Impressionist and Modern Art, Part I
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Impressionist and Modern Art, Part I
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Exact shape of earths orbit. Ellipse or oval? Where is the sun located within the elliptic shape?
Best answer:
Newton showed that the Sun's gravity requires that all planets, including Earth, follow elliptical orbits with the Sun at one focus of the ellipse. An ellipse's "ovalishness" is measured by its eccentricity, the distance bewteen the foci divided by the major axis.In Newtonian gravitation and in the absence of any other planets, the Earth's orbit would be a perfect ellipse, with the Sun in one of the two focal points.
However, in the real world the ellipse rotates, an effect called the perihelion shift. It is very small, but measurable.
The *anomalous* perihelion shift is due to third body perturbations. Solar oblateness, also produces a small contribution.
Add to that the *relativistic* perihelion shift, which is explained by the Sun's mass causing space time to curve. The warped space causes a planet's orbit, that would be a perfect ellipse in flat Euclidean space, to precess.There is nothing 'Ovalish..' or any such about Earth. It is a perfectly elliptical orbit with Sun in one focus (and the other focus empty).
From end to end the 'major axis' (passing through both foci) is split into two equal parts at the center, each is called 'semi-major axis' ('a') and is convenient to us mathematically. Major axis is also known as 'Line of Apses'. From extremities, the sun is at distances of "a (1 -e)" at Perihelion & "a (1+e)" at Aphelion and semi-major axis ('a') happens to be the average distance as measured from focus where Sun is located. So you are right in the observation that "the Earths nearest and furthest points from the sun both occur along the elliptical major axis...". In ellipse the center has no role & only the foci (plural for focus), the pair of points which are 'eccentric' are important. How much apart these two are from the center is the measure known as 'eccentricity' ('e' above). It (e) is the ratio of 'distance between these two foci to the distance between the extremities' (major axis). The ratio is more than zero but less than one for an ellipse.
Another point worthy of note is that the minor axis has extremities too, each at a distance 'a' (semi- major axis) from 'focus' and not from the center that has no relevance.
All these observations were made over several centuries. The Danish Royal Astronomer (in 1500s), Tycho Brahe amassed so much of observational data that it was used by his assistant Johannes Kepler, who laid down the laws governing planetary motion, known as 'motion under central force'. The 3 laws are - that the orbits are 'conic' sections, the line joining the focus (location of the central force) sweeps equal areas in equal times and the square of time period is proportional to the cube of semi-major axis length. 'Oval' is not a mathematical curve as you can't cut a cone or a cylinder to produce this shape.
Earth has a normal orbit like billions of others in space, in this situations with no special properties. Earth (any body) with some energy (in the form of velocity) when attracted by a bigger mass, first falls towards it. Its trajectory would be a vector sum of its velocity & gravitational vector (attraction). Gravitational energy is Potential Energy while velocity imparts Kinetic Energy. The total (PE+KE) remains same through out (constant). As it falls it loses PE but gains KE with increased speed and goes past the attracting body. This time the pull of gravity retards its velocity and it is brought back again. This exchange between PE & KE makes the ellipse possible.
The Longitude Corrected Thrice Method of Indian Astronomy
Article by G 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, theSayana 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.
q1 = Equation of Center, first lunar inequalityq2 = 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 !
Orignal From: Impressionist and Modern Art, Part I
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