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GeoAstro Applets |
Astronomy |
Chaos Game |
Java |
Miscel- laneous |
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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. |
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Select your location from the menu list,
or
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Press the "now"
button to get the current position of the Sun. |
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Select from the
Details menu to show or hide items. |
Check the boxes
to see the construction
by Thales' circle . |
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The azimuth angle is measured eastwards of North: 0° =
N, 90° = E, 180° = S, 270° = W
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You may use the keys
"m", "d", "h", "n" to increase 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:
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 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) 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. |
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Table of Sunrise/Sunset,
Moonrise/Moonset, or Twilight Times for an Entire
Year Astronomical Almanac from 1998 to 2059 Daylight Applet: Table of sunrise, sunset and twilight times |
Please visit my GeoAstro Applet Collection
Last modified: 2023, Oct 07
© 2001-2023 Juergen Giesen