4. Time  The Equation of Time 
From the Astronomical League "Astronomy Before the Telescope" observing program:
4  Two of the above activities have made use of the equation of time but how is it determined? It is one of the things done in the Astronomical League’s Analemma Program. To see how it is done find a telephone pole, street light, other tall straight object, projection on your house, or drive a pole into the ground. You just found a gnomon. Go out weekly for a month or two at exactly the same time of day (Standard Time) and note where the sun casts the shadow of the tip of your gnomon. If possible place a mark for reference so you can see all your points at once. You’ll find the shadow will move the most between observations if you are working in the spring or fall. Graph your results with days (x) and the shadow’s distance from your gnomon (y). Do it for a year and you will get a graph that resembles the picture. Consider following the better instructions on the Astronomical League’s Analemma page and completing the program there.
Here are some photos of my analemma constructed during the month of May 2019. First light marked on Sunday May 26. Gnomon altitude is appproximately 18 cm, maximum Winter shadow calculated to 35 cm at my 40^{o }home latitude. (note 6/9/20: the measured value turned out 34.65 cm at Winter Solstice) Board is 45 cm in length, should accomodate Winter maximum. Marks will be made weekly using white acrylic paint artist marking pen. Weekly readings are to be made on Sunday, weather permitting. Cloudy Sundays readings will be made on the next subsequent sunny day. Reading dates are recorded in a separate journal. Analemma will be covered between readings to protect it from the elements.
Location details: 39.66248 latitude, 104.80444 longitude, 1718 m./5636.5' elevation.
I used a house address number plaque that I had laying around unused. I sanded and coated it with clear acrylic spray. I constructed the gnomon from brass that I use to make my miniature telescopes. I polished the brass parts and clear coated them with brass lacquer.
The completed analemma is mounted to a 4"x4" wooden fencepost that is set in concrete. It is located in a my back yard in a spot were there are no obstructions more than 10^{o} above the horizon at the meridian. The analemma board is attached with two 4" iron angle brackets. It was carefully levelled with a bullseye level. It was conditionally aligned to a northsouth axis using a magnetic compass, then finely aligned at local solar transit (12:56 PM MDT on May 25 at 105^{o} longitude). (note 6/9/20: Solar alignment turned out to be off by only 0.25^{o} based on reading at Winter Solstice.)
Secured to the fencepost with two steel 4" right angle brackets 
Eight weeks into marking one year, July 14 2019. The left side dip in the curve is the Summer Solstice. I see that my NS centerline is a bit off axis, hopefully it will still capture the complete analemma. 
September 13, 2016  Week 16
Waiting for Civil Noon on March 1, 2020  Day 281
I made procedural changes at the time of the Winter Solstice (the top of the plot curve, to the right in this image). The long length of the Winter shadow made the projected crosshairs more fuzzy and indistinct. They were also harder to pick out against the wood background.
I solved these issue by taping a small rectangle of paper to the board in the area of the anticipated path. I abandoned the white acrylic marker in favor of a pushpin. The marker was making a nice visible plot, but the marks were misshapen and imprecise. I thought the acrylic would be water resistant, but you can see where the marks smeared in the lower right area. Even though I kept the analemma covered between observations, I was concerned that a bout of bad weather could potentially erase months of data.
The pushpin is easy to precisely center and leaves a clean point. While not quite as visible, the pushpin plots showed up quite well when I scanned the board. Additionally, the marks are quite permanent.
May 27, 2020  Gathering the Data and Crunching the Numbers
On day 367 I made mark #64, the final mark on the analemma. This completes one full year of observation. Next I had to locate the point on the plane of the analemma directly below the opening in the gnomon. I did this by suspending a small ball chain through the opening (oculus) and marking the point on the board directly below.
After marking the point, I suspended a length of stiff steel wire down through the gnomon opening until it touched the point just marked on the board. I then marked the wire where it passed through the gnomon, and measured the distance as 183 mm. This is the value H that will be used in subsequent calculation.
In order to gather precise data from the analemma, I removed it from the fencepost and removed the gnomon from the board. I was then able to image the board with a flatbed scanner (Epson NX300, nothing fancy). The analemma board is large enough that I had to make two scans and merge them in Photoshop. The pieces of diagonally oriented graph paper help with the alignment of the two images. A metric ruler was scanned alongside the analemma to provide scale.

The Astronomical League Analemma program recommends digitizing the image by either scanning or digital photography. I loaded the scanned image into a scaled vector graphics (SVG) editing program. I'm too poor to afford Adobe Illustrator, so I used an open source freeware editing program called Inkscape.
