Hipparchus and Dulakara Ayanamsha A New Concept on Ayanamsa by Buddhike Sri Harsha Indrasena

Article by Sri Harsha Indrasena, Sri Lanka

Introduction

Sidereal and Tropical Zodiac

A sidereal system is used in Vedic astrology. Among western astrologers tropical zodiac is popular.

In Sidereal astrology, zodiac is defined by the fixed stars in sky round the earth. The zodiac starts undisputedly with Aries, a constellation of stars which is visible in night sky. But in this ‘circle’ of stars of sidereal zodiac the exact starting point or 0° of Aries is debatable.

In Tropical astrology the zodiac is defined by the position of vernal equinox, i.e. the equinox that the Sun passes from south to north. Sign Aries or 0º of Aries starts at vernal equinox and the other signs are named every 30° around the ecliptic in the celestial dome irrespective of the arrangement of the fixed stars.

The equinoxes are formed as follows: As the rotational axis of the Earth is not perpendicular to its orbital plane, the equatorial plane is not parallel to the ecliptic plane, but makes an angle of about 23°26′. The celestial equator and the ecliptic are the imaginarily projected terrestrial equatorial and ecliptic planes respectively out into the celestial dome. The intersection line of the two planes results in two diametrically opposite intersection points, known as the equinoxes, in the celestial dome. The equinox that Sun passes from south to north is known as the vernal equinox or first point of Aries and the opposite point is known as autumnal equinox or first point of Libra.

The starting point of tropical zodiac is definite and can be calculated with high degree of precision. The sidereal zodiac is only a circle of stars in the ecliptic and there is no intersection point as in tropical zodiac. Therefore the location of starting point of sidereal zodiac has to be defined by other means. The aim of this article is to define the starting point of Aries with respect to star Spica.

Precession of Equinoxes

Unlike the starting point of sidereal zodiac, which is fixed, the starting point of tropical zodiac, or vernal equinoctial point, is not fixed because of the slow change in earth’s orientation to the stars. The position of the Sun on the first day of spring (vernal equinox) slowly shifts westward around the sky at a rate of 50″ arc seconds per year with respect to the fixed stars.  This phenomenon is called precession of equinoxes.

Aristarchus of Samos (280 BC) is the earliest known astronomer to recognize and assess the precession of the equinoxes. About 150 years later Hipparchus proposed that the rate of precession of equinoxes was 46″ per year. Hipparchus’s value is a good one compared with the modern value of 50″.

It was Sir Isaac Newton in the 17^th century who produced the first full theoretical explanation of precession and accurately calculated its annual rate (50.29″ arc seconds per year). The Earth wobbles in space like an out-of-balance top. The reason for the slow wobble is that the Earth is not a perfect sphere. The equatorial diameter of the Earth is larger than the polar diameter. Each full wobble takes about 25,772 Julian years.

As a result of moving vernal equinox, longitude of a fixed body, such as fixed stars of sidereal zodiac, defined with respect to vernal equinox will change (increase) slowly. On the other hand, since the stars hardly ever move with respect to the other stars (ignoring the effect of proper motion) the longitude of a fixed body/point (or the point of beginning of sidereal Aries) defined with respect to stars will never change.

Ayanamsha

The range of separation of two zodiacs is commonly known as ayanamsha (Sanskrit – ayanāṃśa: ayana “movement” + aṃśa “component”), or precession. Earlier Greek astronomers like Eudoxus spoke of vernal equinox i.e. starting point of tropical zodiac, at 15° in Aries of sidereal zodiac, while later Greeks spoke of vernal equinox at 8°, then 0° in Aries. The latter is the point of time when tropical zodiac coincided exactly with the sidereal zodiac. This year is known as Zero ayanamsha year (Figure 1).

Fig 1: Diagram showing the westward shift of the vernal equinox among the stars over the past six millennia

The two zodiacs drift apart relative to each other at a rate of about 1.4° per century. The sidereal zodiac was used in Greece before Ptolemy and Hipparchus. Tropical system was introduced by Ptolemy and remains prevalent in western astrology.

Since the point of vernal equinox (starting point of tropical Aries) is precise and unambiguous and can be calculated with certainty, astronomers use this point to calculate planetary longitudes. Vedic astrologers, who use sidereal position of planets for predictions, rather than making their own calculations, alternatively subtract ayanamsha from tropical position of heavenly bodies to obtain the sidereal longitudes of planets.

