Lunar Sundials and the Lunar Analemma.
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Lunar Sundials
In "Jahreschrift 2000 der Deutsche Gesellschaft für Chronometrie", Band 39, is
an article about dials on the moon, written by Heinz Sigmund, in German.
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Date: Sun, 18 Jan 1998 From: "fer j. de vries"
What accuracy do you want?
If you want to convert "moontime" into "watchtime" there are to many
parameters.
But if the main goal is just to find the time roughly there are many solutions.
The main part is a table with average values for the hourangle between the moon
and the sun in realtion to the age of the moon.
Roughly there are 30 days in one period from new moon to new moon. Each day
the hourangle between the moon and the sun changes about 24 * 60 / 30 = 48
minutes.
One solution is to use an equatorial sundial and rotate the dial's plate with n
* 48 minutes in which n is the age of the moon in days from new moon.
Or make on a normal sundial a number of scales for the age of the moon and on
each scale hourpoints with an offset of n * 48 minutes.
Connect all the hourpoints and you get a moondial as in the attached figure.
In that figure the scales for the age of the moon are circles, but that isn't
necessary.
Also the pattern of the normal sundial can be removed.
In the bulletins of the Dutch (92.5), Spanish and British (92.1) Sundial
Societies some about moondials is written.
In the book Orologi Solari by Girolamo Fantoni, Italy, you find a capital about
moondials.
The picture in this mail is scanned from that book. moondial.jpg
Fer de Vries.
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a photograph of a Chinese moondial, probably a few centuries old, was included
by Joseph Needham in Volume III of _Science & Civilisation in China_
(Cambridge: Cambridge University Press, 1959), in Plate XLII, opposite page
309. The briefest of references to this instrument appears on page 311.
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9 Nov 1998 From: Fred Sawyer
You might find the following article on Moon Dialing interesting - I wrote it
for The Compendium : Journal of the North American Sundial Society. I have not
included the graphics or computer program here, but the discussion and example
may prove useful.
Moon Dialing Fred Sawyer Digital COMPENDIUM 1(2) May 1994
One of the more interesting passages attempting to explain or shed light on
the fascination the dialist has with the sundial can be found in Frank Cousins'
introduction to his book Sundials:
"The sundial still remains, despite all advances in the art of time
measurement, the one philosophical tool which combines the two poles of the
human condition most faithfully: transience and arrogance - transience in the
ephemeral nature of the shadow cast and arrogance in the use of the gyrations
of the Earth about its stupendous luminary delicately to define the successive
quanta of human existence." - Frank W. Cousins 1969 Sundials
Transience and arrogance. Indeed, the image of the fleeting shadow has
often been invoked to suggest the transience of our existence and of our works;
the image has a certain natural appeal. However, the arrogance we show by
directly using the sun to clock our daily lives is alluded to far less often.
Perhaps we should make an effort to raise our consciousness and seek ways to
curb our arrogance. I submit that a first step in this direction would be to
consider stepping our arrogance down a notch by using the moon rather than the
sun as our source of light and shadow.
Surely, the moon is a less stupendous luminary - if indeed it even qualifies
as a luminary, offering as it does only the reflection of sunlight. A moondial
would clearly offer us all a lesson in humility and at the same time provide an
answer to the frequent refrain: "What do you do to tell time when the sun goes
down?"
Of course, this revelation is not new, and moondials, though rare, have
existed for at least the last three centuries. A moondial is simply a sundial
together with some mechanism to correct for the difference between solar and
lunar hour angles. This mechanism may take many forms:
1) A simple graph or correction table, such as may be seen here or, in a
slightly different form, on the famous Queen's College sundial in Cambridge
England.
2) A mechanism for rotating the dial to give a direct reading, based on
the current phase of the moon, such as may be seen in Nicholas Bion's well-
known 18th century work, The Construction and Principal Uses of Mathematical
Instruments.
3) A set of spiralling hour-lines designed to replace the usual lines and
to function correctly only in response to illumination by the moon, such as the
arrangement developed by Girolamo Fantoni (Bulletin of the British Sundial
Society 92(1):11-15, February 1992).
