- SUNDIAL CALCULATIONS -

Sunrise, Sunset... 8.4k


  • Averaging Data for the Sun's Position
  • Selecting the Observer's Position
  • Analemmic Gnomon Design
  • Determining Angle ai, and xi
  • Determining Angle bi, and yi
  • Compensating for the Perceived Edge of Shadow.
  • Ephemeris Data and Date Coordinates.

    Averaging Data for the Sun's Position-

          When designing this sundial, I wanted to pick 365 data points to construct the analemmic gnomon that indicate dates as accurately as possible every day each year during the four year leap year cycle. Because February 29th only occurs on leap year I chose to not try to include it on the gnomon. Since there is no February 29th on the gnomon, the sundial's greatest date inaccuracy would be during a leap year on February 29, when the gnomon would be 1/2 day "slow" (indicating February 28th) until noon, and 1/2 day "fast" (indicating March 1st) for the remainder of the day. Because there are actually 365.25 days in the year, each of the gnomon's 365 days will thus be revealed by the sun 1/4 day later each year until the four-year leap year cycle is completed.

          To derive these 365 average data or date points for the gnomon, I used the Ephemeris report data for the position of the sun as observed at noon from a fixed reference point on Earth from the dates March 1, 1997 through February 28, 1999.

          The 730 days were then paired (March 1, 1997 with March 1, 1998, etc.) and a weighted average was then used to calculate the sun's azimuth and altitude position for 365 "average" days of the year. The data for the days closest to February 28, 1998 and March 1, 1998 were weighted highest, and the data for the days closest to March 1, 1997 and February 28, 1999 were weighted lowest. Thus, the average position (PosAvg) for the sun on a given pair of days was calculated by:

         PosAvg = (dA - 1) / 364 x PosA + (730 - dB) / 364 x PosB ; where:
    
         dA = days 1 through 365 (March 1, 1997 through February 28, 1998)
         dB = dA + 365 = days 366 through 730 (March 1, 1998 through February 28, 1999)
         PosA = the sun's azimuth and altitude position on the dates between days 1 and 365.
         PosB = the sun's azimuth and altitude position on the dates between days 366 and 730.
    
    Thus, for March 1:
         PosAvg = (1 - 1) / 364 x PosA + (730 - 366) / 364 x PosB ; where
    
         PosA = the sun's azimuth and altitude position on day 1, March 1, 1997; and 
         PosB = the sun's azimuth and altitude position on day 366, March 1, 1998. 
    
    And for February 28:
         PosAvg = (365 - 1) / 364 x PosA + (730 - 730) / 364 x PosB ; where
    
         PosA = the sun's azimuth and altitude position on day 365, February 28, 1998; and
         PosB = the sun's azimuth and altitude position on day 730, February 28, 1999. 
          Calculating the sun's average position in the above manner provides construction points (date points) for the analemmic gnomon which are most accurate during years which are between leap years. Since there is no February 29th on the gnomon, its greatest date inaccuracy would be during a leap year on February 29, when the gnomon would be 1/2 day "slow" (indicating February 28th) until noon, and 1/2 day "fast" (indicating March 1) for the remainder of the day. The dates indicated by the gnomon would be 1/2 to 1/4 day fast for the remainder of the year, 1/4 to 0 day fast the second year, 0 to 1/4 day slow the third year and 1/4 to 1/2 day slow the fourth year until noon on the next February 29th.


    Selecting the Observer's Position-

          I also wanted the 365 construction points for the analemmic gnomon to indicate the date as accurately as possible at various times during the day. Reading the date on a single sundial at local noon at different locations around the Earth will result in different observed readings of the date. Two identical sundials, one located a few minutes east of the International Date Line and one located a few minutes west of the International Date Line, would agree with each other on indicating the date; but because of the date change across the I.D.L. one would disagree with the local date by 1 day.

          The design of this sundial is such that the date is revealed by moving the edge of your finger above the figure-8 cutout and date markings on the analemmic gnomon... when the edge of your finger's shadow also intersects with the timeline on the dial below, then that edge of your finger's shadow also indicates the correct date mark on the gnomon. If I were to calculate date construction points for the gnomon from data for the sun's apparent position exactly at noon above Greenwich (0 Long.), then if the sundial was located in London, at precisely noon, the shadow intersecting the dial's timeline would be cast from the date mark indicating the exact start of that day. At 6:00pm in London the sundial would indicate that the day is only 1/4 day old. At 6:00am the next morning in London the sundial would still be indicating only 3/4 of the way through the prior day. The date indicated by the sundial would always be 1/2 day slow. Thus, the most desirable location for noting the sun's noon position for calculation of the 365 date construction points for the gnomon would be with the observer positioned 180 around the Earth from the desired sundial location. Then at noon at the sundial's location, the sundial would indicate that the day (date) is exactly half over. This is equivalent to mathematically advancing the date points by 1/2 day from the observed noon position of the sun at the sundial location, so that at local noon, the date is indicated as being 1/2 day complete.

          The accuracy for this sundial could be improved by designing two analemmic gnomon plates. One could be created for use year-round in Asia, and one for use year-round in the U.S. The U.S. analemmic gnomon would be created by plotting the position of the sun at noon from 90 E., in Asia. If this sundial was placed in Greenwich, at 6:00am it would indicate the exact start of the day, and at 6:00pm the sundial would indicate the day is 1/2 over. In the U.S. from 6:00 am to 6:00pm the sundial would indicate that the day is between 1/4 and 3/4 day old, and at noon it would indicate that the day is exactly halfway over.