The correct ayanamsha for a given time is debatable. The official ayanamsha approved by the Government of India since 1950s is that of N.C. Lahiri who believed that zero ayanamsha year was 285 AD. Whereas Fagan – Bradley ayanamsha is popular among astrologers who practise sidereal astrology in west and they believe that the zero ayanamsha year is 221 AD. This is just an example of controversy and there are more than 20 different ayanamshas proposed by different scholars. The sidereal longitudes of planets are directly influenced by variable ayanamshas as shown below for Sun (Table 1).

Table 1:  Comparison of longitude of Sun based on 20 different Ayanamsha Values on 1^st of January 2011 at 00 00 hours GMT

(Z.A.Y. – Zero Ayanamsha Year) (Sag – Sagittarius)

Background

It is my personal opinion that Lahiri ayanamsha is not capable of giving precise sidereal longitudes of heavenly bodies. Horoscope readings and Dasha predictions done based on Lahiri ayanamsha go wrong at many times. Further Lahiri ayanamsha fails significantly in divisional chart analysis and birth time rectification.

The results seem to be better than Lahiri if Fagan-Bradley ayanamsha is used in combination with 360 day Savana year. Even better is Chandra Hari ayanamsha (version 238 AD zero ayanamsha year), which gives much better results in divisional chart analyses and birth time rectifications but still with a few exceptions once in a while. Babylonian-Huber and Babylonian-Mercier ayanamshas are closer to Chandra Hari (Table 1). I believed that the correct ayanamsha should be found somewhere in this range.

When doing an internet search to find out the basis of Babylonian astronomy and the work of Huber and Mercier, I came across the findings of great Greek astronomer Hipparchus.

Hipparchus

Hipparchus was born in Nicaea in Bithynia, but spent much of his life in Rhodes of Greece. His recorded observations span the years 147 BC to 127 BC. Virtually all his writings are lost to date. The Almagest, written by Claudius Ptolemy (90 AD –168 AD), is the source of most of our knowledge about Hipparchus. Ptolemy made extensive use of the work of Hipparchus. Almost all the work of Hipparchus is therefore derived today from Almagest of Ptolemy.

Apart from precession Hipparchus calculated the length of the year to within six and a half minutes, developed a scale to rate the brightness of stars, was the first to record a nova, theorized on the motions of the Sun and Moon, provided high quality planetary observations and created a catalog of 850 stars.

His star catalog, believed to have been produced in 129 BC, is credited with the production of the first known comprehensive catalog of stars in the western world. Hipparchus made the star catalog in ecliptic coordinates. For the naked eye observations of stars he used a self invented instrument called armillary sphere (Figure 2), a model of objects in the sky consisting of a spherical framework of rings centered on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic.

Fig 2: Armillary Sphere

The star Spica is the star that provided Hipparchus with the data which enabled him to describe precession of the equinoxes. Spica (Alpha Virginis, Chitra Nakshatra in Vedic astrology) is the brightest star in the constellation of Virgo, and the 15^th brightest star in the night time sky.

According to Hipparchus’s accounts of the rising and setting of the constellations “beginning of Aries rises when Spica sets” ⁽¹⁾. This finding is the sole basis of this paper.

Principles and Theory

Hipparchus says beginning of Aries rises when Spica sets. Therefore it is needless to say that if the location of Spica is known, 0° Aries can easily be deduced, the two points being situated on either side of the horizon. Since Aries has been defined in relation to a star, this is the sidereal 0° of Aries. At zero ayanamsha year both sidereal and tropical longitudes coincide. Therefore at zero ayanamsha year the tropical longitude of Spica must be as same as (sidereal) longitude of Spica that Hipparchus observed. Precise tropical longitudes of stars are accessible today through astronomical calculations. Therefore the zero ayanamsha year will be the year when tropical longitude of Spica is as the same as the (sidereal) longitude of Spica that Hipparchus observed.

Calculations

What was the longitude of star Spica that was observed by Hipparchus? Spica sets when beginning of Aries rises. Beginning of Aries and position of Spica are related to two ends of the horizon. Since Aries and Libra are opposing signs in the Zodiac one might place it at 0°of Libra which is exactly 180° opposite to 0° of Aries. But this is wrong.

We must understand that what Hipparchus recorded was just what he perceived with the naked eye supplemented by simple instruments available at that time. It was his OBSERVATION or what he saw in night time sky while sitting at Rhodos of Greece at latitude of 39°N. It is assumed here that Hipparchus made his observations at an altitude of zero i.e. at sea level. Due to atmospheric refraction what we see at horizon is not exactly located on a horizontal plane. Light rays bend due to the influence of atmosphere and what we see at the horizon is actually a few arc minutes beyond the exact horizontal plane (Figure 3). The refractive index, which depends on environmental temperature and degree of elevation of the object from horizon, is greatest at horizon.

