According to Kepler's third law of planetary motion, the period of revolution of a planet is related to the planet's what?

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Oct 24, 2017

Kepler's third law relates the orbital period of a planet, or moon to its semi major axis distance.

Explanation:

Kepler's first law states that orbits are ellipses. The third law relates the orbital period $T$ to the semi-major axis distance $a$. The semi-major axis being half the distance of greatest width of the ellipse.

Kepler wrote the law as a proportionality.

${T}^{2} \propto {a}^{3}$

Newton used his theory of gravity to find the constant of proportionality.

${T}^{2} = \frac{4 {\pi}^{2}}{G \left(M + m\right)} {a}^{3}$

Where $G$ is the gravitational constant, $M$ is the mass of the body being orbiting and $m$ is the mass of the orbiting body.

In the case of a solar system the mass of the sun $M$ is significantly greater than the mass of a planet $m$. The Sun contains over 99% of the mass of the solar system. So, the value of $m$ can be ignored to give.

${T}^{2} = \frac{4 {\pi}^{2}}{G M} {a}^{3}$

For our solar system if the value of the period $T$ is measured in years and the value of the semi-major axis $a$ is measured in astronomical units AU then.

$\frac{4 {\pi}^{2}}{G M} = 1$

Hence for everything orbiting the Sun.

${T}^{2} = {a}^{3}$

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