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Sun and Moon rise/set dial and development of second-order Moon anomaly function - November 2016 Representing accurately the motion of the Moon as it appears in the sky in a mechanical fashion upon a dial is a very difficult effort. The combined gravitational influence of the Sun and Earth cause many perturbations. Together these are known as anomalies and over time makes the accumulated orbital positions of the Moon look more like a toroid than a circle or ellipse. In practice there are five major anomalies associated with the Moon's orbital movement (see illustrations below). And there are two anomalies associated with the Earth's tilt and orbit. In theory there are dozens but the main seven will account for 98+% of these. The five anomalies are arranged in their order of the greatest to least orbital perturbations. The degree is the change in measurement from an idealized orbit. The first five corrections will give the degree the moon leads or lags from the mean or average position of the moon in its orbit.
However to truly show when the Moon will rise and set two additional corrections are needed, these are associated with the Earth and its orbit, and to a very minor extent the Sun itself.
Moon's accrued orbital anomaly - one year Moon's accrued orbital anomaly - five years
Moon's accrued orbital anomaly - ten years Moon's accrued orbital anomaly - sixty years These four illustrations depict the Moon's orbit around the Earth for one, five, ten and sixty years respectively. The last illustration shows a rather thick torus, around the Earth. One can readily see why it is so difficult to accurately describe the Moon's orbit! We will correct only for the two greatest anomalies. These are the Great Anomaly at ±.6.58° equaling 26.322 minutes and the Projection (which contains two corrections) at ±2.464° which translates into about ±9.857 minutes in time. So the total maximum error involved is 36.179 minutes. All of the remaining anomalies amount to a combined 9.442 minutes. It is no accident that the dials on the Schwilgué and Festo clocks are large to take advantage of this information. The combined total of all anomalies comes to ±45.621 minutes from a simple rolling moon dial.What has been discussed so far, however concerns the movement of the moon only, an astronomical discussion. We still need an additional correction which is far greater in magnitude that those astronomical anomalies to get a respectable representation of the moon's movement on a dial. These are terrestrial corrections that account for the Earth's tilt to the ecliptic as well as its elliptical orbit around the sun. The cam work needed is latitude-dependent; unlike the equation of time cam which is not. Both draw upon the same characteristics of the earth's tilt and orbit, but the equation mechanism relates to the position of the sun when at its zenith as it relates to an observer on the earth compared to the local time at noon on a clock; the horizon cams relate to when the sun, or in this case the moon appears above or disappears below the earth's horizon in concert to the seasonal variations, hence the need to calibrate the cams for latitude. The total difference in time is 6 hours 12 minutes from the shortest to the longest day. Using the prior two moon anomalies as well as the horizon cam work we have a total of four corrections which will allow for one to see on the dial the position of the moon or sun correspond to what one sees in the sky at the latitude of Chicago, Illinois at 41.88 degrees N. Our Moon rise and set dial is fairly small at just over 3½" or 10cm in diameter so any further corrections would be barely noticeable at this scale. Even thirty six minutes for the two moon anomalies will be hard to discern. This is really more of an exercise to incorporate a classic complication rather than important additional accuracy in the dial. Very few mechanical clocks have incorporated all nine corrections for these anomalies to allow for an accurate representation of the movement of the Moon. A few examples are Jean-Baptiste Schwilgué in his famous astronomical cathedral clock in Strasbourg, France, 1838-1843; Jens Olsen, Copenhagen, Denmark 1945-1955; Rasmus Sørnes, Moss, Norway, 1958-1966; Hans Lang, Essen, Germany, 1982-1986; Festo Corporation, Esslingen, Germany, 1995-2001 (this movement employs ten corrections). There are surely many others I have overlooked, but one interesting item that pops out is the fact that other than Schwigué's all are of recent design and fabrication; all less than 62 years old. This is probably due to the complexity of calculations and fabrication needed to make these corrections. We looked to the few clocks which have these features for guidance. Only the Sornes clock was produced by a single maker and this is also the only clock that is small enough to fit into a domestic setting. The others were very large, institutionally-sized clocks and were the collaboration of many people and the Festo the product of a multi-national corporation. These three were the ones for which there was some documentation of their design and construction. The Schwilgué offered the most information and that was found in both the book written on the clock by Alfred Ungerer, the son of Schwilgué's collaborator on the clock and whose company was the successor to the Schwilgue firm in his book written in 1922, L'Horloge Astromique de Strassbourg, as well as the abstract from that book as represented in Some outstanding Clocks Over Seven hundred Years, H. Alan Lloyd. Other useful information was also contained in Jens Olsen Clock, Otto Mortensen. In all of these examples a cam stack in conjunction with Janvier's variable differentials were used to translate the many difficult calculations involved with the moon's complex motion into a mechanical representation that could be displayed on a clock dial. In Sornes clock he divided a few of the functions between several dials and so it was not as useful to our project as was the Schwilgué clock since here the clock was designed to show the rise and set of the moon as it would appear to a person located in Strasbourg. The Festo clock also accomplishes this along with a few additional very fine adjustments that are far beyond the scope of this project.
