Re: Olber's Paradox

In article <1144379352.864243.157160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Edward Green" <spamspamspam3@xxxxxxxxxxx> writes:
mmeron@xxxxxxxxxxxxxxxxxx wrote:

"Edward Green" <spamspamspam3@xxxxxxxxxxx> writes:

<...>

Hmm... I have one doubt about this reasoning: it seems to depend on
the surface of a star emitting equally through the full solid angle:
some of the star surface we are looking at is seen from glancing
angles. If the surface of stars all radiation normally, then all the
distant surface intercepting our field of vision which did _not_ happen
to be pointing at us -- which is almost all -- should appear black.

Use your head. When you look at a glowing object (the sun will do,
with proper eye protection) does it appear brighter in the middle then
towards the edges?

Umm... I was carried away by the spirit of "paradox", meaning
reasoning contrary to fact? ;-)

The surface emits in all directions, not just just
normal and when you work out the details (such as that when a piece of
surface is viewed at an angle, there is more of a surface enclosed
within a given solid angle), you get uniform brightness from the
surface..

I can't quite see how this works out. It seems to me that the edges
should appear brighter if each point of the surface emits uniformly
through a full solid angle. If we approximate the surface as a uniform
array of point sources, won't we view more sources/solid angle the
smaller the grazing angle?

Using ray optics, I can't see how the surface itself can block any
radiation, unless perhaps when we are sufficiently close to the edge,
and take into account the finite aperture of the eye. But even then we
would seem to predict increasing brightness at least up to some
limiting angle, and a tailing off thereafter -- not uniform brightness.

What am I missing?

In the ultimate account, it comes to the wave theory of light. But,
check out on Lambert's Law. It amounts to "each surface element
radiates in each direction with an intensity proportional to its area
projected on said direction."

But wait: in an infinitely old infinite universe filled with a
uniform density of energy sources (stars), the energy density has been
growing for an infinite time, and should be everywhere infinite!

Yes.

So we're in even bigger trouble than merely expecting the night sky to
be as bright as the surface of a star, which was bad enough. Maybe
Olber didn't have the concept of energy.

Oh, he did as I recall, it is just that the popular version of the
paradox is presented for people who may not have this concept. A
slightly more sophisticated version goes like this:

Taking your current location as a center, divide the volume of the
universe into a set of concentric spherical shells of constant
thickness, meaning the volume between r = 0 and r = R, between r = R
and r = 2R, etc, to infinity. So as not to quible to much about
random fluctuations, lets pick R much greater than the average
separation between stars. Now, the volume of the shell corresponding
to R_n = nR is, to a good approximation, proportional to the square of
the radius thus, assuming constant (on the average) star density, the
number of the stars present in this shell is proportional to n^2. The
radiation reaching us from such shell is proportional to the number of
stars in it, divided by the square of the distance. But the distance
is proportional to n and its square to n^2. So, the average radiation
reaching us from each shell is a constant, independent of the shell
number. But, there is an infinity of such shells. So, the total is
infinite.

Hmm... In that case I might have recourse to the stars having finite
angular size, so that eventually the near stars block the further
stars, so that the limiting brightness of the sky is "merely" the
brightness of a stellar surface. For whatever difference that makes.

Indeed. But it is getting worse. Because, what does "block" mean?
Whatever is absorbed, gets reradiated in the ultimate account.

The "blocking" idea was the first one to be raised to try to explain
the "paradox". It was postulated that dust clouds between the stars
absorb the light. This was dismissed in short order by pointing out
that if the dust absorbs, it heats up and when it heats up, it emits.
In equilibrium it'll be emitting as much as it is absorbing.

So, we could add this as well as other details into the spherical
shells model and after long and tedious calculation reach the same
result. But, we can get it simpler. Remember we assume an infinite
and infinitely old universe which, on large enough scale is isotropic.
You may have local density fluctuations but if you take large enough
chunks of space they all look about the same. So, lets take such a
"large enough" chunk. It'll have to be much larger than typical
galactic size, say a cube few million light years on a side. No
problem, we've an infinite universe at our disposal. OK, lets
consider the energy balance said chunk. specifically lets observe the
change of the radiative energy present within the chunk, over a time
dt. Said change is

dE = P*dt + dE_in - dE_out

Where P is the combined power of all the radiation sources with our
cube, dE_in is energy incoming through the faces of the cube from
external sources and dE_out is the outgoing energy, leaving the cube
through same faces. Note that at this point we did factor absorption
and reemission out of the problem. Sure, these occur but since in
equilibrium the amount absorbed equals the amount emitted, they cancel
for the purpose of the equation above.

Now, lets turn to what's left and lets concentrate on any specific
face of the cube. Consider that the dE_in, incoming through tis face,
is jut the dE_out for the neighboring cube, through same face. Since
overall all cubes are about the same, from symmetry we must've
dE_in = dE_out, on the average (if you want to put in in other words,
in an infinite and large scale isotropic universe, given any plane, on
the average as much energy will be crossing it one way as the other).

So, the outflow cancels the inflow and we're left with dE = P*dt which
is positive and since this is assumed to go forever, we've an infinite
amount of energy on our hands.

Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"
.

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