My goal in the present article is to provide a simple (but computer-aided)
foundation on which the insomniac dialist can begin to retrieve the moondial
from its relative obscurity and dismissal as an inaccurate curiosity. Even
those few writers who have dealt with the moondial have tended to give it short
shrift. Mayall suggests that one is lucky to come within 30 minutes of the
correct time; and he is correct for most earlier treatments of the subject. As
we will see, the formulas produced here and implemented on computer will
virtually eliminate that error.
Basic Theory
How, then, do we adapt a standard sundial for use as a moondial? The basic
theory was outlined by Nicholas Bion in 1723 as follows:
"...the Moon by her proper Motion recedes Eastwards from the Sun every Day
about 48 Minutes of an Hour, that is, if the Moon is in Conjunction with the
Sun on any Day upon the Meridian, the next Day she will cross the Meridian
about three quarters of an Hour and some Minutes later than the Sun: and this
is the Reason that the Lunar Days are longer than the Solar ones; a Lunar Day
being that Space of Time elapsed between her Passage over the Meridian, and her
next Passage over the same; and these Days are very unequal on account of the
irregularities of the Moon's Motion."
"Now when the Moon is come to be in Opposition to the Sun, she will again be
found in the same Hour-Circle as the Sun is; so that if, for example, the Sun
should be then in the Meridian of our Antipodes, the Moon would be in our
Meridian, and consequently would shew the same Hour on our Sun-Dials as the Sun
would, if it was above the Horizon. But this Conformity would be of small
duration, because of the Moon's retardation of about two Minutes every Hour.
If moreover the Sun, at the Time of the Opposition, be just setting above our
Horizon, the Moon being diametrically opposite to it will be just rising,
&c...."
"[To make] the Table...used for finding the Hour of the Night by the Shadow
of the Moon upon an ordinary Dial..., draw 4 Parallel right Lines or Curves of
any length, and divide the [Center] Space...into twelve equal Parts for 12
Hours, and the two other Spaces into 15, for the 30 Lunar Days."
"First observe what Hour the Shadow of the Moon shews upon a Sun-Dial; then
find the Moon's Age, and seek the Hour correspondent thereto in the Table, and
add the Hour shewn by the Sun-Dial thereto; then their Sum, if it be less than
12, or else it's excess above 12, will be the true Hour of the Night. For
example; Suppose the Hour shewn upon the Sun-Dial by the Moon, be the 6th, and
her Age be 5 or 20 Days, against either of these Numbers in the Table you will
find 4, which added to 6 makes 10, and so the Hour of the Night will be 10.
Again, Suppose the Moon shews the Hour of 9 upon the Sun-Dial, when she is 10
or 25 Days old, against 10 and 25 in the Table you will find 8, which added to
9, makes 17, from which 12 being taken, the Remainder 5 will be the true Hour
sought. And so of the others."
- From Edmund Stone's 1758 translation of Nicholas Bion's The Construction
and Principal Uses of Mathematical Instruments, Book VIII, Chapter 6.
Note that this passage is abbreviated because the actual instrument Bion
described to implement this design breaks the cardinal rule requiring that the
gnomon always be parallel to the celestial axis. His use of the instrument
rotates the gnomon out of the north-south plane.
A Different Approach
Given the traditional reliance on knowing the current lunar age (phase) to
use a moondial, it would be natural to think that the key to a more accurate
reading is a more precise determination of the moon's phases. However, more
precision here actually would be of little utility. The traditional
calculation only uses the phase to determine the point of entry in what is
itself a very approximate table of data, assuming an average 48 minutes/day
additional elongation from the sun's position. For more precision, we need to
return to the fundamentals.
Begin by selecting an hour of day to use as a reference time. For
convenience, we use midnight Standard Time - the time when the (fictional) mean
sun crosses the longitude line half-way around the world from the longitude
line at the center of our own time zone. Thus, the reference time for May 1 in
the Eastern Time zone is 2400 EST May 1 (or equivalently, 0500 GMT May 2).
