Question

A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes $55$earth years to complete one orbit around the star. Vega’s mass is $2.1$times the sun’s mass. What is the radius of the planet’s orbit? Give your answer as a multiple of the radius of the earth’s orbit.

Short answer

The radius of the planet's orbit is $4.34\times {10}^5$times of the earth's orbit radius.

Step-by-step solution

Given information

Time period = $55earthyears=1.734\times {10}^{9}s,$mass of the planet =$2.1\times 1.989\times {10}^{30}kg=4.177\times {10}^{30}kg.$

Calculation

According to Kepler's 3rd law, the time period is given by $T^2=\frac{4{\pi }^2{R}^3}{GM}.$

Where, T is the time period, R is the required radius of the orbit, G is the universal gravitational constant and M is the required mass of Vega.

Now solving we get,

$$\begin{gathered} T^2=\frac{4{\pi }^2{R}^3}{GM} \\ {R}^3=\frac{T^2\times GM}{4{\pi }^2} \\ R={\left({\left(1.734\times {10}^{9}s\right)}^2\times \frac{(6.67\times {10}^{-11}N{m}^2/k{g}^2)(4.176\times {10}^{30}kg)}{4{\pi }^2}\right)}^{1/3} \\ R={\left(2.121\times {10}^{37}{m}^3\right)}^{1/3} \\ R=2.77\times {10}^{12}\mathrm{m} \end{gathered}$$

The radius of the earth's orbit is $6.37\times {10}^6\mathrm{m}.$

Radius of the planet is $4.34\times {10}^5$times of the earth's orbit radius.

Final answer

The radius of the planet is $4.34\times {10}^5$times of the earth's orbit radius.