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Gravitational Fields 3

Law #1 - The Orbit Law

The orbit of a planet is in the shape of an ellipse, with the parent star at one focus.

Law #2 - The Area Law

A planet moves such that an imaginary line between it and the parent star sweeps out equal areas in equal times.

Law #3 - The Period law

The square of the orbital period of a planet is proportional to the cube of the mean distance to its parent star.

T - orbital period
rM - mean distance

table

Kepler's 3rd law derived from Newton's Law of Gravitation

The centripetal force F keeping a mass m in orbit is given by:

The angular velocity ω is the angle (in radians) traced out when the mass travels v metres in one second. Stating this definition in an equation and making v the subject:

Substitung for v into our equation for centripetal force:

(i

The centripetal force is provided by gravity.

Hence,

(ii

Equating equations (i and (ii ,

(iii

The period T of the orbital motion is the circumference (in radians) divided by the angular velocity. Making ω the subject of the equation:

and substituting for ω into equation (iii

We obtain the expression:

Now making T 2 the subject:

If we now remove the constants G MS π by making the equation a proportionality:

Hence Kepler's 3rd Law is consistent with Newton's Law of Gravitation.