Question

The time period of earth is taken as $T$ and its distance from sun as $R$. What will be the distance of a certain planet from the sun whose time period is $64$ times of earth?

Answer 1

We are asked to find the distance of the certain planet from the sun. To find this distance we have to use Kepler’s third law. So, recall Kepler’s third law and use it for the earth and the other planet to form two equations. Use these equations to find the value of the distance between the sun and the other planet.

Complete step by step answer:
Given, the time period of earth is $T$. Distance of earth from the sun is $R$. Time period of the other planet is $64T$. Let the distance of the sun from the other planet be $R'$.From Kepler’s third law we have that the square of the time period of a planet to complete one revolution around the sun is proportional to cube of the distance between the sun and the planet.Mathematically we can write,
${T^2} \propto {R^3}$ (i)
where $T$ is the time period and $R$ is the distance between the planet and the sun.

In case of earth, using Kepler’s third law we have,
${T^2} \propto {R^3}$ (ii)
In case of the other planet, using Kepler’s third law we have,
${\left( {64T} \right)^2} \propto {R_0}^3$ (iii)
Now, dividing equation (iii) by (ii) we get,
$\dfrac{{{{\left( {64T} \right)}^2}}}{{{T^2}}} = {\dfrac{{{R_o}}}{{{R^3}}}^3}$
$\Rightarrow {64^2}{R^3} = {R_o}^3$
$\Rightarrow {R_o}^3 = {64^2}{R^3}$
$\Rightarrow {R_o}^3 = {\left( {{4^3}} \right)^2}{R^3}$
$\Rightarrow {R_o}^3 = {\left( {{4^2}} \right)^3}{R^3}$
$\Rightarrow {R_o} = \left( {{4^2}} \right)R$
$\therefore {R_o} = 16R$

Therefore, the distance of the certain planet from the sun is $16R$.

Note: There are three laws of planetary motion given by Kepler which are known as Kepler’s law of planetary motion. Kepler’s first law states that all planets revolve around the sun in elliptical orbits with the sun at one of the foci. Kepler’s second law states that the line joining the sun and a planet sweeps out equal areas in an equal interval of time. And we have discussed Kepler's third law in the above question.