Hello Roger and Frank,
and additonally, shouldn't the plane through the points of the hour
scale be identic with the equatorial plane of the local sky. (I hope my
English formulation is understandable.)
Very best wishes
Helmut
Am 04.05.2013 19:09, schrieb Roger Bailey:
Hello Frank,
Yes, this concept is thought provoking. Here are a couple of other
thoughts on the topic.
1. Orientation of the globe may not be important but the orientation
of the hour scale is. 12 must be on the east and west edge. If the
scale is in the equatorial plane the hours are uniform, 15° each. If
the scale is horizontal, the hours are defined by the azimuth. Any
orientation of the hour scale is possible, defined by the specific
geometry.
2. The projection of the half degree penumbra increases going from
directly facing the sun on to the edge, 90° around. For a flat
surface, this is defined by the cosine. For the curved surface,
calculus of differential cosines would define the spreading of the
penumbra on the sphere. This is why the terminator line is less
distinct than a shadow.
There are math challenges with both these points, starting with good
sketches.
Regards, Roger Bailey
--------------------------------------------------
From: "Frank King" <frank.k...@cl.cam.ac.uk>
Sent: Saturday, May 04, 2013 1:42 AM
To: "Roger Bailey" <rtbai...@telus.net>
Cc: "Willy Leenders" <willy.leend...@telenet.be>; "Sundial
sundiallist" <sundial@uni-koeln.de>; <frank.k...@cl.cam.ac.uk>
Subject: Re: globe on the dome of the Gustaviaum
Dear Roger,
I do enjoy your thought-provoking messages!
I have been thinking over your comments on
the globe dial:
I call sundials like this "Terminator"
sundials as they tell time by the
terminator line between the sunny
and dark sides.
So far, so good. An eloquent description
of how these dials work. You continue:
It is not a sharp line like a shadow
This is what set me thinking...
In general, a shadow is not any sort of
line, sharp or otherwise. On a plane
surface shadows are two dimensional.
Maybe you meant:
It is not a sharp line like the edge
of a shadow
This can't be right either because the
edge of a shadow, the penumbra, is
precisely the part which prevents it
from being sharp!
The terminator is just a posh word for
the penumbral region at the margin of
the shadow on the globe.
You can ponder whether the penumbral
region of this shadow is more or less
sharp than the shadow of, say, a
vertical stick. The truth is that
it is both more sharp and less sharp!
Here's why...
The penumbral region of the shadow of
a vertical stick increases linearly as
you get further from the base of the
stick.
The penumbral region of the shadow of
on a globe has uniform width all the
way round.
Imagine a bug crawling over the surface
of the globe and crossing the terminator
at right-angles...
The bug goes from full shadow to full
illumination by the sun by crawling
an angular distance of about half a
degree On a 10 foot diameter globe
this is about half an inch.
The penumbra of the shadow of a
vertical stick increases from zero
width indefinitely. It will be
half an inch across at about 5 feet
from the base if the sun is low.
If the Earth didn't have an atmosphere,
its own terminator would again be about
half a degree across.
In the Tropics, when the sun sets
normal to the horizon, it would take
about two minutes to go from being
just wholly visible to being just
fully below the horizon.
The atmosphere stretches this a bit
because refraction increases rapidly
as solar altitude heads towards zero.
The atmosphere has a small effect on
the terminator on a globe but it also
has a small effect on the shadow of a
stick. You can ignore this effect.
You continue...
Orientation is important.
Not on a globe it isn't. Globes tend
to look the same whichever way you
orientate them :-)
Very best wishes
Frank
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