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 For the date and time selected by the menus the declination and the local hour angle Sun (LAH) are computed by numerical astronomical algorithms and transferred to the geometrical construction. The altitude and azimuth angle are the results of the graphical construction, drawing Thales' circle using the diameter of the altitude circle (parallel to the horizon) and the diameter of the declination circle (parallel to the celestial equator). Setting the Sun below the horizon the times of sunrise and sunset can be read.

New Sun Plotter version     Select your location from the menu list, or enter latitude in decimal degrees and press return key, enter longitude in decimal degrees and press return key. The latitude may also be set by clicking into the right half of the blue ring. Press the "now" button to get the current position of the Sun. Select from the Details menu to show or hide items. Check the boxes to see the construction by Thales' circle . The azimuth angle is measured eastwards of North: 0° = N, 90° = E, 180° = S, 270° = W You may use the keys "m", "d", "h", "n" to increase the month, day, hour, or minute, or Shift key and "n", "h", "d", "m" to decrease the month, day, hour, or minute.

The observer is located at the centre of his "celestial sphere" with zenith Z above his head and the horizon plane N-E-S-W. The Sun rises at (1), moves along the red arc, passing the meridian NZS at (2), and sets at (3). This path is called the diurnal arc of the sun. Projecting the diurnal arc (circle of declination) of the Sun and the horizontal plane into the NZS plane, they are seen as straight lines intersecting at (1)=(3).

The diurnal arc (declination circle) of the sun varies with the seasons:   Summer Solstice Equinox Winter Solstice

There are two reasons for the difference between the standard time and the solar time. To convert solar time to standard time:

- add 4 minutes per degree west of the time zone meridean, and subtract if east, and take into account daylight saving time. For Berlin - time zone meridian 15° E and longitude 13.41° E, add 1.59*4 min = 6.4 minutes.

- subtract the equation of time, due to the elliptic orbit of the earth around the Sun and the obliquity of the ecliptic (23.44°).  The moment of sunrise is usually defined by the instant when the center of the sun is -0.83° below horizon (taking into account the refraction of light by the atmosphere of the earth and the apparent diameter of the Sun). For a latitude of 50° this equivalent to a time interval of 5 to 7 minutes.

As shown below the construction is in agreement with the equation known from spheric trigonometry:

(1) sin h = sin φ sin δ + cos φ cos δ cos τ

(2) cos Az = (sin δ - sin φ sin h)/(cos h cos φ) φ = Latitude, h = Altitude, δ = Declination, τ = Local Hour Angle LHA

(1) We start computing a, the difference between the vertical lines

a = R sin (90°-φ+δ) - R sin h

a = x sin (90°-φ) = x cos φ
x = r - r cos τ = r (1-cos τ)
r = R cos δ
a =
R
cos δ (1-cos τ) cos φ
sin (90°-φ+δ) = sin (90°-φ) cos δ + cos (90°-φ) sin δ  = cos φ cos δ +  sin φ sin δ
sin h =
cos φ cos δ +  sin φ sin δ - cos δ (1-cos τ) cos φ

sin h = sin φ sin δ + cos φ cos δ cos τ φ = Latitude, h = Altitude, δ = Declination, τ = Local Hour Angle LHA

(2) We start computing b, the difference between the vertical lines

b = R sin (90°-φ+δ) - R sin h = x tan (90°-φ)

x = a - r cos (180°-az)
cos (180°-az) = - cos az

a = R sin (
φ-δ)
r = R cos h
x =
R sin (φ-δ) + R cos h cos az

R sin (90°-φ+δ) - R sin h = [R sin (φ-δ) + R cos h cos az] tan (90°-φ)

sin (90°-φ+δ) - sin h = (sin (φ-δ) + cos h cos az) tan (90°-φ)

sin (90°-φ+δ) - sin h = sin (φ-δ) + cos h cos az] cos φ/sin φ

cos φ cos δ +  sin φ sin δ - sin h = [sin φ cos δ -  cos φ sin δ + cos h cos az] cos φ/sin φ

sin φ cos φ cos δ +  sin2 φ sin δ - sin φ sin h = [sin φ cos δ -  cos φ sin δ + cos h cos az] cos φ

sin φ cos φ cos δ +  sin2 φ sin δ - sin φ sin h = cos φ sin φ cos δ - cos2 φ sin δ + cos φ cos h cos az

sin2 φ sin δ - sin φ sin h = cos2 φ sin δ + cos φ cos h cos az

sin δ - sin φ sin h = cos φ cos h cos az

sin δ - sin φ sin h = cos φ cos h cos az

cos Az = (sin δ - sin φ sin h)/(cos h cos φ)

Both equations (1 and 2) can also be found within the construction of my applet Quadratum Horarium Generale (Regiomontanus Dial).

 Books Kurt Hoffmann: Sterne, Mond und Sonne: Astronomie ohne Fernrohr; Eva Hoffmann Verlag, Stuttgart 1999,  ISBN 3-932001-03-6. Web Links Daylight Applet: Table of sunrise, sunset and twilight times Please visit my GeoAstro Applet Collection 