This diagram is a from Jens Olsen's design papers depicting in graphical form the five anomalies of the moon's orbit as outlined above. These can be directly translated, when curved into a circle, creating the topological surface of each cylindrical cam needed to depict these mechanically. When one tries to research the characteristics of these anomalies on the internet or other modern sources, there is no information as to how these orbital fluctuations are translated into a mechanical form; only the technical characteristic as explained in astronomical terms. We have chosen the two largest terms and when compared with the remaining three are by far the greatest anomalies. The other factor to consider is that the dial diameter in Schwilgué's clock is over five feet. So all of these factors are far more consequential than in our dial which is only about 3 .5", or 10 cm.
These diagrams are from Ungerer's book. The first depicts the complete set of cam stacks. The center one is for the moon's five orbital anomalies for the moon hand, the left two cam stack is for the solar equation for the sun hand and right single cam is for the moon nodes for the eclipse indicator. This arrangement shows the entire set driven from a common 24 hour source. The output of the solar and node stacks are combined into one output since the nodes are reduced to the plane of the ecliptic for dial display purposes and connected to the solar hand. The center stack connects to a separate moon output. Both outputs are driven via the pinions between them. The second diagram shows a detail of the center five moon anomaly cam stack. Each cam is independently driven through the wheel set located below to give the appropriate cycle for each as outlined in the graph above and all cam movements are then combined to create the movement in the connecting rod above the stack.
A photo of the three cam stacks used in Schwilgué's clock.
These diagrams show the moon outputs depicting a front and side elevation of each. These are based on Antide Janvier's design for a variable differential. The first is connected to the combined solar and node stacks and are used in the prediction of solar eclipses. The second is connected to the center moon cam stack and is used to show the daily motion of the moon and is controlled by the perturbations of the moon cam stack. This is also known as the Projection variable differential in this project.
This is a photo of the upper differntial wheel set to the Moon on Schwilgué's astronomical cathedral clock in Strasbourg. The two silver horizontal structures in the back and foreground is the wheel cage. Within the cage is one slant wheel differential combined with a second pair of interactive differentials in the middle portion of the device. Note the sector gear just to the left of the center nested wheel set and just before the last two large output wheels to the left. This is connected to the output of the moon anomaly five cam stack through the center of the three steel rods seen in the background behind the main slant wheel and introduces the complex perturbations to the daily motion of the moon output as represented by this variable differential. 
 The first photo is another view of the output for the daily motion of the moon it is located above the node output set of variable differentials. Notice the center arbor driven by the contrate wheel with four pinions that drive both the upper moon and lower solar and node differentials. The second photo shows the three rods from the cam stacks. The center rod is connected to a sickle-shaped lever which is in turn connected on the same arbor as the sector gear located within the center of the upper differential assembly. This carries the combined perturbations from the center five cam stack. The small weight is also connected to this to give a bias to the sector gear. The other two rods are each connected to one of two long horizontal frames that straddle the lower differential wheel set. Each frame holds an arbor with two pinions, each connecting with a different wheel within the differential. These can also be seen in the lower background in the photo to the left. Both frames are independently movable in a small arc about the differential's central axis and introduce the solar and node cam's perturbations into the differential. This mechanical unit displays a constellation of complex and visually stunning mechanical achievements a tour de force in wheel work design. We owe much to the masters of the past. We now move to the application of this knowledge to the sun/moon rise, sunset module.
These four photos show the first concrete example of what Buchanan is proposing for the Moon corrections. They show an alternative way of applying a differential gear system and are based on Antide Janvier's method to simulate the erratic motions of celestial bodies and that were later employed by Jean-Baptiste Schwilgué. I have always been fascinated by this variable differential gear arrangement and wanted to have it incorporated into the panoply of mechanical contrivances within the machine. The first two photos show the slant wheel in its two extreme positions. In this model Buchanan uses the 23.50 tilt of the earth’s axis, mimicked by the slant wheel to correct for that anomaly in the Projection. The next two photos show the same wheel set with a pair of dials denoted in degrees mounted to the slant wheel as well as the input wheel. In this way one can see how the slant wheel moves in relation to the input wheel. Since we did a Breguet style differential for the equation work I suggested we use this design for the Moon’s motion.
That relationship as represented by the Great Anomaly is plotted on two graphs. The first one above shows the differential changes as a change in radial degrees over one complete rotation. It plots the difference between the pointer on the drive wheel which turns at a constant rate and the readout of the slant wheel, Buchanan calls this the “error from the true pointer”. That true pointer is the one mounted to the drive wheel as seen in the fourth photo above of the differential mockup. The second graph is the same information, but plotted as the mathematical function. This is not unlike the graphs one would get plotting the equation of time which is the difference between the mean solar time, or the time we read off our clocks, and the position of the Sun in the sky. The physical cam used to represent this function in a clock has a similar kidney shape as well as the graphical illustration that looks like a sine wave for the mathematical function. This similarity to the equation of time function is no accident as the Earth’s tilt is one of the two components used to determine the equation of time, the second is the eccentricity of its orbit, which is also contained within the fourth anomaly known as the Great Anomaly, 2nd term.
