# Venus has an average distance to the sun of 0.723 au. Calculate the orbital period of Venus.

## Question:

Venus has an average distance to the sun of 0.723 au. Calculate the orbital period of Venus.

## Kepler's Third Law of Planetary Motion

In his work 'Harmonies of the World', the seventeenth century German mathematician Johannes Kepler showed that there existed a simple relationship between the orbital period ({eq}T {/eq}) of a planet and its semi-major axis ({eq}R {/eq}) raised to the cubic power. Mathematically, we write this (Kepler's third law, as it is now known) as:

{eq}\displaystyle \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} {/eq}.

This law can be derived from Newton's law of gravitation

We can obtain the orbital period of Venus using Kepler's third law and the Earth as the second planet. Recall that the semi-major axis of Earth is defined as 1 a.u. Hence, we list:

{eq}R_2 = 1 \ a.u.\\ T_2 = 365 \ days \\ R_1 = 0.723 \ a.u. \\ T_1 = ? {/eq},

and invoke Kepler's third law, into which we substitute the data above:

{eq}\displaystyle \begin{align} T_1^2 &= (\frac{0.723 \ au}{1 \ au})^3 \times (365 \ days)^2 \\ &= 50350 \ days^2 \end{align} {/eq}.

Taking the square root, we obtain

{eq}T_1 = 224.4 \ days {/eq}