AW: Shadow Sharpener Again

[Arthur Carlson]

Patrick Powers wrote ...
  The basic formula is actually f=(s^2)/(L), where f is
  the focal length, s is the radius of the (infinitely thin!)
  hole and L is the wavelength of the light.

I would express this a bit differently, since a pinhole does not form an
image in the sense that a lens does.

Consider different pinholes imaging the sun on a plane at a fixed distance.
The images are all the same size, but which one is sharpest?  The images
from the biggest pinholes are fuzzy by the size of the pinhole.  But if you
make the pinhole too small, then diffraction takes over and the images get
blurry again.  Patrick's formula tells you about what compromise you need to
make to get the sharpest image.

But remember that sharpness isn't everything.  In particular, the smaller
the pinhole the dimmer the image.  If brightness is a problem, you might
want to make the pinhole a few times bigger than this, particularly for
large dials.

If the pinhole size is fixed and you vary the the projection distance, the
image gets fuzzy at short distances.  At distances above that given in
Patrick's formula, the sharpness doesn't improve much.
near-perfect shadow sharpener should work when used on sundials.

--Art Carlson


-

AW: difference between equinoxes and midsummer

[Arthur Carlson]

A final(?) note on this topic.  By drawing little pictures of ellipses and
crosses and triangles and applying Kepler's Second Law, I was able to come
up with an expression for the difference between the length of summer and
the length of spring to lowest order in the orbital eccentricity
(epsilon=0.017) and the difference between the winter solstice and the
perihelion (delta=13 days):

4 * epsilon * delta = 21 hrs

This is in good agreement within the assumptions I made, so I put to rest my
remaining doubts that this is the correct answer to Willy's question.

While I was at it, I found the corresponding formula for the difference
between the spring/summer half year and the fall/winter half-year:

4 * epsilon * 365 dys / pi = 8 days

Also in good agreement with the information from
http://aa.usno.navy.mil/data/.

--Art Carlson

-

difference between equinoxes and midsummer

[Arthur Carlson]

Dear John,

Let me try it this way.  Take the Earth's orbit as it is and change the tilt
from 23 degrees to 10 degrees, but still pointing in the same direction.
Does this change affect where the Earth is at any particular moment?  No.
Does this change affect the positions on the orbit that correspond to the
solstices and equinoxes?  No.  Therefore it does not change the time
(measured not with a sundial but in seconds since the Big Bang) that the
solstices and equinoxes occur.  The answer are the same if we change to tilt
to 1 degree, or 0.1 degree.  The tilt is needed to define the seasons, but
the amount of the tilt makes no difference at all in the lengths of the
seasons.

The tilt does affect the Equation of Time due to something that I like to
think of as a coordinate transformation.  The trick is that the coordinate
systems for any degree of tilt happen to coincide at the solstices and
equinoxes, which is why this part of the Equation of Time is zero on these
four days.

You wrote:

> I'm sorry but I have to disagree. BETWEEN the Vernal Equinox and the
> Summer solstice the correction due to the tilt is NOT zero. Every day
> EXCEPT at the equinox and solstice the day is a bit shorter (as the
> sun is early) due to this tilt contribution. Summing up these days
> (Solar days which the Civil calendar uses and not Sidereal days which
> astronomers use) leads to a shorter Spring than the summer where the
> days are now a bit longer.

The sundial is fast compared to the clock for every day from April 16 to
June 14, but that doesn't mean that the solar day is always less than 24
hours during this period.  Take the beginning of June, for instance.
Looking at the Equation of Time, we see that one each successive day, the
sundial is about 9 seconds less fast, compared to a clock, than the day
before.  That means the solar day is 24 hr 0 min 9 sec long.  (If you think
I made a sign error, the length of the solar day around May 1 calculated
this way is 23 hr 59 min 52 sec.)

Servus,

Art Carlson

-

AW: AW: difference between equinoxes and midsummer

[Arthur Carlson]

John Shepherd wrote:

Now back to the original question: Why is the difference between the
time between the Vernal equinox and the Summer Solstice different
from the Summer Solstice and the Autumnal Equinox?

This effect is approximately due to the tilt of the Earth's axis

http://www.uwrf.edu/sundial/Eqntime.html ) on the Equation of Time
(EoT), which can be approximated by a sine wave of a period of 6
months and amplitude of 10 minutes. The actual length of a day, as
defined by solar noon to solar noon, is the Equation of Time minus
the EoT. This is what must be integrated over the period involved.
What I meant by averaging is that an integral over a period is equal
to the average over that period TIMES the period. In this case the
average of the half period of a sine wave is 10 mins*2/Pi or 6.37
mins. This is multiplied by 90 (or more accurately 92) days gives
about 10 minutes. The solar time is less than the standard time by
this and we get the same number but of opposite sign for the period
after the solstice. So the difference is twice that or approximately
20 minutes. The elliptical orbital effect is very small on this
difference essentially cancelling.

We're talking about the same question now, but I beg to differ on the
answer.  The tilt of the Earth's axis cannot explain any difference in the
length of the seasons.  The only reason you need to bring the tilt of the
Earth into the discussion at all is to define the equinoxes as the times
when the Earth is on the line through the sun which is perpendicular to both
the axis of the Earth's orbit and the axis of the Earth's rotation.

The Equation of Time itself has nothing to do with the question, but if it
did, the component with the 6 month period couldn't explain the difference
because it is zero at the equinoxes and solstices.

The eccentricity of the orbit, on the other hand, is on the order of 1%, and
1% of a year is a few days, so without doing a detailed calculation, the
average difference ((spring+summer)-(fall+winter)) could be on the order of
the 21 hours cited by Willy.  The magnitude of (spring-summer), since the
perihelion is near the winter solstice, must be much smaller.  Up to five
minutes ago, I was going to insist that the eccentricity of the orbit
explains the effect.  It is certainly true that that contributes a
difference, but can it be that we still don't have the right answer, the one
that explains the lion's share of the 21 hours?  (Or else I still haven't
understood John's answer.  It happens.)

--Art Carlson


-

AW: difference between equinoxes and midsummer

[Arthur Carlson]

John Shepherd wrote:

1. The equation of time gives the difference between the sun time and
standard time. Your difference is cumulative or integral of the daily
difference. The orbital effect has a maximum difference of about 8
minutes (this does not include the inclination effect). Averaging
this approximately sinusoidal variation over 6 months is
approximately 7 minutes per day. 7 times 180 days = 21 hours.

Actually this point works the other way around.  The difference between the
length of any given day and the mean day is only handful of seconds.  These
snippets must be integrated to arrive at the Equation of Time.  Integrating
the Equation of Time doesn't produce anything meaningful.

Actually, I don't think it is possible to directly deduce anything about the
length of the seasons (Willy Leenders' question) from the Equation of Time.
The answer to his question depends on the mismatch between the direction of
the tilt of the Earth's axis (relative to the plane of the orbit) and the
axis of the ellipse of the Earth's orbit.  This is, however, related to the
relative phase of the annual and semi-annual components of the Equation of
Time.

--Art Carlson

-

AW: Ceiling Sundial Limitations

[Arthur Carlson]

Anselmo Pérez Serrada wrote
...
Perhaps the mirror could be made somewhat larger with an X marked from
corner to corner, thus allowing us to find the center of the light spot
even if partly blocked.
...

A design on the mirror surface will not be imaged onto the ceiling.  Usually
you will want to choose the mirror small enough so that its angular diameter
as seen from the dial surface (ceiling) is significantly less than the
angular diameter of the sun (0.5 degree).  In that case, it doesn't make
much difference if part of the mirror's surface is in shadow (or if part is
X-ed out).

--Art Carlson

AW: Caustic and 2 minute limit.

[Arthur Carlson]

Dear Bill, dear John,

I realize that a shadow smeared over 2 minutes can be read to a fraction of
that period (especially if it is symmetircal, as in John's dials), and that
using images can give you a sundial with extreme accuracy.  (What is the
limit?  Except with an azimuthal dial, I expect the first limit you hit
would be the variation in atmospheric refraction.)  The cost is comlexity
(if focussing elements are used) or contrast/ease of reading (if pinholes
are used).  I did some experiments along the lines Bill suggests, although
with pinholes, two years ago and convinced myself that I could determine a
point in time under real-life conditions within 2 or 3 seconds.  Making a
complete sundial capable of this accuracy, however, looked like a difficult
project.

I was just curious if caustics could possibly give you the accuracy of an
image in a way that is intuitive to read.  That is, if you use a simple
image, you have to tell the user whether to use the leading or trailing edge
of the image.  Bill's idea of using a double image solves this problem
neatly and is probably more accurate anyway, due to its symmetry.

--Art

-Ursprungliche Nachricht-
Von: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] Auftrag von
[EMAIL PROTECTED]
Gesendet: Tuesday, January 08, 2002 3:09 PM
An: [EMAIL PROTECTED]; sundial@rrz.uni-koeln.de
Betreff: Re: Caustic and 2 minute limit.


In a message dated 1/8/2002 4:19:42 AM Eastern Standard Time,
[EMAIL PROTECTED] writes:

> Since a caustic is a very different animal from an image, is there any
>  chance of getting around the 2 minute limit on sundial accuracy due to
the
>  sun's angular diameter?

Art, I can't address the issue of caustics, but the 2 minute barrier can be
broken by using two focused images of the sun, side-by-side, separated by a
tiny amount of space.  This space could be, say, 15 seconds of time, and
would serve as the time indicator.  If you have any doubt that this is
feasible, I have a close up photo of my dial which operates using a single
focused image of the sun, and although the image is 2 minutes wide, it is
readable to better than 1 minute.  The edges of the image are razor sharp,
and it is easy to see that a design with two of these images side-by-side is
achievable.  Someday I may make one, but it is not high on my list of things
to do.
This JPEG is available to any who request it.

Bill Gottesman
Burlington, VT
44.4674N, 73.2027W

AW: Polar ceiling sundial

[Arthur Carlson]

Since a caustic is a very different animal from an image, is there any
chance of getting around the 2 minute limit on sundial accuracy due to the
sun's angular diameter?  Does the caustic of an extended object form a line,
or is it also smeared out?  (I suspect there's no free lunch here, but I
thought I could ask.)

--Art

-Ursprungliche Nachricht-
Von: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] Auftrag von Tim Yu
Gesendet: Tuesday, January 08, 2002 3:22 AM
An: Sundial List
Cc: Tim Yu
Betreff: RE: Polar ceiling sundial


[David]
> What is a caustic curve?

See the website:

http://www.cacr.caltech.edu/~roy/Caustic/

A simple Java applet demonstrates how a caustic curve is formed by
parallel light rays bouncing off a cylindrical, reflective surface.


Tim

AW: Ceiling Sundial

[Arthur Carlson]

You likely have a sheet of glass already clamped in place nearby -- the
window.  Couldn't you calculate a vertical dial for the right orientation,
print it on a transparency, tape the transparancy to the window glass, and
mark out the lines with a laser pointer or perhaps with a projector that
casts shadows of the lines onto the ceiling?

--Art

-Ursprungliche Nachricht-
Von: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] Auftrag von Dave Bell
Gesendet: Wednesday, January 02, 2002 8:28 PM
An: J Lynes
Cc: Mailing List Sundial
Betreff: Re: Ceiling Sundial


I like it!! Printing the dial artwork on transparency film should work
well. Rub it down onto a thin sheet of glass supported in a frame, perhaps
with a film of water or the like to keep it in place. The frame would need
to be accurately leveled and oriented, but could easily be clamped in
position, once that is determined. One hitch might come in, if the mirror
is placed on the inside sill of a fixed picture window, making it hard to
get the dial center over the mirror...

Dave
37.29N 121.97W

On Wed, 2 Jan 2002, J Lynes wrote:

> Here's a simpler proposal.
> Transfer the declination lines and hour lines of a horizontal sundial onto
a transparent sheet.
> Mark a small circle on the centre of the mirror.
> Support the horizontal transparent sheet, rotated from north to south,
with its nodus vertically above the centre of the circle, at a distance
equal to the height of the transparent sundial's gnomon.
> Project a laser beam through the transparent sheet onto the centre of the
circle.  Make sure the beam passes through the sundial scale at a point
corresponding to some chosen time and date.
> The reflected spot on the ceiling is the appropriate point on the ceiling
sundial.
> Repeat for other dates and times.
> John Lynes
>

Trigon-Folding

[Arthur Carlson]

Mystery solved.  There are two different ways of carrying out the fold in
the first part of your step F.  Of course, I first did the one that doesn't
work.

--Art

-Ursprungliche Nachricht-
...
> Actually, I wasn't able to follow your instructions, Edley.  I get line 6
> to be parallel to line 3 (45 degrees).  I think there's a mistake, but I
> haven't figured it out yet.
...

AW: Trigon-Folding

[Arthur Carlson]

Neat stuff.

You can have it a bit easier, though, even if not quite so general.  Take a
rectangular piece of paper and lay it in front of you with one the the short
sides near you.  Fold it in half from left to right (the long way) and
unfold it again.  Now bring the lower left corner onto the crease from the
first fold, and crease a second fold through the lower right corner.  The
second crease makes a 30 degree angle with the lower edge.

Actually, I wasn't able to follow your instructions, Edley.  I get line 6 to
be parallel to line 3 (45 degrees).  I think there's a mistake, but I
haven't figured it out yet.

--Art Carlson

-Ursprungliche Nachricht-
Von: [EMAIL PROTECTED]
[mailto:[EMAIL PROTECTED] Auftrag von Edley
Gesendet: Saturday, December 01, 2001 1:48 AM
An: sundial@rrz.uni-koeln.de
Betreff: Trigon-Folding


Dear Membership,

Here is more help for Emergency Sundial Makers.

Trigon - Folding

When you need 15, 30, 45, 60, 75 degree angles to lay out radially
from a gnomon to create hour lines and don't even have a pencil, but
do have something foldable; paper, foil, starched linen, etc., here
is how to fold these angles.

There turns out to be a number of ways to do this, but I'll describe
only one.

It involves trisecting an angle.  I found the method on
http://chasm.merrimack.edu/~thull/geoconst.html

Starting from a scruffy piece of fom (foldable material) with no
straight lines in it's shape.

A.  Fold ...

AW: Lunar ephemerids

[Arthur Carlson]

Fernando wrote:

Without intending to be so meticulous as we think Germans are,
I'd like to do something similar (but much, much simpler), like
observing if seeds sowed in the new moon do any better than
seeds sowed in the waning moon, etc.

I'm afraid you will have to be meticulous if you don't want to waste your
time.  (Leaving aside the question of whether the project is likely to be a
waste of time regardless of how carefully it is done.)  If you want to plant
the seeds outdoors, you will need many (many!) years before you can get
statistically significant results because you have to control not only for
the season but also for the weather in each year.  For example, you need to
compare two sets of seeds, both planted at the equinox, but one set in a
year where the moon was full at the equinox and the other in a year where
the moon was new at the equinox.  But that is not enough because you have to
be sure that the temperature, cloudiness, and percipitation at the time of
planting and several weeks before and after were similar.  Your only hope to
prove an effect would be to plant the seeds indoors and keep the
temperature, humidity, and light at constant levels over several months.
Several plantings would be necessary to be sure the seeds weren't drying out
or something from one planting to the next.  If you could manage to prove a
small but consistent effect it would have no immediate application because
the weather and other effects would certainly be more important in deciding
when to plant in any given year.  On the other hand, an incontrovertible
positive result would be extremely interesting from a scientific point of
view -- precisely because it would contradict so much of what we believe to
understand about the world.

Best regards,

Art Carlson

AW: diameter of reflected sun image

[Arthur Carlson]

The 
classical experiment using a mirror to detect minute rotations is not by 
Michelson and Morley, who used an interferometer, but by Cavendish, who measured 
the universal gravitaional constant in the lab.  But the technique has been 
used often.
 
