Sunrise Sunset Seasonal Markers. Roger Bailey and Mike DeAmicis-Roberts

Transcription

1 Sunrise Sunset Seasonal Markers Roger Bailey and Mike DeAmicis-Roberts NASS Tucson 2002

2 Solar Cycles Daily Rotation Time Annual Orbit Seasons

3 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

4 Solar Orbit and Seasons Winter Orbital Plane Summer Equatorial Plane

5 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

6 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

7 Analemmatic Seasonal Markers Solve the triangle for the distance X North Az Where Sun Rises Cos Az = Sin Dec / Cos Lat Az Z Z = a *Tan Dec * Cos Lat West X X = Z * Tan Az = a * Tan Dec * Cos Lat * Tan Az East When Sun Rises

8 Seasonal Sunrise Marker North Solstice Sunrise Summer Solstice 3 8 May Jul West 6 5 Sunrise Time Sunrise Apr Aug Equinox Sep Mar Oct Feb Jan Nov Sunset East 9.00 Winter Solstice Stand on Marker, sight over date to see where the sun rises Stand on Date, sight over marker to see when the sun sets

9 Test Layout

10 Stand on the Date, sight over the Marker for the time of sunset

11 Stand on the Marker, sight over the Date, to see where the sun sets

12 Seasonal Sunrise/set Markers W jun 1-may 1-apr 1-mar 1-feb 1-jan 1-jul 1-aug 1-sep 1-oct 1-nov 1-dec E Mike s good question leads to a brilliant idea! Seasonal Markers converge almost to a point

13 Fer De Vries s Day Circles W jun 1-may 1-apr 1-mar 1-feb 1-jan 1-jul 1-aug 1-sep 1-oct 1-nov 1-dec 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

14 Seasonal Markers and Lambert s Circles W jun 1-may 1-apr 1-mar 1-feb 1-jan 1-jul 1-aug 1-sep 1-oct 1-nov 1-dec E Seasonal Markers are not the ellipse focus points Circles are the exact solution; marker points are linear approximations

15 Seasonal Markers and Lambert s Circles 7 Circles exact W 6 5 Seasonal Marker Point approximate

16 Seasonal Marker Error Position (m) Latitude Marker Position Differences Focus Equinox Solstice Difference (cm)

17 Seasonal Marker Error Analysis Sigma (m) Minimum error at 20.2 proposed by Helmut Sonderegger Declination

18 Periodic Error in Marker Distance Declination (Degrees) Jan Apr Jul Oct Feb Seasonal Marker Distance (m) -30 Date 2.7 Dec Position

19 Periodic Error Correction Epicycle Declination (Degrees) Declination Jan Apr Jul Oct Feb Marker Distance Date Seasonal Marker Distance (m) Equinox Solstice August February July January East Sunset West Sunrise September March June December Centre April October May November Going around the epicycle, Dec Position 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

20 Correction Epicycle Sunrise Marker Diameter = Center = February & August January & July Sunset Marker is the mirror image March & June & December West September East April & October May & November From the date on the circumference, drop a perpendicular to the axis to establish the precise date marker

21 Reality Checks for Sunset Path 16 SD Ψ 34 Refraction Apparent Horizon Azimuth Correction Theoretical Horizon Time Correction Horizon Corrections Semidiameter 16 Refraction 34 Time Correction 50 x 4 / Sin Ψ Azimuth Correction 50 x Cos Ψ

22 Mike s Seasonal Marker Dial Seasonal Marker +/- 2.8 m for 4.5m dial at Wild X Ranch: Lat 36.92, Long Correction epicycle diameter is 15.7 cm Seasonal Marker distances: Equinox +/ m Solstices +/ m

23 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

24 Diablo Range Horizon at Wild X Ranch Approximate path Ψ of rising sun at different times of the year Angle with horizon Ψ varies from 49.1 to 53.1 and 51.6 on 30 April

25 Sunrise Geometry Vertical Plane Altitude = AzS = 3.56 Horizon Altitude at 6:40:40 on 30 April is 4.5 Azimuth shift from sunrise is 3.56 (4.5/Tan 51.6)

26 Offset Geometry: Summer Horizontal Plane AzS Azimuth Shift 3.56 Offset Az Date Mark Time Shift Sunrise Marker Offset = DM(1-Tan Az/Tan Az+AzS) =1.56(1-Tan /Tan Offset = m

27 Offset Geometry: Equinox Time Shift Offset Sunrise Marker Date Mark Offset = SM x Tan AzS = 2.80xTan 3.56 Offset = m 3.56 Horizontal Plane

28 Offset Geometry: Winter Time Shift Offset Offset = DM((Tan Az/Tan Az-Azs)-1) = 1.56((Tan 60.16/Tan(56.6)-1) Offset = m Sunrise Marker Date Mark Horizontal Plane Azimuth Shift 3.56

29 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

30 Sunrise Marker Offset Offset varies with real horizon altitude and season Correction Epicycle m Φ Offset ~0.2 m North West East Equinox Solstice 2.720

31 Skeptic s Test

32 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

33 Assistance Acknowledgements 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