# Hermes has an elliptical orbit about the sun, with a perihelion distance of 0.622 AU and an...

## Question:

Hermes has an elliptical orbit about the sun, with a perihelion distance of 0.622 AU and an aphelion distance of 2.69 AU. What is the period of its orbit (in years)?

## Kepler's Law:

Kepler's third law of planetary motion describes the relationship between the time period and the distance. It is able to give information about the motion characteristics of the planets orbiting the sun.

Given:

Perihelion distance is {eq}0.622\ AU {/eq}

Aphelion distance is {eq}2.69\ AU {/eq}

Kepler's third law of planetary motion can be used to find the time period and the expression for the time period is given as,

{eq}T=\frac{4\pi ^{2}a^{3}}{MG} {/eq}

• a is the semi-major axis
• G is the gravitational constant
• T is the time period
• M is the mass of the earth

It is clear that,

{eq}2a=(perihelion+aphelion)\\ a= \frac{(perihelion+aphelion)}{2}=\frac{0.622+2.69}{2}=1.656\ AU\\ {/eq}

Therefore, the time period is given as,

{eq}T=\sqrt{\frac{4\pi ^{2}a^{3}}{MG}} {/eq}

{eq}T=\sqrt{\frac{4\pi ^{2}(1.656\times 1.496\times 10^{11}m)^{3}}{6.67\times 10^{-11}Nm^{2}/Kg^{2}\times 2\times 10^{30}}}=6.70\times 10^{7}\ years {/eq} 