--Art 
Carlson

  -Ursprüngliche Nachricht-Von: 
  [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]Im 
  Auftrag von John CarmichaelGesendet: Tuesday, August 14, 2001 
  4:15 PMAn: [EMAIL PROTECTED]Cc: 
  sundial@rrz.uni-koeln.deBetreff: Re: diameter of reflected sun 
  image
  Hi Fritz
   
  Good to hear from you!  What an interesting 
  story.  I seem to remember an experiment by Michaelson-Morley at the turn 
  of the last century where they used mirrors to amplify the small movements in 
  light.  (I think they were trying to prove the the old theory that 
  Einstein later disproved that light traveled through an "either" and that 
  its speed changed).

RE: question on EoT

[Arthur Carlson]

Dear fellow dialists,

I am forwarding this inquiry I received privatly from Yaaqov Loewinger.  It
seems right up our alley.

Regards,

Art Carlson

-Ursprüngliche Nachricht-
>From [EMAIL PROTECTED] Fri Mar  2 10:21 MET 2001
Date: Fri, 02 Mar 2001 10:33:16 +0200
From: "Y. Loewinger" <[EMAIL PROTECTED]>
X-Accept-Language: en-US,hu,de-CH,fr-FR
MIME-Version: 1.0
To: [EMAIL PROTECTED]
Subject: Old equation of time:Equation d'Horloge

This is a multi-part message in MIME format.
--20215C5ACD79E9D29D96B5DD
Content-Type: text/plain; charset=us-ascii
Content-Transfer-Encoding: 7bit

Dear Mr. Carlson,
I read yr.  excellent article on the Equation of Time. I posted
following questions,  in the History of Astronomy Discussion= Hastro
group, and didn't get satisfactory answers. can you help me ? Thanks in
advance! ( you can write me- if you like , in German).

Subject:  setting of clocks prior the introduction of modern mean  time

History of Astronomy Discussion Group
<[EMAIL PROTECTED]>



Dear List members,

Prior the introduction of modern mean time into civil life (round
1780--1830), modern equation of time (= EoT) and modern distribution of
time,  civilian  public clocks were set with the help of sundials.

But as clock time is by definition a mean time, I wonder to which kind
of
mean time were the  public clocks set generally, prior to, say 1780 ?

I see here 3 possibilities:

1. public clocks were simply set always to sundial time , without using
any equation of time tables. It seems that public clocks were set rather

often, say once a week, as their quality was rather poor and as in such
a short interval EoT changes only insignificantly, so it was simply
neglected.

2. clocks were set to "Nov. 3 mean time", as round this date modern
equation of time is maximal (~ + 16 minutes). So, beside  looking up
sundial time, tabular values of "equation of clocks" running from 0 (on
~Nov 3) to ~31( on ~Febr 11) min, were added to sundial time to get
clock time.

This "equation of clock" = in French: "equation de l'horloge" (EdH)" is
the old Ptolemaic equation of nychtemeron(=day and night) of the Handy
Tables.
As this EdH was tabulated in the 18 th century French almanacs
Connaissance des Temps, I guess they were widely used all over  Europe
(otherwise they would not have bothered printing it !).
Beside this EdH the French almanac of, e.g. of 1751, indicated also the
value of modern mean time at true noon, so modern mean time and modern
EoT were also in use. Which one, EoT or EdH were used for setting
civilian clocks ?

Can we assume that the EdH table was really used everywhere in civil
life in Europe, and  can we assume in historical research that time
indications in documents from the 18 th century mean " Nov 3 mean time"
?

See an attached diagramm of Equation d'Horloge.


3. the third possibility to set clocks was, to set them to "Febr 11 mean

time". Round this date modern EoT is minimal, ~ 14 min. If a clock is
set on that date to sundial time, so a "Febr 11 equation of days", which

runs from 0 ( on ~Febr 11) to 31(on~ Nov 3)  min, had always to be
*subtracted* from sundial time, to get clock time. Such equations of
days can be found till the 17 th century in astronomy books ( e.g. the
one of Huygens, near 1640), and it is also of Ptolemaic origin,
standartly used in the middle ages. They seem to be less popular than
EdH, perhaps because adding of EdH  seemed to be easier for clock
setters.

To sum up my question: can we assume that a standard church clock, say
in
Central Europe, was set to November 3 mean time, still in the late 18 th

century?

Best regards


Yaaqov Loewinger, dipl.ing. ETHZ


--
Y.  L o e w i n g e r
mail: P.O.B. 16 229 ; 61 161 Tel Aviv / Israel
tel.: 972-3- 604 61 79; ++ 523 98 33
fax : 972-3- 546 90 76
e-mail  : [EMAIL PROTECTED]


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A

Re: Sun's Apparent Angular Diameter

[Arthur Carlson]

Concerning Bill Gottesman's proposal of a method to measure the solar
diameter:

Well, it's basically a very good idea, but there are a number of traps
to watch for.  First, the slits need to be parallel.  They also need
to be aligned closely to North-South (or rather, perpendicular to the
sun's motion at the time the measurement is made).  Finally, what you
are measuring is the time it takes the sun to change its longitude by
one solar diameter, so there is a correction for the declination of
the sun and another for the motion of the Earth around the sun.  If
all you are interested in is the ratio of the solar diameter at the
two solstices, then the first correction cancels out, but not the
second one.

Also, it may be harder to determine a point in time accurately than a
point in space.  I would favor two parallel "pin-slits" casting images
on a plane normal to the suns rays.  The angular diameter (in the
direction perpendicular to the slits, which should be oriented close
to North-South and used near noon to eliminate any residual effects of
atmospheric refraction) is inversely proportional to the distance
between the slits and the plane when the images just touch.  That is,
the measurement consists of moving the plane until the images on it
just touch and then measuring the distance from there to the slits.

--Art Carlson

Re: Gnomon for Vertical Decliner

[Arthur Carlson]

Mac Oglesby <[EMAIL PROTECTED]> writes:

> It does leave one surprised that apertures are quite commonly 
> installed at an angle to the plane receiving the shadow.

Is this irrational or are they just optimizing to some other feature?
I mean, what's really so great about circular spots?  What you really
want is readability, which is a compromise between brightness and
blurriness for any pinhole.  Assuming you are interested in an
accurate reading of the declination as well as the time, the best
pinhole may be the usual choice of a circle in a plane perpendicular
to the sun's rays (which is itself also a compromise).  Extending this
logic, a vertically elongated pinhole in a vertical plane might have
some advantages over either of the other arrangements.  Hmmm.

Art

Re: Gnomon for Vertical Decliner

[Arthur Carlson]

Dear Tony,

By the time I got your ProjHole.gif it was FUBAR.  Is there any chance
you can send it without any unnecessary layers of propietary encoding
(mac-binhex40)?  Thanks.

Art Carlson

Re: Gnomon for Vertical Decliner

[Arthur Carlson]

[EMAIL PROTECTED] writes:

> Oy vey!  Maybe this will restart the "Shadow Sharpener" thread going
> again!  Sounds like quite a project-good luck.  I would like to
> suggest that if you use a pin-hole, that the aperture be parallel to
> the dial face.  This may seem obvious, but it wasn't obvious to me 2
> years ago.  This way a round hole will always cast a round image,
> and will not spread into an ellipse as the angle of the sun changes.

It is true that parallel rays shining through a circular hole will
produce a circular image on a flat surface parallel to the hole.  But
an extended source shining through a pinhole will produce an
elliptical image (unless the optical axis is normal to the surface).
Consequently, the edges will be fuzzier in one direction than the
other, whether or not you want to consider that to be an elliptical
image.  I suspect that a circular hole parallel to the surface is
still the "best" compromise.

Regards,

Art Carlson

! message from owner-sundial !

[Arthur Carlson]

Daniel Roth <[EMAIL PROTECTED]> writes:

> This message is sent for two reasons: 1st to remind how
> subscribing and unsubscribing works and 2nd to bring into
> discussion again the allowed length of a message including
> attachments.
> ...
> The length of a message is limited to 25 kB. Many subscribers
> still send messages, which are longer. This requires a manual
> intervention by the list owner. Please take into account that your
> attachment has to be downloaded by members, which may have
> only a 14.4 baud modem. Please vote for one of the following
> choices:


I have a connection through a research institute, so I'm not bothered
by any size message.  Still I don't want to vote for "no limit"
because I am also not particularly bothered by expediencies like
requesting a file by email or by browser.  My vote is for something
very close to the lowest common denominator, i.e., if more than two or
three members have a serious problem that can be solved without
terrible inconvenience to the rest of us, they should be accommodated.

--Regards to all, also to our brethren still living in the stone age,

  Art Carlson

Re: Bifilar dial in Genk Sundial Park

[Arthur Carlson]

Chris Lusby Taylor <[EMAIL PROTECTED]> writes:

> "Frans W. MAES" wrote:
> 
> > I know one more case of
> > an interesting bifilar dial. Using a pole style and a specially shaped
> > curve in the equatorial plane, one may obtain a polar dial with
> > straight, parallel E-W date lines, perpendicular to the hour lines.
> > This principle was described in the Bulletin of the Dutch Sundial
> > Society in 1979 by Th.J. de Vries.
> > [...]
> > http://www.biol.rug.nl/maes/zonwyzer/en/zwappi-e.htm
> >
> 
> This is an exciting sundial. Who would have guessed that you could achieve
> straight, parallel, date lines? Brilliant. Is the formula for the curve
> available, please? (Don't tell me - it's a catenary, right?)

The principle is relatively straightforward.  As the description says,
the pole style and base plate together constitute a polar dial.
(Since the second shadow is not needed to tell the time, I would
hesitate to classify this as a bifilar dial at all.)  At any given
time of day, the shadow plane will always cut the edge of the yellow
glass at the same point.  For different dates/declinations, the shadow
of this point will move up and down by the distance L*tan(D), where D
is the declination and L is the distance from the edge of the gnomon
to the edge of the shadow.  At noon, L must equal the height of the
style, H.  The trick is to make L = H for every time of day.  If x is
the distance from the base of the style, measured in units of H, and y
is the distance above the base plate in the same units, then the
equation for the necessary curve is this:  x = (1-y)*sqrt(1-y^2)/y

Have fun proving this!

--Art Carlson

Re: outdoor decor sundial question

[Arthur Carlson]

Dave Bell <[EMAIL PROTECTED]> writes:

> I'd call it a fairly expensive joke!
> 
> Note that a "real" dial should, roughly speaking, have the hours from 0600
> to 1800 in a semicircle, running from East through North to West (in the
> northern hemisphere). This is a clock face, with only room for 12 hours in
> a day!

The sundial at
http://www.shopoutdoordecor.com/cgi-local/SoftCart.exe/online-store/scstore/p-AWS209S.html?L+scstore+wxsc3599ff367336+981445839
is certainly poorly (criminally?) designed in that it gets out of
whack as the declination changes (by about +/- 1 hour, even if
properly mounted).  The simple fact that there are only 12 hours in a
circle does not, however, make it totally useless.  Since the foot of
the gnomon is on the circle rather than in its center, the shadow
falls at about the right spot near sunrise and sunset.

In fact, if the circle were either perpendicular to the gnomon or
elongated to an ellipse along the 6-12 axis, it could be turned into a
perfectly fine sundial.

In the "Sundial Installation Instructions", the company states, "These
sundials are designed for ornamental use and give an approximation of
time. As a very accurate sundial would require constant adjustment and
less ornamentation, these models have been selected to give years of
enjoyment without the aggravation of constant tuning."  I find these
words rather painful, knowing that sundials certainly can be accurate
(limited in most cases by the Equation of Time), and having seen many
examples of the beauty the artisans of this list can bestow on such an
accurate dial.

--Art Carlson

Re: Length of the year

[Arthur Carlson]

Richard Mallett <[EMAIL PROTECTED]> writes:

> >> As for determining the "length of the tropical year ... with a
> gnomon between successive solar solstices", I don't believe this is a good
> method.  One can determine the exact date/time of an equinox much more
> accurately than that of a solstice (although the solstice is conceptually a
> bit easier to deal with). <<
> 
> Can you elucidate please ?  I would have thought that the
> solstices, representing the extremes of solar altitude (measured when the
> Sun crosses the meridian) would be easier to determine.

Suppose you can measure the declination give or take one tenth of a
solar diameter, i.e., to +/- 3'.  Around the equinox, the declination
changes by about 1' per hour, so your measurement would allow you to
pin down the time of the equinox to +/- 3 hrs.

At the solstice, the declination varies quadratically from its extreme
value by about 0.22'/dy^2. In the worst case, you measure a value 3'
below the maximum, so you might actually be right on the solstice, but
you could also be at a date, either before or after the solstice,
where the declination is 6' smaller than the extreme value.  So the
uncertainty in your measurement of the soltice can be as large as
+/- sqrt( (6') / (0.22'/dy^2) ) = +/- 5 days.

--Art Carlson

Re: A Sundial as a Prize

[Arthur Carlson]

> ... A photo of a dial similar to the one made for Patrick Moore can
> be seen on the internet at
>  http://www.lindisun.demon.co.uk/smallest.htm

I have a question for Tony Moss about the dial pictured.  Unless there
is another scale on the back we can't see or the dial plate can be
turned over, this dial can only be used in summer.  That's OK, but
then why do you include the Equation of Time for the whole year?

--Art Carlson

Re: Length of the year

[Arthur Carlson]

Gordon Uber <[EMAIL PROTECTED]> writes:

> The length of the tropical year was determined with a gnomen between 
> successive solar solstices. The length of the sidereal year was determined 
> from successive heliacal risings.
> 
>  From Time in History by G. J. Whitrow.

I have long wondered how to make "accurate observations of the sun
relative to the stars" (as John Sheperd put it).  Given the key word
"heliacal rising", I have been able to find the definition and some
discussions on the Net.  I find it surprising that this could be, as
John Sheperd said, "pinned down to a single day".  Wouldn't this
depend on the brightness of the star and the viewing conditions and
God knows what?  On the other hand, the position of a given star at
sunrise will change by 1 degree from one day to the next, which seems
like a manageable distance.  And I suppose what counts (for present
purposes) is not what the actual relationship between the sun and the
star is, but just the reproducibility of the phenomenon.  Still, you
would need to take years where the meteorological conditions were
comparable.

As for determining the "length of the tropical year ... with a gnomon
between successive solar solstices", I don't believe this is a good
method.  One can determine the exact date/time of an equinox much more
accurately than that of a solstice (although the solstice is
conceptually a bit easier to deal with).

--Art Carlson

Re: Length of the year

[Arthur Carlson]

Allan Pratt <[EMAIL PROTECTED]> writes:

> I guess getting within 6 minutes of accuracy isn't all that hard after
> all.

Hmmm, you're right.  Since the length of the year happens to be so
close to 365 1/4 days, even a blind chicken (e.g., Julius Caesar) can
get a very accurate number without sweating.  Now we really do need a
historian to tell us just how much more clever than this Hipparcus
was.

Art

Re: Length of the year

[Arthur Carlson]

Allan Pratt <[EMAIL PROTECTED]> writes:

> According to a source I read, Hipparchus, a 2nd C BC astronomer
> calculated the length of the year to within six minutes of accuracy.
> Considering that at best he had a sundial and a water clock, how did he
> do this?

I hope a historian will answer this, but I am willing to speculate.

H's minutes were surely defined not with respect to a cesium clock but
as a fraction of a day.  The year is defined by the seasons, i.e., the
declination of the sun.  The declination is most sensitive to the date
around the equinoxes.  Since the equinox is one of the most
fundamental and easily observed astronomical events, it is plausible
that the equinox had been determined and recorded, at least to the
nearest day, for hundreds if not thousands of years before Hipparchus.
If he had available an uninterrupted calendar and a record of an
equinox 240 years earlier, then, by counting the number of days
between that and a contemporaneous observation of an equinox and
dividing by the number of years, he could calculate the length of the
year to a the claimed accuracy:  (1 dy) / (240 yrs X 365 dys/yr) =
1/87,600 = (6 min / 1 yr) / (60 min/hr X 24 hrs/dy X 365 dys/yr).

Alternatively, if he knew what he was about, he could by careful
naked-eye observation determine the time of the equinox to within a
fraction of a day.  If his observations had an accuracy of 0.1 day,
then he would only need observations 24 years apart, easily within a
professional lifetime even in those days.

