The Obliquity of the Ecliptic !

The Earth's Axial tilt is called the Obliquity of the Ecliptic and the is angle between the perpendicular to Orbit and the North Celestial Pole.

The Equatorial coordinate system is based on the 360 degree Celestial Equator Circle. The Ecliptic coordinate system is based on the 360 degree Ecliptic circle.

The mathematical conversion from Equatorial to Ecliptic is effectuated by the equation for the Ascendant

Lagna = arctan ( Sin E / Cos E Cos w - Sin w Tan A )

where Lagna is the Lagna on the Ecliptic andE is the Lagna on the Celestial Equator, the Sayana Kala Lagna. The Sayana Kala Lagna, E, is reduced to the Ecliptic by this equation. The Lagna is the intersecting point between the Eastern Celestial Horizon, the Kshitija with the Ecliptic.

w is the Sun's maximum declination and A is the latitude of the place.

The above diagram is by courtesy of Wikipedia

Computation of Lunar Longitude

w was an important angle in the Munjala Model and the solution to the problem of a difference of 2.5 degrees in the lunar longitude had to be solved. So Munjala brought in an angle, w, angle between the Mean Sun and the Moon's Apogee.

The angle n is the elongation of the Sun from the Mean Moon and so the

Manda Anomaly, Alpha = w + n


This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Epicycle Model of Bhaskara

The Model propounded by Aryabhata is an algorithm. The Khmers drew the diagrams of the Sun by using the epicyle equivalent of the model developed by Bhaskara in the seventh century. Eccentricity is variable in this Epicycle Model.

You can see a relevant animation of Dennis Duke at
http://people.scs.fsu.edu/~dduke/pingree2.html

The True Equant

The Indian astronomers could calculate the Manda Kendra ( The Equation of Center of Western Astronomy ) and the Manda Phala, but a problem presented itself when calculating the lunar longitude.

The Concentric Model and the Epicylic Model could not calculate Moon's longitude at quadrature, even though they could calculate the lunar longitude at the times of New Moon and Full Moon. There was a difference of 2.5 degrees between the longitude computed by the Concentric Equant and Epicyclic Models. So the ancients had to give a correction to the Equation of Center, which reached a maximum of 2.5 degrees.

So the Indian astronomers came out with a solution. They created a new Equant (E'), the true Equant, which moves on an epicycle, whose center is the Mean Equant, E. The epicycle has a radius e, equal to EoE., on the Line of Apsis, OA.

q1 = Equation of Center, first lunar inequality
q2 = Correction, second lunar inequality.

True Longitude = Mean long + Eq of Center + q2

The first lunar anomaly was the Evection and the second, the Variation. The first inequality was the Equation of Center and the Evection and the Variation became the second and third inequalities. Actually Indian Astronomy recognised 14 major perturbations of the Moon and 14 corrections are therefore given to get the Cultured Longitude of the Moon, the Samskrutha Chandra Madhyamam. Then Reduction to Ecliptic is done to get the true longitude of Luna !



This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Concentric Equant Model of Aryabhata

Aryabhata developed a Concentric Equant Model, in the sixth century. The Sun moves on a circle of radius R, called a deferent, whose center is the Observer on Earth. The distance between the Earth and the Sun, the Ravi Manda Karna, is constant. The motion of the Sun is uniform from a mathematical point, called the " Equant", which is located at a distance R x e from the observer in the direction of the Apogee ( e = eccentricity ).


All Indian computations are based on this Concentric Equal Model. The normal equation for computing the Manda Anomaly is R e Sin M and resembles the Kepler Equation, M = E - e Sin E.


This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Lunar Model Of Munjala

The Concentric Equant theory was developed by the Indian astronomer, Munjala ( circa 930 CE ).

The Geocentric theory of the ancient astronomers had the ability to produce true Zodiacal Longitudes for the Moon. But the perturbations of the Moon were so complex, that the early Indian and Greek astronomers had to give birth to complicated theories.

The simplest model is a concentric Equant Model to compute the lunar true longitude.

In the above diagram

M = Moon
O = Observer
Eo = Equant , located at a distance r from the observer , drawn on the Line of Apsis and the Apogee.

A = Apogee, Luna's nearest point to Earth






Angle Alpha = Angle between Position and Apogee
Angle q1 = Equation of Center . Angle subtended at Luna between Observer and Equant
Equation in Astronomy = The angle between true and mean positions.


