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Ancient Theories of the Sun:
1. Eccentric Model Applet

2. Epicyclic 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).


There are two mathematically equivalent models of ancient Greek astronomy explaining the unequal motion of the Sun:
     Apogee
     A
epicycle sun
     P
    Perigee
 Apogee
A
eccentric sun
P
Perigee
The Sun moving on the epicycle with center D rotating on the deferent circle (center C) at the same angular speed.
e = distance from D to Sun
rMin=r-e, rMax=r+e
eccentricity=(rMax-rMin)/(rMax+rMin)=e/r
 The Sun moving on the circle (radius r) centered at C seen from the eccentric point E.
e = CE
rMin=EP, rMax=EA
eccentricity=(rMax-rMin)/(rMax+rMin)=e/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

The rounded number of days in zodiac signs agree with those of Geminus:


 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  14/60 h  48/3600 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 Practice 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.

Gemini Elementa Astronomiae, editit C. Manitius (Greek/German), Teubner, Stuttgart 1974.

Updated: 2023, Oct 07