The observation must not necessarily be of the equinox.  One could use
solar eclipses in a similar way, or simply the date in spring on which
the sun first becomes visible in a notch between two mountains.

Note that you don't even need a sundial or a water clock for any of
this!

--Art

Re: Equinox

[Arthur Carlson]

"fer j. de vries" <[EMAIL PROTECTED]> writes:

> Some members of this list have drawn an east-west line at the september =
> equinox.
> But what accuracy this line will have?
> 
> Assuming a perfect horizontal plane, ...
> ...
> A line between these 2 points has an angle of 0.067 degrees to the =
> east-west line.

It is interesting to note that this source of error isn't limited to
the equinox.  The usual mark-two-shadow-crossings-on-a-circle method
is also affected, except near the solstices.

--Art Carlson

Re: vertical gnomon sundial fotos

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> The first dial I found is a beautiful "bow tie" dial designed by Luke
> Colletti .

I agree emphatically that the design is beautiful, but I am puzzled
about the functionality.  Can this sculpture be used to tell time?  It
seems to me the lines would have to converge on a point behind the
gnomon in order to indicate the time, rather than on the base of the
vertical gnomon, as appear to be the case.  It look likes the dial, if
it had markings, would only indicate the azimuth.  This is clearly
illustrated by the gif submitted by Fer.  There are two crosses below
the "bow tie", the upper one would be the location of the gnomon,
while the hour lines clearly converge on the lower cross.

Art Carlson

Re: Off topic, but not too much

[Arthur Carlson]

SÈrgio Garcia Doret <[EMAIL PROTECTED]> writes:

> 1 - Assume the hours equals exactly 1/24th of the earth revolution time and
> suppose a disguster lover choose to retire into a cave, where daylight is
> entirelly shut off for a period of six months to the minute. ...
> What adjustment does his watch need?

As pointed out by others, the assumption does not even come close to
the actual definition of an hour, but what the heck.  The watch owner
has more important things on his mind.  I see two answers, depending
on the type of watch:

1) If it the usual "stupid" kind, no adjustment will be necessary.  12
   o'clock is 12 o'clock, and the watch can't distinguish 12 noon from
   12 midnight.

2) If the watch has a date display, then it must be adjusted by 12
   hours, and it makes a difference whether you set it forward or set
   it back because you wind up on a different day.  The correct
   procedure is to set it back to give the rotation about the axis
   time to make up for the revolution about the sun.

--Art Carlson

Re: steriographic projection

[Arthur Carlson]

"Patrick Kessler" <[EMAIL PROTECTED]> writes:

> Can anyone recommend an essay on steriographic projection?  In particular I 
> am searching for a proof that circles on the sphere are mapped onto the 
> equatorial plane as circles.

http://www.geom.umn.edu/docs/doyle/mpls/handouts/node33.html
"outline[s] two proofs of the fact that stereographic projection
preserves circles, one algebraic and one geometric".  You should also
be aware of http://www.astrolabes.org/.  Finally, if you are using
search engines, or even a card catalog, you'd better spell
"stereographic" with an "eo".

Have fun.

Art Carlson

Re: Shadow Sharpener

[Arthur Carlson]

I wrote:

> Nevertheless, I have a feeling that it may not be possible to improve
> on a simple pinhole.

Let me reconsider that.

Consider an aperture a distance L from a surface, so that the image of
the sun through an infinitesimal pinhole would have the diameter D =
L*(0.5 degree).

With a circular pinhole of diameter d

Re: Shadow Sharpener

[Arthur Carlson]

It is easy to read a sundial with an accuracy a bit better than the
solar diameter, even if the shadow is from a simple edge.  The worthy
goal of a shadow sharpener is to significantly improve on that
accuracy.  Since we still want to make the reading with a human eye,
the "best" system will be determined to a large extent by
psychophysics.  Human vision is so complex that it is not obvious just
what we are looking for, so the final judge will be experiments.

Nevertheless, I have a feeling that it may not be possible to improve
on a simple pinhole.  The image produced by any of the systems
discussed (a simple pinhole, an annular pinhole, or a classical shadow
sharpener which is a pinhole "downstream" of a conventional gnomon)
will be sharper if the holes are smaller, at the expense of
brightness.  It may be hard to find with arbitrary accuracy the center
of the image produced by a simple pinhole even if it is perfectly
sharp, but one should be able to locate the edge of the image as
accurately as desired.  I would thus suggest to the experimentalists
that they always compare the clever designs with simple pinholes,
where the pinhole diameter should be varied to find the optimum, and
where both forms of reading, from the center of the image and from an
edge of the image, are compared.

A simple pinhole may also be less sensitive to variations in the
distance to the scale and gross variations in the position of the sun
during the course of the day.  On the other hand, reading from the
edge of an image may be less intuitive for the casual user of a dial.

Cheers,

Art Carlson

Telling Directions from the Sun and the Moon

[Arthur Carlson]

One of the things that got me going on sundials was an article in the
magazine of the German Alpine Club on telling directions from the
moon.  I found the procedure impossibly complicated and spent much
time trying to understand celestial mechanics in order to think of
alternatives.  At long last, I have put my thoughts into words, which
may be found at

   http://www.ipp.mpg.de/~Arthur.Carlson/sun-compass.html

This may be of interest to some of you, but even if it is not, I would
appreciate any feedback on its accuracy and pedagogical value.  The
audience is intended to be hikers more than dialists, and I would
like to eventually publish the essay in said magazine of the DAV.

--Art Carlson

Re: Coming equinox

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> It's an interesting thought to use the moon's shadow at sunrise and sunset
> on the equinox to locate your east-west points.  Although this can be done
> with the sun, you would have errors using the moon, unless there is an
> eclipse on the equinox also.  If the moon is not in the same plane as the
> sun, it will not act like the sun.
> 
> Since the moon moves about two minutes/hour eastward in the sky, the only
> way you can do this with precision is to use precisely calculated times and
> lunar coordinates such as those sent to you by Jim Cobb.  First you would
> find the meridian with the moon, using Jim's data, then find east/west.

Of course, the moon doesn't cast shadows during an eclipse, so we are
really talking about taking a sight on the moon. Even during a total
eclipse, without additional information, the errors will be several
times larger than using the sun. If you have additional information,
i.e., the coordinates of the heavenly bodies at various times, then
there is no need to wait for the solstice or an eclipse, you can take
a sight on anything at any time and deduce the cardinal directions
from the result.

I think the purist's way to find directions (a purist being a Druid
with the knowledge and technology of Stonehenge), is to mark the
directions of sunrise and sunset for a few days near the solstice. By
interpolation, you can get the hypothetical direction to the rising
sun for each hour of the period in question, and likewise for the
setting sun. For one of these hours, the directions will be exactly
opposed to each other. This is East/West, and the hour is the exact
time of the equinox. Magic.

Have fun out there with the coyotes!

Art

Re: equation of time

[Arthur Carlson]

Willy Leenders <[EMAIL PROTECTED]> writes:

> The equation of time has two causes. The first is that the orbit of
> the earth around the sun is an ellipse and not a circle. The second
> is that the plane of the earth's equator is inclined tot the plane
> of the earth's orbit.  Please can anyone explain me the second cause
> so that I can conceive it. I am not a astronomer!

I have given this question a lot of thought, but I realized when I was
asked about it a few days ago that I am still not satisfied with my
answer.  I have tried to explain it in detail on my page
"http://www.ipp.mpg.de/~Arthur.Carlson/sundial.html";, but that isn't
the intuitively obvious answer we would all like to have in order to
claim that we "understand" the effect.  If I had to answer in one
sentence, I might say that the effect arises because the sun moves
against the stars (in the Ptolemaic sense) on a circle (the ecliptic)
that differs from the coordinate system we use to define time (the
equatorial plane).  You can see that it is a "mathematical" effect, as
opposed to the "physical" effect of the eccentricity, by considering a
planet that does not rotate, so you can place the poles anywhere you
want.  The hour angle of the sun during the course of the year, except
at the solstices and equinoxes, will depend on your choice.

> You can do it in Dutch (for preference), in French, in German or in English.

I can offer you German, if you have trouble understanding the English.

Art Carlson

Re: GPS, spherical sundial

[Arthur Carlson]

Fernando Cabral <[EMAIL PROTECTED]> writes:

> Just put a vertical stake on the. You can also use a plumb line with
> a node somewhere at a suitable hight. Now, at chosen intervals,
> like 9 o'clock, 10 o'clock... until  noon, mark in the ground where
> the stake shadow ends. For each mark, draw a circle having
> the stake as the center.
> 
> In the afternoon, mark where the shadow is at each corresponding
> hour. That is, 1 pm, 2 pm and 3 pm corresponding, respectively
> to 11 am, 10 am and 9 am.
> 
> Now draw a line connecting each pair of hour. Now pick one of the
> lines and bisect it and draw a perpendicular line. This line should coincide
> with the noon line. Also, if everything is ok, this perpendicular should
> bisect all the other lines.
> 
> If the stake if correctly placed in the vertical you will end up with a very
> precise North/South line from which you can find any azimuth you
> may need. (By the way, the closer the solstice on the "other" hemisphere
> the more price the measuring will be).
> 
> I do think this method is easier and more precise than the compass and
> the GPS (at least in "normal" usage).

I agree that a shadow method is likely to be easier and more precise
than using either a compass or GPS, but I have some comments on your
exposition.  First, the clock time of the readings is not
important. In the afternoon, you want the locations where the shadow
of the nodus crosses each circle, regardless of what your watch says
the time is. Second, the accuracy of the result can be compromised if
the ground is not level from East to West. For high precision, you
will first need to prepare a level surface for the shadows to fall on.

Regards,

Art Carlson

Viviani's pendulum

[Arthur Carlson]

Looking up Foucault's pendulum experiment in Meyers Grosses
Taschenlexicon, I read the claim that Vincenzo Viviani in 1661 was the
first to do the experiment, 189 years before Foucault!  Browsing
through the Web for more details, I was only able to find two further
references: In http://www.newadvent.org/cathen/15183a.htm "Foucault's
pendulum experiment was materially forestalled [sic] by Viviani at
Florence (1661) and Poleni at Padua (1742), but was not formally
understood."  and in
http://www.physik.uni-greifswald.de/~sterne/Observatory/events.html
"Already in the year 1661 Vincenzo Viviani discovered this
phenomenon. It was rediscovered by Leon Foucault in 1850."

I'm hoping some of the erudite contributors to this group can give me
a few more details.  It seems like the experiment, while requiring
some care, should have been within the range of 17th century
technology.  Did Viviani really look for rotation of the plane of
swing of a pendulum?  Did he know it would provide the proof of the
Earth's motion that eluded his mentor Galileo?  Did he get a positive
result?  Why was the experiment forgotten for almost two centuries?

Thanks and best regards,

Art Carlson

P.S.  I come to this question because I am reading "Galileo's
Daughter" by Dava Sobel.  I thought that the interest shown in this
forum for her book on "Longitude" was justification enough for asking
my question here.  In addition, there are some connections with
sundials through the time-keeping aspects of pendulua and through
Galileo's attempts to solve the longitude problem using the moons of
Jupiter.  Apropos Sobel's new book, I'm a third of the way through.
Up to now it's a remarkably straightforward biography of Galileo.  It
certainly won't have the fascination for this list that "Longitude"
did.

Re: OFF TOPIC -- OFF, OFF TOPIC

[Arthur Carlson]

Fernando Cabral <[EMAIL PROTECTED]> writes:

> I've heard the French Assembly has approved
> a "Resolution 495" which determines (so I heard)
> that every public organization in France has to replace
> Microsoft Windows by Linux.

Even if it's not true it's a great rumor, so I have been working to
spread it.  My wife (a journalist) wasn't able to dig up anything, but
a colleague found this "in an article someone posted to the
scientology group":
   
   Windows 2000 starts out against the wind

   Government investigations, bug reports tales of
   horror and strong competition: Linux

   Stuttgart, Germany
   February 22, 2000
   Stuttgarter Zeitung
   
   [...]
   
   Swiss authorities believe Windows 2000 is too expensive and
   they are reviewing Microsoft's pricing politics. EU
   commercial competition commissioner Maria Monti is
   investigating complaints that Microsoft has arranged network
   functions in Windows 2000 so that they will work only with
   software which comes from the House of Microsoft. 

   A French importer is also suing in an EU court over
   competition obstacles. The corporation bought a French
   language Microsoft program in Canada because it was
   cheaper there. Microsoft's French branch company
   prohibited the sale of this import. Last week, an EU court
   stated the importer's complaint was justified, thereby forcing
   the EU Commission to take the case. 

   The French are happy over the decision: one initiative in the
   French Senate aims to have only software with free source
   code installed in all government agencies by the year 2002;
   the Culture Ministry is already converting to Linux. 

> I hope everybody will excuse me for abusing this
> list's patience.

An occasional off-topic post is no problem among friends.

Art Carlson

Re: Declination Table

[Arthur Carlson]

"Gianni Ferrari" <[EMAIL PROTECTED]> writes:

> Since, time ago, I had interested in this matter for the search of a
> development of the declination  in Fourier serie (perhaps some old member
> of this list remembers this)

Gianni,

I would like to have a simple approximation for the Equation of Time
and also for the declination (if the simple sine function is not
accurate enough).  What were your results?  I couldn't find anything
on the Web, and I hope I will not have to take apart source code to
find a formula.

Thanks,

Art Carlson

Re: Declination Table

[Arthur Carlson]

There are at least three aspects of this problem: accuracy, ease of
use, and elegance.  We might be able to agree that dials with a
built-in analemma are conceptually the most elegant because they
utilize to the maximum extent the information from the sun and require
a minimum of external input.  With ease of use, I have my problems
with the analemma.  And if I have problems, how is a citizen going to
react who is not a scientist and possibly has never seen a sundial
before?  Arrows on the analemma may designate unambiguously which
branch should be used, but take too much thought.  Labeling with the
seasons is better in this respect.  This is well and good for a noon
mark, but an accurate sundial should have marks every five minutes or
so.  Analemmas already start to overlap at half-hour spacing.  A
sundial with a hundred analemmas may be a work of art and an accurate
instrument, but it will be impossible to read, especially if the
branches have to be labeled.

And now another word or two about accuracy.  If we could read the
position of the shadow *exactly* (and always knew for sure which
branch we needed), then a dial with analemmas would be perfectly
accurate (though perhaps impossible to read in practice).  The
fundamental disadvantage of a dial without dates becomes evident when
the shadow cannot be read exactly.  During most of the year a dial
with analemmas will be about as accurate as one with date lines
because the analemma lines are nearly "vertical".  The problems arise
at the solstices where they are nearly "horizontal".  Since the shadow
is used to determine the declination as well as the hour angle, the
accuracy decreases with the slope of the analemma.  The more
accurately the shadow can be read, the greater the disadvantage of
pure analemma solutions, because a reading will come closer to trying
to use a purely horizontal mark.

We might ask, for example, at how accurate the shadow would have to be
read before the leap year problem of a dial with 365 day-lines becomes
greater than the solstice problem of a dial with unmarked analemmas.
With knowledge of the date but not the year, the first type of dial
cannot be made more accurate than +/-(1/2) day, which in late December
means +/-15 sec (EoT changes by 30 sec/day).  As explained above, a
dial with analemmas will generally be less accurate than one with
dates, but if the accuracy of reading the shadow becomes better than
+/-15 sec, then the former will continue to improve, but the latter
will be stuck with this uncertainty.  But to reach this accuracy near
the solstice, the analemma would also have to be able to determine the
time of the solstice within +/-(1/2) day, which in turn would require
an angular accuracy of +/-1.6 arc sec [(23 deg)*(2pi*(1/2)/365)^2/4].
If you could determine the position of the sun to this accuracy, then
you could also read your sundial to well better than 0.1 msec!  So
much for the leap year problem.

One way to combine the best of both approaches is to label the
analemma not just with the seasons but with the dates.  The problem
remaining is the ease of use, which can be solved by employing moving
parts.  I have not been able to think of any purely passive design
that simultaneously does not require knowledge of the date, but
permits use of the date to increase accuracy, and is intuitive to use.

