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Gravitational Fields 3

 

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.

 

Kepler's Laws - equation #10

 

where,

 

T is the orbital period

rM is the mean distance between the planet and the parent star

 

 

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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        

 

NB The centripetal force is provided by gravity.

 

For a star mass Ms and planet mass m,    

 

kepler's laws - equation #4                       (ii         

 

Equating equations (i and (ii , (force of gravity = centripetal force)

 

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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