Next, determine the lunar and solar apparent right ascensions for the
reference times. These values can be found from an ephemeris or from the
astronomical software available on many electronic bulletin boards. Although
the solar right ascension does not change significantly from hour to hour,
there is a fairly large variation in the lunar value - whence the need for
selecting a reference time.
Now note that at any moment, the difference between the sun's and moon's
hour-angles equals the difference between the moon's and sun's apparent right
ascensions:
HA sun - HA moon = RA moon - RA sun
Thus, by calculating this difference between the apparent right ascensions,
we obtain a preliminary value for the lunar equation of time. When this value
is added to the moon's hour-angle, which is obtained by reading the shadow on
the moondial, we have the solar hour-angle, i.e. local apparent time.
Now modify this initial value for the equation to account for the usual
solar equation of time with which all dialists are familiar, and for the
difference in longitude between the dial's location and the central meridian of
the time zone. These are both normal adjustments which convert any traditional
sundial reading from local apparent time first to local mean time and then to
standard time.
Finally, for convenience, add (or subtract) 12 hours to (from) the equation
as it now stands. This modification allows us, for example, to interpret what
is normally the 10am line on the sundial as 10pm, and the noon line thus
becomes the hour-line for midnight.
We now have a lunar equation of time calculated for the moment of midnight
standard time in the dial's time zone. This value should work very well close
to midnight; however, the never very well-behaved moon rapidly introduces
errors. To improve accuracy at other times of night, make an additional
adjustment, after the lunar equation has been applied to the reading, by
subtracting 2 minutes for each hour that the result falls before midnight and
adding 2 minutes for each hour after midnight. This adjustment, which works
well enough over short periods, is based on the traditional rule of thumb which
forms the basis for older moondial treatments: the moon's elongation from the
sun increases on average by 48 minutes in each 24 hour period.
All of these adjustments are incorporated into the executable program
included with this issue of the Digital COMPENDIUM. A listing of the
QuickBASIC source code for the program is also included. The program produces
a monthly table for the lunar equation of time and displays a worked example of
its use.
With this equation and final adjustment, the dialist should be able to get
much improved accuracy from a moondial on any bright moon-lit night. Of
course, for about half the nights in any given month, before and after the new
moon, there will not be enough light from the moon even to take a reading
(these nights are roughly those shown in red on the digital moondial chart).
But for those nights when a near full moon shining brightly through the window
wakes you from a sound sleep, you can now use your favorite dial to tell you
how many hours you have before the alarm goes off and the 20th century intrudes
again on your dialing reverie.
Closing Reflection
Having now finished presenting the intended material for this article, I
feel compelled to make a confession to the reader:
Midway through the preparation it occurred to me that a good case could be
made for the charge that contriving a celestial billiard game, banking rays of
light off the moon and down to a dial - all in order to keep track of our night
hours, may yet be a step up in arrogance. Perhaps there is no escaping our
baser nature!
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As an example, suppose on the night of June 21, 1994 we have a moondial
reading of 11:35 a.m. at longitude 72 degrees in the Eastern time zone.
Recall that what would be an a.m. reading on a sundial becomes p.m. on the
moondial. The digital moon chart lists an equation for this date and location
of -1:24, thus correcting the original reading to 10:11 p.m. This adjustment
includes allowance for the solar equation and for the 3 degree (i.e. 12 minute)
difference between the dial's longitude and the central meridian (75 degrees)
of the time zone.
Making a final adjustment of -4 minutes (since the first result is
approximately 2 hours before midnight) yields a time of 10:07 p.m. Eastern
Standard Time.
Reading 11:35 (a.m. converted to p.m.)
Equation -1:24
________
First Result 10:11 p.m.
