Sundials_Basics
This part is brief description of sundial basic for my open source software developed for design of sundials. You can download them from GitHub.
https://github.com/AminKH/SunDials
Sundials Basics
Sundials are instruments for measuring time, with shadow, cast by sunrays. Movement of the sun at the sky, cause to move shadow on dial’s face. Sundials are accurate, because of planet Earth movement are stable. We can design and build very accurate sundials, but we seldom use them. One reason may be is that sundials measure time locally.
Now time unit is second which is defined as 9192631770 number of amplitudes of Cesium 133 atom in stable state. Historically, second defined as 1/86400 of mean day. Today, precise time passage, independent of earth movements, kept by atomic clock in standard centers. International Earth Rotation and Reference system Center, measure earth rotation time, by keep watching far distant stars, which, their location, will not change because earth rotations. From time to time, they add one second to standard time. Because standard time and earth rotation time should not differ more than one second. Earth rotation time around itself, very slowly, because of gravitational force of moon and tides, gets slower. This standard time is called Universal coordinated Time. So, we may notice that sundials sometimes differ only one second per year with atomic clocks.
Other units of time are, minute, which is sixty seconds, and hour, which is sixty minutes or 3600 seconds. Sixty units system, remains from Sumerian civilization. If we consider, 360 degrees of arc minutes of one rotation of planet earth around itself, in 24 hours, then it takes one hour to rotate, 15 degrees of arc minute, which is equivalent to 4 minutes of time for one degree.
Now we measure our daily activities, with standard time. Before this concept of standard time for an area or country, each city has its own time, which was based on solar time. By advances in science and technology, transport became faster, and adjusting time for every city for trains, became a burden. At 1847 England, accepted a universal time based on time of location named Greenwich.
Planet Earth spherical shape is known to Iranian and Islamic world, and there are various measurements for its radius. One of prominent historical techniques was performed by Biruni. After Europeans find American continent the need for more sea travel and transport and advances in mathematics, the need for more accurate maps caused, Europeans to measure earth radius in different locations and noticed that Earth radius is not constant, and Earth is not a perfect sphere, but it is a little, elliptical. Picture below is a model of planet Earth. Meridian are ellipses, which are in any plane that passes through any place on Earth, such as, P, and Earth rotation axes NS. Meridian Ellipses specifications are:
Earth radius at Equator a = 6378137.0 m
Earth distance from center to poles b = 6356752.3141 m
Eccentricity = f = (a-b)/a = 0.0033528106818
Inverse of Eccentricity = 1/f = 298.257222101
First Eccentricity e2 = f(2-f)
Cross section of any plane perpendicular to Earth Axes, NS, is a circle, called parallel. Largest parallel is called equator. Closer we to poles, smaller is the diameter of parallel. If we draw two concentric circle, one with earth radius at equator and one with radius of meridian at poles, with one scale, their difference, as in next picture, will not be noticeable. Seventy one percent of earth surface is covered with water and rest is land, which also has, lakes, rivers. Ratio of highest point on planet, which is Everest to earth radius is 0.1378 percent and ratio of deepest point, Marian, is 0.171 percent. Which means that, we consider, earth as a rather smooth, sphere. In picture below, at point, P, meridian NPP”S make an angle , POP” with Equator . This angle is called geographical latitude, and usually, Greek letter f is used for it. Horizontal and vertical lines are shown. Plumb lines do not pass through the center of Earth, because of ellipsoidal shape of planet. This gravitational angle is called astronomical latitude and Greek letter F is used for it. These angles have differences and in astronomical calculation, this difference is considered. Formula of radius of curvature of any point on meridian, radius of parallels is:
Difference of geographical and astronomical latitudes at sea level:
And for an observer at elevation H:
Largest difference of geographical and astronomical latitude, occur at 45 degrees, which is equal to 11 arc minutes and 3.73 seconds.