These diagrams are by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

Astronomy & Maths In India

The Physics Professor of Florida State University, Dennis Duke remarks

"The planetary models of ancient Indian mathematical astronomy are described in several texts. These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic. "

The mutli step algorithms of Indian Astronomy never approximated any Greek geometrical model. Ptolemy's Almagest was the first book, according to Western Astronomy. We have now the information that Ptolemy did not invent the equant.

Bhaskara II was an astronomer-mathematician par excellence and his magnum opus, the Siddhanta Siromani (" Crown of Astronomical Treatises") , is a treatise on Astronomy and Mathematics. His book deals with arithemetic, algebra, computation of celestial longitudes of planets and spheres. His work on Kalana ( Calculus ) predates Liebniz and Newton by half a millenium.

The Siddanta Siromani is divided into four parts


1)The Lilavati - ( Arithmetic ) wherein Bhaskara gives proof of c^2 = a^ + b^2. The solutions to cubic, quadratic and quartic indeterminate equations are explained.

2)The Bijaganitha ( Algebra )- Properties of Zero, estimation of Pi, Kuttaka ( indeterminate equations ), integral solutions etc are explained.

3)The Grahaganitha ( Mathematics of the planets ).


For both Epicycles

The Manda Argument , Mean Longitude of Planet - Aphelion = Manda Anomaly

The Sheegra Argument, Ecliptic Longitude - Long of Sun = Sheegra Anomaly

and the computations from there on are explained in detail.

4)The Gola Adhyaya ( Maths of the spheres )


Bhaskara is known for in the discovery of the principles of Differential Calculus and its application to astronomical problems and computations. While Newton and Liebniz had been credited with Differential Calculus, there is strong evidence to suggest that Bhaskara was the pioneer in some of the principles of differential calculus. He was the first to conceive the differential coefficient and differential calculus.

Khagola, the Celestial Coordinate System

( Above diagram by courtesy of www.wikipedia.org )

A 360 degree Circle is a coordinate System. And Khagola is the Celestial Coordinate System.

Like the Geographical Coordinate System, the Celestial Coordinate System is another coordinate system, which computes the coordinates of the Khagola, the Celestial Sphere.

The Ascending Sign is called the Ascendant ( Raseenam udayo lagnam ) and is the intersecting point of the Ecliptic ( at the East Point) with the Celestial Horizon. The Descending Sign is the Descendant ( Astha Lagna ) and lies 180 degrees West on the Khshithija, the Celestial Horizon.

Like the Geographical Meridien ( the Prime Meridien ) and the Geographic Equator, the Celestial Coordinate System has a Celestial Meridien ( Nadi Vritta ) and a Celestial Equator.

The Vernal Equinox and the Autumnal Equinox are two intersecting points of the Ecliptic with the Vishu vat Vritta, the Celestial Equator, known as Meshadi and Thuladi.

The Hindu Zero Point of the Ecliptic starts from 0 degrees Beta Arietis, Ashwinyadi, which is the beginning point of the Sidereal Zodiac. This is the Nirayana System, sidereal. The Tropical System, Sayana, also has its adherents in India and starts from Meshadi, 0 degree Aries.






The Galactic Center, the Vishnu Nabhi, lies in Sagittarius. NEP is the North Ecliptic Pole, NGP is the North Galactic Pole and NCP is the North Celestial Pole.

It is raining cats and dogs in Kerala

On Saturday, the heavens brimmed with pessimistic prophecies and then came the downpour. ( Today is 19th Jul 2011 )

The Sun has disappeared and it is now raining cats and dogs here. As a concomitant result, I got cold !

This SW Monsoon, defined as a failure this season, may perk up, compensating for the lack of rains during the earlier Mrigasira and Aridra Solar Periods ( Njattuvelas ). Now Punarvasu Njattuvela is on, as the Sun transits Beta Geminorum.

Now the paddy fields are full of water and it rained heavily at night day before yesterday. The ocean became hostile on Chavakkad Beach and surrounding areas, wreaking destruction.

The Double Epicyclic Model of India

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

We have the Double Epicyclic Model - that of Manda Epicycle and Sheegra Epicycles - in Indian Astronomy, which explain the Zodiacal and Solar anomalies. One Epicycle explains the Zodiacal Anomaly and the other the Solar Anomaly.

