The Idea of Planetary Mass in India

Many ancient cultures have contributed to the development of Astro Physics.

Some examples are

The Saros cycles of eclipses discovered by Egyptians
The classification of stars by the Greeks
Sunspot observations of the Chinese
The phenomenon of Retrogression discovered by Babylonians

In this context the Indian contribution to Astro Physics ( which includes Astronomy, Maths and Astrology ) is the the development of the ideas of planetary forces and differential equations to calculate the geocentric planetary longitudes, several centuries before the European Renaissance.

Natural Strength is one of the Sixfold Strengths, Shad Balas and goes by the name Naisargika Bala. It is directly proportional to the size of the celestial bodies and inversely proportional to the geocentric distance. ( Horasara ).

Naisargika Bala or Natural Strength is used to compare planetary physical forces. When two planets occupy the same, identical position in the Zodiac at a given instant of time, such a phenomenon goes by the name of planetary war or Graha Yuddha,happening when two planets are in close conjunction. The Karanaratna written by Devacharya explains that the planet with the larger diameter is the victor in this planetary war. This implies Naisargika Bala.

The Surya Siddhanta says " The dynamics or quantity of motion produced by the action of a fixed force to different planetary objects is inversely related to the quantity of matter in these objects"

This definition more or less equals the statement of Newton’s second law of motion

M = Fa
or
a = F/M

So it strongly suggests that the idea of planetary mass was known to the ancient Indian astronomers and mathematicians.

Differential Equations used in Siddhantas

Motional strength is one the sixfold strengths, known as Cheshta Bala. This motional strength is computed by the formula

Motional Strength = 0.33 ( Sheegrocha or Perigee - geocentric longitude of the planet ). This motional strength is known as Cheshta Bala.


Differential Calculus is the science of rates of the change. If y is the longitude of the planet and t is time, then we have the differential equation ,dy/dt.

During direct motion, we find that dy/dt > 0 and during retrogression dy/dt < 0. During backward motion of the planet ( retrogression) y decreases with time and during direct motion y increases with time. When there are turning points known as Vikalas or stationary points, we have dy/dt = 0 ( where planets like Mars will appear to be stationary for an observer on Earth ).

The quantity in bracket is the Sheegra Anomaly, the Anomaly of Conjuction, the angular distance of the planet from the Sun. This Anomaly or Cheshta Bala is maximum at the center of the Retrograde Loop. Cheshta Kendra is defined as the Arc of Retrogression and is the same as Sheegra Kendra, Kendra being an angle in Sanskrit. During Opposition, when the planet is 180 degrees from the Sun, Cheshta Bala is maximum and during Conjunction, when the planet is 0 degrees from the Sun, it is minimum

The Motional Strength is given in units of 60s, Shashtiamsas.

Direct motion ( Anuvakra ) 30
Stationary point ( Vikala ) 15
Very slow motion ( Mandatara ) 7.5
Slow motion ( Manda ) 15
Average speed ( Sama ) 30
Fast motion ( Chara ) 30
Very fast motion ( Sheegra Chara ) 45
Max orbital speed ( Vakra ) 60
(Centre of retrograde)

The Nine Oribtal Elements

Mean and true planetary longitudes in the Zodiac is computed by Nine Orbital Elements, in Indian Astronomy.


Mean longitude of Planet, Graha Madhyama , M
Daily Motion of the Mean Longitude, Madhyama Dina Gathi, Md
Aphelion, Mandoccha, Ap
Daily Motion of Aphelion, Mandoccha Dina Gathi, Apd
Ascending Node, Patha, N
Daily Motion of Ascending Node, Patha Dina Gathi, Nd
Heliocentric Distance, Manda Karna, radius vector, mndk
Maximum Latitude, L, Parama Vikshepa
Eccentricity, Chyuthi,e


In Western Astronomy, we have six orbital elements

Mean Anomaly, m
Argument of Perihelion, w
Eccentricity, e
Ascending Node, N
Inclination, i, inclinent of orbit
Semi Major Axis, a

With the Nine Orbital Elements, true geocentric longitude of the planet is computed, using multi step algorithms.

There is geometrical equivalence between both the Epicycle and the Eccentric Models.

The radius of the Epicycle, r = e, the distance of the Equant from the Observer.

Astronomical Units of Time Measurement

We find Yuga cylces mentioned not only in astronomical works, but also in mythological works in India.

Kali Yuga began on the midnight of 17th Feb 3102 BCE and the duration of this Kali Yuga is said to be 4.32 K solar years. Dwapara is 2*Kali Yuga years. Treta is 3*K Y and Krita Yuga is 4*K Y.


Krita Treta Dwaparascha Kalischaiva Chaturyugam
Divya Dwadasabhir varshai savadhanam niroopitham


Thus an Equinoctial Cycle, Mahayuga is equal to 4+3+2+1 = 10 KYs.

