Samvatsar in contrast to Samvat
Samvatsar which earlier meant a year, now does not represent the year any longer. Instead, it represents the time it takes Guru (Jupiter) to move across one rashi. The Jupiter cycle is nearly 12 years. Hence, we get kumbha mela every 12 years which follows this cycle.
The cycle of 60 years (also followed by the Chinese) comes from the fact that a cycle of Saturn takes 30 years and the LCM of 12 (Jupiter’s cycle) and 30 (Saturn’s cycle) is 60. At the end of a 60 year cycle both Jupiter and Saturn will be back to the original position. These years also have names:
1) Prabhava … to 60) Akshara, but the Chinese have given them names of animals which are more popular.
Celestial equator, Latitude, longitude and celestial coordinates systems
As we have seen, the earth is a sphere which rotates around an axis, the two points at which this axis passes through the earth are called the poles (the north and south pole)
The equator is a great circle on the earth equidistant from the two poles. The equator divides the earth into two parts called the north and south hemisphere.
To decide the latitude and longitude of the given point P. We use a secondary circle which passes through the two poles and point P. Let the circle intersect the equator at Q. The arc PQ measured in degrees is called the latitude of P.
To measure the longitude, we need another circle passing through the poles called the prime meridian. In India the prime meridian was taken to the secondary circle passing through Ujjain.
Nowadays it is taken to be the circle passing through Greenwich. Let the prime meridian intersect the equator at O. The arc OQ is called the longitude of P.
Celestial coordinates
This system of coordinates on earth is extended in various ways to the celestial sphere. We imagine that all the stars and planets in the sky are on a very large sphere called the celestial sphere.
The poles are extended to intersect the space as the celestial North and South poles. Consider the plane containing the earth equator. If we imagine that it is extended indefinitely, this plane will intersect the celestial sphere in a circle called the celestial equator.
The Celestial prime meridian is called the first point of Aries or the vernal equinox. It is denoted by y , the greek letter gamma, because that symbol resembles two horns of a ram, and is the zodiac symbol for Aries. Even though this has moved it is still called the first point of Aries.
So a circle through the poles and y also intersects the celestial equator at
y . And the angle y Q is called the right ascension abbreviated to RA (corresponding to the longitude) and the rotation up is called declination abbreviated by dec (corresponding to the lattitude).
A daily table of star and planet position is called an ephemeris. Software C2A can generate the ephemeris table. There are also python packages like pyeph and skyq that can give accurate information on the same.
Ecliptic coordinates
In a second system of coordinates, one uses the ecliptic circle or plane instead of the equatorial plane. The ecliptic plane is around 23’ off from the equatorial place and the vernal equinox (and its diametrically opposite point on the celestial equator) is there the two meet.
The ecliptic plane is the plane in which the earth orbits around the sun or equivalently, the plane in which the sun is observed to move in the sky.
A line from the center of the ecliptic circle, and perpendicular to its plane, intersects the celestial sphere in two points, again called ecliptic poles.
We use a great secondary circle from the ecliptic poles through the point P and let it intersect the ecliptic circle at Q. The angle PQ is called the celestial latitude. The celestial latitude of the ecliptic plane is zero. The prime meridian remains a great circle through the ecliptic poles and the first point of Aries ???? which point is at the intersection of the ecliptic and equatorial planes. The angle ???? Q is called the celestial longitude.
For this purpose, there is a third coordinate system called the ALT-Azimuth system. In this system, the basic plane is the one which passes through the position of the observer on earth and the horizon.
Precession of the equinoxes
This precession of a rotating object is observable in an ordinary top. While the top spins about its axis, the axis itself rotates. This rotational motion of the axis is called precession.
The same thing happens in the case of the earth. Not only does the earth rotate about its axis, the axis itself slowly rotates pointing to different parts of the sky and different times.
Thus, the axis currently points to what is called the pole star. But this star was not the pole star 1500 years ago and will not be 1500 years later.
The precession is rather small 50.3 arc seconds per year (an arc second is 1/60 of a minutes which is 1/60 of a degree) or precession is about 1° in 71.6 years or 360° in 25,772 (or 26000 years the rate of precession may vary a bit)
What the difference does that make?
This extra rotation of the axis of rotation itself adds a small amount to the rotation speed of the earth around the sun.
How much?
