Sun - Equinoxes and Solstices
By definition, the equinoxes and solstices occurs at the instants that the apparent geocentric longitude of Sun is a multiple integer of 90º, since that the Sun's latitude is not real zero, his declination will not also be zero at the time of the equinox.
Approximately instants can be obtain by the following way. First of all we find the time of the mean equinox or solstice, using the expressions in the Table A or in the Table B. The Table A only should be use for the years between -1000 and +1000, and the Table B for the years between +1000 and +3000. Actually the first table can be used for some centuries before -1000 without big errors as also the second one for some centuries after +3000.
It's also important to mentioned that the Y on the top each table must be calculate using an integer value for the year. If any value will be used will produce incorrect values for the time of the equinoxes or solstices.
So we have:
T = (JDE0 - 2451545.0)/36525
W = 35999º.373T - 2º.47
Dl = 1+ 0.0334cosW + 0.0007cos(2W)
Bellow it's a table with the sum S of the 24 periodical terms given in the Table C. Each of one of these terms have the form Acos(B+CT), where the argument of the cosine are given in degrees, in other words:
S = 485cos(324º.96+1934º.136T) +
+ 203cos(337º.23+32964º.467T) + …
So, the wanted time in Julian Ephemeris Day is (JDE):
JDE = JDE0 + 0.00001S/Dl (days)
The final value for JDE can be transformed the normal calendar date (see Time and Calendars), appearing in this way in Dynamical Time.
|
For the years [-1000 to +1000] (Y=year/1000) |
|
March Equinox: JDE0 = 1721139.29189+365242.13740Y+0.06134Y2+0.00111Y3-0.00071Y4 |
|
June Solstice: JDE0 = 1721233.25401+365241.72562Y-0.05323Y2+0.00907Y3+0.00025Y4 |
|
September Equinox: JDE0 = 1721325.70455+365242.49558Y-0.11677Y2-0.00297Y3+0.00074Y4 |
|
December Solstice: JDE0 = 1721414.39987+365242.88257Y-0.00769Y2-0.00933Y3-0.00006Y4 |
Table A
|
March Equinox: JDE0 = 2451623.80984+365242.37404Y+0.05169Y2-0.00411Y3-0.00057Y4 |
|
June Solstice: JDE0 = 2451716.56767+365241.62603Y+0.00325Y2+0.00888Y3-0.00030Y4 |
|
September Equinox: JDE0 = 2451810.21715+365242.01767Y-0.11575Y2+0.00337Y3+0.00078Y4 |
|
December Solstice: JDE0 = 2451900.05952+365242.74049Y-0.06223Y2-0.00823Y3+0.00032Y4 |
Table B
|
A |
B (degrees) |
C (degrees) |
|
485 |
324.96 |
1934.136 |
|
203 |
337.23 |
32964.467 |
|
199 |
342.08 |
20.186 |
|
182 |
27.85 |
445267.112 |
|
156 |
73.14 |
45036.886 |
|
136 |
171.52 |
22518.443 |
|
77 |
222.54 |
65928.934 |
|
74 |
296.72 |
3034.906 |
|
70 |
243.58 |
9037.513 |
|
58 |
119.81 |
33718.147 |
|
52 |
297.17 |
150.678 |
|
50 |
21.02 |
2281.226 |
|
45 |
247.54 |
29929.562 |
|
44 |
325.15 |
31555.956 |
|
29 |
60.93 |
443.417 |
|
18 |
155.12 |
67555.328 |
|
17 |
288.79 |
4562.452 |
|
18 |
198.04 |
62894.029 |
|
14 |
199.76 |
31436.921 |
|
12 |
95.39 |
14577.848 |
|
12 |
287.11 |
31931.756 |
|
12 |
320.81 |
34777.259 |
|
9 |
227.73 |
1222.114 |
|
8 |
15.45 |
16859.074 |
Table C
To determinate the start of the Seasons of the year the result is immediate, since that:
March Equinox = Start of Spring
June Solstice = Start of Summer
September Equinox = Start of Autumn
December Solstice = Start of Winter