Adjustment -0:04
________
10:07 p.m. Eastern Standard Time
Thus, a total correction of -1:28 is applied to the 11:35 reading to
obtain the time 10:07 p.m. In order to see how close this result is to a
precisely calculated value, we can refer to an ephemeris, first, to find that
at 10:07 p.m. on that date the apparent right ascensions are as follows:
Moon 16:42:57
Sun 06:02:08
________
Diff. 10:40:49 =====>> -1:19:11 (by subtracting 12 hours)
Also, at this date, time and longitude, the local apparent time
(corresponding to the sun's hour angle) is found by subtracting from Eastern
Standard Time 1 minute 50 seconds for the solar equation of time and adding 12
minutes for the longitude difference, to obtain the result 10:17:10 p.m. Now
subtracting the difference between the right ascensions yields an hour angle
for the moon corresponding to 11:36:21, very close to the 11:35 reading with
which this example began. Thus, in this case, the procedure produces a value
with an error of less than 1.5 minutes.
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14 Jan 1999 From: Patrick Powers
...there are a few examples of moon dials - the most famous in the UK is the
one at Queens College Cambridge (though I believe the Nuremburg Diptych dials
carry them too).
On the Queens College dial there is a table of 'hours correction' versus
'age of the moon, in days'. This is simply applied to the observed reading to
convert moon time to apparent solar time. One then applies the EoT to get
clock time.
This process is not at all accurate and, except perhaps at the moment of
full moon, is not likely to be something on which you would like to depend for
an appointment!
An interesting point is that the Queens College table is designed so that
the correction is always added whereas I believe that on other moon dials may
have to be added or subtracted.
There is an interesting article in the BSS Bulletin (No.97.1 January 1997,
p37) by Ing JTRC Schepman of the Netherlands entitled "How to determine time by
Moonlight". In this article the author discusses these correction tables
but also describes a simple analogue instrument to convert lunar indication to
apparent solar time - it consists of concentric disks - and you may wish to
refer to that article.
Also, if you are able to get to Cambridge or can write to the Head Porter
to buy a copy, you might wish to read the blurb in the Queens College leaflet
on the dial - that too describes the way in which the simple table works. This
dial was also discussed (albeit rather derogatorily I thought!) by the late
Charles Aked in the BSS Bulletin 94.3 October 1994.
There was another article in a BSS Bulletin (I think - though it may have
been in the NASS Compendium, my memory is not clear! - about the ease (or
otherwise!) by which a 'moon EoT' might be derived but I cannot find that at
the moment.
I suspect that these ideas only work on dials that use hour angle or
azimuth. Patrick
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Incidentally, a Sun- and Moondial would not necessarily have 24 divisions.
Twelve divisions (equally valid for day or night) could be "shared" by the Sun
and Moon - provided one knew what one was doing, which should include keeping
track of both the synodic and draconic months. How early are moondials attested
to?
There is a fine 17th- or 18th-c. solunar dial at Queen's College
Cambridge, though somewhat tarted up by modern restorers:
http://www.quns.cam.ac.uk/Queens/Images/sundial.html
www.quns.cam.ac.uk/Queens/misc/Dial.html
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The Lunar Analemma. By Maurice Bruce Stewart.
As you know, you can photograph the solar analemma by making pictures on
the same photographic plate every day for a year. Likewise the lunar analemma
could be made by making pictures on the same photographic plate every "day" for
a month. The question is what is meant by "day."
To answer this question, let us start with what we know for certain about
the stars. A given star transits once every 86164.161 seconds, what we may call
a star day. The Sun is moving east among the stars and completes one circuit of
the celestial sphere every 365.2564 days, the so-called sidereal year. Because
the Sun is moving east among the stars, a given star must make one more transit
than does the Sun during these 365.2564 days . Thus there will be 366 transits
of the star but only 365 transits of the Sun. Hence, we conclude that what we
may call the sun day will be 366.2564/365.2564 of star day, or 86400.00133
seconds long. In a similar way we may define a moon day. The Moon is also
moving east among the stars and completes one circuit of the celestial sphere
every 27.32166 days, the so-called sidereal month. Because the Moon is moving
east among the stars, a given star must make one more transit than does the
Moon during these 27.32166 days. Thus there will be 28 transits of the star but
only 27 transits of the Moon. Hence, we conclude that the moon day will be
28.32166/27.32166 of star day, or 89317.79300 seconds long. In this way we see
that the moon day is 1.033770736 times as long as the sun day. Putting it
another way, we should expect the Moon to transit once every 24.81049766 hours,
or 24 hours, 48 minutes, and 37.791576 seconds.