The sun revolves around the sky 360° in one sidereal year, of 365.26 days. So, it covers 1° in 365.26/360 days. Therefore, it covers 50.3 arc seconds = 50.3/3600 degrees in 50.3/3600 x 365.26/360 days.
Since 1 day is 1440 minutes, this amounts to 50.3/3600 x 365.26/360 x 1440 = 20.4 minutes. That is because of precession, the time from equinox to equinox = the tropical year is less by 20.4 minutes than the sidereal year or the time it takes for the earth to revolve around the sun.
First point of Aries
As a result of precession not only does the pole star change with time but so do the equinoxes and the constellations. The north celestial pole, decided by the earth’s rotation axis, is always perpendicular to the celestial/earth equator.
So when the axis rotates so does the equatorial plane. The equinox is the point when the sun crosses the celestial equator.
That is, it is at the intersection of both ecliptic and equatorial planes. This intersecting point will change with the movement of the equatorial plane.
That is the vernal equinox or the first point of Aries will change with time. In fact, the vernal equinox is no longer in the Aries constellation. It is currently in the Pisces constellation.
Now what coordinates does one use to measure this movement?
A difference of 0.5° in 35 years may not seem like a lot but especially for space science it can be a huge difference because distances in space are huge.
Therefore, NASA offers a choice of various coordinate systems, J2000 refers to the equinox of the epoch of 2000 CE or one can use the current equinox, called equinox of date etc.
What to do in the context of Indian astronomy and the Indian calendar?
- The first approach is that the “first point of Aries” means what we actually observe as the first point of Aries (mesh). This is called the Nirayana or Sidereal system. Therefore, there is a difference between sidereal and tropical longitudes, the difference is called ayanamsh.
- Western ephemerides usually follow the tropical system with the idea that star positions given in star charts are to be updated every 50 years. The Indian approach has both systems.
Panchang
The panchang has 5 elements, we already learnt three tithi, vara and nakshatra. The other two elements are karan and yoga which we will not be using.
The scientific panchang has only the 5 elements of samvat, maada, tithi, vara and samvatsar.
A karan is just half a tithi so there are 60 karana’s. These are divided into 4 fixed karan plus 7 moving karan to reduce to 11. The 7 moving karan’s repeat 8 times in a chandramas of 30 tithi’s = 60 karan’s. The 4 fixed karans are kimstughna, sakuni, catuspada, nagava. The 7 repeating karanas are Bava, Baklava, Kaulava, Taitila, Garaja, Vanija, visti.
How to calculate karana
This is a simple process of dividing the difference in longitude of moon and sun by 360’ (minutes corresponding to the 6° as it is half of a tithi which is 12°) and then adding one in the shukla paksha and diminishing 1 in the krishna paksha.
Yoga
Yoga (pronounced Yog)here means sum (not yoga which means union of atman and brahman, but misunderstood as hatha-yoga = physical exercises). The yoga in astronomy/astrology is the sum of the celestial longitudes of the sum and the moon. Modulo 360° and divided by 13°20’ to get a number from 1 to 27 as in the case of Nakshatra’s.
That is. It is the time in which the sum of the longitudes of the sun and moon modulo 360 increase by 13°20’.
Yoga mentioned in the Panchang is not found in Surya Siddhanta. Aryabhatiya or Laghu Bhaskariya etc. Its scientific significance is not clear.
The Drk Panchang (Lahiri)
There are two ways to correct the texts of Indian astronomy one is to attempt to calculate all that has changed. The other is to use the understanding and relate it to the current observations (e.g. of the pole starts, etc)
The word Drk comes from Drshti i.e. relates to current observation. This panchang is based on the ephemeris (position of sun, moon etc) actually observed. The starting point is the ephemeris published by NASA Jet Propulsion Lab (JPL).
That is a tropical (sayan) ephemeris which must be converted to a nirayana using one of the various systems of ayanamsh.
This JPL ephemeris had been packaged for astrology by the “Swiss ephemeris”. The original release of the software in 1997 was based on the DE405/406 ephemeris. Since release 2.00 in February 2014, it is based on the DE431 ephemeris released by JPL in September 2013.
The authors of the package Swiss Ephemeris are Dieter Koch and Alois Treindl sold by the company Astrodientist AG, Switzerland.
It also has a table of the Lahiri Ayanamsa used in the Indian national panchang one of whose authors was N. C. Lahiri a mathematician, and part of the Indian calendar reform committee set up after Independence.
