Re: More Conservapedia Blather

In article <1172872201.911809.27850@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Robert Carnegie <rja.carnegie@xxxxxxxxxx> wrote:
Robert Grumbine wrote:
The crispness of the shadows depends on what's between you and the
sun. If you're in vacuum, they're perfectly sharp. If it's a dry
atmosphere, as for, say, the Thar Desert by Jaipur (or over in Egypt,
Greece, ...) the shadows are quite sharp. I was surprised by just
how sharp when I visited Greece last year on my honeymoon. I've
lived in cloudier, more humid areas, so am accustomed to fairly
fuzzy shadow edges.

Hmm - I thought the size of the sun's disc would be the main blur-
factor, and independent of atmospheric conditions, but you tell me
otherwise. I was assuming, therefore, that something more than a
simple shadow was involved.

I'm still sceptical of the astonishing precision of architectural
astronomical instruments.

Hm. Well, some skepticism is warranted.

Let us consider the width of the disk. It's 0.5 degrees limb to limb.
On the other hand, at the limbs, there's essentially no length (I'm thinking
of strips perpendicular to the solar equator -- the strip at the limb heads
to zero length, the strip through the poles has the full 0.5 degrees).
So the brightness goes to zero there and it doesn't contribute to the
shadowing. The transition zone in the shadow from no sun to full sun is,
therefore, more rapid than this consideration will show.

Now consider the length of the shadow, and, more importantly, the spread
of width due to a small angle. l = h*cot(theta) where l = length of
shadow, h = height of gnomon (I'll take 10 meters for illustration), and
theta is the angle of the sun above the horizon. dl, then, = h/sin^2(theta)
* dtheta, dl = width of shadow, dtheta is angular size of body. Take
theta of 45 degrees, to take a middling angle, and the 0.5 degrees are
approximately 1/120 radians. For h = 10 meters, the width of the shadow
transition (extreme value) is 83 mm. It's probably a factor of several
smaller than that in practice, given the sensitivity of human eyes.

Conversely, let us examine how much the angular position of the sun moves
in 1 second. It's 4 (clock) minutes per degree of angle, so 1 (clock) second
is an angle of about 0.25 arc minutes, about 100 times smaller than the
(maximum) width of the solar disk. This leaves us with about 1 mm motion in
the shadow.

To the extent that you try to see the motion of the shadow along the blade
it casts on the ground, it'll probably take several seconds, perhaps tens
of seconds, to see the motion versus the width of the transition zone. On
the other hand, out at the tip of the shadow, gnomons usually being made to
have a pointy end, ... probably an easier matter.

Of course this also ignored atmospheric turbulence, refraction, and a host
of other matters (just how flat _is_ the ground?).

By the way, at
http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20070301.shtml
which should stick around for a while - there is an audio discussion
of optics. But I haven't listened to all of it yet.


--
Robert Grumbine http://www.radix.net/~bobg/ Science faqs and amateur activities notes and links.
Sagredo (Galileo Galilei) "You present these recondite matters with too much
evidence and ease; this great facility makes them less appreciated than they
would be had they been presented in a more abstruse manner." Two New Sciences

.

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