Differential Equations and Planetary Mass
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
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
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
This definition more or less equals the statement of Newton’s second law of
M = Fa
a = F/M
So it strongly suggests that the idea of planetary mass was known to the
ancient Indian astronomers and mathematicians.
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
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)