To calculate the Ayanamsa it is necessary to set the starting date and time. Lahiri ayanamsha = 23° 15 ’00’’.658 on 21 March 1956, 0:00 TDT (Terrestrial dynamical time). Reference star Spica, its J2000 longitude is 170° 58 ’58’’. Then calculate precession for any time from that value and the given model of precession.
To calculate the panchang for a given Gregorian date. The procedure is to take a Gregorian date and convert it to Julian day which is easily related to Ahargana as explained above. Ahargana = jd – 5888465.5.
For that Ahargana we can immediately calculate the Kali samvat or the number of Kali years elapsed using the duration of the sidereal year,
Kali samvat = integer part of (ahargana / duration of sidereal year)
Now we have precise position/longitudes of sun and moon from the ephemeris for the Julian date. We then convert it to tithi.
Traditional method of calculating tithi
Traditional Indian astronomy texts give the revolution numbers for the various planets. Example as stated in the Laghu Bhaskariya 9-14
Calculate the Ahargana as explained in Laghu Bhaskariya
Different authors have different values (Table from Rao)
These mean (average) motions are obtained as follows,
Example 1: As per table on Aryabhata, if in 4320000 (sidereal) year there are 1577917500 civil days, then in 1 (sidereal) year there are 157791 7500 ÷ 4320000 = 365.258680 days compared to the modern value 365.25636 days.
Example 2: If the moon makes 57753337 revolutions in 157791750 civil days, then it makes 1 revolution in 1577917500 ÷ 57753337 = 27.321668 days = sidereal month, modern value 27.321661 days.
We can easily get the mean motion of the moon. If in 27.321668 days the moon moves 360°, then in 1 day it moves 360/27.321668 = 13.176355°, very slightly different from its modern value 360/27.321661 = 13.176358°. Hence, in 1 day the moon approximately covers 1 Nakshatra = 13.33°.
Likewise if in 365.258680 days the sun covers 360° then in 1 day it covers 360/365.258680 = 0.9856°. Hence, it covers 1 rashi = 30° in 30/0.9856 = 30.438 days.
In these revolution numbers, given in all traditional astronomy texts, the revolutions of the sighrocca of the inferior planets Mercury and Venus are given. This corresponds to their revolutions about the sun.
Since the sighrocca as seen from earth occurs at “superior conjunction” when their relative velocities of rotation around the sun are most different.
The mean motion obtained above enables the calculation of mean longitude of sun and moon and planets for a given ahargana. These are mean values and the planets do not move with uniform speed as the very existence of mandocca etc. shows.
Obtaining the truth or values required a series of corrections called the manda and sighra corrections. These use the epicycle model to explain who the planets are slowed down or speeded up.
In this epicycle theory, the mean planet moves with its mean velocity in a circle round the earth (from west to east), the true planet moves in a smaller circle with its center on the mean planet, in the opposite direction from east to west. The faster and slower motions are then explained by adding the two velocities.
Indians had a clear idea of the cause of eclipses
Aryabhata stated this explicitly
However, many historians insist that Indians were superstitious and thought that eclipses are due to demons rahu and ketu.
In fact, Lalla in his 20th chapter denies the demonic theory of eclipses very explicitly. Stating if they were indeed demons then why is it that we can predict accurately when these are going to happen.
Solar eclipses
A solar eclipse is caused by the moon coming between the earth and the sun, obstructing our view of the sun so that the shadow of the moon falls on some point of the earth.
When does an eclipse occur?
This can happen only when the sun, moon and earth are very nearly in a straight line. This does not happen at every full moon (purnima, lunar eclipse) or new moon (amavasya, solar eclipse)
Moon’s orbit and ecliptic
Because, the plane of the lunar orbit around the earth is at an angle to the plane in which the sun appears to revolve around the earth. The angle is about 5° 7’ 47.9’’.
Why a solar eclipse does not happen every amavasya?
A solar eclipse does not happen at every amavasya, but can take place only when both sun and moon are near one of the moon’s nodes called Rahu and Ketu; these are the two points at which the orbit of the moon intersects the ecliptic.
Conclusion:
In this session, we explored the ideas of C. K. Raju, who showed us that it’s okay to think differently and question old ideas. His work in mathematics, especially with time and calculus, teaches us that learning is not just about accepting what’s already known, but about exploring new possibilities. As we finish, let’s take inspiration from C. K. Raju and always be curious, willing to learn, and open to new ways of understanding the world around us.