The first step in Inkscape was alignment and centering. I created a centerline starting at the Winter Solstice mark and extending to the point directly below the gnomon oculus. I then was able rotate the image so the centerline was parallel to the background coordinate grid. This only required a 0.25^{o} clockwise rotation.
Next I moved the image so that the gnomon oculus point was directly over the (0,0) point on the x,y scale of the document grid.
Then I overlaid a transparent layer and marked the 64 observation points plus the gnomon origin point. Next I created additional layers to number the points for later reference. I also added a 10 cm grid for visual perspective. In the image below, the original scanned image is still there in the background layer, but it has been rendered invisible for the sake of clarity.
The position of the plot points were adjusted so that the lower left corner of the plot point bounding box was centered over the mark on the board. This corner of the bounding box determines the coordinates of plot point. These x,y coordinates are displayed on the tool bar. The result is that on the visual representation, all of the plot points are slightly offset up and to the right. You can see this by zooming in on the plots for the gnomon point and the Winter Solstice. Visually, these plots appear to be marked to the right of the centerline. This offset has no effect on the subsequent calculations.
Next I generated a Microsoft XL spreadsheet to record and tabulate my data. A representation of this spreadsheet can be viewed here. (Opens in a new window. This will enable you to toggle easily between the data and the explanations.)
To convert the graphic physical points on the analemma to a digital format and database, I had to individually record the x,y coordinates of all 64 observations. I did this by selecting each data point as shown above, then recording the toolbar coordinates to the spreadsheet.
In spite of all my tinkering with Inkscape, I could not get the coordinates to display in anything other than pixel scale. I used the scanned scale reference ruler to calculate a conversion factor. I constructed a line (green, above) superimposed on the ruler. Zooming in I was able to set the endpoints precisely at 0 and 22 cm. This was the greatest whole digit span in the image. The toolbar read the length of this line as 917.986 pixels. Dividing 917.986 pixels by 22 centimeters you get 41.727 pixels per centimeter, or 4.1727 pixels per millimeter. This is used for a conversion factor. Columns F and G of the spreadsheet are the pixel values of the x,y coordinates; Columns H and I convert these values to millimeters. This conversion is necessary for subsequent calculations.
"Graph
your results with days (x) and the shadow’s
distance from your gnomon (y). Do it for a year and you will get a graph that resembles the picture." 
I set up column J of the spreadsheet to calculate the distance of each plot point from the gnomon point ( 0, 0 ). I did this by entering the formula =SQRT(POWER(H9,2)+POWER(I9,2)) in the spreadsheet formula bar. This is the pythagorean theorem for calculating the hypoteneuse of a right triangle translated in Microsoft XL. By dragging this formula down the length of column J, it automatically calculates the distance from the gnomon for each data point.
I used the XL spreadsheet Chart Wizard feature to generate the graph above. Plotting the distance of the data point from the gnomon (y axis) against the date (x axis) did not generate the graph expected by the activity.
Again using the Chart Wizard, I generated a plot of date (x axis) versus height of the datapoint above or below the centerline (y axis)column H of the spreadsheet. The centerline is based on a line drawn from point P (0,0,H) to the mark at the Winter Solstice. This graph approximates the expected plot. The curve of both plots cross the centerline at the Solstices. The maximum displacement (expressed in centimeters) occurs either side of the Winter Solstice. Smaller peaks occur either side of the Summer Solstice. This displacement represents the sun time such as you would read on a sundial running fast or slow. In a following exercise, that displacement will be translated into minutes and seconds. With this information, a fairly accurate correction factor for sundials is obtained. The reason for the displacement is the fact that the Earth circles the Sun in an elliptical orbit, not a perfect circle. This will be covered in more detail below in activities #3 and #4 of the AL Analemma observing program.
This completes the activity requirement for Astronomy Before the Telescope section 4, The Equation of Time. I shall now continue using these data to complete the required activities for section 11, measuring solstices.
"Measure the Sun’s altitude as it crosses the meridian on the solstices using a gnomon."
The table below shows my Summer Solstice observation results calculated in the spreadsheet. Note that no observation was made on the Solstice date due to cloudy weather. The solar altitude is changing very slowly from day to day at the time of either solstice. The observations I made several days before and after the solstice show a fairly accurate representation of the actual Summer Solstice elevation. The value predicted by Stellarium is 73.8^{o}.