Every point on Planet Earth can be depicted by two components. One is latitude, which already defined, as angle between parallel and equator. Equator is largest parallel and is known. Second component is called longitude, and it is an angle between meridian of location and a refence meridian. There is not any specific meridian such as equator. countries Before 1881, referred to their own reference meridian, which usually pass their capitals. In 1881, at an international conference at Washington, Greenwich meridian chosen to be the international refence meridian. Meridian east of this meridian, considered to be positive from zero to 180 and west of it, negative.
In every place, local coordinates are based on horizontal and vertical or plumb line. Angle of any celestial object with horizon is called Elevation angle and angle with North-South direction called Azimuth angle.
As already said, Earth rotates at pace of one hour for fifteen degrees. This means, at meridian fifteen degrees east of Greenwich sun rises and sets one hour earlier than Greenwich. Time zones are areas which are usually within meridians with fifteen degrees apart and their mean difference with Greenwich time is multiple of one hour. For example, Iran in time zone with 3.5 hours difference with Greenwich time. Greenwich time now is called. Universal Coordinated Time. Meridian for Time zone of 3.5 hours is 52.5 (3.5x15=52.5). So official time is based on this meridian. Locations east of this meridian, such as Mashhad at longitude of 59.57672 has longitudinal difference of 7 degrees and 4,5 arc minutes which is equivalent to 28 minutes and 18 seconds. Therefore, 12 O clock is afternoon at Mashhad. Instead, Tabriz with longitude of 46.28915 is at west of meridian 52.5 and has difference of 6 degrees and 12.65 arc minutes which correspond to 24 minutes, and fifty seconds of time difference with solar and official time. Therefore 12 O clock is before noon at Tabriz. One reason for difference solar and civic time, is adaptation of official time based on one location.
It is true, that sundials are accurate, but their time lapse is not uniform as of human’s time measuring machines. First, rotation of earth around itself, is not uniform, during a year, and it is not exactly 24 hours. Second based on Kepler second law, Plant earth sweeps same area, during its rotation around sun. Earth rotation path around sun is not a circle, but it is an ellipse., which sun is at one of its foci. From spring equinox to fall equinox, distance of earth and sun is greater than distance from fall equinox to next spring equinox. Mean distance of earth and sun is 149.6 million kilometers. Long distances result to longer time. It takes 186 days from spring equinox to fall equinox, but from fall equinox to next spring equinox, takes 179 to 180 days. Combination of these effects, results in time difference of solar time and civic or official time. This time difference is call Equation of Time.
Time difference of local civic time and solar time is algebraic sum of two parts. One is time difference due location, which is constant and depends on longitudinal differences and, equation of time which changes during a year.
Time difference of local civic time and solar time = (Longitude of location – Time Zone X 15) X 4 – Equation of Time
Equation of time constantly changes during a year. Table below is mean equation of time from 2020 to 2025. When equation of time is negative means solar time is slower than civic time, and when it is positive, it means solar time is faster than civic time.
Table of mean equation of time for years from 2020 to 2025
If we draw these times and days as graph, we get graphical representation of equation of time. As the graph below
We can see that in four days of year, equation of time is zero, which are in following table.
There are days of maximum and minimum.
We can conclude that there is not any day, which sun paths the same position at the same time of previous day. The reason is that earth path around the sun has angle with plane of equator. This angle is called obliquity of ecliptic angle and equals to 23.44 degrees. if in one year, for a specific civic time of day, in adequate intervals such every week or so, one takes picture of sun, at end whole pictures of sun will look something like 8, which is called analemma. Because sun does not stay in one location of sky, therefore, shadows will change their position at dial face.
