Sunset Sundial from Scratch

Roger BaileyNASS, Aug 2008St Louis, Missouri

Algarve Coast, Portugal

November Sunsets

Sunset Sundial from Scratch• Sunset views south west over Atlantic Ocean• West facing wall on sundeck• Design a sunset sundial, Reverse Italian

Hours• Limited tools:

– No texts, files computers, programs, internet– Math: geometry, spherical trigonometry– Navigators Equation: Sine Sine Sine Cos Cos Cos– NASS Calculator: scientific & programmable

Wall (Meridian) Plane: West 90º

• Project Equatorial Disc EE’ for time from solar noon

• Angle to Horizon = Co-Latitude

• Mark 15º segments projected to GxTan t on EE’

• Draw hour lines perpendicular

HorizonH H'

P

P'

EE'

Figure 1: Wall (Meridian) Plane

E

G

1

23

45

Navigation Triangle

• Latitude, Altitude & Declination Angles from Horizon on Great Circles

• Navigation Triangle contains time and azimuth angles

Horizon

Zenith

Pole

t

AzEquator

SunLatitude

Altitude

Declination

Co-Angles in Navigation Triangle

• Navigation Triangle sides are complementary angles (90- x)

• Solve for Altitude with Cosine Rule, 2 sides and contained angle known

• Solve for Azimuth with Sine Rule

Horizon

Zenith

Pole

t

Az

Altitude

Equator

SunDeclination

Latitude

CoLat

CoDec

CoAlt

Design Technique for Sunset Sundial

• Calculate polar sundial using equatorial disc• Solve for sunset times at solstices

Cos t = Tan Lat x Tan Dec

• Tabulate t = sunset time – N hours (15º/hour)• Calculate Altitude for t knowing Lat & Dec

Sin Alt = Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t

• Calculate Azimuth for t knowing Altitude– Sin Az = Cos Dec x Sin t / Cos Alt

• Project point onto wall knowing declination of wall, altitude and azimuth

• Plot the results and join the points for hours to sunset and solstice declination lines

Horizontal Plane: Azimuth True

G

G'

AzN SWall(x,y)

H

Sun

x = G/Tan Az

H = G/Sin Az

Figure 2: Horizontal Plane

Solar Plane: Altitude True

y

H Alt

Sun

G'

y = H tan Alt and H = G / Sin Az

then y = G Tan Alt / Sin Az

NASS Programmable Scientific Calculator Programs:

Workshop at NASS Conference in Chicago 2005

Calculator Functions:Trigonometry: Sine, Cosine, Tangent, Inverse-1

Angles: DEG, DMS, TimeMemories: M, K1, K2Programs: LRN, COMP, HLT, (x)

 

Solar Position:Noon, Sunrise, SunsetPrime VerticalAltitude, AzimuthDeclination Lines

Great Circle Distance and Direction (Qiblah)

NASS Programmable Scientific Calculator Programs:

 

Sundials Design Programs:

Horizontal: Time & Hour Angles OffsetsVertical: Time & Hour Angles, OffsetsVertical Declining:

Substyle Distance, Substyle Height, Difference in LongitudeTime, Polar & Hour Angles, Offsets

Analemmatic: Hour Points, Zodiac Date Line, Seasonal Markers Longitude Correction 

NASS Calculator: Sunset Time

Latitude and Declination into the K Memories.  Latitude: 41.88 2ND X->K1, Declination: 12.78 2ND X ->K2

Sunset Time: Cos t = - Tan (Lat) x Tan (Dec)

Program: ON/C K1 TAN x K2 TAN = +/- 2ND COS-1

This gives you t, the time as an angle from noon. (101.73)Divide by 15 for hours. ÷15 = (6.782)For Sunset* add 12 for noon + 12 = (18.782)Convert to Hours:Minutes:Seconds with 2ND ->DMS

Sunset Time is 18:46:56

 

NASS Calculator: Altitude & Azimuth

Sin Alt = Sin Lat x Sin Dec + Cos Lat x Cos Dec x Cos t Program: 2ND LRN MR COS x K1 COS x K2 COS + K1 SIN x K2 SIN =

2ND sin-1 2ND HLT The display shows the Altitude of the sun at 1 pm (58.1064º). Continue for Azimuth using the equation:

Sin Az = Cos Dec x Sin t / Cos Alt  COS 2ND 1/X x MR SIN x K2 COS = 2ND sin-1 2ND LRN 

