Universal Sundials
This part is examples of sundials. Each example consists geometrical explanation, calculation and use of my open-source software developed for design of sundials. You can download them from GitHub.
https://github.com/AminKH/SunDials
In this section universal sundials are discussed. These sundials regarding, designed for solar time, can be installed any place , only their plane should be adjusted to proper angle.
Equatorial Sundial
Dial plate of this sundial as it is implied by its name, is parallel to the Earth equatorial plane. Gnomon is a rod normal to dial face. Picture below shows an equatorial sundial and various planes, including equatorial plane, at earth surface.
The sundial in the picture is at this place: https://www.atlasobscura.com/places/cranmer-park-sundial
This kind of sundial, as Kehui Deng claims, had been in use in ancient China. Karlheinz Schaldach from remains of probably a sundial at temple in Greece, 330 BC, concludes that, it is an equatorial sundial and may be copied from and Egyptian one. This ancient sundial is also investigated by Irina Tupikova & Michael Soffel , in paper “Modelling sundials: ancient and modern errors” and they concluded that sundial has errors, and it is not designed for that specific location. Suppose Sundial is located at Earth center or one Earth poles. Here I have copied Equatorial plate in equatorial coordinates from blog sundial basics.
At equinoxes sun shines directly to the earth, therefore sunrays are parallel to equatorial planes. We can create a model of movements of the sun. We take sun daily moves on the sky with following hour angles.
And sun yearly location, because of obliquity of ecliptic, as this, table is in degrees:
|
23.44 |
20.15 |
11.47 |
0.0 |
-11.47 |
-20.15 |
-23.44 |
To make a model, I used OpenSCAD. OpenSCAD is an open-source software which can be downloaded from their website. https://openscad.org/downloads.html
Model of the sun is a sphere with radius of 0.25 is chosen. The angle we see the sun from earth is about half a degree. We assume the sun and the earth distance as 90.0, because we measure elevation of the sun in degrees. For sunrays we take cylinder to represent them. For equatorial sundial we must consider two side of dial for summer and winter. Picture of sunrays and shadow lines at equinoxes are as in the picture below:
Top View from North Pole East-west view
Because, at sunrays are parallel to equatorial plane, therefore shadow of gnomon tip also will be parallel to plane and shadow length of vertical gnomon became very long. We can do the same for summer and winter solstice. This time shadow length is much shorter, due angle of sunrays with equatorial plane.
View from South pole, winter solstice Side view, South-West
The shadow lines directions do not change from equinox to solstice, only their length varies due to angle of the sun with equatorial plane. We can conclude that shadow lines are in direction of sunrays, so shadow lines angles are equal to hour angles, and the shadow path at each day is a circular arc with the gnomon base as center of arc. We can find length of shadow, Ls, or radius of circle by trigonometric equations from triangle seen from side view at noon time.
Ha is hour angle of the sun. This is a model at equatorial coordinates, we can make a model for horizontal coordinates. For latter model, equatorial plane has angle of colatitude with horizon, and we must use elevation and azimuth of the sun instead of hour angle and declination. Both models are available from here. In equations of shadow length and hour angle there is any argument from geographical locations such as latitude, that is why equatorial sundial is considered as a universal sundial if they are designed for solar times. It is only necessary to align dial plate with polar axis, that is dial plate has an angle with horizon equal to colatitude of location.
In Excel worksheet Sundials_English we enter for following location at sheet “Enter”.
Then at sheet “HourLines” at table under name of “shadow angles” enter 90 at colored cell. Shadow angles will be displayed at rows. You will notice, they are same as hour angles
At sheet “Enter” enter year, select calendar type, select Equatorial sundial, select solar time, and enter gnomon height, and then click button “Calculate Sundial”. Coordinates of x and y of shadow tip displayed at sheet “Equatorial” A simple graph in the sheet, which shows both face of sundial.
