Horizontal Sundials

1. Horizontal Sundials Reference: Sundials, Their Theory and Construction by Albert Waugh, Dover Paperback, 1957 R.L. Kellogg, PhD

2. Identifying The Parts of a Sundial Gnomon Base Dial Plate Hour Lines

3.  Dial Plate Gnomon Top Edge Points North Gnomon North Dial Plate Points To The Celestial Equator Points to the North Celestial Pole (NCP) in the Sky Top View  = latitude of dial (e.g. Los Angeles 34°) Gnomon North South South Point

4.  Dial Plate Sun Through the Seasons Summer Solstice ~ 21 June Equinox ~ 21Mar and ~ 21 Oct  = 23.5° Winter Solstice ~ 21 Dec NCP  = 0°  = - 23.5° The angle of the sun from the celestial equator is called the sun’s “declination” Gnomon North South South Point

5.  Dial Plate 6 pm Noon and 6am/6pm Lines South Point 12 pm At 6am local sun time the shadow is due west At 6pm local sun time the shadow is due east At noon local sun time the shadow is due north Note: cheap dials may not have a straight line between the 6am and 6pm hours. And if the construction is hasty, the line does not meet the South Point of the gnomon! 6 am 6 pm Gnomon 12 pm South Point 6 am

6. Finding the Sundial Equation O D Shadow Triangle Shadow Dial Plate  Sundial Shadow Angle O D West East A P  Latitude Angle A  O A Gnomon

7. Finding the Sundial Equation Shadow Triangle O D To Meridian And Celestial Equator P E Latitude Angle   Sundial Shadow Angle O A Gnomon A

8. Finding the Sundial Equation Sun’s Meridian Local Meridian Gnomon Plane of Celestial Equator E E D H Plane of Celestial Equator Shadow Line O O D  H is the Sun’s Hour Angle A

9. Finding the Sundial Equation • From the dial plate and the celestial equatorial plane, we can obtain the tangents of the shadow angle  and sun’s hour angle off the meridian • Take their ratio • But from the gnomon triangle that has the latitude angle , we recognize that hence or

10. 6 pm 2 pm Dial Lines – The Math  12 pm tan() = sin() tan(H)  = dial hour angle measured from 12 pm noon H = sun “hour angle” is the distance of the sun away from the noon meridian. The sun moves 15° per hour, so 9 am gives H = - 45° (morning ) 2 pm gives H = +30° (afternoon) 6 am 12 pm 2 pm  6 am 6 pm South Point Example:  = 40° (latitude) H = 30° (hour angle of sun = 2 pm) gives tan() = sin(40°) x tan(30°) tan() = .6428 x .5774 tan() = .3711  = atan(.3711)  = 20.36° “sin” is the sine trigonometric function “tan” is the tangent trig function “atan” is the arctangent (arctan) trig function These functions can be found on scientific calculators, Excel spreadsheet functions, etc.

11. Draw A Sundial Hour Line 12pm tan() = sin() tan(H) 2 pm  = 20.36° for H = 2pm (30°) and latitude  = 40° gnomon  6 am 6 pm South Point

12. A Complete Dial Here’s what the 2pm shadow might look like 9 am 10 am 11 am noon 1 pm 2 pm 3 pm 8 am 4 pm 15 cm 12 cm 7 am 5 pm 6 am 6 pm For a dial with a 6 cm high gnomon cut at an angle of 40°, it’s base is about 12 cm long and the dial fits nicely on a 15 x 17 cm plate.

13. Measuring The Latitude of A Sundial If you have a sundial, then you can use a protractor to measure the gnomon’s angle and determine the dial’s latitude. Commercial dials usually have a “one size fits all” approach, using a generic latitude of 40 or 45 degrees. Specially built sundials have a gnomon tailor made for their placed location. If the dial is moved to a different latitude, the dial no longer keeps precise solar time. Some dials have “reworked” gnomons for their new, displaced homes. The owner mistakenly things that by just altering the angle of the gnomon, the dial will tell correct time at its new latitude. But as you now know (see previous vugraphs for the math), the dial plate is also made for a specific latitude.  = latitude

14. Measuring the Latitude from a Dial Plate Although we could measure the various dial hour line angles and work our mathematics backward, there is a simple way both to test dials and to create new ones. The tool is called Serle’s Ruler. A copy of the ruler reproduced by the North American Sundial Society (NASS) is shown below. Make a copy of this page and cut out the ruler for your use.

15. Serle’s Ruler – Step One Start with the Dial Plate (or a copy transferred to paper). Align the ruler so that the ends always lie on the noon line and the 6pm hour lines (arrows) Carefully tilt and slide the ruler keeping the end points on the noon and 6pm hour lines until the hour line scale marks from 1pm to 5pm match up with the corresponding 1pm to 5pm dial hour lines. When aligned, mark the point where the ruler touches the 6pm hour line (red X). Noon x 6pm Noon 6pm

16. x Serle’s Ruler – Step Two Noon Now place Serle’s Ruler along the dial’s 6pm line, with the latitude scale starting at the dial’s south point. At the mark on the 6pm line read the dial’s latitude (this dial here has a reading of about 34°). The measurements of the gnomon angle and the dial plate latitude should agree. If not, it could be a “generic” dial that was commercially assembled for quick and low cost sale; Or the dial could have been moved from its original site and the gnomon refitted (under the false assumption that reshaping corrects the dial’s ability to tell time … there are a number these “discordant” dials with non-matching gnomon and dial plate, and usually an interesting story behind the dial and its owners. 6pm