Fig 3: Effect of atmospheric refraction on setting Sun

To a person at O watching sunset, Sun (S) appears to be above the horizon (S’), apparent sun, even though it has actually already descended below the horizon, true Sun (S).

The blue line indicates horizon which if extended to the opposite direction will represent the degree of Ascendant or Lagna.

Curved light black line represents light rays emanating from Sun influenced by atmospheric refraction.

Dark black line represents the position of the image (S’), the apparent Sun which is visible to O, of true Sun (S).

At 39° north of equator in Rhodes the average atmospheric temperature in March at present is 13.6°C. For these values the refractive index at sea level is 33.37′ ⁽²⁾. When calculating the degree of ascendant or Lagna point for astrological purposes the refraction is not considered (see notes in Figure 3). Therefore at the point of time when calculated 0° of Aries rises in East we cannot expect one to see with naked eye, a heavenly body located at 0° of Libra to descend at the Zenith. It will be 33.37′ arc minutes beyond the horizon. Therefore what Hipparchus saw on the western horizon when Spica was setting was actually NOT 0° Libra BUT 29°26.63′ Virgo or 29°26’37.8″ Virgo.

Results

Since this is an observation made in relation to the actual position of stars in sky this can be considered as sidereal longitudes of Aries and Spica. The zero ayanamsha years is defined as the time of coincidence of zero points of both tropical and sidereal zodiacs. That is both sidereal and tropical longitudes will be the same and the ayanamsha on that day will be zero. Since the starting point of sidereal zodiac is not known precisely, and the purpose of this article is to find out the starting point of sidereal zodiac, we cannot rely on any available sidereal longitudes of star Spica. Rather we must depend on the tropical longitudes devised with a high degree of precision by modern astronomers. At the time of zero ayanamsha year the sidereal longitude must be as same as tropical longitude. Since the starting point of tropical longitude is the vernal equinox, the exact time of coincidence will be on the day of vernal equinox of a given year.

To calculate the tropical longitude of star Spica the author of this paper used ZET 9.1 software and opted for Swiss ephemeris. Vernal equinox was calculated using the same software. It was found that at the time of vernal equinox in 244 AD, i.e. March 20, 244 AD at 18.46.54 hours GMT, the tropical longitude of Spica was 29°26’23.84″ Virgo. This is almost as same as the sidereal longitude of Spica obtained above from observations of Hipparchus i.e. 29°26’37.8″ Virgo, just a difference of 14″ arc seconds.

In 245 AD the tropical longitude of Spica is 29°27’10” Virgo, a difference of 32″ from the Hipparchus’s observed value. In 243 AD it is 29°25’35” Virgo, a difference of 63″. It is only in 244 AD that tropical longitude of Spica is closest to the observed longitude of Spica by Hipparchus.

Conclusion

Therefore it can be concluded that the zero ayanamsha year is 244 AD.

See the Appendix for ayanamsha figures for a period of 120 years based on 244 AD zero ayanamsha year. The table has been prepared with the help of Zet 9.1 Lite software (Please note that yearly rate of precession is not static because Nutation of zodiac has been taken into account).

Discussion

Sun-Jupiter Conjunction Cycle

There is a discrepancy of 14″ arc seconds between the tropical longitude of Spica in 244 AD (29° 26’23.84″ Virgo) and the sidereal longitude of Spica that was observed by Hipparchus (29°26’37.8″ Virgo). Since the evidence of Hipparchus comes from an observation made more than 2000 years ago, and if his observations were of rough approximations, it is possible that the Zero ayanamsha year is either 243 AD or 245 AD or any other year around 244 AD. This doubt can be sorted out considering Sun-Jupiter cycle.

According to an article circulating on the internet ⁽⁴⁾ “like Sun and Moon opposition and conjunction form the natural cycle for a month, Jupiter and Sun conjunction / opposition create a natural cycle defining not only a year but also the entire precessional cycle of 25800 years”. Accordingly at the time of coincidence of two zodiacs Sun and Jupiter must be either in conjunction or opposition. Since at vernal equinox Sun is at the beginning of Aries, Jupiter should be either at the beginning of Aries or Libra.