Now for city of Mashhad:
We for sundial with civic (Official time) we will have:
You will notice that shadow angles for civil time have different angles with shadow angles of solar time. Difference is 7.07 degrees. This is the difference of longitude of Mashhad and longitude of 52.5 degrees corresponding with time zone (15x3.5=52.5). Select civic time, click button “Calculate Sundial” and you will notice, from the graph, sundial is rotated, and 12 O’clock is afternoon in Mashhad, or noon time is before 12.
By rotating sundial to amount of difference of longitude of official time and longitude of location, time difference of local solar time and official time, is compensated. But still time difference of equation of time remains. To find how equation of time, affect hour lines, run program sundial.exe and enter appropriate answers to the questions, program asks:
Enter Name of Location: Mashhad
Enter Longitude in decimal: 59.5756721496582
Enter Latitude in decimal: 36.310432434082
Enter Altitude in decimal: 1065
Enter Time Zone in decimal: 3.5
Please Select Type of Calendar
Please type I or i for Iranian Calendar:
Please type G or g for Gregorian Calendar: g
Please Select Kind of Operation, or Enter for Sundials
Enter- Sundials or Sun graph
2- Official Time Sun Data
3- Equation of Time
4- Apparent Declination of Sun
5- Local Noon Time
Please Select:
Please Select Type of Planar Sundial or Sun graph
0 -For Sun Graph:
1- For Equatorial dial (Polar Gnomon):
2- For Polar dial (Polar Gnomon):
3- For Horizontal dial (Polar Gnomon)
4- For Vertical dial (Polar Gnomon)
5- For Plane Dial (Polar Gnomon)
11- For Bifilar Horizontal
12- For Bifilar Vertical
21- For Equatorial dial (Vector)
22- For Polar dial (Vector)
23- For Horizontal dial (Vector)
24- For Vertical dial (Vector)
25- For Plane General Gnomon dial (Vector)
31-For Armillary dial
Please Select: 21
Please Select Kind of Time
1- For Local Solar Time:
2- For Local Solar Time + Analemma:
3- For Solar Time considering Civil Time:
4- For Solar Time considering Civil Time + Analemma:
Please Select: 4
Please Select: 4
Enter Gnomon height, perpendicular distance of Gnomon tip and dial Plane
Please Enter: 10.0
Do You want to add Half an hour line:
1- Enter 1 To Add Half an Hour
For last question just enter because we do not want half hour lines. In type sundials there are two options for equatorial sundials, we selected option 21, to calculate by geometric algebra method and elevation and azimuth of the sun. Option 1 calculates coordinates by trigonometric method and declination of the sun and hour angles. Option 1 assumes constant declination for whole day, so lines are straighter, especially at early mornings and evenings.
Calculation results printed out at computer monitor and whole data will be saved at file with extension .ASC.
File: Mashhad__Equatorial_Vector_G10_Y2022.ASC Saved
1- To start from beginning:
2- To select different dial:
3- To quit
Type 3 to quit program.
FreeCAD is an open-source software for design, it can be downloaded from their website.
https://wiki.freecadweb.org/Download
Open FreeCAD, select Create New, from drop down menu select Draft. Click on Macro, and in drop down menu, click Macro ….
In opened window, click on square with …. On it.
In file explorer window, find location, which you have created, and sundial file are located. Files of Python script will be displayed, select FreeCADsunDial.py and click Execute. From windows explorer window find the file Mashhad__Equatorial_Vector_G10_Y2022.ASC, After a few seconds, design will be displayed.
This rough or first design of both faces of equatorial sundial. Black lines are solar time hour lines, which adjusted for location. Blue lines are Analemma, or location shadow tip regarding equation of time. The span of equation of time at south face is larger than north face. This span reaches more than thirty minutes at winter for northern hemisphere. Select day line (red line) and from Modification drop down menu, select Wire to B_Spline. Examine both lines, if transformation is perfect, then delete the first wire line. Repeat this for other lines. From Edit drop down menu, select Box Selection and select north face, upper part, the from File drop down menu, select Export, you can choose the format of your desired, CAD or graphic program. I choose SVG, export it as Mashhad_Equatorial_Vector_G10.svg.
