sunset sundial from scratch roger bailey nass, aug 2008 st louis, missouri
TRANSCRIPT
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