At the time of vernal equinox of 244 AD, when the tropical Sun was at 0°00’00” of Aries, Jupiter was located at 5° 4′ in Aries, just 5° apart. The next Sun-Jupiter conjunction in Aries would be either in 232 AD or 256 AD which are far apart from 244 AD. Neither in 243 AD nor 245 AD at vernal or spring equinoxes is there any Sun – Jupiter close conjunction or opposition.

Atmospheric Refraction

As explained earlier atmospheric refraction causes astronomical objects to appear higher in the sky than they are in reality (Figure 3).

The sole basis of this article is longitude of star Spica observed by Hipparchus with naked eye. This longitude was refined against atmospheric refraction by the author.

Was Hipparchus aware of atmospheric refraction? The answer is No. By analyzing the difference of 5 hours of predicted time of equinox by Hipparchus and the observation made on Alexandria’s large public equatorial ring on 24 March 146 BC it has been concluded that Hipparchus was not aware of atmospheric refraction ⁽⁵⁾. Many scholars have identified the errors caused by atmospheric refraction in the work of astronomical observations made by Ptolemy ⁽⁶⁾.

In my humble opinion, not only Hipparchus and Ptolemy but also Varahamihira (505–587 AD), who lived about 700 years later than Hipparchus, and even other ancient astrologers and astronomers were not aware of atmospheric refraction. The law of “Refraction” was scientifically discovered only in early 1600s AD ⁽⁶⁾. Thus the longitudes of any planet or star measured and recorded by any ancient astronomer by observation is bound to be erroneous slightly by a few arc minutes (in a range from 0′ to 34′ depending on elevation above the horizon) ⁽²⁾ from the exact position of celestial objects in sky; therefore all such ancient knowledge and figures must be interpreted and used with caution.

Dulakara Ayanamsha

It is interesting to note that an ayanamsha by his name of Hipparchus is already in use (Table 1). This had been devised by Raymond Mercier based on Ptolemy’s accounts in Almagest. Rather than taking Spica, he used star Zeta Piscium as the reference point. However this ayanamsha differs quite significantly from the ayanamsha that I propose. According to Mercier’s ‘Hipparchus ayanamsha’ the Zero ayanamsha year is 545 AD. Therefore to differentiate from Mercier’s Hipparchus ayanamsha, I would like to coin the term “Dulakara Ayanamsha” for what I found. The literal meaning of ‘Dulakara’ in Sinhalese language (mothertongue of the author) is ‘Sun’.

Given below (Table 2) is a comparison of Dulakara ayanamsha with other ayanamshas proposed based on star Spica.

Table 2 Comparison of Dulakara ayanamsha with other ayanamshas

(Z.A.Y. – Zero Ayanamsha Year) Ayanamshas given are for 1st January 2011

It is interesting to note that Dulakara and Kuglar 2 ayanamshas are almost the same. The year of coincidence of Dulakara ayanamsha is 244 AD, and according to Kuglar 2 ayanamsha it is 243 AD. The raw data for the former comes from the work of Hipparchus whereas for the latter it is from Babylonian planetary tablets. Babylonian astronomy is older than Greek astronomy. It is possible that Hipparchus used Babylonian astronomical material, including methods as well as observations, to some extent in his studies. Certain historians actually believe that Hipparchus’s work provides a link between Babylonian and Greek astronomy. Therefore it is not a surprise that Babylonian – Kuglar 2 and Dulakara ayanamshas are almost the same because the sources for the calculations are probably having a similar origin.

When compared with Lahiri ayanamsha, which is commonly in use among Vedic astrologers, Dulakara ayanamsha is 0°34’29” more than Lahiri ayanamsha. Thus Dulakara ayanamsha for a given date can be easily obtained by adding 0°34’29” to Lahiri ayanamsha.

It is my sincere and humble belief that Dulakara ayanamsha is the most precise ayanamsha that has ever been proposed. It is not just another ayanamsha but it is the ayanamsha that everybody has been in search for for centuries.

However readers need not stop here but should continue to research on this important topic either to refute or confirm Dulakara Ayanamsha, or to come out with a better one.

About Author: I started reading on Vedic astrology when I was 15 years old. I learnt by self study. In addition to classic Vedic astrology I am also interested in Western astrology, Magi Astrology and a few modern variants of Vedic astrology. I incorporate and blend all branches of astrology known to me when analysing charts which I do as a hobby at leisure time and am a consultant surgeon by profession. The best single word that I can find as of today to describe astrology is “Astrology is a Science”, and as such we should not stick ourselves only to ancient texts.