Computation of the great Jupiter-Saturn perturbation !

In Indian Astronomy, the great Jupiter - Saturn perturbation, is calculated using the formula given below.

The Kali year, ky, is calculated and 4660 is deducted from it to get the kyb, the kali year balance. Its Bhujamsa,x is calculated using the Equation of Bhuja and then its Bhujajya ( Sin x ) is calculated to get the value of Manda Phala, a.

Mathematically, it can be written as

Y, (Year of Birth) + 3102 = ky, the Kali Year of Birth

kyb ( kali year balance ) = ( ky - 4660 )

x = kyb*360/918

b, bhujajya = sin(x)

Since the duration is 918 years and perturbation amplitude is 1187 seconds,

Manda Phala, a = sin(x) * 1187

( Example - My year of birth is 1955. Add 3102 and we get 5057. Less 4660 is 397. 397*360/918 is 155.6 and its Bhujamsa is 25 degrees roughly ( 180-155). So Sin (25 ) is 0.41. Manda Phala, therefore is 0.41*1187 seconds or Vikalas. )


This value, Manda Phala, a is added to Jupiter's mean longitude, if Jove's long is less than 180 and subtracted if Jove's long is greater than 180.

In the West, this discovery is attributed to Laplace. The duration of the perturbation of 918 years is known as Laplace period. Its amplitude of .332 degrees is similar to the Indian value of 1187 seconds.

Laplace gives the equation 2 nj - 5 ns, bringing to light the 5:2 Resonance in the orbits of the two celestial giants. 2 revolutions of Saturn = 5 revolutions of Jupiter = 60 years !

Computation of Ahargana

Ahargana is defined as the no of days elapsed after the start of the Iron Age, the Kali Yuga, which began on 3102 BC.

Kali Era = English Era +3102

The no of days elapsed from Kali Start was 1822903 on 01/01/1890.

Hence

Ahargana = ( xdate - 01/01/1980 ) + 1822903

where xdate is the date of birth.

This Ahargana is printed on every V A Horoscope. All calculations in Indian Astronomy is based on Ahargana. Ahas means day in Sanskrit and gana means calculation.

The present Kali Era is therefore

3102 + 2010 = 5112

The difference between the Malayalam & English eras are 825. Malayalam Era is 1185 now and if you add 825 to Malayalam Era, you get English Era.

Definition of Ascendant

The Ascendant is defined as the intersecting point between the Ecliptic and the Celestial Horizon and is the Eastern Celestial Horizon. It is represented by the formula

Tan L = Sin E / cos E cos w - Sin w Tan A

where A is Latitude of the place, w is the maximum declination of the Sun, E is the Right Ascension of East Point ( Sayana Kala Lagna ) & L - the Lagna.

Raseenam Udayo Lagnam - Udaya Lagna, the Ascendant, the pivot of the horoscope, is the Eastern Celestial Horizon. In Astrology everything is based on Lagna, Sarvam Lagnepi Chinthayel.

180 degrees opposite to the Ascendant is the Descendant, the Astha Lagna and is the Western Celestial Horizon.

THE EQUATION OF BHUJA

Bhujamsa is the degrees traversed by the planet in the Zodiac. In the mighty 360 degree Circle, the planet may be anywhere and its position is the mean longitude of the planet. The planet actually is in the ellipse and its position is the true longitude of the planet.

If the planet is say at 42 degrees, then the Equation of Bhuja states that its Bhujamsa is 42 degrees. That is in the first Oja Pada, in the first quarter of 0-90 degrees, the Bhujamsa is the same.

If the planet is say at 110 degrees, then the Equation of Bhuja states that its Bhujamsa is 70 degrees. That is in the first Yugma Pada, in the second quarter of 90-180 degrees, the Bhujamsa is 180 - bhuja.

If the planet is say at 200 degrees, then the Equation of Bhuja states that its Bhujamsa is 20 degrees. That is in the second Oja Pada, in the third quarter of 180-270 degrees, the Bhujamsa is bhuja - 180.

If the planet is say at 300 degrees, then the Equation of Bhuja states that its Bhujamsa is 60 degrees. That is in the second Yugma Pada, in the fourth quarter of 270-360 degrees, the Bhujamsa is 360 - bhuja.

