What is proof of Kepler law of planetary motion that all planet has elliptical orbit? And why they have elliptical orbit instead of a circular orbit?


The orbit of a planet is an ellipse where one focus of the ellipse is the sun.

An ellipse is defined by two focii and all points for which the sum of the distances are the same. The semimajor axis (a) is the long distance from the center to edge of the ellipse. If r1 and r2 are the distances from the focii to any point on the ellipse then r1 + r2 = 2a. The short axis is called the semiminor axis.

How “elliptical” an orbit is can be described by the eccentricity(e). The eccentricity is equal to the distance between a focus and the center (c) of the ellipses divided by the semimajor axis (a). That is, e = c/a. 

See the elementary proof: View

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