This is for civic time. If we take time with sun’s own time, sun movement in the sky will be more linear, but not stationary in one place. That is because of ecliptic angle. This angle causes seasonal changes. From an observer at center of equator, sun position changes from positive 23.44 at north hemisphere summer to minus 23.44 at north hemisphere winter. Sun elevation reaches to its maximum, which is called summer solstice, and to is minimum at winter solstice. Two days of year, sun’ s angle to equator became zero. These days are called spring equinox and fall equinox. Sun’ s angle with equator is called sun declination. Next picture shows, declination changes when earth rotates around the sun.
A, Spring Equinox, PAQ, Celestial Plane, UAV, Ecliptic plane, BC, Declination, AC, Right Ascension, AB, Sun Longitude,
ACB angle = 90, CAB, obliquity of ecliptic angle
When sun pass meridian of any location, we call it noon time. At noon time elevation of sun is equals to colatitude of location. colatitude equals to 90.0 minus latitude of location (90 – latitude). The following picture tries to show this concept.
The picture at right side is from Wikipedia. Angle d is latitude of location. Sunrays reach us in plane of ecliptic. An observer in mid latitudes of north hemisphere, notice that at noon time elevation of sun in highest at summer solstice and lowest at winter solstice. The picture below, tries to depict this concept geometrically.
In above picture, line pq represents equator, line AB as meridian of location, line AF as sunray and UV represents ecliptic plane. Line OA is perpendicular to line AB, because plumb line is perpendicular to horizon. Angle Aoq, has a constant value and equals to latitude of location. Angle between equatorial plane and ecliptic plane is constant and equals to obliquity of ecliptic, that is 23.44 degrees. In one year, angle qoB changes and it is equals to sun declination d in equatorial coordinate. Lines AF and UV are parallel and angles FAB and Abo are internal angles of these two lines crossed by a line, therefore, they are equal. Angle FAB is elevation of sun at location. In rectangular triangle of oAB, for summer and winter solstices, we will get:
And for equinoxes:
Hour angles are angles which sun travels in one hour in observer’s sky and are measured from noon time. Angles before noon considered negative and after noon are positive. Hour angle equals 15 degrees for one hour, because of earth 360 degrees rotation in 24 hours around itself. From these consideration, one can conclude that, elevation of sun at noon time for any location is:
By calculating declination of sun for every day, one can find elevation of sun. below is table of mean declination of sun for every day of months from year 2020 to year 2025.
We can see in above table that, declination of sun changes from twenty something seconds of arcs to less than one second of arc in one day. Because sun shines only some part of day, we can assume, declination remains constant during a day.
The mean distance of earth and sun is about 149.6 million kilometers. Diameter of sun is about 1.4 million kilometers. For a scale of one million kilometers to 1 centimeter, Sun will be a circle with diameter of 1.4 centimeter and earth will be a point with diameter of 0.12 millimeter at distance of 150 centimeters. We can assume that, changing location of a sundial on earth if its designed angles do not change, will not affect its performance. With these assumptions, now we will look at sundials.
Planar sundial types differ by the angle they have with horizon. Horizontal Sundials and vertical sundials are obvious. Equatorial sundials plane has an angle equal to colatitude of location with horizon and polar sundials plane has an angle equal to latitude of location. To define sundials in general, we to specify two angles to define plate of a sundial. Picture below shows a plane plate in three-dimension coordinate at horizon. The vector n is normal to plane and v and h are local coordinates on dial plate.
Plate declination, Sd, is angle between the south-north direction and projection of normal n on horizon. Plate inclination, Sz ,is angle between, horizon and plate, or angle between normal n, and vertical direction. These angles for some typical sundials are as following:
|
Plane Type |
Inclination, Sz |
Declination, Sd |
|
Horizontal |
0.0 |
0.0 |
|
Vertical, Due south |
90.0 |
0.0 |
|
Vertical, Due East or west |
90.0 |
-90.0 or 90.0 |
|
Polar |
Latitude of location |
0.0 |
|
Equatorial, North |
90- Latitude of location |
0.0 |
|
Equatorial, South |
90+ Latitude of location |
0.0 |
We can define orientation of plane in respect to equatorial plane. Picture below, shows orientation of planes at point, P, respect to earth center.
