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Sunset Sundial from Scratch Roger Bailey NASS, Aug 2008 St Louis, Missouri

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Algarve Coast, Portugal

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November Sunsets

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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

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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

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Navigation Triangle Latitude, Altitude & Declination Angles from Horizon on Great Circles Navigation Triangle contains time and azimuth angles

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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

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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

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Horizontal Plane: Azimuth True

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Solar Plane: Altitude True

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NASS Programmable Scientific Calculator Programs: Workshop at NASS Conference in Chicago 2005 Calculator Functions: Trigonometry: Sine, Cosine, Tangent, Inverse -1 Angles: DEG, DMS, Time Memories: M, K1, K2 Programs: LRN, COMP, HLT, (x) Solar Position: Noon, Sunrise, Sunset Prime Vertical Altitude, Azimuth Declination Lines Great Circle Distance and Direction (Qiblah)

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NASS Programmable Scientific Calculator Programs: Sundials Design Programs: Horizontal: Time & Hour Angles Offsets Vertical: Time & Hour Angles, Offsets Vertical Declining: Substyle Distance, Substyle Height, Difference in Longitude Time, Polar & Hour Angles, Offsets Analemmatic: Hour Points, Zodiac Date Line, Seasonal Markers Longitude Correction

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NASS Calculator: Sunset Time Latitude and Declination into the K Memories. Latitude: ND X->K1, Declination: ND 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

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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 2 ND HLT The display shows the Altitude of the sun at 1 pm ( º ). 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

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NASS Calculator: Sample Output Altitude & Azimuth Time t º Altitude Azimuth Noon º 0 º 1:00pm 15 X->M: ON/C COMP ( º ) COMP ( º ) SW 2:00pm 30 X->M: ON/C COMP ( º ) COMP ( º ) SW 3:00pm 45 X->M: ON/C COMP ( º ) COMP ( º ) SW 4:00pm 60 X->M: ON/C COMP ( º ) COMP ( º ) SW 5:00pm 75 X->M: ON/C COMP ( º ) COMP ( º ) SW 6:00pm 90 X->M: ON/C COMP ( º ) COMP ( º )*NW 7:00pm105 X->M: ON/C COMP (-2.31 º )** COMP ( º )*NW 11:00am -15 X->M: ON/C COMP ( º ) COMP ( º ) SE Chicago Latitude (41.88 º N), Aug 18 Declination (12.78 º )

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Times from Noon and Sunset Equinox Line

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Solstice Hours from Noon Hour and Declination Lines

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Sunset Sundial: Winter Solstice

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Sunset Sundial: Summer Solstice

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Final Sunset Sundial Design

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