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)))
n: number of Days after summer solstice day
np: the position of perihelion seen from
in Days. I determined them using SkyMap
Pro 9 (these values are different then in Ruggles [1999,
tropicalyear: length of tropical
anomalistyear: length of anomalistic
d: declination [°]
The below figure provides insight in the length of the astronomical
on the above formula:
Following modern definitions of the astronomical seasons:
From summer solstice (max. declination) until autumn equinox
From autumn equinox (declination=0) until winter solstice (min.
From winter solstice (min. declination) until spring equinox
From spring equinox (declination=0) until summer solstice (max.
Definition of equinox
A few definitions of equinox are possible over (pre-)history:
- 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.
- the date that the sun's rise (or set) azimuth is half way the
(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).
- the date that the sun is again at the same location after 182/183
days (is relatively easy to determine). Method used by Thom (, page 109). This will have the
'equinox' declinations at around 0.5º
- 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).
- 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.
- 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."
- 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
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
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).
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:
- If I take a change of 2.3 msec per century for Solar day, the
(light blue curve) for
former years gets closer to the SkyMap results.
SkyMap Pro 10
(like Cassidy) uses Stephenson & Houlden  for
n-dot of -26 ["/cy2],
which can explain the 2.3 msec per century. SkyMap Pro could be
changed in future (pers. comm. C. Mariott ).
- The above sinusoidal function (green
values) seen in the vernal-equinox year seems to be somewhat
out of phase and/or period. This needs more investigation.
- As Cassidy says; the vernal-equinox year
(green line and blue squares) is quite stable over the present years (while the
other event years
change more) and that is why he thinks that vernal equinox could have
been chosen as the start of the year.
Looking at the below graph, winter-solstice (pink line) was also quite
neolithic times, so could this have been the start of the year in
neolithic time... So I doubt if Stonehenge was following the
vernal-equinox start of the year as discussed by Cassidy.
But to be
honest, I don't know if neolithic man was able to determine this
temporal stableness of the mean
winter-solstice/vernal-equinox year without having accurate clocks.
The vernal-equinox and winter-solstice years do not quite follow the
average event years of course (=mean tropical year):
- The above formula has been checked with Meeus' formula, using
information from Irv
Bromberg [Pers.comms 2006]:
There is not much difference (max. 17 sec [1 sigma]) between my formula
Meeus's. This difference is somewhat larger then the normal variation (5*60/100 = 3 sec)
of a year.
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Last major content related changes: May 20, 2004