Another important item is called gnomon, or style of sundial. Gnomons are usually, either normal to sundial plate, such as vector n in above picture, or they are parallel to earth axis. In latter case, it is called polar style and their direction in north hemisphere is toward Polaris.
Tracking shadow of gnomon on dial face, because of the sun movements at observer ‘s sky, is the bases of sundial time keeping. The sun moves daily from east to west in circular path. This daily movement of fifteen degrees per hour causes shadows daily path. The sun’s yearly movement at observer’s sky, because of obliquity of ecliptic angle, causes, the shadow daily path differ each day from previous day. For each hour corresponding to each hour angle, yearly movement of the sun, makes hour lines. Picture below, tries to show this concept.
Hour Angle, Ha, is measured from noon time. Declination angle of the sun equals the yearly movement of the sun at sky. Considering equatorial plane, the sun shines directly to plane, at equinoxes and shines over the plane from spring equinox to fall equinox and shines below the plane from fall equinox to next spring equinox.
Sundial design is to calculate, coordinates of shadow of gnomon tip, P(x,y), on sundials face and to find hour lines angles. Length of style, GS, is U and its height from dial plane is h. Coordinate origin, O, is placed at base of style or gnomon, OG. Style angle with plane is y and angle of its projection on plane with x axis is b. Three general methods considered to calculate Above items from dials plane angles with horizon and Gnomon height.
Trigonometric or geometrical method from chapter 58, Calculation of Planar Sundial, of Jean Meeus book “Astronomical Algorithms”. This chapter is written by R. Sago and D. Savoie. Style is polar and Px, Py , U and y calculated from a set of formulas. In this chapter simpler formulas derived for special cases of equatorial, Horizontal, and Vertical sundials. I add two more special cases:
Polar sundials. Getting values of Sd and Sg from previous table for polar sundial and after simplifying, we will get:
Vertical sundials facing East or West:
Sz = 90.0 , Sd = 90.0
We can find angle of hour lines with horizontal axis. Because we want to find hour line angles, we consider them constant for each hour angle and take differences by changing declination.
By diving above equations and simplifying, we get:
It shows that all hour lines angles are independent of hour angles and are constant equal to latitude of location, therefore they are parallel. We can find their distance from base of gnomon, or coordinate origin. Because hour lines are parallel, we get the distance at equinox, when declination is zero.
In above relations, h, is gnomon height, Ha, is hour angle, f, is latitude of location and d is declination.
Geometrical or vector algebra method by Giuseppe Matarazzo presented in website article “Sundial Design”. In this article origin of three-dimensional coordinates is with unit vectors of I, j, and k is located at gnomon base. Gnomon height is hg Normal vector, n, is Characteristic vector of plane. Sunrays and gnomon considered as vectors S and G. For vertical gnomon, G is hg.n. There are relations for polar and general style components in three-dimensional coordinates. There are formulas to calculate components of sunray vector, S, from elevation and azimuth of the sun. By knowing S and G article gives formulas for x and y coordinates of shadow tip. Instead of taking formula for shadow tip, I consider normal vector n is detrimental in calculating shadow tip coordinates. We can find angle between, sunray vector S and vector n by vector product (or cress product) of two vectors. By finding the angle, we can calculate shadow length.
By knowing shadow length on sundial plate, we can find its angle with horizontal coordinate unit vector of plate, h. On horizontal sundial, horizontal coordinate unit vector is i. For other cases, as paper indicates vector h is cross product of normal vector n and unit vector k.
Now we have shadow vector sn and its length. we find angle between vectors sn and h by dot product of them.
To find coordinates of S(x0, y0) I take same approach. We can find angle between normal vector n and style vector G by dot product of them. If we know this angle, then we can calculate, style length, Lgs and its projection on dial plat, Lgp.
