Dear Chris,
I am a bit behind with my reading and I have only
just read your comments, and corrected comments,
on the umbra discussion.
Subject to your corrections, I concur with almost
all you say but I feel a little amplification of
one of your follow-up remarks is needed. You
say...
> If the gnomon is ... a conventional wedge-shaped
> gnomon with two style edges, then you can adjust
> the hour lines, by a little under a minute.
I am very nearly happy so far!
> This adjustment is definitive, in the sense that
> it is irrespective of the time of day, time of year,
> size of sundial, type of sundial (horizontal or
> vertical) or latitude.
Yes, I am still happy [subject to your qualification
later that "the sun moves faster across the sky at
the equinoxes than at the solstices"].
My need for amplification is confined to the
50 seconds figure here...
> It is an angular change (about 50 seconds of time),
> not a linear change.
It is indeed an angular change; it's the 50 seconds
figure which I feel needs a bit of amplification...
At the equinoxes the sun obligingly trundles along
the celestial equator (well close to it) at a rate
of 1 degree every 4 minutes of time or 1 arc-minute
every 4 seconds of time.
Consider a bug, sitting on the 2pm hour-line (say),
looking at the sun (via special eye protection) as it
approaches the business edge of a fat wedge gnomon.
There will be a period lasting just over two minutes
(to be justified below) of significant interest:
1. Since sunrise the bug has been able to see the
entire solar disc and has been basking in full
sunlight. Then, about 13:59 sun time, the sun
makes first contact with the edge of the gnomon
and, soon afterwards, the bug notices a drop in
light.
2. The solar disc steadily slips behind the edge
and, about 14:01 sun time, we have second contact.
Thereafter, the sun is wholly behind the edge and
the bug is in maximum shadow.
Taking the angular diameter of the sun to be 32' the
time taken from first contact to last contact will
be 128 seconds. This, measured in time, is the full
width of the umbra.
The time from 14:00 to second contact is 64 seconds
and I pondered how to reduce this to 50 seconds...
[Aside: of course I agree that you divide by the
cosine of the declination if you are not at an
equinox and the angular diameter varies a little
from 32'. These, as you say, are minor matters
though the first INCREASES the 64 seconds.]
You later say, and again I concur, that "most people
seem to judge the edge very close to the dark side."
The key words here are "very close".
To reduce the true 64 seconds to the apparent
50 seconds you seem, implicitly, to be saying
that "very close" translates into 14 seconds of
time, or 3.5 arc-minutes, about 10% of the solar
diameter.
This feels about right but I should love to see
the results of some properly set-up experiments.
I can imagine that the results would be different
for going from shadow to light (morning times)
from going from light to shadow (afternoon times).
Do you have some mathematical justification that
has escaped me for the missing 14 seconds or is
this just a sensible estimate?
Best wishes
Frank
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