Question

If the earth revolves around the sun in an orbit of double its present radius, then the year on earth will be of
A. $\dfrac{365}{4}\: days$
B. $365\times 2\sqrt{2}\;days$
C. $\dfrac{365}{2\sqrt{2}}\;days$
D. $365\times 4\;\;days$

Answer 1

Hint: Use Kepler’s third law of planetary motion.

Formula used: ${{T}^{2}}\propto {{R}^{3}}$
 Here,
$T$ is the time period of orbital motion
$R$ is the radius of the orbit

Complete step by step solution:
If the radius is doubled,
$R'=2R$
Then,

\[\begin{align}
  & \dfrac{{{{{T}'}}^{2}}}{{{{{R}'}}^{3}}}=\dfrac{{{T}^{2}}}{{{R}^{3}}} \\
 & \dfrac{{{{{T}'}}^{2}}}{8{{R}^{3}}}=\dfrac{{{T}^{2}}}{{{R}^{3}}} \\
 & T'=\sqrt{8{{T}^{2}}}=2\sqrt{2}T \\
\end{align}\]

Therefore, the year on the earth will be $365\times 2\sqrt{2}\;days$
The correct answer is option B.

Additional information:
Kepler’s laws of planetary motion are three fundamental laws which describe planetary motion in the solar system. The laws can be stated as:
1^st law: All planets revolve around the sun in elliptical orbits with the sun as one of the foci of the ellipse.
2^nd law: The radius vector connecting the planet and the sun sweeps equal areas in equal intervals of time.
3^rd law: The square of the time periods of revolution of the planets is directly proportional to the cube of their average distances from the sun.
These laws are also applicable for natural and artificial satellite motions and other stellar systems. They can also be used to predict the orbits of asteroids and comets. They were crucial in the discovery of dark matter in the Milky Way Galaxy. They do not take into account perturbation and other gravitational effects.

Note: Kepler’s law inspired Newton in his formulation of laws of gravitation. He showed that the bodies moving under gravitational force can also follow parabolic or hyperbolic trajectories apart from an elliptical trajectory. The path (any open conic curve) taken will be determined by the total energy of the body.