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Ancient Theories of the Sun:
2. Epicyclic Model Applet

1. Eccentric Model Applet

3. Eccentric and  Equant Model Applet

 data

Select from the Details menu.

time
                  interval
Select the time interval.
year
The angular position of the apogee of the Sun is slowly moving with time (about 1.71° per cencury, this is not the precession of the equinoxes).

Eccentric Model Applet

There are two mathematically equivalent models of ancient Greek astronomy explaining the unequal motion of the Sun:

epicycle
                  sun



eccentric
                  sun
The Sun moving on the epicycle with center D rotating on the deferent circle (center C) at the same angular speed.
The Sun moving on the circle (radius r) centered at C seen from the eccentric point E.
(eccentricity e = CE/r)

Lenghts of the seasons:



Hipparchus (90 BC-120 BC)
Ptolemy (90 AD-168 AD)
applet
100 BC
Meeus
0
Meeus
2000
Spring
vernal equinox
 -
summer solstice
94 1/2 d
94.51 d
93.96 d 92.76 d
Summer
summer solstice
-
autumnal equinox
92 1/2 d
92.55 d
92.45 d
93.65 d
Autumn
autumnal equinox
-
winter solstice
88 1/8 d
88.11 d
88.69 d
89.84 d
Winter winter solstice
-
vernal equinox
90 1/8 d 90.07 d 90.13 d
88.99 d
Sum
365 1/4 d
365.24 d 365.23 d 365.24 d

seasons


data

Rounded number of days in zodiac signs:


Ari Tau Gem Can Leo
Vir Lib Sco Sag Cap Aqu Pis
days 31 32 32 31 31 30 30 29 29 29 30 31 365
days 95 92 88 90 365


Seasons Applet


equation of the center


The equation of the center (EoC) is the difference between the actual position of the Sun and the position it would have if its angular motion were uniform.
From apogee to perigee the actual Sun is behind the mean Sun (EoC negative), from perigee to apogee the apparent Sun is in advance (EoC positive).
The maximum value of the
equation of the center is at 90° from apogee:

maxEoC = arcsin(e)

e=1/24.0    maxEoC = 2° 23.3'
e=1/24.1    maxEoC = 2° 22.7'
e=1/24.04    maxEoC = 2° 23.0'

Ptolemy:
e=1/24.04    maxEoC = 2° 23'

In the appendix 2 "Calculation of the Eccentric-Quotient for the Sun" of Thurston's book, e is computed to be 143/3438 = 24.04, using the lengths of the seasons and 365 d  14/6048/3600 min = 365.2467 d for the length of the tropical year given by Ptolemy.

Web Links

Hipparchus: Orbit of the Sun (Wikipedia)

Ptolemy (Wikipedia)

Jahreszeit (Wikipedia)

Gemini Elementa Astronomiae, editit C. Manitius (PDF, Greek/German)

Des Claudius Ptolemäus Handbuch der Astronomie (Übers. Karl Manitius)

Books
James Evans: The History and Prctice of Ancient Astronomy,
Oxford University Press, 1998, Chapter Five: Solar Theory.

Hugh Thurston: Early Astronomy, Springer, Berlin/New York 1994.

Jean Meeus: Astronomical Tables of the Sun, Moon and Planets. 2nd ed., Willmann-Bell, Richmond 1995.

Updated: 2012, Jul 15