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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




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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