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Ancient Theories of the Sun:
1. Eccentric Model Applet

2. Epicyclic Model Applet

 Select from the Details menu. Select the time interval. 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      P     Perigee Apogee A 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

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

 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) 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