Date  Mark#  Day  Remarks  Elevation 
6/16/2019  4  22  73.66^{o}  
6/17/2019  23  
6/18/2019  24  
6/19/2019  25  
6/20/2019  26  
6/21/2019  27  Summer Solstice  
6/22/2019  28  
6/23/2019  29  
6/24/2019  5  30  73.75^{o} 
The weather cooperated enough for me to make a reading on the day of the Winter Solstice. This is in within one degree of the Stellarium prediction of 26.9^{o}.
12/21/2019  33  210  Winter Solstice  27.84^{o} 
This completes the activity requirement for Astronomy Before the Telescope section 11, measuring solstices. I will now use the above data to complete the calculations required by the Astronomical League Analemma observing program.
The Astronomical League Analemma Observing Program
Here is an outline of the four required activities:
(6/10/20) Progress on these calculations will be posted as completed.
Activity #1  Calculating the tilt of the Earth's Axis and your observing latitude
With reference only to your analemma and measured dimensions of your observing apparatus, calculate (1) the tilt of the Earth’s axis off the normal (i.e., the perpendicular vector) to its orbital plane, and (2) your observing latitude.
I followed the eight step procedure outlined in the Analemma Program Appendix F to perform these calculations.
Step 1  setup a coordinate system
with:
 P(0,0,0) at the opening of the enclosure / tip of the gnomon.
 the xaxis as east / west (positive being eastward).
 the yaxis as north / south (positive being northward).
 the zaxis as up / down (positive being upward).
This has
already been accomplished as described in the
above section.
Step 2  Locate the point both directly below the opening in the enclosure and in the plane containing the analemma. This also has been accomplished as described above.
Step 3  Make the following three measurements from P(0,0,h) (determined in Step1 above):
Distance from the opening in the enclosure to the plane in which the analemma lies; in Appendix A, this distance is referred to as “h” This has already been measured as 183mm as described above.
Distance to the Summer Solstice point. This is the point on the analemma curve closest to the xaxis This value is calculated in the spreadsheet cell (column: J, row: day 30), and that value is 53.35mm. Note that this data is actually 3 days past Summer Solstice. Unfavorable weather prevented a reading to be made on the solstice, so I used the closest reading.
Distance to the Winter Solstice point. This is the point on the analemma curve farthest from the xaxis. Once again from the spreadsheet (column: J, row: day 210), this value is calculated as 346.51mm.
Step 4  Calculating the altitude of the Sun at the Summer and Winter Solstices. This value was calculated by inserting the recommended formula (AltSolstice) = arctan(h / (DistanceToSolsticePoint) into column K (Solar Altitude) of the spreadsheet. The calculated values are: Summer Solstice  73.75^{o}, Winter Solstice  27.84^{o}. This compares to the Stellarium generated values of: Summer Solstice  73.38^{o}, Winter Solstice  26.94^{o}
Step 5  Specify the relationships between the altitude of the Sun at Summer Solstice, the tilt of the Earth’s axis, and the Observer’s Latitude.
( Observed Solar Altitude at Summer Solstice) + (Observer's Latitude) – (Tilt Of Earth's Axis) = 90°
The the diagram above and formula are from appendix F of the observing program. I could have simply inserted my observed values of the solar soltice altitudes into the equations provided in step 4 and 5 and come up with values for the observer's latitude and the tilt of the Earth's axis. I wanted to go beyond plugging values into an equation. I was having difficulty understanding how the above relationship formula was derived from the diagram. I was finally able to visualize the relationship by constructing a crude model and my own diagrams to work through the geometry.
The observer is standing at an unknown latitude in the northern hemisphere at Summer Solstice. The tilt of the Earth's axis is also unknown. The yellow stick represents the observer's measured angle of of the solar altitude (angle) above his horizon.
A new red line is constructed parallel to the observer's horizon while intersecting the center of the Earth. A new yellow line is constucted parallel to the observed altitude angle and extended to the Earth center.
From this visualization I was able to construct a set of diagrams. I added green lines to represent the Earth's equator and axis of rotation, and a green circle representing the observer's meridian.
The observed solar altitude above the horizon is represented by the yellow
segment.
It is the angle between the incoming sunlight and the equator.
The tilt of the Earth's axis at Summer Solstice is unknown, shown here
as the
green angle segment between the incoming sunlight and the equator.
The observer's latitude (unknown) is the red segment, the angle between
the observer's zenith and the equator.
The quadrant between the equator and the observer's zenith is 90^{o} by definition.