As long as we are talking about the limits of accuracy at special
times, I would also like to mention the refraction of the atmosphere.
This can introduce an error of a minute or so for most designs.  The
only design which is not affected by refraction is an azimuthal dial
(in the strict definition).  If the terms of John Davis' "design
challenge" were accuracy from sunrise to sunset, rather than from 9am
to 3pm, this would be another argument against a dial using analemmas.

--Art Carlson

Re: Design challenge

[Arthur Carlson]

"John Davis" <[EMAIL PROTECTED]> writes:

> I have a question/challenge to all you sundial designers:   what is the most
> accurate design for a Standard Time dial?
> ...
> As a starter, the "Singleton" dial recently discussed here would seem to be
> a reasonable candidate.  It's main limitation, common to all dials which
> incorporate an EoT correction, is that it is drawn for a some MEAN EoT
> curve, and no allowance is made for the leap year cycle and the other minor
> variations.  Is there some geometry of dial plate and style which minimises
> the time error caused by small year-to-year variations in the mean daily
> declination? If this is achieved, then the small change in the EoT over a
> single day may be allowed for.

The maximum rate of change of the EoT is about 30 sec/day toward the
end of December.  Averaging over leap years can be done to make the
chart "wrong" by at most half a day, or 15 sec.  The diameter of the
sun is 0.5 degree, or 120 sec of time.  Before you worry about the
leap year problem, you first need to find a way to locate the center
of the shadow edge 8 times more precisely than the degree to which it
is smeared.  We (e.g., John Carmichael and I) have discussed here some
designs which might be capable of this accuracy, but they tend to be a
bit hard to use.  If you insist, one possibility is a camera obscura
with a slit (ideally oriented parallel to the Earth's axis).  This
gives a sharp line image of the sun, which can be used to read the
time from a series of date lines like we have been discussing.  If
you're really worried about leap years, you can pile four years' worth
of dates on top of each other.

The other approach advocated by some, namely determining the EoT
directly from the declination, rather than the date, will always
suffer near the equinoxes.  For example, if you determine that the
declination is 23 deg 11 arcmin +/- 15 arcmin, the EoT can vary over a
range of 11 minutes!

--Art Carlson

Re: Declination Table

[Arthur Carlson]

Daniel Lee Wenger <[EMAIL PROTECTED]> writes:

> The reading of standard time via a sundial may be accomplisted by
> mearly reading the declination of the sun and using an analemma,
> determining standard time. At no point is the current date needed to
> do this.

Way, way back I explained why I was not totally satisfied with this
method, essentially because there are (almost always) two values of
the EoT for each value of declination.  At the solstices there are
even an infinite number of values (in some technical sense).
Consequently, if you are interested in relating the sundial reading to
clock time, you always need some knowledge of the current date.

Art Carlson

Re: Azimutha Sundial (once more)

[Arthur Carlson]

"fer j. de vries" <[EMAIL PROTECTED]> writes:

> Back to the bifilar dial : A bifilar dial can be constructed in such
> a way that the hourlines ( for local suntime ) are equi-angular
> spaced.  Than it is also possible to correct for EoT and/or
> longitude by rotating the hourscale.  So we have at least 2
> possibilities to correct for EoT with bifilar sundials.

Is there a resource on the Web with the theory of bifilar sundials, or
at least a picture or some info on constructing them?  I spent many
idle hour trying to come up with a sundial that would allow an easy
mechanical correction for the equation of time.  The best I could
devise was using a gnomon tilted halfway between the Earth's axis and
either the vertical or the meridian.  This allows the hour marks to be
placed evenly around the circumference of a circle, so that the dial
can easily be set forward or back by rotating the circle.  The catch
is that the center of the circle has to be moved to match the
declination.  (I assume that also this invention of mine is old hat
and has a name that someone will kindly tell me about.)  It sounds
like the bifilar dial is the solution I was not clever enough to find
myself.  Given time, I would be able to work out the theory on my own,
but I'm also willing to forgo some of the fun on this one and just
read about it.

Art

Re: Singleton's azimuthal

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> >...  Why not follow John Singleton's notion (p. 51, BSS Journal
> > for Feb 2000) and use your normal taut wire pole style?
> >
> >Have I missed something in the discussion?
> 
> Maybe we all have.  I think John Singleton's azimuthal will not work (except
> at noon, sunrise and sunset).  I know this is a rather bold statement to
> make, but I think there is a general misconception that azimuthal dials can
> work with either a vertical gnomon or a polar axis gnomon as was originally
> suggested in an earlier discussion.  This has always bothered me because it
> seemed impossible.  If a polar axis works, then it would certainly solve the
> gnomon height problem.  
> 
> Rather than speculate, I did a simple experiment. Using a Spin drawing of an
> azimuthal for my location and an icepick for the gnomon, I quickly found out
> that the dial worked correctly when the icepick was vertical and became
> progressively worse as I tilted it towards the celestial pole.

A dial with date rings (neither "azimuthal" nor "Dali" is quite the
right name) can be designed for any gnomon, in particular for either a
straight vertical gnomon or a straight polar gnomon, but any given
dial plate will only work with its own gnomon.  Your mistake was
trying to use a "vertical drawing" with an polar gnomon.

--Art

Re: thumbs down on azimuthals

[Arthur Carlson]

John Carmichael listed the pros and cons of azimuthal dials and
concluded that "it is NOT an appropriate design for me to build".

Of his "pro" arguments:

> 1. It looks different, original and pretty (especially if you like the
> Batman logo!)
> 2. It can be made to tell Standard Time
> 3. It requires a simple vertical gnomon
> 4. It can be designed by Fer's Spin program
> 5. It is horizontal (usually), and horizontals are very commercial
> 6. It tells time from sunup to sundown

I have always placed great weight on number 2.  Perhaps because I'm a
physicist, I hate to see a machine that is an order of magnitude less
accurate than its inherent possibilities (+/-15 min EoT compared to
+/-1 min (of time) solar radius).  An EoT chart is an awkward remedy.
I got onto azimuthal dials (before I knew what they were called) as a
way to build sundials that accurately show clock time, but also saw
great possibilities for "different, original and pretty" designs (pro
argument 1), and I offered my Arizona dial as an example.

Now look at John's "cons":

> 1. It requires an absurdly tall gnomon at middle and lower latitudes which
> would make the sundial look odd and would have severe shadow fuzziness
> problems in the summer.
> 2. To avoid using a tall gnomon, the shadow must be artificially extended by
> visual guesstimation or by a string shadow extender, both of which would
> make the dial less precise.  Also, changing the date ring order complicates
> calculations and makes the dial even harder to read.
> 3. It is inherently hard to read even with just one hour time lines,
> especially for the novice, without instructions.
> 4. It is very difficult to make this dial precise with small time line
> divisions.(For fun, try Spin using five minute time increments (step
> hour=5/60=.0833, and you'll see what I mean)
> 5. Small time increments make the dial even harder to read.
> 6. There is severe time line compression on the inner date rings, making
> engraving and reading difficult. 
> 6. If the geniuses on the Sundial List have a hard time understanding it, I
> doubt my customers ever will!

These revolve around the short shadows of vertical objects at some
times and places and the difficulty of reading the wildly curving
lines.  I think it is still possible to have the best of both worlds
(except pro 3), specifically by using a polar gnomon.  (Some other
contributors are already playing with designs with concentric dates
rings and non-vertical gnomons.)  This would immediately eliminate
cons 1 and 2 (too tall gnomon).  It would be much easier to read,
understand, calculate, and manufacture (the remaining cons, except
perhaps the first of the sixes) because the would look nearly like a
conventional sundial.  The hour lines would be nearly straight since
they only have to accommodate the EoT, not the declination.  You can
tell at a glance about what time it is (as with an uncorrected dial),
or you can look for the date ring and tell the time within a minute or
two.  The flexibility of choosing the shape and location of the date
rings remains (pro 1), so an Arizona dial, for example, is still
possible.  (Words, words, words!  Will one of you that has been
posting azimuthal dial plate designs please plug in a polar gnomon for
me?)

Are you interested in such a compromise, John?

Regards,

Art

Re: azimuthal gnomon length problem

[Arthur Carlson]

Chris Lusby Taylor <[EMAIL PROTECTED]> writes:

> Another approach would be to have a ring for each value of the
> EoT. That turns the figure-of-eight hour lines into a
> figure-of-infinity date line. Having found today's date one follows
> the ring it's on to read the mean time directly from the shadow.
> This should be easy to use. The hour lines could be almost straight.

This would work with an axial gnomon, where the direction of the
shadow does not depend directly on the declination.  I haven't done
the math(s) in detail, but I don't think it would work with a vertical
gnomon because the azimuth of the sun depends not only on the hour
angle (Equation of Time) but also on the declination.  In other words,
the direction of the shadow at a particular time depends on the date
for two reasons (EoT and declination) with different dependences on
the date.  Consequently, it would be a great exception if two dates
could share the same time markers.

Cheers,

Art

Re: Metric v's Imperial.

[Arthur Carlson]

Gordon Uber <[EMAIL PROTECTED]> writes:

> Let's face it: The Babylonians got it right when they developed the base-60 
> system.  It was applied to the sixth of a circle (one sixtieth of this 
> being a degree) and the hour, of which we still use the first and second 
> minutes.   Third minutes (sixtieths of second minutes) are not in common 
> use, although I would note that the third minute of an hour is the period 
> of U.S. power main standard 60 Hz alternating current.  Coincidence?

Is this the origin of our (English, at least) names for units of time?
"Seconds" because it result from dividing an hour by 60 twice?
("Min'-ute", I assume, is related to "mi-nute'" and "mini".)

Is it known whether the Babylonians, when they chose 360 degrees to a
circle, were more concerned with the convenience of numbers divisible
by 2's and 3's or with the fact that there are 360 days in a year
(within a percent or two)?

--Art

Re: Diverging Light Rays

[Arthur Carlson]

Andrew James <[EMAIL PROTECTED]> writes:

> My idea is this: is it possible to combine the two points made?  Arrange,
> say, two sets each of four posts with three 0.4 mm gaps between, one set
> having slightly wider posts but with the same gap, so as to make three light
> rays the outer two of which diverge by the same small amount - say 0.2
> degrees - in each direction from the inner.  Then balancing the appearance
> of the outer rays should give a rather more accurate estimation of the angle
> of the centre of the solar disc.Any takers?

I'll buy it.  I did a lot of thinking and some experimentation last
summer.  I used a slit and two pinholes and tried to balance the
intensity of light on the two sides.  I found I could judge the moment
of symmetry within a second or two of time, which corresponds to one
arc minute or better of angle, which I found very respectable.  The
principles are these: (1) your eye can judge symmetry better than just
about anything else, and (2) the light passing through
lenses/pinholes/slits varies most sensitively if the apparatus is
aligned with the limb of the sun.  My slit produced a line image of
the sun.  Both the diameter and the separation of my pinholes were
about equal to the width of this line image.

--Art

Re: metric

[Arthur Carlson]

Peter Tandy <[EMAIL PROTECTED]> writes:

> ... Of course, for some specialised work,
> metric measurements are no better and no worse; atronomers for instance do
> better with the numbers they need to measure huge distances, when in a
> metric form, and physicists with the numbers they need to measure minute
> atomic distances. But neither of these is a measurement that us ordinary
> folk use on a day-to-day basis - and for those, Imperial with its greater
> number of divisors is far better.

The way to cut the Gordian knot is to throw out everything and start
over with a base 12 numeral system.  Then the scientific calculations
and the everyday divisions by 2, 3, 4, and 6 are *both* easy.

(Time measurements with base 12 is another kettle of fish. 12 months
in a year is good, but the 7 day week is still a killer.  24 hours in
a day is close, but there's that pesky divisibility by 5 when
splitting hours into minutes or minutes into seconds.)

--Art

Re: optical resolution tables

[Arthur Carlson]

I just wrote:

> ...You will find that you can make a beam anywhere within a few tens
> of a degree.  (To be precise, 0.5 deg at sunrise and sunset, closer
> to 0.3 deg near noon.)

I got that backwards.  The sun subtends a larger azimuth when it is
higher in the sky, so the beam can be formed to point in any direction
in a range of something like 0.7 degrees near noon (at mid-latitudes).

--Art

Re: optical resolution tables

[Arthur Carlson]

Tony Moss <[EMAIL PROTECTED]> writes:

> This brought to mind an informal experiment I carried out this morning
> with an enlarged version of my Meridian Alidade recently completed for
> a US client.  The instrument consists of pairs of vertical cylindrical
> metal posts spaced 0.4mm apart. (see my webpage for a picture of the
> standard model) The posts stand astride a fine centreline on a brass
> protractor plate with one pair at each end of the line.
> 
> This one-off custom version has extra high posts for use by a client
> at low latitudes.  In use, a sunray passing through the gap is aligned
> with the central line and the other pair of posts is used as a sight
> to lay out a meridian line after certain calculations are performed.
> Because of the height-extended posts and low winter sun the sunray,
> which diverges slightly for the reasons Bill has mentioned, passed
> right across the protractor plate and between the second pair of
> posts.  Result?...a non-diverging line of light that crossed the whole
> table in a very distinct line.
> 
> There's a useful principle for a heliochronometer here I think.

Careful! It is true that you can produce an (almost) non-diverging
line of light this way. (By the time the beam has traveled ten times
the separation between the post pairs, it will have broadened by about
ten times the gap between the posts, in this case to 4 mm.  Simple
geometrical optics.)  But this line is not unique.  You will get such
a line if the instrument is aligned toward any part of the sun's disk.
Do your experiment again (Or was Friday the one sunny day allowed in
Britain this year?), but this time wiggle the instrument a little once
you have a beam.  You will find that you can make a beam anywhere
within a few tens of a degree.  (To be precise, 0.5 deg at sunrise and
sunset, closer to 0.3 deg near noon.)

--Art

Re: drawing hour lines using gnomon

[Arthur Carlson]

Arthur Carlson <[EMAIL PROTECTED]> writes:

> [EMAIL PROTECTED] (John Carmichael) writes:
> 
> > Let's say ...
> > ...  Will this technique produce the same shape hour lines at any
> > time of the year?
> 
> Yes.  The hour lines will always have the same shape.  This is even
> true if the gnomon is not aligned with the axis of the Earth, as long
> as it is straight.

Some other respondents have touched upon the question of the
orientation of the gnomon.  I stand by my answer to the question as
stated: John's technique will "produce the same shape hour lines at
any time of the year", for any straight gnomon.  If you want to label
these lines for clock time, the labels will have to change during the
year.  Or you can put labels on them that are valid for some day of
the year and read corrections to these labels from a table.  The
advantage of a polar gnomon is that these corrections are just a
function of the day of the year (the familiar Equation of Time).
With, say, a vertical gnomon, the correction will depend both on the
day of the year and also the time of day.  (Obviously, I've been
thinking too much about my Dali dial.  I better go cool off my brain.)

Art

Re: drawing hour lines using gnomon

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> Let's say you want to build a large sundial using the ground as the dial
> face.  The ground is somewhat irregular and not quite horizontal.  You
> decide to draw the hour lines, not by calculation, but by using the
> technique of building the gnomon (style) first and then marking the position
> of the shadow on the ground at selected time intervals using clock time and
> correcting for EOT and longitiude.
> 
> You draw the hour lines from the edge of the dial face to the dial center,
> tracing the shadow.  Since the ground is irregular and not flat, you notice
> that the shadow line is not straight, but irregular also, depending on the
> terrain.  This produces hour lines that are not straight.
> 
> Because the sun's declination changes during the year, changing the angle
> that the sun strikes the style, will this technique produce the same shape
> hour lines if it is done at any time of the year or is the declination
> irrelevant?  Will this technique produce the same shape hour lines at any
> time of the year?  I think it should. but I'm not sure.