( Zodiacal Anomaly - That all planets move slower at Aphelion and faster at Perihelion.
Solar Anomaly - The astronomical phenomenon of Retrogression. Backward Motion. When a planet changes its course from perihelion to aphelion, it retrogrades in order to gain the Sun's celestial gravity )

Dennis Duke, of Florida State University, says " We have only to conclude that Ptolemy did not invent the equant. " If Ptolemy did not invent the equant, as Westerners widely believe, then who did ?

"The bisected Indian equant model is pre-Ptolemaic' says he. Other Greek books, prior to Ptoemy, may have influenced Indian Astronomy,says he. Then what are those books, prior to the Almagest, which had influenced the Indian system? The answer is "unknown sources".




Remarks Duke " Indeed, since the very earliest investigation of the Indian models by Western scholars it has been presumed that the models are somehow related to a double epicycle system, with one epicycle accounting for the zodiacal anomaly, and the other accounting for the solar anomaly (retrograde motion) This perception was no doubt reinforced by the tendency of some Indian texts to associate the manda and sighra corrections with an even older Indian tradition of some sort of forceful cords of air tugging at the planet and causing it to move along a concentric deferent . Since our goal in this paper is to investigate the nature of any connection with ancient Greek planetary models, it is only important to accept that the models appear in Indian texts that clearly pre-date any possible Islamic influences, which could, at least in principle, have introduced astronomical elements that Islamic astronomers might have derived from Greek sources. ( "The Equant in India: the Mathematical Basis of Ancient Indian Planetary Models" By Dennis Duke, Florida State University )

Computation of Geocentric Distance, Sheegra Karna

In the diagram above, the geocentric distance, EQ called X here , the distance of the planet from the Earth is calculated by the equation

X^2 = EQ^2(EP+PL)^2 + QL^2

or = EN^2 + QN^2

In a trignometric correction, called Sheegra Sphashteekarana, this equation is given by Bhaskara.

where

E = Earth
P = Planet in its Orbit
Q = Planet on the Epicycle
QL = Sin
PL = Cos


We have said that Sheegra Kriya reduces the heliocentric postions to the geocentric.

According to this oscillating Epicyclic Model of Bhaskara, EP = R ( Called Thrijya ), PQ is the Sheegra Phala, QL is the Bhujaphala and PL is Kotiphala.

The Hindu algorithms for the computation of mean and true celestial longitudes seems to be totally different from the Western, from the methods adopted by Kepler, Laplace and Co. Hence the Hindu Planetary Model is original and not influenced by Greco Roman sources, as some Western scholars believe.

Sheegra Kriya for inferior planets

Different equations have been given for superior planets ( Mars, Jupiter and Saturn ) and inferior planets ( Mercury and Venus ) in Astronomia Indica.

In the case of Mercury, an inferior planet in the diagram above, the center of the Sheegra Epicycle is located on the straight line running through the Sun and the observer, on the geographical parallel of the observer.

The above diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

The Sheegra Phalam, x, in the equation 1/2 Tan ( A -x ), where A is the Elongation or Sheegra Kendra, obtained is deducted from the Sun's longitude, to get the geocentric longitudes of Mercury and Venus.

Indian Astronomy Pre-Ptolemaic

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, Saturn, a superior planet, is on the circumference of the Sheegra Epicycle, where it is met by a radius drawn parallel to the direction of the Sun from the observer.

To the Western scholars, Indian Astronomy is mysterious. Let us see what astro scholars have said about IA.

Dennis Duke, of Florida State University suggests that Indian Astronomy predates Greek Astronomy

"The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic" says he.

The earliest Indian Planetary Models are two sets from the writer Aryabhata, both dating from 6th Century AD.

1) The Sunrise System , after the Epoch, which is taken from the sunrise of 18th Feb 3102 ( Arya Paksha ). It appears first in Aryabhatiya

2) The Midnight System, after the Epoch, which is taken from the midnight of 17/18 FEB 3102 ( Ardha Ratri Paksha ). It appears first in Latadeva's Soorya Siddhanta

The Local Meridien is taken as Lanka, Longitude 76 degrees, Latitude 0 degrees.

Of Manda and Sheegra Epicycles

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given.