By cross product of normal vector n and style vector G, we get a vector Gp, which lies in dial plate. By dot product of this vector and dial plate horizontal unit vector, h, we get angle between them, a.
Even if a normal style is considered on dial plate, S(x0, y0) is a point which all hour line converge.
Vector Algebra method form the book “Sonnenuhren” by Ottmar Beucher. This method is in equatorial coordinates and style is polar and the aim is to find hour lines angles. For each hour corresponding to each hour angle, yearly movement of the sun happens in a plane, such as the noon time happens in meridian plane, as shown in picture for plane at equatorial coordinates. Hour lines are cross sections of dial plane and these planes, which we call them hour planes. I took this approach, and it is: by cross product of normal vector of dial plane and normal vector of hour planes, we can find a vector in direction hour lines hours. Meridian plane is the plane which sun pass at noon time. Hour angle of noon is zero, therefore normal vector Meridian is plane of Ha = 0 and in equatorial coordinates is:
We can consider other hour planes as rotated meridian plane by rotation matrix.
By cross product of any perpendicular two vectors in any plane we can find normal vector of that plane. For dial plane, we can define to vector v and was coordinates of dial and cross product of them will be normal plane of dial. In “View of local meridian” all planes are common in unit vector e1 therefore we can take vector e1 as vector unit w. Picture below shows an arbitrary plane with unit vector e1 and vector unit w in common with others. It is as this plane rotated around axis e1 to e degrees. Therefore, we can take unit vector v as v=De.e2. De is rotation matrix around e1. By further investigating the picture, we find that normal vector of dial plane is nz=De.e3.
Cross product of vectors nH and nZ give a unit vector in direction of hour line. Dot product of this vector and vectors v and w gives the angles between hour line and coordinate vectors.
Sa is hour line angle with coordinate. Above equation for hour lines angle is general formula for planes of horizontal, vertical, polar, equatorial and all plane with inclination but no declination that is Sd = 0.0
To consider other angles of dial plane with horizon, book find equations for rotation of horizontal plane about South-North direction and rotation of vertical plane around plumb line. Here we take more general approach and get the same results for above conditions.
Rotation around East-West axis, will not change vector v, therefore, it will e same as before:
Next rotation will be around this vector, so again, v would not change, but w will be:
Which Dq is rotation matrix of rotation of plane around axis v of by angle q.
Now we can get the nz vector of dial plane.
And vector s, of hour line:
By dot product with v and w we get the hour line angles
Style is polar, therefore e3 is its unit vector. We can get the angels of style with dial plate, r, and meridian, b, with same technic.
The above equations for horizontal and vertical planes are:
Software
I tried to explain briefly, underlying basics of sundial design, I have used in developing this software. Plus, calculation of location of the sun at observer’s sky is based on algorithms of Jean Meeus book “Astronomical Algorithms”. This software is sundial.exe, build on Fortran and an Excel worksheet Sundial_English.xlsm. All tables in this article are created by this two software. The following is the Starting page of Sundial_English.xlsm
Start with sheet “Enter” and enter name, longitude, latitude, and elevation of location. Elevation is for calculating atmospheric pressure and temperature, if you do not it, enter zero. Select, calendar type, enter year. Now you can select one of data types:
· Noon, Local time
· Equation of Time
· Mean Declination of Sun
· Elevation of the sun at noon
· Apparent Declination of the sun
And click “Calculate Data” button. Data will display at the table. Select any sundial type and Time. For sundials, Horizontal, Polar, Equatorial, specify Style height. For Vertical, Horizontal Bifilar and Vertical Bifilar, specify Style height and dial declination. For General type sundials specify, style height, dial declination and dial inclination. Click the button “Calculate Sundial”, x and y coordinates of shadow tip, will be in sheet with name of sundial type.