If the angle of the tilt of the Earth's axis is subtracted from the observed
solar altitude,
the remaining segment between the equator and zenith is equal to the
red segment, or observer's latitude. Therefore:
(Summer Observerved Solar Altitude) + (Observers Latitude)  (Tilt of Earth's Axis) = 90^{o}
This equation describes the relationship
between the altitude of the Sun at Summer
Solstice,
the tilt of the Earth’s axis, and the Observer’s Latitude.
Step 6  Specify the relationships between the altitude of the Sun at Winter Solstice, the tilt of the Earth’s axis, and the Observer’s Latitude.
Here are the diagrams which I constructed to describe the relationship of the observed altitude of the Sun at Winter Solstice, the tilt of the Earth's axis, and the observer's latitude.
The observed altitude diagram shows the angle (yellow segment) between the ecliptic (yellow line, incoming sunlight) and the observer's horizon.
The observers latitude (an unknown) is the red segment, the angle between the observer's zenith and the equator
The tile of the Earth's axis (an unknown) is the green segment, the angle between the equator and the ecliptic.
The fourth diagram illustrates that the sum of the three angles is equal to 90^{o}, the angle between the Earth's equator and the observer's zenith.
This relationship may be expressed by the equation:
(Observer's Latitude) + (Tilt of Earth's Axis) + (Observed Solar Altitude) = 90^{o}
^{ }
^{Step 7  Specify the Observing Latitude by adding the equations from Step 4 and Step 5, above: }
Adding the two relationship equations from steps 6 and 7
will remove the unknown variable
for the Tilt of Earth's Axis and allow a solution for the Observer's Latitude:
(Summer Observerved Solar Altitude) +
(Observers Latitude)  (Tilt of Earth's Axis) = 90^{o}
+
(Winter Observerved Solar
Altitude) + (Observers Latitude) + (Tilt of Earth's Axis) = 90^{o}

=
(Summer Observerved Solar Altitude) + (Winter Observerved Solar Altitude) + 2X(Observers Latitude) = 2X(90^{o})
Inserting values for observed altitudes and rearranging and solving for Observer's Latitude:
(73.75^{o}) + (27.84^{o}) + 2X(Observer's Latitude) = 2X(90^{o})
2X(Observer's Latitude) = 2X(90^{o})  (73.75^{o})  (27.84^{o})
2X(Observer's Latitude) = (180^{o})  (73.75^{o})  (27.84^{o})
2X(Observer's Latitude) = (78.41^{o})
Observer's Latitude = (78.41^{o}) / 2
Observer's Latitude = 39.205^{o}
Observer's Actual Latitude per Google Maps = 39.662^{o }an error of 0.457^{o}
Step 8  Specify the Tilt of the Earth’s Axis by subtracting the equation from Step 4, above, from the equation from Step 5, above:
Subtracting the relationship equation of Step 4 from the
equation of Step 5 will remove
unknown variable for the Observer's Latitude and allow solving for the Tilt of
the Earth's Axis
Step 5:
( Observed Solar Altitude at Summer Solstice) + (Observer's Latitude)  (Tilt Of
Earth's Axis)
= 90°

Step 4: (Observed Solar Altitude at Winter Solstice) + (Observer's Latitude) + (Tilt of Earth's Axis) = 90^{o}

=
(Obs. Solar Altitude at Winter Solstice)  (Obs. Solar Altitude at Summer Solstice)  2X (Tilt of Axis) = 0^{o}
Inserting values for observed altitudes and rearranging and solving for Tilt of Earth's Axis:
(27.84)  (73.75)  2X(Tilt of Axis) = 0^{o}
2X(Tilt of Axis) = 73.75^{o}  27.84^{o}
Tilt of Axis = (73.75^{o}  27.84^{o}) / 2
Tilt of Axis = 45.91^{o} / 2
Tilt of Axis = 22.95^{o}
Accepted value of current tilt = 23.43^{o}, an error of 0.480^{o}
The accepted value of the tilt changes slowly over time due to the obliquity of the ecliptic
Activity #2: with reference only to your analemma and measured dimensions of your observing apparatus, calculate the Sun’s path in the sky and produce a sketch or plot to depict that path.
Activity #3: with reference only to your analemma and measured dimensions of your observing apparatus, calculate the Equation of Time curve across the year.
This activity was duplicated in the Astronomy Before the Telescope  Equation of Time activity above, bookmarked:
Activity #4: with reference only to your analemma and measured dimensions of your observing apparatus, calculate the eccentricity of the Earth’s orbit