Math of Astrology - 2: The Sinhala/Hindu New Year

Today is the first day of the new year according to the traditions in Sri Lanka (or rather the traditions of Buddhists and Hindus in Sri Lanka). I personally find the traditional new year in SL interesting as unlike many traditions around the world where the new year begins with a new day, it begins at a particular time of the day marked by the movement of the Sun from Pisces to Aries.

The traditional new year also carries along with a lot of "traditions", astrological and otherwise. There are specific times defined by astrologers ("nakath") which denote the beginning of the new year, the beginning of Sun's transition in to the new year and the times defined as "auspicious" for various tasks (starting your profession for the new year, first meal of the new year etc.).

As mentioned in the earlier post, the objective of this attempt is not to find scientific proof of the astrology, but to analyse the mathematical basis of astrology. In this one, let's try to dissect the math behind the Sinhala and Tamil New Year. For the purpose of this post, I will refer to the system as Vedic astrology unless it is necessary to identify between the Sinhalese and Hindu tradition.

1. Definition of an year
Definition of an year is both simple and complex. The most simple (and scientifically accepted) definition of an year is that it is the time taken for the Earth to complete a full cycle around the Sun. While we can easily picture this movement and measurement of the position easily, it is not easy for an observer on the surface of Earth to measure this. The only mechanism an observer on Earth can measure the movement of Earth around the Sun is to observe the movement of Sun along a background coordinate set.
There are two methods of measuring the the duration of the year:
1. Sidereal Year
2. Tropical Year

A sidereal year refers to the period of time when the Sun starts off from a position relative to a distant star and returns to the same position. This is the "true" year we would observe if we were outside the solar system.  The tropical year is the time period for the Sun to start off from a position relative to the Earth, and to end up at the same position (imagine the longitude/latitude coordinate system we have on Earth being projected to the space. Tropical year is the time required for the Sun to start off from a given longitude and return to the same longitude). The sidereal and tropical years differ as the tropical year is affected by the slow turning of Earth's own axis, precession (i.e. the projected longitudes also shift slowly as time passes). This is equivalent to you trying to catch up with a moving target around a circle. The difference between the sidereal and tropical years  amounts to a few hours.

The calendar used in Vedic astrology uses a sidereal year and measures the movement of Sun from the first point of Aries to first point of Aries (a full cycle). Since the measurement of the year is relative to a fixed point in the sky (as opposed to a fixed number of day), the system does not require an adjustment of leap years and additional days. Due to measurement of the movement of the Sun to a fixed point in the space, the new year can occur at any point of the "clock" as opposed to mid night.

2. Definition of the starting point
As mentioned above, the basis of year on the Sinhala and Hindu new year is the time taken for the Sun to start from the first point of Aries, travel around the sky and back to it. One very important question that needs answers is the definition of the first point of Aries.

The first point of Aries is generally accepted to be the longitudinal (or rather the RA) line that follows across Hamal or the brightest star in Aries in western astrology.

In Hindu astrology, the actual point is debated to be either Hamal (the brightest start of Aries) or to refer to Sheratan and Mesarthim (Beta and Gamma Arietis). The confusion is based on the fact that there are no surviving star charts from the text that refer to astrology. The argument for the stars Beta and Gamma Arietis stem from the fact that according to Vedic astrology, the Aswini nakshatra (or "house") begins here; In Hindu mythology, Aswins refer to twin gods.

(As can be seen from the next point of discussion, there is an even more interesting observance. Either the star Hamal marks the end of the Aswini nakshatra or the effect of precession has taken its toll that now the astrologers are referring to a wrong point in space.)

3. Time of new year
According to the traditional almanacs, the time of eve of new year for 2013 is on 13 Apr 2013 19:59:00 UTC. At this time, the Sun is at RA:01h28m12s, a point that is safely inside Pisces. The new year is however referred to as the movement of Sun from Pisces to Aries.

Dawn of traditional New Year in SL

Could it be that the ancients had a different demarcation of constellations as well? That is a definite, but given that the starting point has not changed (i.e. the first point of Aries), the actual boundary of the constellation does not matter.

We know for a fact that the precession of the Earth's axis causes the position of the stars to move. A simulation run on Stellarium shows that the Hamal was actually aligned with the Vernal Equinox in 373BC. The Vernal Equinox point is important because it defines the "Greenwich Mean Line" of the sky. On each sidereal year, the stars move slightly in relation to the equinox because of the precession.

