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How Aryabhata could calculate the conjunction of the Moon with zeta Piscium at the beginning of Kaliyuga

Convention : strictly speaking, the new Moon is the conjunction between Moon and Sun but we'll call "new Moon" the period between the last visible waning crescent and the new visible waxing crescent (more or less three days).

"Firstly, we have to calculate the location of the mean Moon and its age on 18/02/3102 BCE at 00:00 (beginning of Kaliyuga) from the conjunction of the mean new Moon (age = 1.48 days) of 28 / 02 / 499 CE with zeta Piscium (at 02:42) by using the data of Aryabhata.

The latter date is the beginning of the lunar month including the reference date of Aryabhata (21 / 03 / 499 CE)

Elapsed Julian days from 18/02/3102 BCE at 00:00 to 28 / 02 / 499 CE at 02:42 = 1,314,910.112

Note : the ancients were able to count the number of days between two events. (An aside: the precise number of days between 18/02/3102 BCE at 00:00 and the crowning of King Alphonse X of Spain was calculated in the Alphonsine tables of 1483.)

Elapsed sidereal periods = Nsid = 1,314,910.112 / 27.32167368 = 48126.99717

(with 27.32167368 days = lunar sidereal period of Aryabhata)

As the number of sidereal period is practically an integer, the Indian and Khmer astronomer could be sure the mean Moon was in conjunction with zeta Piscium on 18/02/3102 BCE at 00:00.

Elapsed sidereal years = Y = 1,314,910.112 / 365.25875 = 3599.941444

(with 365.25875 days = sidereal year of Aryabhata)

Elapsed synodic periods = Nsyn = Nsid - Y

Nsyn = 48126.99717 - 3599.941444 = 44527.05573

By using directly the true synodic period:

Nsyn = 1,314,910.112 / 29.53058912 = 44527.05317

The difference is negligible (3.7 minutes)

It should be noted 0.05573 corresponds to an extra-age of 1.65 day (0.05573 * 29.53) with respect to the age of the Moon on 28 / 02 / 499 CE (1.48 days)

So, the age of the Indian mean Moon of February 3102 BCE was theoretically 1.48 + 1.65 = 3.13 days.

It is possibly the "new Moon" mentioned in the Surya Siddhanta. As the age of the true Moon was 0.71 day, the alleged error of the ancients would have been 2.42 days. For a period of 3600 years, it is understandable.

However, it is possible the new Moon's date was calculated from the Moon's eclipse which should have been observed in India during the same lunar month, ie on 04/03/3102 BCE at 19:24 (meridian of Lanka). This alternative of course requires that the date of the eclipse had been accurately recorded.

A related date (21/03/499 CE at Lanka's noon) was determined by Aryabhata by counting very accurately 3600.00 sidereal years from 18 / 02 / 3102 BCE at 00:00 by using most probably the elapsed days between the two dates. It is worth noting all those calculations could be made without knowing the season (ie the date in a solar calendar) of the remote time of reference. The number of elapsed days was enough.

Secondly, we calculate in the Aryabhata way (from the event of 3102 BCE) the location and the age of the mean Moon of 11 / 04 / 967 CE at 01:43:36 (Lanka time) when it crossed zeta Piscium.

That means we apply to the Angkorian epoch the method used previously by Aryabhata circa 499 CE. The following calculations may have been made some centuries before 967 CE. Our theory indeed shows some components of the planetary diagrams related to April 967 were conceived around 800 CE.

Elapsed Julian days from 18/02/3102 BCE at 00:00 to 11/04/967 CE at 01:43:36 = JD = 1,485,889.072

Elapsed sidereal years = Y = JD / 365.25875 = 4068.04511

Elapsed sidereal lunar periods = Nsid = JD / 27.32167368 = 54384.99447

As this number is practically an integer (error = 3.6 hours), the Moon was supposed to cross the ecliptic meridian of zeta Piscium (as a matter of fact, it was)

Elapsed synodic periods = Nsyn = Nsid - Y = 50316.94936

By using directly the true synodic period:

Nsyn = 1,485,889.072 / 29.53058912 = 50316.94647

The difference is negligible (4.3 minutes)

It should be noted that (1 - 0.94936) of a synodic period corresponds to a decrease in age of 1.5 day (0.05064 * 29.53) with respect to the age of the Moon on 18/02/3102 BCE (3.13 days).

So, the age of the Moon was theoretically (from Aryabhata data) : 3.13 - 1.5 = 1.63 days. This time corresponds to the usually observed, first crescent Moon.

The true age was 27.86 days (last crescent). As the Khmer astronomers  probably observed this last crescent (after designing all the ancient diagrams), they should have realized their calculations were not perfect but it is something they had most probably predicted and, anyway, the expected event occured around new Moon.

We noted two important things:

- The lunar equation of centre was negligible on 11 / 04 / 967 CE at 01:43:36. This means the Khmer astronomers would have calculated the same sidereal position for the mean Moon and the true Moon

- The precise longitude of the true Moon of 15 / 04 / 967 CE at 23: 58 (true midnight on Angkor meridian) was used to design the lunar diagram of Angkor Wat.