Thoughts on Ayanamsa and Precessional Cycle
Length of Tropical and Sidereal year
Mean Length of Tropical year = 365.242189 Civil days (Wikipedia)
Mean Length of Sidereal year = 365.25636 Civil days (Wikipedia)
Difference between the two years: Sidereal year is longer than the tropical year by 0.014171 civil days comprising of 24 civil hours or the period between two midnights (or Sunrises).
Reason behind the Difference in Year Length
The sidereal year is longer than the tropical year because, after one full revolution around the zodiac in a tropical year i.e., period between two vernal equinoxes, Sun has to travel additional angular distance to make up for the angular displacement between the sidereal Aries and Tropical Aries.
If we superimpose the tropical zodiac over the Sidereal zodiac, we can see that the tropical zodiac is rotating in an zodiacal direction relative to the sidereal zodiac. This means that the sidereal zodiac will appear to the receding backward, what is known as the Precession of the Equinox, or the Ayana Chalana in Sanskrit. The angular displacement is called Ayana-Amsa, the degree elapsed due to the Ayana-Chalana.
Thus, in current age while the Sun appears to be in Virgo on 25th of August, it truly is in Leo.
Thus lets say, we started with Sun in Zero degree Tropical Aries (the day of Vernal Equinox) on March 20, 2015 at 22:36:52 UT when. After one full rotation around the zodiac, Sun reaches the same position on March 20, 2016 at 4:21:54. This duration is 365.2396 days. Thus the length of Tropical Year in 2015–2016 is 365.2396 days.
However, if we look at the sidereal zodiac, on the start day, Sun’s longitude is 5 Pi 56' 43.16" (335.933533148148148 degrees). After one complete tropical rotation of the Sun around the zodiac, on March 20, 2016 at 4:21:54, the Sun’s sidereal position is 5 Pi 55' 58.57" (335.916937824074074 degrees). Thus Sun will need additional time to make up for the angular displacement of 0.0165953240741 degrees, which is equivalent to 59.7432 arc-minutes. Thus we can say that as on the year 2015–2016, the zodiac is receding at the rate of 59.7432. Thus in 21,692.88 years, the prevision will complete 360 degree rotation of the sidereal zodiac.
Precessional Cycle based on Difference between Mean Sidereal and Tropical Year
Mean Tropical year = 365.242189 Civil days
Mean Sidereal year = 365.25636 Civil days
Difference: 0.014171 days
Now coming back the mean differences of the Sidereal and Tropic year of 0.014171, let’s try to find out the rate or precession and the duration of one complete precession cycle. If Sun takes 365.242189 days to complete 360 degrees of the zodiac, how much angle will be covered in 0.014171 days? It is 360 / 365.242189 * 0.014171 = 0.01396761 degrees. This is equivalent to 50.28339 arc-minutes. This means that the mean duration of a complete Precessional Cycle is 360 / 0.01396761 = 25,774 years.
It is clear that the rate of precession is not constant, but varies with time. What is known as Ayanamsa is the mean value of the precession taken over a long period.
Deriving the Precessional Cycle based on Siddhantic Texts
Taking the figures of Aryabhatt and Bhaskaracharya, we note that in a period of 43,20,000 sidereal years, we have 1,57,79,17,500 civil days. Which means that the mean length of a sidereal year is 1,57,79,17,500 / 43,20,000 = 365.258680556 civil days. This is longer than the mean length of a sidereal year of 365.25636 civil days as on J2000 epoch. This means that the Mean Precessional Cycle (360 rotation of Equinox in Sidereal zodiac) in a yuga is higher than what we arrived based on mean day length as per J2000 epoch data.
If we use the mean length of the tropical year as 365.2421896698 and compare that with the mean sidereal year length in a Mahayuga i.e., = 365.258680556, we find a difference of 0.0164908862 days. The angular displacement per year is 360 / 365.2421896698 * 0.0164908862 = 0.01625419845 degrees = 58.51511442. This gives us the Precession of Mean Precessional Cycle as 360 / 0.01625419845 = 22,148 years (sidereal years).
My Gut feeling (difficult to verify)
I am inclined to assume that there are 200 mean Precessional Cycle in a Mahayuga. And it is likely that I am not correct in my assumptions. If say, it is correct, this means that the mean length of the Mean Precessional Cycle is 21,600 (43,20,000/200).
If we break the 200 cycles into 10 parts, we should have 80 cycles (4 parts) in Satya yuga, 60 cycles (3 parts) in Treta yuga, 40 cycles (2 parts) in Dvapara yuga and 20 cycles (1 part) in Kali yuga. Thus the total period of Kali yuga is 20 * 21,600 = 4,32,000 years. This gives us a mean Ayanamsa of 57.34513 arc-minutes per year.
We observe that, this is very close to the current figure of Precession Cycle of 21,692.88 years which gives an Ayanamsa of 59.743 arc-minutes per year.
If we accept 21,600 to be the duration of Mean Precessional Cycle, then, we can see that one cycle contains 60 years of the Gods, taking 1 solar day of god (1 degree angular motion of the Sun) = 1 year of Human. This means that 1 year of the gods = 360 sidereal years of the humans.
Does the 60 years divine cycle hold any special significance? I don’t know. But I presume, this can be equated with the 60 years Jovian cycle that we observe in our calendar. Perhaps after every divine year or 360 human years, there is a big shift in the world events.
I will examine that in future and share my findings.