          The Asian analemmic gnomon would be created by plotting the position of the sun at noon from 90 W., in the U.S. Thus at 6:00am on the International Date Line the sundial would indicate the exact start of the day, and at 6:00pm the sundial would indicate the day is 1/2 over. In Asia from 6:00 am to 6:00pm the sundial would indicate that the day is between 1/4 and 3/4 day old, and at noon it would indicate that the day is exactly halfway over.

    Observer's Position... 27.3k

          For these calculations for a "U.S." gnomon, I used RedShift® 2 to generate Ephemeris data for the 90 East Long. noon position of the sun as observered from the North Pole (90 N Lat.), for the 720 days from March 1, 1997 through February 28, 1999. An Azimuth reading of 180 indicates that the sun is directly 90 East. An Altitude reading of 0 indicates that the sun is directly on the horizon from the North pole, in the Earth's equatorial plane.


    Analemmic Gnomon Design-

    Gnomon Design... 55.1k       The analemmic gnomon is a flat plate with a Figure-8 pattern cut out of the center. This gnomon plate is suspended in the exact center of the Equatorial Dial Ring and can swivel about the North Celestial Pole axis of the sundial. At 12:00 noon, a solar azimuth of 180 and altitude of 0 corresponds with the exact center of the gnomon plate.
         Angle ai = (180 - Azmi) , and is measured in the equatorial plane.
         Angle bi = Alti ; a positive value is north of the equatorial plane.
         r = the desired radius of the sundial.
         Azmi and Alti are the apparent Azimuth and Altitude of the sun at noon on day i.
         xi and yi are (x,y) date coordinates from the center of the gnomon plate, calculated by:
    
    		(xi,yi) = (r×sin(ai), r×cos(ai)×tan(bi))
    
          The calculations for xi and yi were derived as follows:

    Determining Angle ai , and xi -

    Determining xi... k
          Above is a top view (from directly above the North Celestial Pole axis) of the sundial's analemmic gnomon plate and a portion of the Equatorial Dial Ring. The gnomon plate is suspended in the exact center of the Equatorial Dial Ring and can swivel about the NCP axis of the sundial (about the axis (x0,y).) At 12:00 noon, a solar azimuth of 180 corresponds with the exact center (x0) of the gnomon plate.

          When reading the sundial, the gnomon plate is always swiveled about the NCP axis to directly face the sun, casting the largest possible figure-8 shadow onto the dial ring below. In the above diagram, because the sun at noon is off by Angle ai from the direct overhead position, the gnomon plate is also rotated by Angle ai to directly face the sun. Thus:

    		xi = r×sin(ai) ; where:
    
         r = the radius of the sundial, measured from the equatorial dial to the front of the gnomon plate.
         Angle ai = (180 - Azmi) , and is measured in the equatorial plane.
         Azmi is the apparent Azimuth of the sun at noon on day i.
         x0 is the horizontal coordinate at the center of the gnomon plate, where Azm0 = 180.
         xi is the date coordinate's horizontal distance from the center of the gnomon plate.
    

    Determining Angle bi, and yi -

    Determining yi... k
          Above is a side view (from within the equatorial plane) of the sundial's analemmic gnomon plate and a portion of the Equatorial Dial Ring. The gnomon plate is suspended in the center of the Equatorial Dial Ring, and at 12:00 noon, a solar altitude of 0 corresponds with the exact center (y0) of the gnomon plate.

          Although the gnomon plate can swivel about the NCP axis to directly face the sun, it is always perpendicular to the equatorial plane. Because it swivels about the NCP axis to directly face the sun, the actual distance from the equatorial dial to the front of the gnomon plate shortens from r, to r×cos(ai). Thus:

    		yi = r×cos(ai)×tan(bi) ; where:
    
    	r×cos(ai) = the distance from the equatorial dial to the front of the rotated gnomon plate.
    	Angle bi = Alti ; a positive value is north of the equatorial plane.
    	Alti is the apparent Altitude of the sun at noon on day i.
    	y0 is the vertical coordinate at the center of the gnomon plate, where Alt0 = 0.
    	yi is the date coordinate's vertical distance from the center of the gnomon plate.
    

    Compensating for the Perceived Edge of Shadow-

          Viewed from Earth's orbit, the sun has an apparent width of about 0.48. Because the sun has width and is not a single-point light source, objects in sunlight will cast a full shadow (umbra) with a partially-illuminated outer edge (penumbra.) Because of the way the human eye and brain process light information, on a clear sunny day we perceive a sharp visual edge of a shadow located in the penumbra region immediately near the umbra. On a hazy day we perceive a gradual edge of a shadow spread more across the penumbra region, away from the umbra. And of course on a cloudy day we can't even distinguish a visual edge of a shadow at all. Using empirical measurements on sunny and some lightly hazy days, when one is more likely to use a sundial, a reasonable average of (a fudge factor for) the visually-perceived edge of a shadow was found to be about 0.2 inward (toward the umbra) from solar center. To compensate for this, the edge of the sundial gnomon must be moved 0.2 outward to move the visually-perceived edge of shadow onto the proper time mark.

          Slightly off topic, but can our perception really be deceived by shadows so easily? Yes, it can.

    Ephemeris Data and Date Coordinates-

    Last Updated: May 4, 2010
    Send Mail to: pete@swanstrom.net
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