About Hindu Astrology, Philosophy , Cosmology and Cosmogony
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 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 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 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 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 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
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
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
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 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