Yes.  The hour lines will always have the same shape.  This is even
true if the gnomon is not aligned with the axis of the Earth, as long
as it is straight.  If the shadow passes through a point P, then it
will pass through all the points of the plane containing P and the
gnomon.  If it does not pass through P, then it will pass through none
of the other points in the plane (except those of the gnomon).  The
intersection of this plane with the ground (the shadow line) is just a
subset of the points of the plane, so the same holds there.

Art

Re: gnononistically challenged

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> Thanks for taking the time to explain the Dali dial to us gnomonistically
> challenged dialists.  I think I'm beginning to understand it, but will have
> to think about it some more.  What threw me off was that I was thinking that
> a Dali dial would be draped over the edge of a table!

It could be, hence the thread title.

Perhaps it's best to start by thinking about a "normal" sundial with a
horizontal plate and a paraxial gnomon.  Design such a dial to read
clock time without any EoT correction.  Then design another dial to
read clock time plus 5 minutes, another to read clock time plus 10
minutes, and so on for +15 min, -5 min, -10 min, and -15 min (always
without any EoT correction).  Now, to read the time from such a dial
you don't need the whole plate.  Any ring will do.  So cut a ring out
of each of the 7 dials you just made, but of different radii so that
the rings do not overlap.  Properly position all the rings around a
single gnomon.  Now if you want a +5 min correction, you just read the
time from the +5 min ring.  For a -10 min correction, use the -10 min
ring.  Label each ring with the dates that its correction is valid,
and you have a version of my sundial.  At any given hour, the shadow
will always point in about the same direction, but will be shifted
right or left a bit due to the EoT.

If you move away from a paraxial gnomon, the concept becomes a bit
trickier because a plate with radial hour lines will only be accurate
for a particular declination of the sun, that is, a particular date.
But, look!  You are only using each ring for a particular date anyway,
so it doesn't matter.

Someplace around here we realize that most of what we have learned
about sundials doesn't matter any more either.  The gnomon could be
bent.  The plate could be warped.  You just have to make sure that the
shadow of the gnomon always crosses the date line at exactly one
point, and that this point is never in the shadow of something else
(like a fold in the plate).

I used this extreme flexibility to propose a sundial with date lines
in the shape of Arizona, but only after I checked a map to see that
Tucson was conveniently located (somewhat south of the middle) and
that no line radiating from Tucson intersects the border more than
once.

So.  I think you have caught on to the concept by now.  One thing I
like about some dial designs is their inevitability.  The plate *must*
be horizontal if it is to always be illuminated when the sun is above
the horizon.  The gnomon *must* be paraxial to minimize the difference
to clock time over the course of the seasons.  This concept, in
contrast, has very few constraints, although it is in a sense more
accurate than the conventional form.  How can this freedom best be
used?  Can we require the dial to do something new, that other dials
can't?  Is there an especially esthetic form?

Have fun with it.

Art

Re: Dali dials

[Arthur Carlson]

[EMAIL PROTECTED] (John Carmichael) writes:

> I'm trying to understand your letter.  Your design sounds very intrigueing.
> In fact, I've often thought of carving a map of the state of Arizona onto
> the dial face, with Tucson at the center of the dial.  All the hour lines
> would radiate out from Tucson.  With my vertical pointer in the center and
> the correctly oriented map on the face, you could point to any place within
> the state, like a boyscout does with his map and compass.   I could even put
> latitude and longitude lines on the map.
> 
> But I was going to use my cable coaxial gnomon, with a sphere on the cable
> to serve as the nodus for date readings instead of a vertical style with
> nodus sphere at tip.  Your design would place Tucson north of center, and
> the hour lines would radiate from a point south of Tucson (probably in
> Mexico!). So with a vertical gnomon you would lose the ability to use the
> poiner to take a compass bearing.

A neat idea to use the hours lines to show the azimuth, but that won't
work with my idea.  My hour lines wind up being wavy like the Equation
of Time, and maybe bowed as well, so you can't use them to point
anywhere.  The main misunderstanding is that my design does not have a
nodus.

> >Methods which use the declination of the sun, either by using a
> >specially shaped gnomon or by observing the shadow of a nodus, rather
> >than an edge, are perhaps more esthetic, but they are inherently
> >ambiguous at the solstices and double-valued the rest of the time.
> 
> What do you mean by "inherently ambiguous at the solstices and double-valued
> the rest of the time?

The locus of the shadow cast by a nodus at, say, noon through the
course of the year is the figure-eight-shaped analemma.  When it is
noon on any day, the shadow will fall on the analemma.  But on most
days, it will cross over the analemma twice, and you have to know
which of these two crossings to use to tell time.  You have to know
whether the current date is in the first half or the second half of
the year (which shouldn't be a big problem, even for absent-minded
types like myself) and on that basis decide which branch is currently
valid (which always is confusing, even for the mathematically and
astronomically inclined).  That's the double-valued part.  The
ambiguous part comes in because at the solstices, the shadow traverses
the ends of the analemma loops tangentially, so it is hard to decide
exactly when the crossing occurs.  You don't have to choose between
two distinct but well-defined candidates as during the rest of the
year, but your one candidate becomes rather fuzzy, extending over
several minutes.  The only dates that are free of these problems are
April 13 and August 31, the crossing point of the figure-eight.

> >If we make the user do this work instead of the
> >nodus, the figure-eight can be unfolded and made unambiguous.
> Are you talking about a moveable nodus?

No.  No nodus.  None.  Like with an analemmic dial, the time is read
by looking at the intersection of a shadow line and a time line.  In
the present case, there are a multiplicity of time lines, one for each
day of the year (or as many as you have room for).  In contrast, in a
conventional dial, the shadow line coincides with the line marking the
time.  With a nodus, the position of the shadow point tells the time.

> And from here on I'm completely lost!  I can't imagine what the face might
> look like, or the gnomon, let alone how you would calculate such a dial.
> Wish I could see a picture!

Here, here!  Maybe someone can whip up a picture with Fer's program to
give us something to point at while we're talking.  (My PC is in the
shop just now.)  My idea is basically the azimuthal dial he mentions.
My only contribution is pointing out that such a dial will still work
even if the time lines are not circles and the gnomon is not vertical
(or even necessarily straight).

> If you design it, I'll build it (if it works!)

It will look rather bizarre, but should be accurate and easy to read.
I'm hoping you or Fer will be intrigued and do the calculations.
(Unless you are willing to do a lot of dot-to-dot drawing, the first
thing you will need is a mathematical representation of the shape of
Arizona!)  If not, I'll put it on my pile and get to it manana.

Regards,

Art

Dali dials

[Arthur Carlson]

A "normal" sundial has the gnomon coaxial with the Earth.  This is
done to keep the errors with respect to clock time to a minimum during
the course of the year.  If we have the ambition to make our sundial
read clock time to better than +/- 15 minutes, then we have to correct
for the Equation of Time.  There have been many public discussions
here and private ones in my head about the best way to do this.
Simply reading a table or graph is inelegant and subject to errors of
sign.  Methods which use the declination of the sun, either by using a
specially shaped gnomon or by observing the shadow of a nodus, rather
than an edge, are perhaps more esthetic, but they are inherently
ambiguous at the solstices and double-valued the rest of the time.

One way of thinking about the nodus methods (which has come up here in
discussions of the EoT with respect to leap years) is that the
declination tells you what the date is, and the figure-eight-analemma
allows you to find (with the restrictions mentioned above) the EoT for
that date.  It seems reasonable to suppose that everybody has a pretty
good idea of the date already, so we are making the sundial do
unnecessary work.  If we make the user do this work instead of the
nodus, the figure-eight can be unfolded and made unambiguous.  (I am
sure I have seem such a dial design somewhere, but I can't remember
where.)  For example, the date-lines can be made concentric circles,
from Jan 1 innermost to Dec 31 outermost, and the EoT for each hour
marked as a nearly radial wavy line.  We could even trivially
accommodate the change between standard-time and daylight-saving-time.
As a practical matter, I think it would be easier to read clock time
(corrected for EoT) from such a dial than from any alternative.  The
freedom opened up by this arrangement is astounding: The date-lines
can have (nearly) any shape, and they could all have different shapes
(as long as they don't cross).  The gnomon need no longer be parallel
to the Earth's axis; it doesn't even have to be straight!

I could envision a Salvador Dali sundial, but maybe I should start
with something for John Carmichael: Draw the outline of Arizona many
times at different scales and put them inside of one another, but so
that all the Tucsons overlap, and of course properly oriented with
respect to the compass.  Put an obelisk at the location of the
Tucsons.  Label each outline with a date and calculate (the hard
part!) where the shadow of the obelisk will fall across that outline
for that date and each hour of the day.  For each hour, connect the
points for all the dates and label the resulting wavy line with the
hour.  Voila!

I'd love to do the design myself, but realistically I know I won't
find the time any time soon, so I'd rather through the idea out to the
world.  Is the description clear enough?  (The idea is probably
between 500 and 2000 years old anyway.)

Regards,

Art Carlson

Re: Twisted band sundial

[Arthur Carlson]

I can think of three ways to incorporate the Equation of Time into a
twisted band dial:

(1) A correction can be made in the hardware by simply turning the
band around its axis. Since it is hung up on a polar support, this is
easier to accomplish than with some other designs like horizontal
plates. The drawback, of course, in addition to mechanical complexity,
is that somebody has to constantly readjust the dial.

(2) When the sun is at different declinations, the line of the shadow
will be at an angle across the strip. Multiple lines can be put on the
strip and you only have to read from the one with the angle matching
the shadow. The drawback is the inherently bivalent nature of the
analemma, i.e., for most dates there will be two lines that match and
you still have to figure out which one to read. Furthermore, the
accuracy is not great near the equinoxes because the EoT is changing
but the declination is not.

(3) The shadow is a line, but only a point is needed to tell the
time. This opens up the possibility of adding date lines down the
length of the strip and making the time lines the same wiggly shape as
the Equation of Time graph (plus, ideally, an additional correction
for the changing slope of the shadow mentioned above). To use the
dial, you find the intersection of the shadow with the line for the
current date and read the corrected time from the wiggly line passing
through that point. I like this solution. It is easy to manufacture
because all the time curves have the same shape and the same
separation, and it is easy and accurate to use.

Regards,

Art Carlson

Re: FAQ commentary

[Arthur Carlson]

Jim_Cobb <[EMAIL PROTECTED]> writes:

> I've thought of another tip for spotting worthless horizontal sundials
> (such as is sold in garden shops, etc)--if the shadow of the gnomon
> crosses the hour lines it's no good.  This test requires only
> horizontal positioning, not polar alignment, and a lot will fail this
> test because the gnomon for cheap dials often does not intersect the
> dial plate at the convergence point for the hour lines.

Actually it doesn't require horizontal positioning either, or even a
shadow. For each hour line, you should be able to find a position for
your eye such that the edge of the gnomon is superimposed on the hour
line. If they ever "cross", i.e., if you can ever see part of the hour
line above the gnomon but not all of it, then the gnomon will not
intersect that line in the dial plate, and the dial is "worthless".

--Art Carlson

Re: Anodising Afterthoughts

[Arthur Carlson]

"The Shaws" <[EMAIL PROTECTED]> writes:

> How times have changed...
> <<...aluminium is cheap...>>
> ..reminds me of the story that Napoleon had an extensive monogramed dinner
> service made from aluminium - just because it was the most expensive metal
> of his day.

I heard the story differently, that Napoleon issued each of his
soldiers an aluminum mess kit to take advantage of the reduced weight,
despite the horrendous cost.

--Art Carlson

Re: SEND QUESTIONS!

[Arthur Carlson]

Dear John,

The question that got *me* interested in sundials was the Equation of
Time. That the yearly variation of the orbital angular velocity could
give rise to an yearly correction made sense, but I had trouble
understanding where the half-yearly correction came from.

I also remember a usenet exchange with someone who thought the gnomon
was slanted just to make the shadow easier to see, so even the most
basic principles should be covered in a FAQ.

I am myself still uncertain of the basic classification of types of
sundials.

How accurate *can* a sundial be is also a question that a novice might
ask, but which is subtle enough that it has sparked extended
discussions on this list.

Of course, another favorite is "What is the biggest sundial?"

Good luck with the project. It's important to us who love sundials.

Regards,
Art

Re: cylinder dial

[Arthur Carlson]

john hoy <[EMAIL PROTECTED]> writes:

> I've have made a form interface to the dial which got me
> started with PostScript in the first place, i.e., the 007 capuchin dial,
> so now you can avoid editing the PostScript.
> 
> They both may be found via http://axum.tripod.com

You've got some problems with your links.

> On another note, what sort of sundial installation would be particularly
> well suited to mark the 45th parallel? I like the idea of using a
> vertical-horizontal pair and a two sided equatorial dial, something like
> the one in the sunclocks book. The idea is to get something that really
> shows that there is something special about 45 degrees lat.

I think a square standing on one corner would most clearly show the
symmetry involved.

--Art Carlson

Re: Equation of Time Graph wrong way up Down Under ?

[Arthur Carlson]

"Ron Anthony" <[EMAIL PROTECTED]> writes:

>  If I gave you a nice looking but somewhat erraticatic watch (the dial), and
> said, "It works good but its 10 minutes fast".   You would have no problem
> making the mental calculation to get the correct time.

I think you're right that native English speakers would readily
understand a table labeled "dail fast/slow". Non-native speakers might
have more trouble. In German, zum Beispiel, a clock is not said to be
fast ("schnell"), but rather to run ahead ("geht vor"). Technically,
the dial is not running faster than a mechanical clock in November and
December, but slower. That is why the amount the sundial is ahead of
the clock is decreasing. I agree that the EoT on a sundial should give
the correction needed to convert sundial time to clock time, and that
it should avoid relying on any plus/minus conventions, but I would
suggest saying "dial ahead/behind" rather than "dial fast/slow".

--Art Carlson

RE: 16th century navigation/surveying

[Arthur Carlson]

I emailed my ranger and now have his reply, which I have attached to this
missive. I hope it is not considered too long for this list, but I think
some of you will find it as interesting and helpful as I did. There were two
main issues bothering me. One was how Cabrillo could still believe the
Pacific was small even after Magellan had sailed across it. There were, of
course, many factors, but basically the cartographers of the day believed
the North Pacific was much narrower than the South Pacific. The other issue
was the conflict between the ranger's claim that the eclipse measurement in
Mexico City was off by 25 degrees and Jim Morrison's claim that the error
was only a single degree. It turns out there was an early, inaccurate
measurement in 1541, just before Cabrillo set sail, and a later, accurate
one in 1577.

Thank you all for the various insights.

--Art Carlson

P.S. A few months ago there was an extended discussion here on the green
flash. I stumbled on some good web sites explaining the phenomenon and
showing some good photos: http://mintaka.sdsu.edu/GF/

From: "George Herring" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Cc: "Terry DiMattio" <[EMAIL PROTECTED]>
Subject: Re: A Question of History -- related article
Date: Mon, 9 Aug 1999 16:55:30 -0700
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 Mr. Carlson

 Below is the draft-article I refer to and promised in my response to
 your e-mail.  Again, if you would like to see the maps mentioned
 please send a mailing address and I'll see that a copy of the
 newsletter containing the article is sent to you.


 George

 _


 Not Far To The West

 Christopher Columbus was lost to his dying day. In fact, between 1492
 and 1542, all Europeans in the New World, in a sense, were lost. Why?
 Because most Renaissance mapmakers accepted the classical belief that
 China existed approximately 180 longitudinal degrees east of Europe1..
  It's actually 120 degrees east.  In this article I hope to convince
 you with early 16th century maps why Columbus, Juan Rodriguez
 Cabrillo, and others thought Asia lay just beyond the western horizon.

 Let's begin with this 1503 woodcut-map, drawn by Gregor Reisch, is
 based on the work of Claudius Ptolemy, a Roman-Egyptian geographer of
 the 2nd century AD2..  Ptolemy collected the geographic work of other
 scholars and the tales of travelers to fill in the details of this
 map.  A 1478 version of the exact same  map is known to have been in
 an Atlas owned by Christopher Columbus. The important feature of this
 map to note relative to our story is the extent of it's eastern
 extreme.  Asia is depicted as extending more than 180 degrees to the
 east of Spain!