NASS Calculator: Sample OutputAltitude & Azimuth

Time tº Altitude AzimuthNoon 0 60.90º 0º1:00pm 15 X->M: ON/C COMP (58.106º) COMP (28.537º) SW 2:00pm 30 X->M: ON/C COMP (50.941º) COMP (50.700º) SW 3:00pm 45 X->M: ON/C COMP (41.384º) COMP (66.793º) SW4:00pm 60 X->M: ON/C COMP (30.712º) COMP (79.191º) SW5:00pm 75 X->M: ON/C COMP (19.609º) COMP (89.769º) SW 6:00pm 90 X->M: ON/C COMP ( 8.492º) COMP (80.414º)*NW 7:00pm 105 X->M: ON/C COMP (-2.31º)** COMP (70.520º)*NW 11:00am -15 X->M: ON/C COMP (58.106º) COMP (-28.537º) SE

Chicago Latitude (41.88º N), Aug 18 Declination (12.78º)

Times from Noon and Sunset Equinox Line

EquinoxDec = 0 Sunset = 6:00:00 Latitude = 37.0897 Gnomon= 1Time from Time Time to Altitude Azimuth x = G/ Tan Az y=G*Tan Alt/Sin AzNoon Angle Sunset

6 90 0 0.000 90.000 0.000 0.0005 75 1 11.915 80.821 0.162 0.2144 60 2 23.506 70.803 0.348 0.4613 45 3 34.337 58.907 0.603 0.7982 30 4 43.695 43.752 1.045 1.3821 15 5 50.400 23.956 2.251 2.977

Noon 0 6 52.9103 0 Infinity Infinity

Solstice Hours from NoonHour and Declination Lines

Winter Solstice Hours from noon

Time from Time Altitude Azimuth x yNoon Angle

6 90 -13.880 70.922 0.346 -0.2615 75 -2.893 62.542 0.520 -0.0574 60 7.241 53.221 0.748 0.1593 45 16.118 42.477 1.092 0.4282 30 23.199 29.940 1.736 0.8591 15 27.842 15.578 3.587 1.967

Summer Solstice Hours from noon

Time from Time Altitude Azimuth x yNoon Angle

7 105 2.893 62.542 0.520 0.0576 90 13.880 70.922 0.346 0.2615 75 25.424 78.885 0.196 0.4844 60 37.288 87.070 0.051 0.7623 45 49.235 83.499 0.114 1.1672 30 60.892 70.566 0.353 1.9051 15 71.230 47.560 0.914 3.987

Sunset Sundial: Winter Solstice

Winter Solstice Dec=-23.44 Sunset 70.86578 4.72438533 4:43:28 PM

Time to Time from Time Altitude Azimuth x = G/ Tan Az y=G*Tan Alt/Sin AzSunset Noon Angle

0 4:43:28 PM 70.86578 0.000 60.088 0.575 0.0001 3:43:28 PM 55.86578 9.833 50.421 0.827 0.2252 2:43:28 PM 40.86578 18.275 39.211 1.226 0.5223 1:43:28 PM 25.86578 24.750 26.152 2.037 1.0464 12:43:28 PM 10.86578 28.610 11.362 4.976 2.769

4:43:28 12:00:00 AM 0 29.470 0.000 Infinity Infinity

Sunset Sundial: Summer Solstice

Summer Solstice Dec=23.44 Sunset 109.1342 7.27561467 7:16:32 PM

Time to Time from Time Altitude Azimuth x = G/ Tan Az y=G*Tan Alt/Sin AzSunset Noon Angle

0 7:16:32 PM 109.1342 0.000 119.912 0.575 0.0001 6:16:32 PM 94.13422 10.785 68.677 0.390 0.2042 5:16:32 PM 79.13422 22.201 76.699 0.236 0.4193 4:16:32 PM 64.13422 33.999 84.745 0.092 0.6774 3:16:32 PM 49.13422 45.951 86.314 0.064 1.0365 2:16:32 PM 34.13422 57.744 74.718 0.273 1.6436 1:16:32 PM 19.13422 68.642 55.665 0.683 3.0977 12:16:32 PM 4.13422 75.895 15.749 3.546 14.663

7:16:32 12:00:00 PM 0 76.350 0.000 Infinity Infinity

Final Sunset Sundial DesignHorizon

Equi

nox

Hours from Noon

Winter Solstice

Sum

mer

Sol

stic

eHours to

Sunset

4

3

21

5

6 PM7 PM

5 PM

4 PM

3 PM

2 PM

1 PM

2 PM

1 PM

3 PM4 PM