Sunrise SunsetSeasonal Markers

Roger Bailey and Mike DeAmicis-Roberts

NASS Tucson 2002

Solar CyclesAnnual OrbitDaily Rotation Time

Seasons

Earth�s Rotation and Time�Sun�s position, Earth�s rotation is the essence of time

�1 day, 24 hours, 360°

�Time Angle 15°/Hour

�Shadow on sundial tells time

�Time from Noon / Midnight

�Time from Sunrise Sunset

Solar Orbit and Seasons

Orbital Plane Summer+23.5°

Winter-23.5°

Equatorial Plane

Sundials as Seasonal Markers

� Mike�s first good question� �Can a sundial be used to

follow the sun through the seasons?� Yes!

� Solstices and equinoxes� Declination Lines� Analemmatic Zodiac Table� Sunrise/set Azimuth� Analemma: EQT vs Dec

Analemmatic Seasonal Markers

� Mike�s second good question.� �Is there a point in an analemmatic sundial

that can be used with the Zodiac date line to show where the sun rises?�

� Solve the triangle for the marker position� Zodiac date line Z= a *Tan Dec * Cos Lat� Sunrise Azimuth Cos Az = Sin Dec / Cos Lat

Analemmatic Seasonal Markers

Az

AzZ = a *Tan Dec * Cos Lat

Cos Az = Sin Dec / Cos Lat

X = Z * Tan Az = a * Tan Dec * Cos Lat * Tan Az

Solve the triangle for the distance X

North

West

Where Sun Rises

When Sun Rises

East

Z

X

West East

North

Jan

Feb

Mar

Apr

MayJul

Aug

Sep

Oct

Nov

Winter Solstice

Summer Solstice

Equinox

5

7

8

910

11 12 12

3

4

5

66

7

9.00

2.72 2.72

5.413.09

Sunrise Sunset

Solstice Sunrise

Sunrise Time

Seasonal Sunrise Marker

2.80 2.80

Stand on Marker, sight over date to see where the sun risesStand on Date, sight over marker to see when the sun sets

Test Layout

Stand on the Date, sight over theMarker for the time of sunset

Stand onthe Marker,sight over the Date,to see wherethe sun sets

1-jan1-feb

1-mar

1-apr

1-may

1-jun 1-jul1-aug

1-sep

1-oct

1-nov

1-dec

5

6

7

8

910

11 12 1314

15

16

17

18

19

W E

Seasonal Sunrise/set Markers

Mike’s good question leads to a brilliant idea! Seasonal Markers converge almost to a point

Fer De Vries�s Day Circles

1-jan1-feb

1-mar

1-apr

1-may

1-jun 1-jul1-aug

1-sep

1-oct

1-nov

1-dec

5

6

7

8

910

11 12 1314

15

16

17

18

19

W E

Fer de Vries “hor_analem3.bas” Delta Cad macro shows the exact solution, Day or Lambert’s Circles through the date line and ellipse focus points to the time of sunrise/set

Seasonal Markers and Lambert�s Circles

1-jan1-feb

1-mar

1-apr

1-may

1-jun 1-jul1-aug

1-sep

1-oct

1-nov

1-dec

5

6

7

8

910

11 12 1314

15

16

17

18

19

W E

Seasonal Markers are not the ellipse focus pointsCircles are the exact solution; marker points are linear approximations

Seasonal Markers and Lambert�s Circles

Circlesexact

Seasonal Marker Pointapproximate5

6

7

W

Seasonal Marker Error

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80Latitude

Posi

tion

(m)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

Diffe

renc

e (c

m)Marker Position

Differences

Focus

Equinox

Solstice

Seasonal Marker Error Analysis

0

0.005

0.01

0.015

0.02

0.025

0.03

10 15 20 25

Declination

Sigm

a (m

)

Minimum error at 20.2° proposed by Helmut Sonderegger

Periodic Error in Marker Distance

-30

-20

-10

0

10

20

30

00-Jan-00 09-Apr-00 18-Jul-00 26-Oct-00 03-Feb-01

Date

Decl

inat

ion

(Deg

rees

)

2.7

2.72

2.74

2.76

2.78

2.8

2.82

2.84

2.86

2.88

2.9

Seas

onal

Mar

ker D

ista

nce

(m)

Dec Position

-30

-20

-10

0

10

20

30

00-Jan-00 09-Apr-00 18-Jul-00 26-Oct-00 03-Feb-01

Date

Decl

inat

ion

(Deg

rees

)

2.7

2.72

2.74

2.76

2.78

2.8

2.82

2.84

2.86

2.88

2.9

Seas

onal

Mar

ker D

ista

nce

(m)