Bhuja Jya is Sin ( M ) and is the sine value of the Kendra. Bhuja Jya is used in the calculation of Epicycles, in calculating the Equation of Center for Moon, Jupiter and Saturn.

The Manda Phala, the Equation of Center is given by the formula

Equation of Center, Mandaphala =Circumference of the Epicycle ( Parama Phala ) * Manda Kendra Jya ( Sin M ).

This is the equation used in calculating the 14 perturbations of the Moon, the five of Jupiter and the six of Saturn.

The Auxiliary Circle, the Vikshepa Vritta

In order to compute the celestial longitudes of planets, first the Graha Madhyamam, the mean longitude of the planet is computed.

We have to understand that the planets traverse in elliptical orbits. If their orbits are circular, then there is no need for jya samskaras ( trignometric corrections ).

Once the mean longitudes of the planets are ascertained, then we first start with the First Jya Samskara, the first trignometric correction. Manda Jya means Sin M in Western Astronomy.

The Kepler Equation is M = E - e Sin E, where e is eccentricity and E is the Eccentric Anomaly, an auxilary angle in Kepler's equations.

Like Kepler who brought in an auxiliary angle ( E, the Eccentric Anomaly ), Indian Astronomy uses Vikshepa Vritta, an auxiliary circle. The mean longitude of a planet reduced by Manda Kriya is the Vikshepa Vritteeya Sphuta, the once corrected longitude of the planet.

While Western astronomers compute the celestial longitudes using the formula Theta = v + w ( Celestial Longitude = True Anomaly + the Argument of Perihelion ), Indian astronomers use the Triune Trignometric Method. Longitudes are corrected thrice using Manda Kriya, Parinathi Kriya and Sheegra Kriya.

The perturbations of planets

All planets have perturbations. Moon has 300 perturbations, of which 14 are major. Hence for 14 perturbations, 14 jya samskaras have to be done ( Chatur Dasa Jya Samskara). The largest of them is the Evection. There are others like the Variation, the Annual Equation and the Parallactic Equation. When 14 trignometric corrections are done, we get the Reduced Longitude of the Moon, the Samskritha Chandra Madhyamam.

Jupiter has five major perturbations ( guror pancha kendrani bhavanthi ) and Saturn has six. So five jya samskaras and six jya samskaras have to be done for Jupiter and Saturn, before commencing the Triune Trignometric Method.

The four systems of Astronomy

There are four major methods of calculation in Astronomy

They are

Longitudes calculated along the Zodiac or Ecliptic - The Ecliptic System
Longitudes calculated along the Celestial Equator - The Equatorial System
Longitudes calculated along the Celestial Horizon - The Horizontal System
Longitudes calculated along the Celestial Meridian - The Meridian System


Longitude measured along the Kranti Vritta, the Ecliptic or Bha Chakra , the Zodiac is known as Kranti Vritteeya Sphuta, the true longitude of the planet.

Longitudes measured along the Vishuvat Vritta, the Celestial Equator is known as Vishuvat Vritteeya Sphuta, Right Ascension.

Udaya Lagna, the Ascendent and Astha Lagna, the Descendent are measured along the Celestial Horizon,The Kshitija

And the Madhya Lagna, the MC and the Patala Lagna, the IC are measured along the Celestial Meridian, the Nadi Vritta.

About Sine, Cosine and Reverse Sine

Jya ( Sine ), Kotijya ( Cosine ) and Utkram Jya ( Versine ) are the three trignometric functions introduced by the Indian astronomers and mathematicians.

In order to compute the celestial longitudes of planets, these functions were used by the trinity of Indian Astronomy, Bhaskara, Brahnmagupta and Aryabhata.

Thrijya, the Radius or R

If 360 = 2 Pi r
then, r = 360/2Pi in degrees

One degree is 60 minutes and one minute is 60 seconds and hence one degree is 3600 seconds.

R in seconds will be ((360/2pi)* 3600 ) Vikalas or 206265 seconds. This figure 206265 is known as the Magic Figure of Astronomy.