After further editing of the SVG file in graphic software, such as open-source Inkscape, this will be the result for gnomon height of 10.0, for me, of course you may prefer different design
Tabriz is at west of meridian 52.5 degrees (Time Zone 3.5x15) with longitude of 46.2891502380371 degrees. The difference of longitudes is -6.21084762 degrees. We do same procedure for design of equatorial sundial, but this time in selection of sundial type, we select option 1. Noon time at Tabriz is after 12 O’ clock of civil time.
Above is both face of sundial for Tabriz
Armillary Sundials
This is another universal sundial. Armillary sundials are like equatorial sundial, but instead of two side plate, a cylindrical surface parallel to polar axis is used.
Sunrise and sunset happen when upper edge of sun is at horizon such as this photo of a few seconds after sunrise. Atmospheric refraction affect on sunrise and sunset maximum of one minute, but geographical terrain could delay sunrise for some minutes.
Hour angle of sunrise and sunset is
H0 is hour angle of sunrise and sunset, f is geographical latitude, and d is declination of sun. The sun is seen within angle of about 0.5 degree or 0o 16’. Therefore, sunrise or sunset is when sun center is 0.266 degree below horizon. Atmospheric refraction affects, sunrise and sunset time. Atmospheric refractions definition can be found form these sites: Wikipedia or Here. Atmospheric refraction cause, sunrise happens sooner than, if there were no air. To take this effect, value of -0.5666 degrees considered for to compensate it. These considerations are for horizon. Sunrays reach us after overcoming, obstacles, such as mountains as in above photo.
To make a model of armillary sundial, I select, Ai Khanoum. This location now is at Takhar province of Afghanistan and ancient like armillary sundial, 145 BCE or later, has been found there. Ai Khanoum which means “Lady Moon” has enigmatic history.
· Wester historian claim, this city had been built by Greek Alexander at 320 BCE and brought civilization to this area, or by Seleucus or his son Antiochus, between 300 to 285 BCE (Wikipedia).
· Rachel Mairs, University of Oxford, in her paper THE FOUNDER’S SHRINE AND THE FOUNDATION OF AI KHANOUM, claims that, based on archeological remains, such as Achaemenid place and temples which does not resemble, Greek temples, was Achaemenid city, which Greeks after occupations, changed building and added gymnasium.
The site location from Wikipedia:
We use solar elevations and azimuth of horizontal coordinates for equinox and solstices. A half cylinder with radius 110.0 and length 95.38 parallel to polar axis, in block with width of 334.0 and height of 220.
Next picture is dial block with sunrays and shadows at equinoxes. Blue lines are shadow lines. Sunrays and shadow lines are in radial form and normal to style. Shadow lines angles are equal to hour angles
Adding sunrays and shadow lines for summer and winter solstices, and looking direct to cylinder, (view) , we notice that for all days, shadows for each hour angle are in one direction and all coincide at tip of style.
Looking closely inside cylinder and shadows, we notice that shadow tips for any specific hour, in one year, move along a straight line. Length of these lines are:
R is cylinder radius, and e is ecliptic angle (23.44 degrees).
Shadow lines distance form noon time, on cylinder surface is:
These are for solar times. To compensate for time difference due to location and official time, like equatorial sundial, must rotated accordingly, either clockwise, or anti clockwise. To design with analemma for hour lines, run sunDila.exe
And enter location data, and in sundial selection, select 31 and enter radius of cylinder.
Armillary sundials can show equinox and solstices days. When shadow is under style tip, is equinox and when shadow reaches to edge of dial cylinder, is solstice.
