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Kepler's laws #1

Kepler's laws #2

Kepler's laws #3

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

 

 

Kepler's 1st. Law - The Orbit Law

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

 

Kepler's 1st law

 

 

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Kepler's 2nd. Law- The Area Law

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

 

Kepler's 2nd Law

 

 

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Kepler's 3rd. Law - The Period law

 

Kepler's 3rd 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

Kepler's Laws - equation #10

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

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

Kepler's Laws - equation #1

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:

kepler's Laws - equation #2

Substitung for v into our equation for centripetal force:

kepler's laws - equation #3        (i        

The centripetal force is provided by gravity.

Hence,    

kepler's laws - equation #4                       (ii         

Equating equations (i and (ii ,

Kepler's Laws - equation #5                         (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:

Kepler's Laws - equation #6

and substituting for ω into equation (iii

Kepler's Laws - equation #6b

We obtain the expression:

kepler's Laws - equation #7

Now making T 2 the subject:

Kepler's Laws - equation #8

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

Kepler's Laws - equation #9

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

 

 

 

 

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