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