Horizontal Sundials
R.L. Kellogg, PhD Reference: Sundials, Their Theory and Construction by Albert Waugh, Dover Paperback, 1957
Identifying The Parts of a Sundial
Dial Plate
Gnomon
Hour Lines
Base
Gnomon Top Edge Points North
Points to theNorth Celestial Pole (NCP)in the Sky
= latitude of dial(e.g. Los Angeles 34°)
Top View
North
North
Dial Plate
South
South Point
Gnomon
Dial Plate
Points To TheCelestial Equator
Gnomon
Sun Through the Seasons
NCP
North
Dial Plate
South
Summer Solstice~ 21 June
Winter Solstice~ 21 Dec
Equinox ~ 21Mar and ~ 21 Oct
= - 23.5°
= 23.5°
= 0°
Gnomon
The angle of the sun from the celestialequator is called the sun’s “declination”
South Point
Noon and 6am/6pm Lines
At 6am local sun timethe shadow is due west
At 6pm local sun timethe shadow is due east
At noon local sun timethe shadow is due north
Dial Plate
6 pm
6 am
6 pm
6 am
12 pm
12 pm
South Point
Note: cheap dials may nothave a straight line betweenthe 6am and 6pm hours. And if the construction is hasty,the line does not meet the South Point of the gnomon!
Gnomon
South Point
Finding the Sundial Equation
Dial Plate
Shadow
A
O D
GnomonO A
P
Latitude Angle
EastWestA
O D
ShadowTriangle
Sundial ShadowAngle
Finding the Sundial Equation
OA
OE)sin(
GnomonO A
P
E
Latitude Angle
To MeridianAnd
CelestialEquator
A
O D
ShadowTriangle
Sundial ShadowAngle
OA
OD)tan(
OA
OE)sin(
Finding the Sundial Equation
ShadowLine
GnomonPlane ofCelestialEquator
A
O
D
E
LocalMeridian
Plane ofCelestialEquator
O D
Sun’sMeridian
E
H
H is theSun’s
Hour Angle
OE
ODH )tan(
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
OA
OD)tan(
OA
OE)sin(
OE
ODH )tan(
OA
OE
OD
OE
OA
OD
OE
OD
OA
OD
H
)tan(
)tan(
hence )sin()tan(
)tan(
H
or )tan()sin()tan( H
Dial Lines – The Math
tan() = sin() tan(H)
= dial hour angle measured from 12 pm noonH = 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 pm
6 am
12 pm
2 pm
6 am 6 pm
12 pm
2 pm
Example:
= 40° (latitude)H = 30° (hour angle of sun = 2 pm)
gives
tan() = sin(40°) x tan(30°)tan() = .6428 x .5774tan() = .3711 = atan(.3711) = 20.36°
South Point
“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 scientificcalculators, Excel spreadsheet functions, etc.
Draw A Sundial Hour Line
6 pm6 am
12pm
tan() = sin() tan(H)
gn
om
on
= 20.36°for H = 2pm (30°)and latitude = 40°
2 pm
South Point
A Complete Dial
1 pm 2 pm 3 pm
4 pm
5 pm
6 pm
noon11 am10 am9 am
8 am
7 am
6 am
For a dial with a 6 cm high gnomon cut at anangle of 40°, it’s base is about 12 cm long and the dial fits nicely on a 15 x 17 cm plate.
15 cm
12 cm
Here’s what the2pm shadow mightlook like
Measuring The Latitude of A SundialIf you have a sundial, then youcan use a protractor to measure the gnomon’s angle and determine the dial’s latitude.
Commercial dials usually havea “one size fits all” approach, using a generic latitude of 40 or 45degrees.
Specially built sundials havea gnomon tailor made for theirplaced location. If the dial ismoved to a different latitude, thedial no longer keeps precise solartime.
Some dials have “reworked” gnomons for their new, displaced homes. Theowner mistakenly things that by just altering the angle of the gnomon, thedial will tell correct time at its new latitude. But as you now know (see previousvugraphs for the math), the dial plate is also made for a specific latitude.
= latitude
Measuring the Latitude from a Dial Plate
Although we could measure the various dial hour line angles and workour mathematics backward, there is a simple way both to test dialsand to create new ones.
The tool is called Serle’s Ruler. A copy of the ruler reproduced by theNorth American Sundial Society (NASS) is shown below.
Make a copy of this page and cut out the ruler for your use.
Serle’s Ruler – Step OneStart with the Dial Plate (or a copytransferred to paper). Align theruler so that the ends alwayslie on the noon line and the 6pmhour lines (arrows)
Carefully tilt and slide the rulerkeeping the end points on thenoon and 6pm hour lines until the hour line scale marks from 1pm to 5pm match up with thecorresponding 1pm to 5pm dial hour lines.
When aligned, mark the point where the ruler touches the 6pm hour line (red X).
6pmNoon
Noon
6pmx
Serle’s Ruler – Step Two
x
Now place Serle’s Ruleralong the dial’s 6pm line, withthe latitude scale starting atthe dial’s south point.
At the mark on the 6pm lineread the dial’s latitude (thisdial here has a reading of about 34°).
The measurements of thegnomon angle and the dialplate latitude should agree.
If not, it could be a “generic”dial that was commerciallyassembled for quick and lowcost sale;
Or the dial could have been moved from its originalsite and the gnomon refitted (under the falseassumption that reshaping corrects the dial’s abilityto tell time … there are a number these “discordant” dials with non-matching gnomon anddial plate, and usually an interesting story behind the dial and its owners.
Noon
6pm
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