 Why does the map depict Asia extending so far to the east?  Because
 Ptolemy said so, that's why.  Renaissance scholars had no means with
 which to measure longitude and only crude estimates of distance
 covered by ships.  They had no reason, or evidence, to refute the
 knowledge of Ptolemy.

 Columbus, like most explorers after him for 100 years, accepted
 Ptolemy's estimate3..   And, like many amateur scientists, he
 manipulated his data.  Columbus interpreted the writings of Marco Polo
 to indicate that the mainland of Asia extended east 253 degrees, and
 he cited Polo's accounts of a large island named Zipango (Japan)4.
 said to lay an additional 30 degrees east of China.  This left, in his
 estimates, about 77 degrees of ocean to cross from the Canaries to
 Japan5..

 When Columbus bumbled into the West Indies he had traveled about 60
 degrees, but he estimated he'd traveled about 75 degrees.  Far enough,
 he thought, to have reached the eastern fringes of the Indies.  Since
 the people he encountered appeared to be "Indian," it was natural for
 him to conclude he was in the "Indies".  He maintained that Cuba and
 the other islands of the Caribbean were the East Indies until his
 death in 1506.

 Scholars and explorers throughout the early 16th century argued about
 just what Columbus had found and how it fit with the writings of
 Ptolemy and the beliefs of the Church. Consistently they persisted in
 the belief that China lay less than 180 degrees west of Spain (60
 degrees or less west of Mexico.)  Furthermore, according to the Bible
 the apostles had spread the gospel to all the peoples of the world.
 For this to be feasible it was necessary for mapmakers to depict the
 America's as accessible from Asia. Virtually all maps from the 16th
 century, therefore, depicted Asia and the Amer

RE: 16th century navigation/surveying

[Arthur Carlson]

Thanks for the many responses to my questions. I think am starting to get a
grip on the situation. In the first place, the time between the discovery of
America and the measurement of the exact longitude using a total lunar
eclipse was not all that long - 85 years in an age when a single voyage
could take years. The alternative of using the position of the moon relative
to the stars would have been just too complicated for the knowledge of the
time. Of course, chronometers and telescopes (to observe Jovian eclipses)
did not exist. And it's not like the Europeans had a lot of time on their
hands to measure the longitude of Asia either. Vasco da Gama did not reach
India until 6 years after the discovery of America.

What I still don't understand is this: Even without an exact measurement of
the longitude, how could Cabrillo still think it was only a hop, skip, and a
jump from Mexico to Asia 20 years after Magellan had taken 4 months to cross
the Pacific? Magellan was sailing for Spain, so the knowledge of the rough
size of the Pacific must have been available to Cabrillo (as if you could
keep such a story secret anyway).

--Art Carlson

16th century navigation/surveying

[Arthur Carlson]

Dear dail-list-ists,

I visited Cabrillo National Monument Saturday and talked with Ranger George
about the Spanish attempts to finish what Columbus started. I am full of
questions but feel sure that some of you can help me.
For decades after Spanish colonies had been established in Mexico, they
still believed the Pacific was only a third as big as it is. I don't
understand this. Of course, finding the longitude for purposes of navigation
is tough because the accuracy required is high and the conditions are
terrible, i.e., a rocking surface that is not in the same place for any two
observations, the need for a good fix greatest when the observing conditions
are worst, and no access to libraries of reference material and
mathematically-minded consultants. But how can it be that hard to determine
the longitude of, say, Mexico City with an accuracy of at least 10 degrees,
even given only 16th century technology? I would imagine sending a couple
grad students over to record the time of day (night) that various stars
disappear and reappear behind the moon, sending the tables back to Spain
where similar observations were made, and setting the brightest mathematical
minds of the empire to work comparing the two sets of observations to come
up with a fix on the longitude.
Apparently something similar was eventually done using a lunar eclipse, but
the result was still off by 25 degrees. A lunar eclipse is a rather sluggish
event. Is this really a better method than occultations? Was the final error
due to poor observations or poor theory, or is that the inherent accuracy of
the method?
Ranger George seemed to think that the expectations that the Earth was
smaller greatly inhibited the discovery of the actual size. The ancient
Greeks had believed in a rather small value, and who could question their
authority? It was also an important point of theology at that time that the
Gospel had spread to all mankind, so the Americas logically had to be
connected by land to Asia. I have trouble believing that hard-headed
economic interest in knowing the truth would not win out over these
prejudices, but then, I am definitely a child of my age.
Knowing where Mexico is is only half the problem, but finding the longitude
of Asia could be solved by the same method, unless the Spanish did not have
a presence there. When did the first European ship reach Asia? The first
Spanish ship? Was it sailing from the West or the East? Was Magellan's fleet
the first to cross the Pacific or only the first to accomplish a
circumnavigation in one piece?
Thanks for any elucidations.

--Art Carlson

RE: Off topic but interesting enough to share

[Arthur Carlson]

I don't think the Bernoulli effect can explain an attraction in this
geometry. I would look more at the friction from the water moving in the
powerful eddy the jet sets up in the bucket. If this is correct, then a
stick held in the center of the bucket near the hose should also be pulled
in, and one held near the edge, where the water is rising will be pushed
out. The force pushing the stick up should be somewhat smaller than that
pulling it down because the friction with the wall and the greater area
available will cause the upwelling to be slower than the downdraft. If the
shape and size are similar, the stick should be pulled in with more force
than the hose, because the centrifugal force of the water in the curved
hose, which pushes it up, should still be there, cancelling some of the
force of the circulating water on the hose. I would be surprised if the hose
is actually pulled in, as opposed to be pushed out more weakly. Which way
does it move when you let go of it?

--Art Carlson

RE: signal mirrors with attachment

[Arthur Carlson]

Dave Bell wrote:

> Well, as best I can make it out, the spheres act as lenses with focal
> length equal to their diameter. If they are embedded in a reflective
> background, an incoming bundle of parallel rays is focussed to a point
> (disregarding spherical aberration) on the rear surface, reflected back
> (must be along complementary paths!), and recollimated into a parallel
> bundle again - directly back along the entry path. From the diagram, a
> small part of the light from the sun passes through the hole in the
> silvered layer, reflects off the rear glass surface onto the beads, and
> returns to the same point on the rear surface. Some of the return light is
> reflected back towards the sun, and some penetrates the rear surface, to
> the user's eye. He sees a diffuse, much dimmed virtual solar image,
> aligned with the direction the major part of the reflected light takes,
> towards the target.

Thanks for the explanation. That's about what I figured out in the meantime.
The key is the combination of retroreflection from the beads with a
(partial) reflection from any of the glass surfaces parallel to the mirror.
Clever.

--Art

RE: signal mirrors with attachment

[Arthur Carlson]

Tony Moss wrote:

> >I found a couple of web sites on signal mirrors.
> >http://www.equipped.com/signal.htm#ReflectionsOfLight
> >describes how they work and how to use them. I also made a sketch of how
> >I think it works. The attached bit map file shows this.
>
> I understood the principle immediately from your diagram.  'One picture
> is worth 1 words'

Well, I'm sorry, but I haven't caught on yet. Bob refers twice to an image
of the sun, but I don't see from his diagram how or where any image of the
sun should be visible. The web site explains, "The fireball is produced by
retrodirective reflection from small metalized glass spheres adhered to a
mesh grid or cloth disk with a center hole." I could understand it if the
reflective material consisted of needles or flakes perpendicular to the
plane of the mirror, but spheres don't seem to be direction specific enough.
Wouldn't they look bright regardless of the orientation of the mirror?
Please try to share your insight!

--Art Carlson

RE: Heliograph

[Arthur Carlson]

Bob Haselby and Tony Moss dialoged:

> >This sounds like a "signal mirror" ... It uses double internal
> >reflection in the hole to give a virtual image of the sun
>
> Any chance of a diagram or somesuch to show how this works Bob?

It could work like this: Set up two sheets of glass and a mirror so they are
all perpendicular to one another. There will be a faint image of the sun
reflected in each sheet of glass, but also a still fainter image due to a
reflection from both sheets. The direction of this third image is also the
direction the sunlight will be reflected from the mirror, so if you tilt the
assembly until the faintest image is superimposed on your target, they will
see the light.

This does not yet sound like a practical piece of emergency equipment, but
maybe it will give somebody enough of an idea to figure out how real signal
mirrors work.

--Art Carlson

RE: Heliograph

[Arthur Carlson]

Tony Moss wrote:

> In my impecunious searches of WWII 'surplus' stores back in the
> 1950s I came across a "Portable Heliograph Set' in a pouch.   It
> was simply a mirror about four inches across with a sighting hole
> in the middle.  A length of cord attached it to a short rod with
> a bead on top.
>
> In use the mirror was held in one hand near to the operator's
> eye. The cord was then stretched tight and the 'bead' used to
> 'sight' the target.  If the mirror was then rotated until a
> sunray coincided with the bead above the other outstretched hand
> a flash of sunlight would be directed at the target.

I learned a different method in Boy Scouts: While looking through the hole
at the target, you will also see an image of your face in the back side of
the mirror. There will be a spot of light on your face where the sun shines
on it through the hole in the mirror. If you tilt the mirror until the image
of the spot coincides with the hole in the mirror, the sun will be reflected
toward the target.

This method might be considered less intuitive than the
stick-string-and-bead method, but I actually find it simpler. I am fairly
certain it is also more accurate. It also takes less equipment, so it can be
carried out without preparation with any two-sided reflecting surface.

And while we're on the subject, I would be interested in learning how the
heliograph in Peter Mayer's jpeg is aimed. It's not as easy as aiming a
laser or a search light because information on the position of the sun as
well as that of the target is needed.

--Art

RE: Easter ( a bit off topic)

[Arthur Carlson]

Any rule for calculating the celebration of Easter depends on whether you
are interested in the Western or Orthodox holiday. Furthermore, any
calculations for the future will become wrong if the rules are changed. See,
for example,
   http://www.smart.net/~mmontes/pr.wcc.19970324.html

--Art Carlson

RE: rotating protractor section

[Arthur Carlson]

John Carmichael wrote:

> I've been thinking that, using plywood, I could cut a curved section (20
> degrees) of a protractor's scale and tying both ends to the dial's center
> (w/ two long non-strechy strings), I could rotate this pie-shaped thing,
> like a drawing compass, about the center to mark the timeline angles.  Tom
> Semadeni did something similar in builing his "Celeste".  This rotating
> protractor section would be very simple and probably easier than the
> coordinate method.  No problems with perpendiculars and measuring tape
> (which takes two persons).  Also the marking of each time lime
> only involves one measurement instead of two.

It still sounds like a lot of work. As long as you're only laying out hour
lines, not doing something more difficult like drawing analemmas or setting
up an analemmic dial, why not do this:

With a tape measure or non-stretchy string draw a (half) circle of known
radius centered on the base of the gnomon. Each hour line should intersect
this circle in a particular point. Calculate the distance from these points
to the intersection of your north-south line with the circle. Using a tape
measure pegged to the intersection of the north-south line with the circle,
measure out and mark the distances to the points on the circle. Connect the
dots. Voila!

--Art Carlson

RE: Giant sundial meridians

[Arthur Carlson]

John Carmichael wrote:

> You know Art, you really write GREAT answers!

That's probably sarcastic, but my ego refuses to entertain such a thought.
My objective side would advise you to listen to all the advice from others
who have actually built large dials.

> I thought that I could use either or both of the methods
> described by Mayall
> pg.75-77. I'll call these" the concentric circle method" and" the apparent
> noon method"  Wouldn't both of these methods show true north very
> precisely
> if the vertical gnomon was very tall?

They can. Even here you should consider sources of error. Near the
equinoxes, for example, the change in declination during the course of one
day could be enough to skew your north-south line by a good fraction of a
degree, possibly more than the precision with which your dial can be read.

> Once I know the true meridian, all of the right angles to it are easy to
> mark using plane geometry.

In geometry class I learned to construct a perpendicular to a line. The
method can be rigorously proven to be exact, although the lines as they end
up on my paper may not be. I recently learned how carpenters make a large
rectangle: They lay out something that looks about right, then compare the
length of the diagonals, then tweek things until they are really right. The
point is that there may be methods which are not rigorous in the sense of
high school geometry but are easier to use and may even give more
satisfactory results.

> Well, I will certainly take great pains to see that this does not happen!
> Using water level as a guide, it is possible to produce a perfectly level
> surface. )This is how the egyptians leveled the base of the
> pyramids.  If I
> orient the azimuth of my gnomon using Mayall's methods, and I
> know that the
> face is level, then using simple geometry, it is easy to set it at the
> proper angle or height.

This will work, though it gets complicated if you need a slope for drainage,
your site is deliberately on a hillside, you are installing a dial on a
pre-existing surface, etc. I have given some more thought to the previous
thread involving how to set a sundial using the time method to minimize
errors, but anything like that is probably out of the question for a civil
engineering sized dial.

> But which day of the year is the best for doing this?  Please
> tell me if I'm
> wrong in my thinking, but wouldn't you want a day where EOT=0 because then
> you could quickly mark the timelines by a clock without corrections. (I
> suppose you could also set the clock off by the EOT amount on the day of
> hour line marking). This would give a longitudinally corrected dial.  You
> would also want to do it near the solstices when solar declination is
> changing the least.  Right?  So the best day of the year would be Dec. 25.
> What do you think Art?  Is this correct?

It doesn't seem any harder to me to take measurements at 3 min 46 sec past
the hour rather than on the hour. Why are you worried about the solar
declination? If it's a question of the change of the EOT during the course
of your day of measurements, then December 25 is the worst possible day (15
sec in 12 hours).

> >Isn't a giant protractor just a piece of non-stretchy string?
>
> No, a piece of non-strechy string is just drawing compass that hasn't been
> nailed down yet!

You can tell just how long ago that geometry class was! A curved scale is
hard to make, and I don't see any real advantage over linear measurements. I
would establish a few primary reference points, e.g., with the concentric
circle method or EOT corrected shadow measurements, and then map several
secondary reference points from these using a tape measure. Work down from
there to the smallest scale you need. Always use measurements from at least
three reference points to reduce errors (and also get an idea for how
accurately your work is proceeding). I am presuming that you know enough
analytic geometry to calculate the distances from the reference points and
have a computer to help with the numbers.

Keep us up to date on any large projects you start!

--Art Carlson

RE: Multiple sunsets

[Arthur Carlson]

John Pickard wrote:

> I saw the green flash two or three times in 1980 when I wintered at
> Davis in Antarctica (68 degress 38 minutes S). I would ahve to look
> at my diaries to determine the date etc. But on one oocasion, we ran
> up the hill to get another look, and sure enough, there it was again.
> Pure magic!

Of course, sunset lasts longer at polar latitudes, so the flash should last
longer and the opportunities for mulitple observations should be enhanced.
This raises the interesting question of whether the green flash might last
minutes, hours, or even days at the proper time of year for some locations
within the polar circles.

--Art Carlson

RE: Analemma Link

[Arthur Carlson]

Roger Bailey wrote:

> I would like to pass on a recommendation in the latest Sky and Telescope
> magazine (June 99). Go to  for the best
> explanation of the analemma that I have ever seen.

Awesome. 100% correct. Explanations easy to understand. Presentation
professional, with appropriate use of html features. A rarity.

--Art Carlson

RE: Multiple sunsets

[Arthur Carlson]

Frank Evans wrote:

> Regarding Fernando Cabral's multiple observation of sunsets at one
> setting I recently wrote to the "Marine Observer" (British Met. Office)
> concerning a meteorologist who claimed to have seen the sunset green
> flash three times at one setting by climbing to successive decks of an
> ocean weather ship as the sun sank.  This elicitated the following note
> to the same journal (Marine Observer, April 1999) from Dr. R. J.
> Livesey, Director of the Aurora Section, British Astronomical
> Association, whose text I acknowledge:
>
> "I was in Holyrood Park, Edinburgh, on the lower slopes of Arthur's Seat
> with a good clear sky.  The sun was setting behind the top of David Hume
> tower block of Edinburgh University, which is about a mile away, when I
> saw a green flash.  Realising the geometry of the situation I ran back
> up the slope until the sun was again visible above the tower block.
> Again as it set behind the building there was a green flash.  I was able
> to repeat the phenomenon by a second run up the slope.  One was required
> to move only so much as would bring the upper limb of the sun just above
> the tower block to cause a repeat as the sun sank down."

