The Longitude Corrected Thrice Method of Indian Astronomy                    

By Govind Kumar                                                                                       

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.

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

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.

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. 
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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, theSamskrutha Chandra Madhyamam. Then Reduction to Ecliptic is done to get the true longitude of Luna !