can also see that each such X point, the bundle of rays has a nonzero extent
in the dX/dZ direction; specifically it has extent theta, where theta is the
angle subtended by the sun, i.e. about half a degree.

So at the plane of the lens, we have a rectangle of phase space, of size D times
half a degree.

It is an easy (but very enlightening) exercise to show that for any thin lens,
phase space is conserved. If you use a telescope to form a large image of the
sun (large extent in X) the rays from the lens will hit the image plane with
a small range of angles (small extent in dX/dZ).
-- By induction, you can show the same conclusion applies to any *combination*
of thin lenses.
-- By Liouville's theorem, the same conclusion applies to *any* physical system.
That includes non-thin lenses, mirrors, and everything else.

===

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. This would
immediately violate the second law of thermodynamics, since you could in principle
run a heat engine using the focal spot as the "hot" side of the engine, and the
surface of the sun as the "cold" (!) side, thereby producing work using only one
heat bath (the sun) rather than the conventional two.

====================

Funny story: When I was in junior high school, I was interested in amateur telescope
making. I specifically wanted to build the ultimate rich-field telescope, one that
gathered so much light that it would be possible to see beautiful nebulae and other
deep-sky objects with the naked eye.

I tried really hard to design such a system. I went through reams of paper. If I
made the aperture bigger at constant focal length, the exit pupil became too big
(bigger than 7mm, the diameter of the dark-adapted eye). If I made the exit pupil
smaller, it always seemed to come at the cost of more magnification than I wanted,
meaning the image was too spread out, i.e. too few photons per unit area per unit
time, i.e. too dim.

Eventually I gave up; I figured I just wasn't smart enough to design the desired
system.

Years later I learned about Liouville's theorem, and realized that no such system
is possible. Each of the deep-sky objects I was interested in is intrinsically so
dim that even if you travelled there and stood in the middle of the object, you
couldn't see it with the naked eye. (The beautiful pictures you see come from
long time exposures, which the eye cannot do.) No system of lenses (or anything
else) can produce an image with more surface brightness than the original object.

=======================

Liouville's theorem can be taught at many levels. At the simplest level, it
readily lends itself to discussion in terms of examples and pictures. For
instance in one-dimensional ray tracing (as dicussed above) it is as easy as
pi to sketch the parcel of phase space at critical points in the diagram.
a) What does passing through a lens do to the parcel?
b) What does propagation through free space do to the parcel?

Hint: Part of the answer to (b) is "shear" : a square will get sheared into a
parallelogram........

Phase space has a place of honor in the physicist's toolbox. You don't need
it every day, but when you need it, you reeeeally need it.