E C = 10 KYs.

A Greater Equinoctial Cycle ( Manvantara ) = 71 Equinoctial Cycles

There are cusps happening in between Manvantaras, each equal to a Krita Yuga in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas.

Hence 14*71+6 = 1000 Mahayugas = 4.32 Billion Years

Sahasra yuga paryantham
Aharyal brahmano vidu
Ratrim yugah sahasrantham
The Ahoratra vido janah ( The Holy Geetha ).

This is one Cosmological Cycle, called Brahma Kalpa.

Chaturyuga sahasram indra harina dinam uchyathe


From one second, it can be logarithmically shown, upto 10^22 seconds. This is what the above diagram shows. This diagram is by courtesy of Wikipedia.

From 10^0 it goes upto 10^22 seconds. One day of Brahma is 4.32 billion years and 100 years of Brahma therefore is 311.04 trillion years, which is shown logarithmically above.

One Asu is 4 seconds, one Vinadi is 24 seconds and one Nadi is 24 minutes. 60 such Nadis make up one day. This is the Sexagesimal division of a day into 60 Nadis ! In Astronomy, one degree is sixty minutes and one minute is sixty seconds. Hence sexagesimal division is justified ! 365.25 such days constitute a year and Hindu calculation goes upto 311.04 trillion years !

Astronomical Units of Time Measurement

We find Yuga cylces mentioned not only in astronomical works, but also in mythological works in India.

Kali Yuga began on the midnight of 17th Feb 3102 BCE and the duration of this Kali Yuga is said to be 4.32 K solar years. Dwapara is 2*Kali Yuga years. Treta is 3*K Y and Krita Yuga is 4*K Y.


Krita Treta Dwaparascha Kalischaiva Chaturyugam
Divya Dwadasabhir varshai savadhanam niroopitham


Thus an Equinoctial Cycle, Mahayuga is equal to 4+3+2+1 = 10 KYs.

E C = 10 KYs.

A Greater Equinoctial Cycle ( Manvantara ) = 71 Equinoctial Cycles

There are cusps happening in between Manvantaras, each equal to a Krita Yuga in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas.

Hence 14*71+6 = 1000 Mahayugas = 4.32 Billion Years

Sahasra yuga paryantham
Aharyal brahmano vidu
Ratrim yugah sahasrantham
The Ahoratra vido janah ( The Holy Geetha ).

This is one Cosmological Cycle, called Brahma Kalpa.

Chaturyuga sahasram indra harina dinam uchyathe


From one second, it can be logarithmically shown, upto 10^22 seconds. This is what the above diagram shows. This diagram is by courtesy of Wikipedia.

From 10^0 it goes upto 10^22 seconds. One day of Brahma is 4.32 billion years and 100 years of Brahma therefore is 311.04 trillion years, which is shown logarithmically above.

One Asu is 4 seconds, one Vinadi is 24 seconds and one Nadi is 24 minutes. 60 such Nadis make up one day. This is the Sexagesimal division of a day into 60 Nadis ! In Astronomy, one degree is sixty minutes and one minute is sixty seconds. Hence sexagesimal division is justified ! 365.25 such days constitute a year and Hindu calculation goes upto 311.04 trillion years !

India slump to their heaviest defeat

We had warned in our columns that India is facing the adverse 10th transit of Jupiter. Dhoni is under the 8th Jovian transit.

But that did not deter him to score 74 not out, but all his heroics were in vain, as India slid to an inglorious defeat.

The 10th adverse Transit of Jove refers to "loss of position and wandering about". This is true of the One day World Champions, who will get brickbats and not the laurel, when they return home.

Sending a second XI to the West Indies and a first XI to England, well, things wont work out that way. England was preparing hard and smashed India 3-0!

The Geometric Model of Paramesvara

The Indian astronomers were interested in the computation of eclipses, of geocentric longitudes, the risings and settings of planets,which had relevance to the day to day activities of people.

Did not Emerson say?

"Astronomy is excellent, it should come down and give life its full value, and not rest amidst globes and spheres ".

They were not bothered about proposing Models of the Universe and gaining publicity. But then they did discuss the geometrical model, the rationale of their computations.

The above diagram explains the Geometric Model of Parameswara, another Kerala astronomer. Paramesvara and Nilakanta modified the Aryabhatan Model.

By Sheegroccha, he meant the longitude of the Sun." Sheegrocham Sarvesham Ravir bhavathi ", he says is his book Bhatadeepika . For the interior planets, the longitude of the Sheegra correction is to be deducted from the Sun's longitude, Ravi Sphuta to get the Anomaly of Conjunction.

The Manda Prathimandala is the mean angular motion of the Planet, from which the trignometric corrections are given to get the true, geocentric longitude.