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

1 Answer
Oct 24, 2017

Answer:

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 prop a^3#

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

#T^2 = (4 pi^2)/(G(M+m))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 = (4 pi^2)/(GM)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.

#(4 pi^2)/(GM)=1#

Hence for everything orbiting the Sun.

#T^2 = a^3#