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                      applet, analemma, Sun, equation of time
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                    applet, analemma, Sun, equation of time
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                    applet, analemma, Sun, equation of time
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The Analemma of the Sun, and the Eccentricity and Obliquity of the Earth.

         

A publication of Thomas Hebbeker about the analemma of the Sun encouraged me to write an interactive Java applet drawing the analemma of the Sun depending on several parameters:
  • orbital eccentricity of the Earth,
  • obliquity of the Earth's axis,
  • location of the observer,
  • hour of observations.





Applet: instructions, and details:


latitude
                obliquity eccentricity
enter decimal values,
(southern latitude negative,
western longitude negative)
adjust the timezone to your longitude.

input parameters
Then hit the button "Apply input".
obliquity of axis
Use the buttons to decrease or increase the obliquity angle.
orbital
                eccentricity
Use the buttons to decrease or increase the eccentricity.
steps menu
The step size of the buttons (obliquity, eccentricity) may be changed by the menu items.
view
There are some more view options.
sun position
                horizon
sun data
Check the box to see the current position of the Sun (if above horizon), and more data.
azimuth
Azimuth: minimum, current, maximum.
altitude
                elevation
Azimuth at max. and min. altitude.
sun altitude
Current altitude of the Sun,
without atmospheric refraction.
equation of
                time graph
Hit this button to open a window showing the equation of time
(diagram and table).
sphere

resize
Select this item to get a spherical projection of the sky.

Drag the right side to resize the window.
  key    key
You may use the key "h" to increase the hour, or shift key and "h" to decrease the hour.


Description of my calculations:


Elliptic motion:
T = orbital period
τ = time pf perigee passage
t = time of observation
M = mean anomaly
E = eccentric anomaly
ν = true anomaly
ε = obliquity of earth's axis
e = orbital eccentricity
Ecliptic system:
L = ecliptic longitude
ω = ecliptic longitude of perihelion
Equatorial coordinates:
α = right ascension
δ = declination
H = local hour angle
GST = Greenwich sidereal time
Horizon system:
Φ = geogr. latitude
λ = geogr. longitude
h = altitude
az = azimuth
formula

equ. of time = ω + M - α
Kepler's equation is solved by an iteration.

Due to the rectangular coordinate system of the applet the analemma figure is distorted for high altitudes. A point at 90° of altitude is streched to a line parallel to the horizon.

The equation of time (red curve) is the combined effect of eccentricity (period 1 year, blue curve), and  obliquity (period half a year, green curve).


obliquity inclination

At present, the mean obliquity currently decreases by 0.0130° (46.8'') per century.
Near the year 12030 a minimum will be reached (22° 36' 41 '').
About the year -7530 there was a maximum inclination (24° 14' 07 '').
(Meeus: Astronomical Algorithms, Chapter 24)


eccentricity variation change

According to an expression for the eccentricity by Simon et al. in 1994 (Meeus: More Mathematical Astronomy Morsels, Chapter 33).


Obliquity Applet

Orbital Eccentricity of the Earth


The analamma figure (50° N, 10° E, 11:00 UT) will change only slightly from 2013 (red, ε=23.44°, e=0.01675) to 8000 (black, ε=22.77°, e=0.01376):

overlay 2013 8000 change variation



Location: 50° N, 10° E,
local time: 12:00 (11:00 UT)


1.  ε = 23.44°, e = 0.01675

eps_e    equation of time diagram


2.  ε = 0°, e = 0.01675:

eps0_e   equation of time

3.  ε = 23.44°, e = 0:

eps_e0   analemma equation of time


4.  For ε = 23.44° and e≈0.045 (or greater) there is no intersection (knot):

no intersection   analemma


5.  For e = 0.016751 and ε≈13° (or less) there is no intersection (knot):

ellipse   ellipse


The analemma as seen on the equator:

analemma equator



The amplitude A of the equation of time depends on obliquity and eccentricity:
A = maximum+|minimum|

equation of time obliquity        eccentricity equation of time

For obliquity 23.44°, eccentricity 0.016751:
maximum = 16.5 min, minimum = - 14.2 min, amplitude = 30.7 minutes.


The accuracy of my values (for 2013):
Reference: MICA (Multiyear Interactive Computer Almanac, U.S. Naval Observatory)


mean abs. error
L
0.005°
RA
0.003°
declination
0.004°
LHA
0.008°
azimuth
0.012°
altitude
0.003°
equ. of time
0.015 min



Sources and Links


Thomas Hebbeker: Die Sonne in der Achterbahn, wie das Analemma entsteht; Sterne und Weltraum, March 2013, p. 80-87; ISSN 0039-1263
The autor determines the orbital parameters (obliquity, eccentricity) from his analemma photo.

Analemma (Wikipedia)

www.analemma.com

Position of the Sun

position of sun in sky

Kepler's Equation (Wikipedia)

Jörg Meyer: Die Sonnenuhr und ihre Theorie; Verlag Harri Deutsch, Frankfurt/Main 2008; ISBN 978-3-8171-1824-3

Jean Meeus: Astronomical Algorithms; Willmann-Bell, Richmond Virginia, First English Edition 1991; ISBN 0-943396-35-2


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Updated: 2023 Oct 05