After entering specification of location at sheet “Enter”, they will be copied to all other sheets. By clicking “Calculate” button at sheet “EqSol”, azimuth, elevation, and declination of sun, for four days of spring equinox, summer solstice, fall solstice, and winter solstice from sun rise till sun set with one-hour steps, will be at page.
At sheet “Hourlines” Hour lines angles are automatically calculated and are in appropriate tables. To can change rotation angles, by entering desired angle in colored cells. You see results immediately. There is a table and graph for analemmatic sundial in this sheet. You can change height of gnomon and see the results.
Sheet “Average” is to calculate average of either equation of time or declination of the sun, for period of years. Type of depends on data type at sheet “Enter”. You can determine period of years by entering years at “from year” cell and “To year” cells. Click button “Calculate Mean”. Calculation time depends on number of years.
Sheet “VectorDial” is for designing sundials by vector algebra. This method uses elevation and azimuth of the sun. Location copied from sheet “Enter”. Select dial type, calendar type. For horizontal, polar, and equatorial dials, specify only, style height. For vertical dial specify style height and dial declination. For general dial, specify all.
The software Sundial.exe is based on the same mathematical methods. It is built on Fortran. For precession, nutation and obliquity I have used subroutines from IAU SOFA. The results are more accurate than Excel spreadsheet. I recommend, to transfer it to a folder specific for sundials. The result of sundial calculation is saved in a text file with extension .ASC. If you run this program, at first it asks for specification of location.
Enter Name of Location:
Enter Longitude in decimal:
Enter Latitude in decimal:
Enter Altitude in decimal:
Enter Time Zone in decimal:
And then selecting, type of calendar, for Iranian calendar, enter “I” or “i” and for Gregory Calendar, enter “G” or “g” .
Please Select Type of Calendar:
: Please type I or i for Iranian Calendar
Please type G or g for Gregorian Calendar:
In next selection, for sundials just enter, for other options, type the number and enter.
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:
If you have selected sundials, then next selection menu will be displayed.
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:
Based on your selection, a series of selection menu, will be displayed, like kind of hour lines on dial face.
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:
Option 1 is for sundial with solar time only. It creates straight hour lines. Option 2 is the same as option 1 with analemma curves added. Option 3 is sundial with hour lines respect to civil time and option 4 is sundial same as option 3 with analemma curves added.
Then Specification for style and dial will be asked.
Enter Gnomon Hight, perpendicular distance of Gnomon tip and dial Plane
Please Enter:
The following question will be asked for vertical or general sundials.
Enter Plane Declination, that is the angle perpendicular to Plane and
Southern meridian toward west
Please Enter:
Enter Plane Inclination, that is the angle perpendicular to Plane and
Zenith, for Horizontal Sundial = 0.0 and vertical Sundial = 90.0
Please Enter:
In case for sundial with not a specific inclination or declination (general type):
Enter Gnomon (Style) Declination, that is the angle perpendicular to Plane and
Southern meridian toward west
Please Enter:
Enter Gnomon (Style) Inclination, that is the angle perpendicular to Plane and
Zenith, for Horizontal Sundial = 0.0 and vertical Sundial = 90.0
Please Enter:
Do You want to add Half an hour line:
1- Enter 1 To Add Half an Hour
And if you half an hour lines to be added to sundial, enter 1.
In entering numbers, please enter decimal point, even if the number has no decimal part. The results are saved in a asci text file with extension ASC. using FreeCAD software we can change these data to sundial lines and curves, by the help of macro ability of FreeCAD and FreeCADsunDials.py. From menu, select Marco-> Macro …, then in opened window click on square button with …. on it to find, address of folder which FreeCADsunDials.py in located. Select the file in list displayed, click Execute button. Next time in submenu Macro, select Recent Macros, and click on FreeCADsunDials. In FreeCAD, you can select the design and export it as SVG or dxf format, for further optimizing the design in your favorite CAD or graphic software. More description in my next post for examples of sundial design. You can download my open-source sundials software from GitHub:
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