This surprises me greatly. I thought the green flash was a phenomenon
related to the passage of the sun's light through long distances of
atmosphere. Can it really be observed when the sun goes behind buildings?!
Then it should not be such a rare phenomenon after all. It should also be
visible if you move your head so the sun disappear behind the side of a
building. I expect myriad reports by tomorrow morning.

The vertical motion of the sun at sunset is about omega ~ 1e-4 rad/sec. If
the sun is setting behind an object only 1 km away, one must only climb at a
leisurely 0.1 m/sec to keep up with it. In contrast, the height required to
see the setting sun above a flat horizon is h = R*omega^2*t^2/2 ~ 0.03
m/s^2. If your top climbing speed were 1 m/sec, already at 8 m above sea
(horizon) level, you will not be able to keep up any more. If you climbed
twice as fast but stopped now and then to observe the flash, which takes 3
or 4 seconds, you cannot have time for more than 3 or 4 observations.

I am currently living in San Diego, where bluffs overlook the ocean to the
west. I will try the trick myself, when circumstances permit.

--Art Carlson

RE: Urgent request.

[Arthur Carlson]

For the benefit of Tony Moss, a search on
http://bible.gospelcom.net/cgi-bin/bible in "KJV" for "every thing
beautiful" yielded:

He hath made every thing beautiful in his time: also he hath set the world
in their heart, so that no man can find out the work that God maketh from
the beginning to the end.

... Of course, the "authorized" version is always what the client wants.
Whether the Bible is infallible or not, the customer certainly is.

--Art Carlson

RE: Sundial for downed pilots

[Arthur Carlson]

Jim Cobb wrote:

> Except for a new (which you can't see) or full moon, you can use the
> terminator as an indicator of a perpendicular direction to the plane
> of the ecliptic.  Follow the implied ecliptic to either horizon to get
> a sense of east and west.  If you're familiar with astronomy and can
> anticipate whether the ecliptic should be north or south of the
> celestial equator [at the two horizon points ] for the current date
> and time you can refine this indication of east or west direction.

That's exactly what I had in mind. This is a rule that can be easily
understood and remembered, as opposed to remember to ADD nine hours to the
clock time for a three-quarter moon, if it is WANING. What I would like to
figure out is the errors involved in both methods, given orbital parameters.

Ron Doerfler wrote:

> > On my list of
> > things I would like to do and know how to go about but haven't found the
> > time is to investigate telling directions from the moon. I read an
article
> > in the magazine of the German Alpine Club a few years ago on this topic
and
> > found it incredible.
>
> Wow!  Do you still have a copy of the article.  I would
> _love_ to read it.  If you do have it, I'll be glad to
> send a SASE.

Maybe I have a copy or can get my hands on one, but not before September
when I return to Munich. As I said, or meant to say, I was not impressed.
The basic technique recommended was to figure out from the phase of the moon
how many hours its position was ahead or behind the sun, then to assume east
at 6 AM, south at noon, west at 6 PM, and everything else interpolated. It
had no mention of errors or corrections for the seasons, the equation of
time, or the orbital plane of the moon. Even so, the instructions were so
complicated I'm convinced that no one who read the article could correctly
recall them six weeks later.

--Art Carlson

RE: Sundial for downed pilots

[Arthur Carlson]

Roger Bailey wrote:

> I recommend the old Air Force Survival Manual (AFM 64-5).
> ...
> Find your location (Latitude and longitude) and
> direction (north) using the shadow of a stick. Make a sextant from fishing
> line or parachute cord. Find your latitude from the length of daylight, or
> by measuring solar  and Polaris altitudes with a Weems Plotter. A
> fascinating book;
> ...
> On my backcountry explorations, my survival kit does not contain a
> parachute with all that useful cordage and fabric. My weapons are limited
> to a Swiss army knife and a big stick. Most of the advice in the manual is
> therefore not applicable.
>
> Accuracy is limited with such navigational techniques. For
> example it would
> be difficult to determine whether you came down in Kosovo, Macedonia or
> Serbia. This could be important!

I'm willing to (brashly) bet there was never a pilot who ever used these
techniques or even took them seriously. Knowing your latitude and longitude
without a map is useless, and if you have a map it is a lot easier and
quicker to find your location from the terrain. If you can't find your
location from the terrain because everything looks alike, an estimate of
your latitude and longitude will not help much either. Mountains or rolling
hills, possibly even forests, are likely to make a measurement of the length
of the day too inexact to be useful. The only thing that is truly useful is
finding directions, but then you should pack a compass, not a sextant. (That
isn't to say none of this is "fascinating".)

I do think being able to look at the sun and estimate directions could be
useful (in case you forgot to pack a compass, shame on you!). On my list of
things I would like to do and know how to go about but haven't found the
time is to investigate telling directions from the moon. I read an article
in the magazine of the German Alpine Club a few years ago on this topic and
found it incredible. With a Ph.D. in physics I think I can figure out how
many hours to add or subtract in which direction to convert moon position to
sun position and then to direction, but I bet very few people dumb enough to
get lost at night without a compass can. But even without a watch, if you
see the moon rising, you know that's east. And if the shadow is oriented
straight up and down, then the moon is in the south. You don't need to know
much more than that to find the nearest road.

--Art Carlson

RE: Sundial with a Second Hand

[Arthur Carlson]

Bill Walton wrote:

> To get the desired accuracy the "pin-holes' themselves must be very
> accurately aligned (not true if the free "pin-hole" technique is used and
> the hole moved back and forth until the shadow of the gnomon is centered,
> and on the hour mark, at the same time)

They would not have to be more accurately aligned than the hour line itself.
In fact, you could drill a hole in the center of each hour line and use that
as your pinhole. You're "on the money" when the image of the gnomon bisects
the image of the sun, regardless of where those images are projected.

--Art Carlson

"Who knows what evil lurks in the hearts of men? The Shadow knows!"

sundial with a second hand

[Arthur Carlson]

Discussions here and experiments of my own have established that "shadow
sharpener" techniques allow a shadow position to be read with accuracy on
the order of one second of time. This led me to look for a configuration
that allows a continuous readout with this type of accuracy, not just the
determination of one point in time (e.g., noon). Furthermore, in the
"hands-off" spirit of sundials, I wanted to read the time by just looking,
without having to fiddle with any instruments. Given a gnomon, ideally
subtending an angle a bit less than that of the sun, a properly placed
pinhole allows a very accurate determination of the time when the center of
the shadow passes over the pinhole. Obviously, many such pinholes could be
used, say one for each second of each minute. More elegant is to place this
series of pinholes so close together that they overlap, resulting in a slit.
The slit projects the sun onto a line of well-defined width. The shadow of
the gnomon falls on a short section of this slit and blocks the sunlight.
The result is a sharp-edged band of light intersected obliquely by a
sharp-edged shadow. The position of the intersection moves along the band of
light at a speed which allows, with proper set-up, resolution on the order
of 1 second of time. With an appropriate scale, this can function as an
indicator of seconds. To keep the dial compact, after a suitable period of
time, e.g., 5 minutes, the shadow could pass onto a parallel slit that
starts the process over again. This dial would be complex to build, and
adjusting for the equation of time to 1 second accuracy would be an ordeal,
but I think it must work and would be an intriguing project. I experimented
a little with the principle using my clipboard on my window sill, but it is
now too late in the afternoon for the sun to shine through my window. Looks
like I'll have to go back to work.

--Art Carlson

FW: Shadow Sharpener

[Arthur Carlson]

Roger Bailey wrote:

> I tried your Shadow Sharpener test today and was amazed at the result.

Me, too! It was easy, just using the shadows falling on my desk. My pinhole
was made by sticking a paper clip through a Post-It, which I stuck to the
edge of a clip board on my window sill. The gnomon was first a seam down the
middle of the window, then I changed to another Post-It stuck to the window
so I could adjust the width of the shadow. With the image of the gnomon a
bit smaller than the image of the sun, I watched the brightness of the spots
on each side where the sun was shining through. There was only about a two
second interval when they looked balanced. That means even a quick and dirty
set-up can yield an accuracy approaching +/- 1 sec! (I am in San Diego near
midday, so the shadows may be running a bit faster than usual.)

--Art Carlson

RE: Shadow Sharpener

[Arthur Carlson]

Bill Walton writes:

> The much increased precision of the Shadow Sharpener is obtained
> by casting a pin-hole image of the gnomon (wire, ball or building edge) in
> the center of a pin-hole image of the Sun  (as described by Charles).
> ...
> A wire or a bead that subtends 1/2 degree (1/8" at a distance of 14") when
> viewed from the scale or surface on which the shadow falls, will just
> obscure the Sun (as pointed out by Art Carlson).  It will cast a tiny,
dark
> umbra that can be closely read, but not as closely as the centered image
of
> the Shadow Sharpener.

I believe the configuration that can be read with the most accuracy would be
a wire or bead subtending just under 1/2 degree, viewed through a pinhole.
Perfectly aligned, the image would be a uniformly illuminated ring. The
slightest misalignment would result in one side being brighter than the
other. This should be much easier to detect than the deviation of a thin
shadow from the center of the sun's image disk.

--Art Carlson

RE: a peculiar sharpener

[Arthur Carlson]

John Carmichael wrote:

> The design which worked the best was a 1/8 inch spherical bead, suspended
by
> thin brass crosswires, in the exact center of a 1/4 inch round hole. (The
> style was about 24 inches from the analemma).

> A very curious thing happens with this type of style. The bead alone, by
> itself, casts a shadow that was twice as big as the bead; but when the
1/8th
> in. bead is in the center of a 1/4" hole, with a space of 1/16th of an
inch
> between the bead's edge and the hole edge, the bead's shadow miraculously
> sharpens into a tight, dark shadow that is only 1/16th of an inch in
> diameter, smaller than the bead itself  The wires which keep the bead
> suspended in the middle of the hole are so thin that they don't cast a
> visible shadow onto the analemma.

And Richard M. Koolish calculated:

> The linear diameter of the diffraction spot (Airy disk) produced by
> a pinhole of a given diameter is:
>
> spot = (2.44 * wavelength * focal_length) / diameter
>
> The optimal size is where spot = diameter, so:
>
> diameter * diameter = (2.44 * wavelength * focal_length)
>
> diameter = sqrt (2.44 * wavelength * focal_length)
>
> An example of a pinhole for a distance of 100 mm and a wavelength of
> 550 nm is:
>
> diameter = sqrt (2.44 * .000550 * 100) = sqrt (.01342) = .366 mm

Using a distance of 24 inches = 610 mm, this becomes 0.9 mm = 1/32 inch,
still several times smaller than John's hole. I think the explanation lies
in simple geometrical optics. Imagine putting your eye where the shadow is
being cast and looking back toward the style and the sun. I would like to
suppose that the distance to the style was something closer to 14" (subject
to objection and correction from John), so that the image of the sun would
be just eclipsed by the 1/4 inch bead, giving a black shadow at the center.
Just a little off-center, an arc of the sun would show around the bead, so
the brightness would grow, but only until the disk of the sun runs into the
edge of the hole. Thereafter the brightness would decrease slowly until the
sun is entirely outside the hole. This would lead to a shadow with a
diameter-at-half-brightness of about 1/16 inch, within a diffuse bright
field with diameter on the order of 1/4 inch. The size of the shadow is
reduced at the cost of reducing the contrast with the surrounding lighted
area. The principle is much the same as a sundial that images the sun
through a pinhole: a sharper image is a dimmer image.

--Art Carlson

RE: A GIANT PRECISION SUNDIAL

[Arthur Carlson]

John Carmichael wrote:

>   Does this mean that there is no upper limit for the size of a
> sundial? *

Seems obvious to me. The limitation in most configurations is the fuzziness
of the shadow, which also implies that size doesn't improve precision.

> If this is true, then one second time line markings could be placed on the
> dial face, couldn't they?  I haven't done the math, but if the one second
> lines at high noon ,when they are closest, were spaced at an easy to read
> distance of about  a 1/2 inch apart on a giant horizontal
> sundial, then the
> height of the style and the diameter of the face could be determined.  It
> would be a large sundial indeed!

The Earth rotates at 360 degrees/day = 0.073 mrad/sec. Divide this into 1
cm/sec and you get the scale of the sundial, 140 m. Monumental, but doable.

> It has long been my dream to design and construct such a sundial,
> maybe not
> with one second markings, but with 30, 20, or 10 second time lines.  (What
> are the time divisions on the large sundial in Japur India, does anyone
> know?)  I'd like to use the same basic design that I use for my horizontal
> string sundials (see website).  The sundial face could be located
> in a park
> and people could walk on it. The cable style would reach way up
> to a pulley
> attached to a building roof edge or southern wall.  A very heavy
> counterweight suspended from the cable would apply tension,
> making the cable
> straight. The diameter of available stranded metal cable may be
> the limiting
> size factor here because if the sundial were too large and the cable too
> narrow then the shadow would completely disappear (like telephone lines do
> on the ground).

The diameter of the sun is 8.7 mrad, so the style would have to be at least
140 m X 8.7 mrad = 1.2 m thick to provide an umbra. Consider using a thinner
cable with a 1.2 m ball attached, so that the date can also be read with
great accuracy. You should be able to determine the exact day of the
solstice with that precision, and the equinox within 15 minutes!

I think you will never be able to locate the position of the shadow this
accurately, however, without imaging optics. The most convenient lens to use
for dialing purposes is that in your eye. You can get a very accurate
reading if you look for the place or time where the image of the sun
disappears behind an appropriately sized object. The biggest drawback of
this approach, particularly in a public park, are the hundreds of people who
will be blinded by looking too long into the sun. One solution that would be
appropriate for a park would be a shallow, arc-shaped pond, preferably with
the bottom black except for the dial markings. The visitors would walk along
the pond until the reflection of the sun is blocked by the reflection of the
style. This also solves another problem of a configuration where the style
is viewed directly, namely the error due to different eye levels.

This would be a sundial the builders of Stonehenge could be proud of!

--Art Carlson

RE: "accurate" vs. "precise"

[Arthur Carlson]

> Speaking of barleycorns reminds me that one can have a lot of fun with
> units.  My favorite combination has components
>
> atmosphere = 101,325 newton/m^2
> yard = 0.9144 m
> barn = 1 x 10^(-28) m^2
>
> Combining these we get the
>
> barn yard atmosphere = 9.265158 x 10^(-24) joule
>
> a unit of energy.

Just to relate this to our everyday experience, I would like to point out
that the barn yard atmosphere is also a convenient unit of temperature,
lying just between the Fahrenheit and Celsius degrees.

I once heard that the mass of the electron in pounds is exactly 2.00 X
10^-30, but I don't know whose pound you need to use to get this. (When the
Germans say "pfund", they mean half a kilo.)

--Art Carlson

RE: double blue moon

[Arthur Carlson]

> > Will two full moons always occur in a March that follows a Febuary with
> > no full moon?

> I think it pretty well follows.
...
> I:31/01   23:50
> plus  29  12:00
>   60/01   35:50
> corr  61/01   11:50
> corr -31 (leap year)
>   30/02   11:50 30/02   11:50
> corr -28   -29
> II:   02/03   11:50 01/03   11:50
>   29  12:00 29  12:00
> III:  31/03   23:50 30/03   23:50

But if the time between two full moons can sometimes be even 1 minute more
than 29.5 days, then it would be possible to have no full moon in February
and only one in March. I hope someone has the definitive answer, but I
believe that the length of the month is sufficiently variable.

Art Carlson

Re: eleven days

[Arthur Carlson]

Martin <[EMAIL PROTECTED]> wrote:
Regarding Franks mention of simple folks cry of "give us back our 11
days" Well I would be pretty riled too if the rent was due 11 days
early as I'm sure evil land lords would have used the change in the
calender as a good excuse to ring money from the masses. I bet they
didnt get paid earlier!!!