In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent.

The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

Vyasardha, the Radius of the Circle

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha, the Radius of the Circle is 3438 minutes and Arc is 5400 minutes.

Circumference = 2 Pi R
R = 360/2 Pi
R = 57.3 degrees
R = 57.3 * 60 = 3438 arcminutes
R = 3438 * 60 = 206265 arcseconds

Half Chord of 90 degrees = 90*60 = 5400 arcminutes.

In his astronomical treatise, the Aryabhatiya, he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry.

In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below.


मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||


Aryabhata's Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences of the values of trignometric sines expressed in arcminutes. Because of this, this Table is referred to as the Table of Sine Differences.

The Epicyclic Theory of Indian Astronomy

All planets traverse in ellipses and epicycles and this came to be known as the Epicycles Theory.

In the above diagram, the circle A is the mean orbit of the planet. P is the mean Position of the planet and the small circle P is the epicycle.

The small epicycles traversed by a planet are calculated and the mandaphala, the equation of center is computed and added if the Manda Kendra is in between 180 and 360 and subtracted if the M K is < 180. Manda Kendra is the angle between Position and Aphelion.

From the perspective of the Epicyclic theory & the Hindu astronomers, the radius of the epicycle was given instead of PQ and the circumferences of the epicycles. Both circumferences and radii are given in degrees, minutes and seconds, so that the equation of the center may be computed in deg min and secs. The epicycle in the case of the Equation of Center is given as Manda Nicha Uccha Vritta.


Manda - Manda Phala or Equation of Center
Uccha - Apogee
Nicha - Perigee

Manda Kriya is a Jya Ganitha Kriya, a trignometric reduction of the mean longitudes and distances of the planets to their heliocentric longitudes and distances.

Kranti, the Sun's declination

The declination of the Sun is computed by the formula

Sin J = Sin L Sin w

where J is the Sun's declination of that particular date and time, L is the tropical longitude of the Sun and w, the Sun's maximum declination, which is 23 degrees and 27 minutes.

The Sun's maximum declination will be reached during Karkyadi ( The First Point of Cancer ) and Makaradi ( The First Point of Capricorn ). At Karkyadi, it will be +23 d 23 m and at Makaradi, it will be -23 d 27 minutes.



And at Meshadi ( The First Point of Aries ) and Thuladi ( The First Point of Libra ), the solar declination will be zero. Days and nights will be of equal duration and hence they are called Equinoxes.

Yada Mesha Thulayo varthathe thada ahoratranam samanani bhavanthi

Chara, the Ascensional Difference

In Indian Astronomy, Chara is the Ascensional Difference and is the difference between Right Ascension and Oblique Ascension.


And is calculated by the equation

Sin ( Chara ) = tan ( decl ) tan ( latitude ).



Here Theta is Chara.

Sin C is called Chara Jya


Right Ascension means the longitudes measured along the Celestial Equator, the Vishu vat Vritta.

Oblique Ascension means the longitudes measured along the Ecliptic, the Kranti Vritta.

Chara is used in the computation of Sunrise and Sunset.

The formula for Sunrise = 6 H + Equation of Time - Chara - Refraction Correction

The formula for Sunset = 18 H + Equation of Time + Chara + Refraction Correction.

Reduction to Geocentric Coordinates

In the above diagram,

m is the Sheegra Anomaly
and the angle EJS is the Sheegra Phalam

where E = Earth, S = Sun and J = Jupiter


In order to get the geocentric longitude, the longitude of the Sun is deducted from the Ecliptic longitude and then we get the Sheegra Kendra, the angle between the planet and the Earth Sun plane.

Ecliptic longitude - Long Sun = Sheegra Kendra

Arka Sphutnoniham Kheda Manda Sphutamihodhitham

Sheegra Phalam is the angle EJS ( Earth, Jupiter, Sun ) and this Sheegra Phalam is deducted to get the true, geocentric longitude. ( Added if Sheegra Anomaly > 180 and deducted if Sheegra Anomaly < 180 ).


Like Kepler, the Indian astronomers may not have said that all planets move fastest at perihelion but this principle was known to them as they called perihelion Sheegrochha. Sheegra in Sanskrit means fast. Similarly they called Aphelion Mandochcha, manda meaning slow, and it implies that all planets move slowly at Aphelion !