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ramyani

Newton's law of gravitation can be used to give a formula for the planets

The force of gravitational attraction between the Sun and a planet is equal to the centripetal force required to keep the planet in its orbit:

$F=G rac {m_1 m_2}{d^2} = rac{m v^2}{d}$

The period of the planets orbit can be given by:

$Time taken = rac {distance}{velocity}$

Where the distance is the circumference of a circle, $2 pi imes d$ (note that d is distance from Sun, and is therfore the radius, not the diameter). This gives us:

$T = rac{2 pi d}{v}$

Which we can re-arrange to make v the subject and substitute into

v 2

in the centripetal force equation:

$v = rac{2 pi d}{T}$

$v^2 = rac{4 pi^2 d^2}{T^2}$

$G rac {m_1 m_2}{d^2} = rac{4 pi^2 m d}{T^2}$

Eliminating m, the mass of the planet, and tidying up:

$G rac{m T^2}{4 pi^2 d^3} = 1$

And finally,

$T^2 = rac{4 pi^2}{Gm} d^3$

We now have a formula in the form $T^2 propto d^3$

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