Actually, they were just having to pay the rent for the extra leap days that
had been "incorrectly" added to the calendar century after century.  They
can be glad they didn't have to pay interest on the back rent.

I suppose a progressive pope might have decreed that the eleven days stuck
should include the day rent was due, letting everyone live some three weeks
rent free.

It is interesting to note that the days of the week were not skipped, so
that the day following Thursday, October 4, 1582 became (in Catholic
countries) Friday, October 15, 1582.  Otherwise there would have been
additional difficulties determining a week's wages.

All in all, it would have been a lot easier, from a practical point of view,
to not drop the 10 or 11 days all at once.  The same effect could have been
achieved by just declaring that none of the next 40 or 44 years would be
leap years.

-- Art Carlson

Re: Internet Time

[Arthur Carlson]

John Carmichael writes:


>Hey, did anyone see the CNN story last night about the watch company
>,"Swatch" that is now selling timepieces which tell "Internet Time"?  I
>can't remember exactly, but they said one minute of normal time=about 1 1/2
>minutes Internet Time, and that the idea behind it is to facilitate
>timekeeping around the world for internet users.  Everybody everywhere
(even
>on Mars?) will be using on the same time!
>
>Arthur C. Clarke believes that the current timezone system will be
abandoned
>and everyone will use Universal Time in the future.  I agree with Arthur.
>Or am I wrong, will we all be using Internet Time instead?


You're both wrong.  In the future everyone will use local solar time.

This is certainly the most natural time for any living thing.  The need to
physically transport time to find the longitude and the need to provide
timekeeping at night and on overcast days led to the rise of mechanical
clocks.  Unfortunately, these were not sophisticated enough to tell the true
time, but had to be satisfied with an unnatural uniform scale.  It is, of
course, trivial for any microprocessor to convert its clock pulses to solar
time and also to convert the time at any other place to the local solar
time.

If I am in Germany and want to arrange a meeting with someone on Japan, I
would say, "After lunch would be good for me, say 2 PM?".  My email or voice
mail would not only be translated to Japanese, the time would also be
translated to the local time of my correspondent, something like 11 PM.  The
airlines use this principle already, in that the arrival and departure times
on the ticket are always the local times.  The radio transmitters used now
to synchronize clocks will in the near future be ubiquitous and short range,
so that you will never have to adjust your watch while traveling.

In a similar way, the attempt to change timekeeping to a base ten system is
an anachronism.  The decimal system is a lifesaver if you have to do
complicated (scientific) calculations in your head or on paper.  For simple
(everyday) calculations, a system based on 12 (or 24 or 60) or possibly 16
is much more convenient.  The processors which are taking over all but the
simplest calculations for us have no trouble with 12 inches in a foot and 24
hours in a day.

The implications this has on the demand generated and respect tendered for
the skills preserved in this mailing list are obvious.

-- Art Carlson

Re: lunar eclipse

[Arthur Carlson]

Jim_Cobb <[EMAIL PROTECTED]> writes:

> I noticed that this time disagrees with the time given in the almanac,
> so I thought I should provide more information so as not to impugn the
> reputation of the excellent xephem program.  The 16:08:17 UT time is
> what xephem computes as the time of the full moon.  I do not know how
> to get it to reveal the maximum eclipse time.

Well, that's interesting.  I would have defined "full moon" as the
time when the moon is most nearly opposite the sun, which would be the
same as the time of maximum ecclipse.  How else can it be defined?
There must be something like a projection into the ecliptic.

Art Carlson

Re: sundial setting

[Arthur Carlson]

An analemmic dial would be insensitive to refraction effects, wouldn't
it?

Art Carlson

Re: sundial setting

[Arthur Carlson]

[EMAIL PROTECTED] (Philip P. Pappas, II) writes:

> Thank you for your thoughtful comments.  I make the statement that the "time
> method is the prefered method for setting a sundial if and only if the
> sundial is properly designed, constructed and leveled (correcting for the
> EOT and longitude of course).

I would say that it is the preferred method *especially* if you
suspect you have a poorly made dial.  If you set it up by the time
method, then at least you know it is accurate at one time for two days
of the year.  This is not guaranteed to, but is likely to reduce the
errors on average.

> >4.  This has just occurred to me and is probably not
> >relevant but it has got my mind wondering.  As we know, the earth is a
> >flattened sphere.  Gravity, from which we derive a vertical (and
> >subsequent horizontal) reference comes from the centre of the earth's
> >mass.  This is presumably right in its centre, assuming that differences
> >in local density do not move it by much.  But as we move towards the
> >flattened poles the angle to the centre of gravity will no longer be a
> >true vertical.  But even so, it is this centre of gravity which is the
> >true reference point for the earth in its orbit around the sun.  
> >Then there is the centrifugal force due to its rotation.  Will
> >this effect a true vertical?  At the equator - no, but imagine a point
> >at 45 degrees latitude, where the centrifugal force must have some
> >effect on any plumb line/spirit level.  I guess that all of these
> >effects are so tiny as to be irrelevant, but I would like to know how
> >much they modify the results.

These effects are one and the same.  The Earth is flattened at the
poles *because* centrifugal force pulls it out around the equator.  At
the Equator and at the poles the vertical passes through the center of
the Earth, inbetween it doesn't, but that doesn't affect the accuracy
of a properly designed dial.

Just for fun, the radius of the Earth is 6,378 km and the difference
between the the equatorial and the polar semiaxis is 21.4 km.  This
makes the maximum discrepancy in the angle about (2*21.4/6378) = 0.4
degree.

Art Carlson

Re: speed of light

[Arthur Carlson]

John Carmichael writes:

> We could make this question even more complimented if we consider the speed
> of light.  When we see the sun's center on the horizon we are seeing light
> that left the sun about 8 minutes earlier.  The sun really has already set.
> (of course this has no practical effect on sundial time, but is fun to think
> about!)

What does that mean, "The sun really has already set."?  I would say,
"By the time the light now leaving the sun gets here, I will have
moved behind the edge of the Earth."  But the sun really is located in
the direction from which the light I see is coming.  (There's a teeny
tiny shift in the direction if I am moving perpendicular to the line
of sight, but that is not the case at sunset.)

Art Carlson

Re: "Analemmatics" on a Gradient

[Arthur Carlson]

[EMAIL PROTECTED] (Mr. D. Hunt) writes:

> In relation to the recent question/replies, regarding detecting/correcting
> 'errors' in the setting of sundials - is there any feasible way of varying
> the layout of an "Analemmatic" dial, to cope with it being on a GRADIENT ?
> 
> My own thinking is that this is just NOT possible, if the dial has to tell
> 'correct' time (disregarding EOT effects) at all times of DAY, plus at all
> seasons of the YEAR - but will welcome any comments/confirmation, on this.

The gnomon, whether vertical or not, together with the direction to
the sun, defines a plane.  The intersection of this shadow plane with
the ground plane, whether horizontal or not, defines a line.  If you
think of the celestial sphere as being a finite size and centered on
the base of the gnomon, then the position of the sun projected along
the direction of the gnomon onto the ground plane will lie on the
shadow line.  The orbit of the sun during the course of a day is a
circle, generally not centered on the base of the gnomon.  The
projection of the orbit on any day will be an ellipse, though the
center of the ellipse will move from day to day.  An analemmic sundial
is designed by rescaling all the ellipses to the same size, then
translating them to lie on top of each other, which implies that the
gnomon must also be translated to a particular position for the
projection to be accurate on that date.  The upshot is, an analemmic
sundial properly designed for sloping ground will be just as accurate
as one on the level.

Re: Help needed with unusual sundial

[Arthur Carlson]

Dear Bob,

Fun problem.

1. If I were setting the thing up, I would turn the existing disk so
that the local longitude pointed up, not that of Greenwich.  That way
the observer can see at a glance "where in the world" he is, as well
as the approximate time anyplace else in the world.  There is a
"blemish" on the longitude disk in your last photo; it looks as though
it could be a locking screw.

2. To trace out the path of the sun, the hole would have to rotate
around the axis, so I would want the "hour angle arc" to be similar to
the one you have drawn, but rotatable.  The hole would be in the
slide, which moves up and down the arc at the middle +/- 23.5 degrees.

3. Either the hour scale must be attached to the hour angle arc and
move under a fixed pointer or the pointer must be attached to the arc
and the scale fixed.  I like the idea of having the scale on the
globe, since it reduces the number of parts.  If the globe moved with
the hole, however, it would be easier and more accurate to align the
sun spot up with a line on the globe, rather than judging when the
spot is round.  On the other hand, if the globe is fixed, then it
would be simpler and more elegant to read the time directly with the
spot of light, rather than with an extra pointer.  Furthermore, the
"globe" can then sensibly be an actual globe, with a map of the world
on it, rather than just a sphere.  (But then why bother with a
separate longitude disk?)

Good luck, and lots of fun.

Art Carlson

Re: Best angle to catch sun light - off topic

[Arthur Carlson]

Fernando Cabral <[EMAIL PROTECTED]> writes:

> Now I am planning to build a house for a small farm I have. I've
> been thinking on how to take the best advantage of the solar
> power. This includes where to have a garder with a nice sundial and
> where to place the solar panels for water heating as well as
> (perhaps) electricity (at least in Brazil solar panels for
> electricity are very expensive).
> 
> At 19 37' 57" S, it is clear that the panel should be facing North.
> But what is the best angle with the horizon. And, if I can have
> several panels, is there a practical to calculate the best angle of
> each so as I can guarantee the highest possible insolation level?
> 
> Say, if I have three panels, is it best to place them side by side,
> with the same inclinatation and declination? Perhas if one is a
> inclined towards the East with a certain angle and the other to the
> West with a proper angle I can capture more light?

Goods questions, to which I don't have the answers.  I would even
question the basic assumption that the panels should face north.  You
may want to have fresh hot water as soon as you can in the morning,
especially if you shower then, in which case the panel (or one of
them) should face east.  At midday and in the evening you can use the
heat that has been collecting all day.  By the same token, it is
probably better to point the panel low to the horizon (angle between
the normal and the vertical equal to 23.5 degrees minus the latitude,
since you are in the tropics) because you will want to produce more
hot water with less sunshine in the winter.  (You may not have enough
of a winter that that matters, but there may be similar considerations
for rainy/dry season, afternoon thundershowers, etc.)

If you determine the optimum angle simply by integration of the
sunlight over some period, then that angle will be the same for every
panel.  If your use patterns are different and the storage
characteristics poor, then you might want to do something like point
one panel to the east for morning hot water and one to the west for
evening hat water.

Do you need some inclination to drive convection through the
collectors?  Or to prevent rain water from collecting on the panel?

Since you say there is a great variety of orientations of panels in
the city, can you get answers to some of your questions by
interviewing residents with different orientations?

Art Carlson

Re: Invention to tame moon monsters

[Arthur Carlson]

Roger Bailey <[EMAIL PROTECTED]> writes:

> I was experimenting with the shareware program "Astronomy Lab". One
> calculation that this program plots is the "Moon Angular Speed" in degrees
> per day. This is the lunar equation of time we have been looking for.  In
> minutes rather than degrees, the variation is up to 14 minutes on top of
> the 48 minute average daily correction that we have been quoting. 
> 
> The moon's equation of time is the variation on that average angular speed.
> The graph shows this well as the sum of two periodic cycles. The major
> cycle is the monthly lunar cycle. The moon speeds up when it is closest to
> the earth (perigee) and slows down when it is most distant (apogee). The
> cycle ranges from about 11.8 degrees (47 min) to 14.2 degrees (57 min). A
> yearly cycle is added to that giving maximum peaks of 15.4 degrees (61min)
> when the full or new moon (lunation) is in phase with the lunar orbital
> cycle. Arthur C. noted the connection between the lunar and solar (year)
> cycles.

I don't understand why the position of the sun should have an effect
on the angular velocity of the moon.  Does the yearly cycle
superimpose another oscillation (like making the moon run generally
slower in summer than in winter) or does it modulate the amplitude of
the monthly cycle (like making the moon run at a more nearly constant
speed in summer than in winter)?

Art Carlson

Re: moon monsters

[Arthur Carlson]

Dear John,

Your explanations sound like about the right level for a users'
manual.  Maybe because I'm a scientist, I think it is important to at
least mention the major sources of error.  In my opinion, the biggest
problem is determining the exact phase of the moon by looking at it.
(Of course, you can get pretty accurate by looking it up in the
newspaper.)  I would guess a misjudgment of the phase by up to one day
is common without a lot a practice.  That will result in up to 48
minutes of error.  The EOT might be considered small compared to that,
but I think I would mention that it should be used.  Some of your
users may get a kick out of an accurate measurement.

The other errors, like those due to the tilt (5 degrees) or
eccentricity (0.0549) of the Moon's orbit, I expect to be on the order
of the square of the parameters, which is under 1%.  But 1% of what?
1% of a day is 14 minutes, so I would need to give this some more
thought.

As far as your eclipse observation goes, I suspect the eclipse either
took place when the correction was small, or you misjudged when the
center of the eclipse was.  Since a lunar eclipse takes a fairly long
time, the "moon time" at the start and the end can differ by several
minutes.

Art Carlson

Re: moonlight readings

[Arthur Carlson]

John Carmichael writes:

> >I have a section which tells how to tell time by using moonlight and a
> >sundial.  I provide a table of corrections from which the time can be
> >estimated if one knows the age (the phase) of the moon.
> >
> >One question though:  Is it nessary to correct moontime with the Equation of
> >Time ?
> >Since the Equation of time is due to the eccentricty of the earth's orbit
> >around the sun and the tilt of the earth's axis, it seems to me that  this
> >has nothing to do with the moon and should not be considered in the
> >corrections.  Am I right?

Roger Bailey <[EMAIL PROTECTED]> writes:

> Hello John,
> 
> My advice is "Don't go there. There be monsters!" *

Good advice.

> The motion of the moon is quite complicated and the "equation of time"
> shortcut will not work. You were right is concluding that the solar
> "equation of time" does not apply, and the eccentricity and obliquity of
> the ecliptic were the determinants of the equation of time.

I wouldn't agree the Equation of Time does not apply, just that
other corrections are much larger.  John does, after all, want to
correct for the phase of the moon, so the position of the sun is
relevant.

> The major
> problem with the moon is the time between new moons (lunation) is 29.53
> days, different from the orbital period of 27.32 days. This means the
> declination cycle, connected with the orbital period, is out of phase with
> the lunation cycle.

This makes it sound like these are two separate orbital parameters.
They are simply connected by the length of the year:

   1/27.32 - 1/29.53 = 1/365

In fact, the time between any two particular adjacent lunations will
have a correction closely related to the Equation of Time.

> For night time checks, I use a "nocturnal" and determine the time based on
> the rotation of the big dipper around Polaris. The date / sidereal time
> correction is easier to build into the instrument. 

Even easier than correcting a sundial for the Equation of Time.

I have been interested for some time in the related problem of finding
directions from the moon, possibly given watch time.  I haven't
formulated the mathematics yet.  To quantify the error of various
methods I will need some more information on the distribution of the
relative positions of the sun and moon.  This is certainly known.  Is
it also readily available in a comprehensible form?

Art Carlson

Re: Definition of Time?

[Arthur Carlson]

"Paul Murphy" <[EMAIL PROTECTED]> writes:

> > September 11-24 , 1752

> Unfortunately, Warren, even this depends where you were at that time! Had
> you been in a place where the Gregorian Calendar had been accepted in 1582,
> quite a lot might have happened. On the other hand had you been in Russia,
> you would have to wait until 1917 to find the lost days!!

I wonder something every time I hear about idiot savants who can tell
the day of the week of any calendar date.  Do they ever make the
switch from the Julian to the Gregorian calendar?  If so, when?  I
suspect the psychologists examining them don't know enough about the
calendar to realize there is an issue.  It's like claiming they can
recognize any prime number instantly without asking, say, if the
product of two particular ten digits primes is prime.

Art Carlson