Dec Position

Equinox

Solstice

Periodic Error Correction EpicycleEast SunsetWest Sunrise

Centre

SeptemberMarch

April October

MayNovember

JuneDecember

JulyJanuary

AugustFebruary

Going around the epicycle, twice per year, generates the sine curve. Drop a perpendicular from the date on the circumference to the axis to find the exact marker position

Declination

Marker Distance

Sunrise Marker Epicycle

West East

Diameter =

Center =

0. 157

2. 799

June & December

January& JulyFebruary

& August

March &

September

April &October

May &November

Sunset Marker is the mirror image

From the date on the circumference, drop a perpendicular to the axis to establish the precise date marker

Correction Epicycle

Reality Checks for Sunset Path

� Horizon Corrections� Semidiameter 16�� Refraction 34�

� Time Correction � 50� x 4 / Sin Ψ

� Azimuth Correction� 50� x Cos Ψ

TheoreticalHorizon

Apparent HorizonAzimuth Correction

Time Correction

16� SD

34�Refraction

Ψ

Mike�s Seasonal Marker Dial

• Seasonal Marker +/- 2.8 m for 4.5m dial at Wild X Ranch: Lat 36.92, Long 121.35

• Correction epicycle diameter is 15.7 cm• Seasonal Marker distances:

– Equinox +/- 2.877m– Solstices +/- 2.72 m

Seasonal Marker Offset• Mike’s third good question.• �Can you offset the seasonal marker to correct

for the real horizon?� Yes, but….• Azimuth shift depends on altitude of real

horizon and path of the rising sun Ψ where• Cos Ψ =Sin Lat / Cos Dec• Azimuth Shift AzS = Alt / Tan Ψ• Offset is always to the north but distance

depends on sunrise geometry. It is not constant for constant horizon altitude and azimuth shift

Approximate path Ψ of rising sun at different times of the year

53.1°

49.1°

49.1°

Angle with horizon Ψ varies from 49.1 to 53.1 and 51.6 on 30 April

Diablo Range Horizon at Wild X Ranch

51.6°

Altitude = 4.5°

AzS = 3.56°

Sunrise Geometry

Altitude at 6:40:40 on 30 April is 4.5°Azimuth shift from sunrise is 3.56° (4.5/Tan 51.6)

Horizon

VerticalPlane

Offset Geometry: Summer

SunriseMarker

AzimuthShift 3.56°

Date MarkOffset

Offset = DM(1-Tan Az/Tan Az+AzS)“ =1.56(1-Tan 60.156/Tan 63.71

Offset = 0.217 mTimeShift

Horizontal Plane

Az

AzS

Offset Geometry: Equinox

SunriseMarker

3.56°

Date MarkOffset

Offset = SM x Tan AzS“ = 2.80xTan 3.56

Offset = 0.174 mTimeShift

Horizontal Plane

Offset Geometry: Winter

SunriseMarker

AzimuthShift 3.56°

Date Mark

Offset

Offset = DM((Tan Az/Tan Az-Azs)-1)“ = 1.56((Tan 60.16/Tan(56.6)-1)

Offset = 0.233 mTimeShift

Horizontal Plane

Seasonal Marker Offset

• Offset can approximately correct for the real horizon

• Offset north is ~ 20 cm or 8” to correct for the Diablo range east of Wild X ranch

• Horizon altitude is more near the equinox as the Diablo Range is closer

• This may correct somewhat for the lower offset distance near the equinox

Sunrise Marker Offset

West

Equinox2.877

Solstice2.720

Offset~0.2 m North

East

0.157 m Φ

Offset varies with real horizon altitude and season

CorrectionEpicycle

Skeptic’s Test

Conclusions

� Seasonal Markers are an excellent addition to analemmatic sundials

� Seasonal Markers show when and where the sun rises and sets throughout the year

� Markers are linear approximations� Errors increase with latitude� Exact solutions are available with the correction

epicycle, but not necessary� Offsets to correct for the horizon are possible, but

not necessary

AssistanceAcknowledgements

� Daniel Roth�s �Sundial Mailing List�� Mike DeAmicis Roberts �good questions�� Fer De Vries �Delta Cad macros & day circles� � Helmut Sonderegger �spreadsheet improvements and error

analysis�� References: Roger�s tool box

� �How Long is My Shadow� NASS 99 RTB� �Analemmatic Sundial Design Spreadsheet� RTB & HS� �Sunrise Phenomenon� NASS 99 RTB� �Hor_analem3SME.bas by Fer De Vries, & RTB