Question

If the radius of the earth’s orbit around the sun is $R$ and the time period of the revolution of the earth around the sun is $T$, the mass of the sun is​
(A) $\dfrac{{G{T^3}}}{{4{\pi ^2}{R^2}}}$
(B) $\dfrac{{4{\pi ^2}{R^3}}}{{G{T^2}}}$
(C) $\sqrt {\dfrac{{4{\pi ^2}{R^3}}}{{G{T^2}}}} $
(D) ${\left[ {\dfrac{{4{\pi ^2}{R^3}}}{{G{T^2}}}} \right]^{\dfrac{1}{3}}}$

Answer 1

Hint: To find the mass of the sun, we need to know about the Kepler’s laws. The Kepler’s third law defines an expression, using which we can find the expression for the mass of the sun.
Formula used: In this solution we will be using the following formula,
$\Rightarrow {T^2} = \dfrac{{4{\pi ^2}{R^3}}}{{GM}}$
Where, $T$ is the orbital period, $R$ is the mean distance of the planet from the sun, $M$ is the mass of the sun, and $G$ is the Universal gravitational constant.

Complete step by step solution:
Kepler's third law states that, The square of the orbital period of a planet is proportional to the cube of the mean distance from the Sun. So, the expression for the third law is given by the formula,
$\Rightarrow {T^2} = \dfrac{{4{\pi ^2}{R^3}}}{{GM}}$
Here $M$ is the mass of the sun
Now we can take the variable $M$ from the denominator of the RHS to the numerator of the LHS
So we get the value of the mass of the sun as
$\Rightarrow M = \dfrac{{4{\pi ^2}{R^3}}}{{G{T^2}}}$
Thus, the correct answer is option (B).

Note:
There are numerous applications of Kepler's laws. It describes the orbits of planets around the sun or stars around a galaxy in classical mechanics. They have been used to predict the orbits of many objects such as asteroids and comets, and were pivotal in the discovery of dark matter in the Milky Way. The other two laws are the first law is the law of ellipses, which explains that the planes are orbiting the sun in an elliptical path. The second law is the law of areas which gives the speed of the planet in the orbit.