Illustrated above is the initial design layout for the Moon complication. Here we have made manifest the two largest anomalies represented by the two differential wheel cages 3 and 4. These represent the Projection and the second the Great Anomaly, the acceleration and deceleration as it approaches and move away from the Sun. Note that the angle in the first differential is in error at 9.29 ° but should be 6.29°.The first differential represented by box ‘4’ supplies its output to the next differential, box ‘3’. These all rotate along with the Moon once every 29.53 days. Each differential will be approximately 3 inches or 7.5 cm in diameter.
This drawing shows one
of the two anomaly correction, slant wheel differentials. The two slightly lenticular, biconvex circles on the front
elevation section of the drawing, left, are showing the two wheels that are
slanted in relation to that elevation and the rest of the wheel work.
In this photo are shown
the areas where the wheel assemblies will be located. The twin anomaly
differentials are anticipated to each be just over 3” in diameter and 2 ¼”
long (7.5 cm, 6 cm). The pair with their associated drive gearing will be 5”
(12.5 cm) wide. This will nicely balance the density seen in the calendar
mechanism on the left hand side of the machine.
The first drawing
depicts the gear trains for the moon’s phases as well as its cycling
around the dial. Next is an overlay with paper disks that depict the wheel
train controlling the sun rise and set shutters.
Here we have a completed drawing with both the moon and sun gearing
represented respectively drive wheels all revolve around the dial's axis
according to the input of the two anomaly differentials. Their input is the
first small red wheel at 12 o’clock
(1). That wheel is fixed and moves the second large red wheel
(2),
upon which is mounted a nested pair of wheels
(3, 4),
which are fixed together. The largest of the pair
(3), meshes with the center
wheel (5), which is fixed. The
smaller of the pair (4), meshes
with wheel (6) and then to
(7) and
(8). Wheel
(8) delivers rotation to the
Moon globe through a pair of bevel wheels. The rotation of the input wheel feed from the anomaly differentials, by rotating the large red wheel upon which the rest of the moon mechanism is mounted induces rotation of the nested pair of wheels engaged with the fixed center wheel; thus causing rotation throughout the rest of the wheel train through to the moon globe. These gears are represented by the calculator shown above for a four
stage wheel set to obtain the accuracy of the moon’s rotation to seven
decimal places. The other thing the calculator does besides giving
extraordinary accurate results is that it allows Buchanan to perform a trial
and error exercise with various numbers and tooth counts almost instantly.
Thus he could try out a five stage to see how it might look vs. a four but
still retaining the correct results. It not only saves time but allows for a
much better design both mathematically as well as esthetically. The sun itself revolves around the dial once per day and is mounted to
the center green wheel (a). That
wheel drives the next four wheels
(b, c, d, e) which then turn a pair of cams
(f, f’). These cams each in turn
have roller follower arms that rotate upon their edges and are attached to
set of sector gears. Those in turn control the two shutters for the sun rise
and set horizon. Only one set of sector gears is shown as they are
superimposed upon each other in this view.
Front and side elevation of the Sun and Moon rise/set dial
This is how the pair of
differentials will fill out behind the dial work. They are drawn to the same
scale as the dial work, These will reach almost to the first strike hammer
mechanism as shown by the diagramed box in a prior photo above.
Buchanan sent the
drawing I requested to see how far the slant wheels as designed would
project beyond the dials. In this front elevation the anomaly differential is
about 2” in diameter. We later decided that it would be esthetically more
advantageous to expand that wheel to 3” (7.75 cm). This will allow more
space to be taken up by this device, yes there still is a bit of limited
real estate left in this machine! But more importantly, the larger wheels
combined with a delicately thin cage design will make the entire pair of
differential wheel presentation look more delicate and appear to ‘float’ as
they rotate within their turning cages.
A schematic showing the full assemblage for the Sun and Moon rise/set dial complication. Note how close the actual accuracy of each system comes to the ideal target.
The proposed design for the Sun and Moon rise/set dial. This is a good illustration of one of the design principals behind this project. We are trying to pack as much information, and by analogy, complications, into the fewest number of dials; thus allowing the viewer to see the beauty of the mechanics that create the information. It would be easy as many past clockmakers have done, to create a dial for each complication in an effort to "show off" the many dials that represent complications. The result, in my opinion is a cluttered look or perhaps looking like the cockpit of a large jet liner, distracting from my main objective which is the machine. The following information and thus complications can be read from this dial and thus add 15 complications to the astronomical clock:
We also have a pair of
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