Coincidence of Vernal Equinox and Hamal in 373BC

Hindu astrology does provide an adjustment for that. It is assumed that a westward correction of 54" (called Ayanamsa) should be added each year as a compensation for the precession. This number although close, is different to the rate agreed by the scientific community, 50.3". Over 100 years, this refers to a difference of approximately 1.54 degrees. While the number is wrong and the actual identity of the person who introduced this correction (the text Surya Siddhanta mentions this correction, but whether this was introduced by the original author or another astronomer -Bhaskara II - at a later time, remains a question), identification of this stellar drift is an important factor.

If we were to add the correction of 54" from the point where the equinox and the first point of Aries were aligned, then the correction up to now (2013) would add 35.79 degrees (the actual position of Alpha Arietis is shifted only  31.97 degrees). Therefore even if the correction is added, the Sun is about 13.75 degrees behind the required point.

What if Alpha Arietis marks the end point of Aswini? If that is the case, Aswini would begin somewhere around RA 01h13m45s. If it the end point was Beta Arietis, then Aswini would begin somewhere around RA 01h02m03s. If this is the case the error in calculation happens to be around 3.6125 degrees and 5.6125 degrees respectively. Although these numbers look visibly small, the error is actually large. An error of 3.6125 degrees refers to an error that is about 6.77 times the disc of the Sun and it takes the Sun approximately 3.92 days to move that distance.

The difficulty in asserting which of the definitions of Aswini is true lies in the fact that most astrological computations rely on almanacs as opposed to actual observations. While I have to leave the question open at this point of time, I intend to find more information on this and update this post at a later time.

4. Nonagathaya
Nonagathaya (Sinhalese) appears to be a tradition specific to the Sinhalese. It is believed that nothing auspicious should be started during this time and that one should get involved in religious activities. Due to this, the time period is also called "Punya Kalaya" (meaning: time for good deeds).

Etymology of the term "nonagatha" is not confirmed. While some attribute this to "No-Nakath" (meaning: without nakshathra) , others believe this evolved from "No-Anagatha" (literally meaning "not-haven't arrived". I.e. to mean that it has not arrived, but has left). Both parties however agree that the period refers to the time where the Sun completes the move across the "boundary line" that marks the new year. The Nonagathaya period spans a total of rougly 13 hours (12h48m) and is divided in to Purva (pre) and Apara (post). The demarcation of the pre and post period is the new year eve.

Let's try to verify the claim that this period refers to the time taken for the Sun to cross the boundary line between the old and new year. The apparent disc of Sun has a mean angular diameter of 32'. Therefore if it is to cross a boundary completely, it should travel a distance equal to the angular diameter. The Sun completes a full circle of 360 degrees in a period of 365.25 days (i.e. 8766 hours). Therefore the time required to travel a 32' is equal to:
     Time required to travel 1 degree           = 8766/360 = 24.35hrs
     Time required to travel 32'                    = 24.35*32/60 = 12.98hrs = 12h58.8m

The error on this may appear to be small (just 10 minutes), however it adds up to a significant number. It appears that in according to astrological methods, it is assumed that the Sun moves by 1 degree in a period of 24hrs as opposed to 0.9208 degree. This error causes the number of days in an year to be 360 and not 365.25.

Could this be an observational error? Let's try to reverse calculate from the answer of 12h48m for the Sun to move a complete disc and see how it works out.
Assume the disc of the Sun is of an angular diameter of x minutes:
     Time required for x minutes of arc                       = 768m
     Time required for 360 degrees                              = (768/x)*60*360
     Actual time taken for 360degrees  (in minutes)    = (24*60)*365.25
                                                            768*60*360/x = 24*60*365.25
                                                                                 x = (768*60*360)/(24*60*365.25)
                                                                                    = 31.54'
Now this, is a more plausible error. In reality, the angular diameter of the Sun varies between 31.6' to 32.7'. Therefore it is possible that without precise measuring equipment, an error of this scale (120th of a degree) was made.

With that, I intend to conclude the second post on the Math of Astrology. While writing this, I realised that I should cite my sources better. Hopefully in the last post, I will cite all the references that I used in this research so that they can be referred and verified and interpreted.

Well that's all for now, and Happy New Year!