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Kepler Planetary Motion

The Kepler’s laws of planetary motions are the direct consequence of planet’s motion in the central force if gravitation by Sun.

1st law: Each planet moves in an elliptic orbit about the Sun, with Sun at one focus.

The above law follows from the solution of equation of motion of the planet. The equation of motion of the planet moving under the gravitational force of Sun is,

where, C = GM_s m. Following the steps, we find that the path of the planet is given by,

which admits the possibility of an elliptic orbit if e < 1. The constant p is given by,

where L is the (constant) angular momentum of the planet in its orbit.

The total energy of the planet moving in an elliptic orbit is given by

Thus, the energy of a planet moving in an elliptic orbit is equivalent to its energy in a circular orbit of mean radius a.

IInd law: The line joining the Sun and a planet sweeps out equal areas in equal times.

The above is true statement of constancy of aerial velocity. It directly follows from the fact that planets move under central force.

IIIrd law: For each planet, the square of the period of revolution is proportional to the cube of the mean distance of the planet from the Sun.

Let us prove the above statement for an elliptic orbit of the planet. The total area of the ellipse is given as,

S_tot = π ab

The constant aerial speed gives the area swept by planet per unit time.

Hence, the time period of planetary orbit around the Sun is,

where we put the value of b and by squaring the above equation, we get

That is, the square of time period T is proportional to cube of semi-major axis a of the planetary orbit. Note that a represents the mean distance in Kepler’s statement of his IIIrd law.