There is another way to consider equation of time, in armillary sundials and that is to change style to shape of analemma, such as this sundial: https://www.helios-sonnenuhren.de/en/chronos
In 1970 archeologist in Ai Khanoum site among other artifacts, found two sundials. These sundials investigated by Rene Rohr and Denis Savoie. They conclude that there are errors in this sundial. This sundial is made of stone and its sizes are: 446 millimeters tall, 343-millimeter width and 150-millimeter depth. A cylinder of radius 110 millimeter carved in block. Lower section is beveled to make an angle 53 degrees with horizon. Inside cylinder there are hour lines for both side of block for summer and winter. There are thirteen-hour lines, which are not parallel, but diverge as sun moves from spring equinox toward summer solstice and converge from fall equinox toward winter solstice. At that time unlike today Egyptian divided day time to twelve equal sections. Now one hour is constant 3600 seconds, therefore daytime at summer solstice at north hemisphere has more hours than winter solstice. In ancient Egypt and probably Greece, twelve sections where constant and “hour” length was variable. There is a hole on top to place a metal rod. The block must be anchored from bottom or metal rod at top acts as support, to prevent collapsing of the block. There are two circular day line inside cylinder, which are about one third of depth, which can be correspond to solstice days at either side of block face. The distance should be about 47 to 48 millimeters. Unfortunately, no indication of how precise the distance are these lines from the face of block. From this evidence, authors conclude than, style attached to the metal rod, should be a polar one.
Ai Khanoum Sundial
Details of Hour lines and day lines, from D Savoie paper
To make a model of ancient sundial, daytime at summer
solstice divided to twelve parts, such as table below, and a model created for
these, azimuth, and elevation angles of the sun. Daytime at equinoxes is in
twelve hours. Side view picture of sundial model, and its cylinder front view,
shows that shadows equinox and solstice is not in one direction, because, hour
length is different. Noon line is common for both, but as time moves toward
sunrise or sunset, difference angle between them for one of twelve sections
increase. Ancient hour lines are not parallel with cylinder axis, but they an
oblique angle with it. This angle increases as time difference of noon and any
time increases.
view shows how shadows of equinoxes and solstices differ as time of day. Blue lines are shadows at equinox and pink lines are at summer solstice.
Rohr and Savoie in their papers claim that the hour lines do not match with times of Ai Khanoum, but a location with latitude of 23.5 degrees. At Ai Khanoum, sunrise at 4: 40 and sets at 19:23, that makes daytime length 14 hour and 43 minutes, whereas at location of 23.5 degrees, sunrise at 5:14 and sets at 18:49 which makes daytime of 13 hours and 35 minutes therefore, daytime at this location is one hour and half shorter than daytime of Ai Khanoum at summer solstice day. This sundial in unique and no similar one has not been found. It seems at that time, they did not have enough knowledge to calculate these angles, and these lines made with ratios of Egyptian astronomers found for the location of about 23.5 Degrees.
Polar Sundial
Polar planes are parallel with polar axis. These planes used extensively in solar heating and power generation because they absorb more power from sunrays than other orientations. Inclination of polar plane, that is their angle with horizon equals to geographical latitude. Same as other sundials, a made is made.
Consider a plane parallel to polar axis with gnomon height of h.
If we connect points of shadow on plane, we get a hyperbola for each day, except for equinoxes, which is a straight line.
By help of trigonometric, we can find x, and y of coordinates of shadow points.
You can find graphical representation of polar dials from here or here.You can find graphical representation of polar dials from here or here. Albrecht Dürer (1471-1528) Studied sundials.
http://www.albrechtdurerblog.com/melencolia-sundial-part-1/
Run the sundial_English.xlsm and enter geographical coordinates of any place and select, solar times, you will get this graph.
But if you select civil time, such as for Mashhad, this will be the graph
And for Tabriz:
Now if run sundial.exe and enter for Mashhad, selecting, either option 2 or 22 and selecting civil time with analemma, you will get:
You may notice, low angle of elevation of sun at sunrises and sunsets, intensifies, span of analemma due to equation of time.
You can download software and models from GitHub.
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