Bhujajya = R Sin
Kotijya = R Cos

The arc sine of the angle is Bhujachapa
The arc cosine of the angle is Kotichapa
The arc tangent of the angle is Sparshachapa

By Jya, Bramhmagupta meant 5 degrees of a sign of 30 degrees. Hence a Zodiacal Sign consists of 6 Jyas ( 30 degrees ) . The Zodiac of 360 degrees was divided into 4 quarters of 90 degrees each. Three Jyas of 30 degrees each becomes a quadrant of the Zodiac and was called Thrijya. Thrijya is also the Radius, the Vyasardha.

His magnum opus, the Brahmasphuta Siddhanta was translated by the Arabs as As Sind Hind. Jya became jiba and Kotijya became kojiba in Arabic. It was translated into Latin as sinus ( meaning " bosom " ). So Sinus and Co-sinus when translated into English became Sine and Cosine !

The Aryabhateeyam of Aryabhata was translated by the Arabs as Al Arjabhat. Trignometry is derived from the Sanskrit Thrikonamithi and Geometry from Jyamithi !

The Celestial Meridian, the Nadi Vritta

All celestial bodies appear to rise in the east, travel westward and set in the west, because of this relative diurnal motion. When the axis of the earth's rotation is extended, it meets the Celestial Sphere on two diametrically opposite points called Celestial Poles. The one in the direction of the Earth's north pole is called the North Celestial Pole and the one in the opposite direction is called the South Celestial Pole.

The Celestial Horizon, the Kshitija

To the observer standing at a place on the surface of the earth and looking around describing a full circle, the earth appears to meet the Heavens along a circle. From the observers' frame of reference, this circle is called the Celestial Horizon, the Khshitija.

The Celestial Meridian, the Nadi Vritta

Almost 90 degrees to the Celestial Horizon, you will find another circle ( the line perpendicular to the plane of the horizon ). This Circle is the Celestial Meridian. The highest point on this Celestial Meridian is called Zenith, the MC and the lowest, the Nadir or IC.

The 18 Siddhantas !

Astronomical Knowledge is known in Sanskrit as Siddhanta. Transcendental Knowledge is Vedanta.

There are 18 astronomical treatises compiled by 18 Seers. Each Siddhanta is named after its author. They deal exhaustively with Astronomy.

They are

Surya Siddhanta
Pitamaha Siddhanta
Vyasa Siddhanta
Vasishta Siddhanta
Atri Siddanta
Parasara Siddhanta
Kashyapa Siddhanta
Narada Siddhanta
Garga Siddhanta
Marichi Siddhanta
Manu Siddhanta
Angira Siddhanta
Lomasa Siddhanta
Paulasa Siddhanta
Yvana Siddhanta
Chyavana Siddhanta
Bhrigu Siddhanta


These are the main astronomical treatises. There are others like Arya Siddhanta by Aryabhata II, Brahma Sphuta Siddanta by Brahmagupta and Maha Bhaskareeya by Bhaskara.

The Celestial Sphere, the Khagola !

When we look up at the heavens on a clear night, we find a multitude of celestial bodies, illuminating the sky with their radiance. It looks like a large hollow hemisphere, with the observer at the center. The planets and the luminaries seem to be scattered thorough the heavens at large distances.

This picture of the observer, the sky as a large crystalline hemisphere , is a very convenient model for the study of the Heavens.

This hemispherical model of the heavens is called Khagola, the Celestial Sphere. Another geocentric model, the Khagola is an imaginary sphere of large radius.

Astronomy, Maths and Astrology ( Siddhanta, Samhita and Hora ) are considered to be Apaurusheya, divine in origin. They were revealed to the Rishies in higher states of Consciousness and hence are revealed sciences. Descendit e Caelo, they cometh from Heaven !

Their date of compilation is believed to be 12th century BCE

Rotation, Revolution & Diurnal Motion

The earth rotates about its own axis, from west to east in the course of a day. Due to this rotation, the observer is carried eastward. But the observer is unaware of his motion in space. To the observer, the Celestial Sphere with all the heavenly bodies, is seen as rotating from east to west. This apparent westward motion of the heavenly objects is known as the diurnal motion.