Kepler's Laws of Planetary Motion

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First law

Kepler's first law

The path of each planet around the sun is an ellipse with the sun at one focus.

If F₁ and F₂ are the two foci, P is the planet and S is the sun:

F₁P + F₂P = constant.

Second law

Kepler's second law

A line joining the sun to the planet sweeps out equal areas in equal intervals of time.

If the time taken by a planet to travel from P₁ to Q₁ is equal to the time taken to travel from P₂ to Q₂, the areas covered are equal (shaded region).

Third law

A planet moves around the sun in such a way that the square of its time period is proportional to the cube of the semi-major axis of its elliptical orbit.

If T is the time period of revolution and 'a' the semi-major axis then,

for circular orbits a = r (radius)

Derivation of Newton's law of gravitation from Kepler's laws

Let m be the mass of a body in an orbit of radius r, velocity v and angular velocity 'w'. the centripetal force

If T is the time period then,

According to Kepler's third law,

If M is the mass of the sun and the body of mass m is revolving around the sun, then the constant of proportionality is assumed to be GM, where G is a constant.

which is Newton's law of gravitation.