Sunrise Sunset Seasonal Markers - Walking Shado · Sunrise Sunset Seasonal Markers Roger Bailey and ... Ł fiIs there a point in an analemmatic sundial ... Seasonal Marker Error
Post on 17-Jun-2018
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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
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
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 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
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
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