In making these simple calculations we have assumed that the Sun and the
Moon move only eastward, not at all north or south, and that they move at a
constant rate. Neither of these assumptions is true, neither for the Sun nor
for the Moon. Therefore the "days" whose lengths we have calculated are better
called mean sun days and mean moon days. The departure of the real Sun from the
mean Sun produces the well-known solar analemma. So, in an exactly similar
manner, we expect that the departure of the real Moon from the mean Moon should
produce what we may call the lunar analemma.
Lunar analemmas for successive lunations.....It is fascinating to watch
these dots being generated on the computer screen because then you can see the
form of the lunar analemma changing from month to month, in much the same way
that we could see the solar analemma change if we could watch for a geological
time span. Maybe I could simulate the computer experience with some sort of
flip cards.
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The Lunar Analemma, Further Detail. by Maurice Bruce Stewart
The Sun which appears in the sky every day is known to astronomers as the
Apparent Sun. If you look for the Apparent Sun at the same time every day you
will see that its altitude and azimuth change from day to day, but repeat
themselves, more or less, after a year has gone by. Astronomers have invented
another Sun, the Mean Sun, which behaves in a much simpler way than the
Apparent Sun. The Mean Sun, being fictitious, is of course invisible, but if
you could see it, you would see that the Mean Sun has the same altitude and
azimuth at the same time of day every day. This simpler behavior of the Mean
Sun is because our ordinary clocks and watches are synchronized to follow the
motion of the Mean Sun. Dennis di Cicco made his famous photograph of the solar
analemma by fixing a camera in his back yard, waiting until the Mean Sun was in
the center of the field of view, and then making a photograph of the Apparent
Sun. He explains all the details of how he did it in his article on page 536 of
Sky and Telescope for June 1979. You can see his photograph of the solar
analemma on the web at http://sundials.org/links/local/pages/dicicco.htm. It
helps to keep in mind that the solar analemma is an artifact, not a natural
phenomenon. It is only because di Cicco took the pictures at the same time
every day that he captured the analemma shape. If you take pictures at the
correct times on the correct days, you can make a picture of the Apparent Sun
spelling out RCA.
Suppose you want to make a photograph of the lunar analemma. You can just
copy di Cicco's technique by substituting the Moon for the Sun. There will be,
of course, two Moons: the Apparent Moon and the Mean Moon. Since you can't see
the Mean Moon, you must calculate when it has the same altitude and azimuth
again and again so that you know when to make the photographs of the Apparent
Moon. If you try to calculate the altitude and azimuth of the Moon directly you
enter a thicket of spherical trigonometry, but there is a simpler way. The Mean
Moon goes east completely around the celestial equator with respect to the Mean
Sun once every synodic month of 29.530589 days. Thus you will "see" the Mean
Moon pass through your field of view only 28.530589 times while the Mean Sun
passes 29.530589 times. Hence, Mean Moon will be have the same altitude and
azimuth every (29.530589 / 28.530589) * 24 hours or 24.841202 hours.
Armed with this knowledge you can make a di Cicco style photograph of the
lunar analemma, but only after your patience is severely tried. Sometimes you
will be photographing the Apparent Moon at night and sometimes in the daytime.
Sometimes the Apparent Sun will get in the way, not to mention the clouds.
However, you can easily see the lunar analemma by using one of the many popular
computer planetarium programs. Just choose the local horizon view, fix the moon
in place, set paths to be shown, and make steps of 24.841202 hours. You
probably don't need more than six digits of accuracy, but five are not enough.
Unlike the subtle changes from year to year in the shape of the solar analemma,
the shape of the lunar analemma changes markedly from month to month. Just
compare the lunar analemmas beginning on 2001 July 3, 2002 July 2, and 2003
November 20. To see the lunar analemma change in size, compare starting on 2005
September 25 with starting 1996 November 5.
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Lunar analemma:
http://www.geocities.com/macsida/sundial_lunar_calendar.html
home page: http://home.europa.com/~telscope/binotele.htm