Differential Equations and Planetary Mass

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.

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