# The Kali Epoch and Ahargana: Part 5: Examining the Mahayuga Elements given by the Siddhantic Texts

Sidereal Year: A sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It equals 365.25636 SI days for the J2000.0 epoch. (Wikipedia)

J2000.0 Epoch: A Julian year is an interval with the length of a mean year in the Julian calendar, i.e. 365.25 days. This interval measure does not itself define any epoch: the Gregorian calendar is in general use for dating. But, standard conventional epochs which are not Besselian epochs have been often designated nowadays with a prefix “J”, and the calendar date to which they refer is widely known, although not always the same date in the year: thus “J2000” refers to the instant of 12h on 1 January 2000, and J1900 refers to the instant of 12h (midday) on 0 January 1900, equal to 31 Dec 1899.10 It is also usual now to specify on what time scale the time of day is expressed in that epoch-designation, e.g. often Terrestrial Time. (Wikipedia).

#### Deriving Mahayuga Elements from the current Duration of Sidereal Year

1. Mahayuga = Solar revolutions (sr)= 43,20,000 Solar years (Sidereal years)
2. Solar months (sm) = 43,20,000 12 = 5,18,40,000
3. Duration of 1 Solar year = 365.25636 Savana years (Civil Years) in current age (Wikipedia).
4. Civil days (cd) = Mahayuga Duration of 1 solar year = 43,20,000 365.25636 = 1,57,79,07,475.2 = 1,57,79,07,475 days (dropping the decimals). Compare this to the one proposed in Aryabhatiya, we get 1,57,79,17,500. There is a difference of 10,025 civil days, which introduces an error in the Ahargana computation.
5. Lunar days (ld)= Almost all the Siddhanta Texts agree that there are 1,60,30,00080 lunar days (Tithis) in a mahayuga. The mean synodic period in current age is 29 days, 12 hours, 44 minutes and 3 seconds (Wikipedia). 30 tithis = 29.5305902778 Civil day. Thus 1 Lunar day = 0.98435300926 civil days. Thus in a period of Mahayuga containing 1,57,79,07,475 civil days, the number of Lunar days = 1602989435.859207 = 1,60,29,89,436 days (Tithis). The number of lunar days as per our computation is 10,644 days lesser than one mentioned in the Siddhantic texts.
6. Lunar month (lm)= Lunar days (ld) / 30 = 1,60,29,89,436 / 30 = 5,34,32,981.2 = 5,34,32,981. In contrast with the Siddhantic texts which specify 5,34,33,336 months, our computation of the lunar month is lesser by 355 days.
7. Lunar revolution (lr) = Solar revolutions + Lunar months = 43,20,000 + 5,34,32,981= 5,77,52,981.
8. Intercalary months (im) = Lunar months (lm) — Solar months (sm) = 5,34,32,981- 5,18,40,000 = 15,92,981
9. Omitted Lunar Days (od) = Lunar days (ld) — Civil days (cd) = 1,60,29,89,436–1,57,79,07,475 = 2,50,81,961
`The Summary-No. of Rev. of the Sun    : 4320000No. of Rev of the Moon    : 5,77,52,981No. of Civil days         : 1,57,79,07,475No. of Solar Months       : 5,18,40,000No. of Lunar Months       : 5,34,32,981No. of Intercalary Months : 15,92,981No. of solar days         : 1,55,52,00,000No. of Lunar days         : 1,60,29,89,436No. of omitted Lunar days : 2,50,81,961`

If we use these elements to compute the Ahargana of Chaitra Shukla Pratipada, Saka 1938 coinciding with 23 March 2016, we get the following:

`Solar years = 1938 + 3179 = 5117Solar Months = 5117 * 12 = 61404Intercalary Lunar Months = 15,92,981 / 5,18,40,000 * 5117 = 157.2392703896605Lunar months = 61404 + 157.2392703896605 = 61561.23927038966Lunar days = 61561.23927038966 * 30 = 1846837.17811169Omitted Lunar days = 2,50,81,961 / 1,60,29,89,436 * 1846837.17811169 = 28897.44438387394Ahargana = 1846837.17811169 - 28897.44438387394 = 1817939.733727816`

This is very far from the truth as the Ahargana on that day is 1869021. This could be due to the fact that the mean no. of civil days (365.25636) in a sidereal year and the mean length of one revolution of the moon (29.5305902778 Civil days) may be true at current age. However, when we use large scale timeline such as a Mahayuga, the mean value changes. So instead of inventing rules based on the current values, we can stick to the values given by the Siddhantic Texts.

We observe here that the Ahargana arrived by using the Mahayuga elements mentioned by Aryabhata and Bhaskara yield the closest possible results. However, it is still not 100% accurate. In the next article, we will see how do we arrive at a higher level of accuracy.