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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: enter decimal values, (southern latitude negative, western longitude negative) adjust the timezone to your longitude. Then hit the button "Apply input". Use the buttons to decrease or increase the obliquity angle. Use the buttons to decrease or increase the eccentricity. The step size of the buttons (obliquity, eccentricity) may be changed by the menu items. There are some more view options.  Check the box to see the current position of the Sun (if above horizon), and more data. Azimuth: minimum, current, maximum. Azimuth at max. and min. altitude. Current altitude of the Sun, without atmospheric refraction. Hit this button to open a window showing the equation of time (diagram and table).  Select this item to get a spherical projection of the sky. Drag the right side to resize the window.  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 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). 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) 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): Location: 50° N, 10° E,
local time: 12:00 (11:00 UT)

1.  ε = 23.44°, e = 0.01675  2.  ε = 0°, e = 0.01675:  3.  ε = 23.44°, e = 0:  4.  For ε = 23.44° and e≈0.045 (or greater) there is no intersection (knot):  5.  For e = 0.016751 and ε≈13° (or less) there is no intersection (knot):  The analemma as seen on the equator: The amplitude A of the equation of time depends on obliquity and eccentricity:
A = maximum+|minimum|  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 Updated: 2013 Mar 11