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The length of the astronomical seasons

The declination of the sun over the tropical year is determined by a slightly adjusted formula of Ruggles [1999, page 54] and Thom [1967, page 24]:

d = asin (sin e * cos (360*n/tropicalyear + 2*ec*180/pi * sin(360*(n-np)/anomalisticyear)))

With:
n: number of Days after summer solstice day
np: the position of perihelion seen from summer solstice in Days. I determined them using  SkyMap Pro 9 (these values are different then in Ruggles [1999, page 54]):

Year
Perihelion (np)
[Days]
2000 CE
197.26
2000 BCE
126.04
4000 BCE
90.41

tropicalyear: length of tropical year
anomalistyear: length of anomalistic year
e: obliquity [°]
ec:ecccentricity [-]
d: declination [°]

The below figure provides insight in the length of the astronomical seasons based on the above formula:

Following modern definitions of the astronomical seasons:

Definition of equinox

A few definitions of equinox are possible over (pre-)history:
  1. the date when set and rise azimuths of sun (on one day) are opposite (180º) to each other. No counting of days is needed (is easy to determine). Will come close to modern definition of equinox, if horizon altitude is the same for set and rise positions.
  2. the date that the sun's rise (or set) azimuth is half way the rise (or set) azimuths of winter and summer solstice (is quite easy to determine). If the rise (or set) altitudes are the same for solstices/equinoxes, this will provide an equinox date close to the modern definition (although it does not account for the variation due to climatic precession changes).
  3. the date that the sun is again at the same location after 182/183 days (is relatively easy to determine). Method used by Thom ([1967], page 109). This will have the 'equinox' declinations at around 0.5º
  4. the date half way the summer and winter solstices dates. Remember half way the solstice dates is difficult to determine due to slow movement of sun around these solstices (thus not easy to determine).
  5. the date when the sun rises due east or sets due west. In this case the concept of east/west must be known and to reach the modern equinox definition the horizon altitude must be 0º.
  6. From DIO web-site:
    "Equinoxes are not (and were not anciently) determined “near sunrise and at sunset” as ... asserts, and that cannot be true for the equinox of 132 AD, which was allegedly observed at 2 PM. Ptolemy says his equinoxes were determined by noon observations using a ‘meridian” transit instrument (Almagest 3.1’s reference to I.12; Toomer p 61) which only can observe on the meridian."
  7. the modern definition (declination = 0), will be quiet unlikely for neolithic man, because the concept of the ecliptic was not yet known (I think).

Vernal equinox year

The vernal equinox year length is slowly changing. In the below picture one sees the Vernal equinox year in Solar days.
SkyMap Pro 9 (purple) values (see for its theoretical backbone on this page) are compared with computed (green) values which are calculated with above formula and with (dark blue) values of Simon Cassidy. The SkyMap Vernal equinox year is calculated as the average over 100 Vernal equinox years (the 4th order polynoom (black) values are also provided based on values up to 5000 CE).
Vernal equinox year over time

The calculated (green) values (with a change of 1.7 msec per century for Solar day, Stephenson [1997, page 514]) are quite close to SkyMap Pro 9's values (around 2 sec for the period 2500 CE to 400 BCE). This graph is quite different from the SkyMap values when looking before 400 BCE.
Two things can be seen:

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Last major content related changes: May 20, 2004