Question

A planet \(P_1\) is moving around a star of mass \(2M\) in an orbit of radius \(R\). Another planet \(P_2\) is moving around another star of mass \(4M\) in an orbit of radius \(2R\). The ratio of time periods of revolution of \(P_2\) and \(P_1\) is:

• A. \( \frac{1}{2} \)
• B. \(2\)
• C. \(4\)
• D. \( \frac{1}{4} \)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Kepler's third law states \(T^2 \propto R^3\). However, this proportionality assumes the central mass is constant. When the mass of the star \(M_{star}\) varies, the generalized law is \(T^2 \propto \frac{R^3}{M_{star}}\).
Step 2: Key Formula or Approach:
1. Time period \(T = 2\pi \sqrt{\frac{R^3}{GM_{star}}}\).
2. Ratio: \(\frac{T_2}{T_1} = \sqrt{\frac{R_2^3}{M_2} \times \frac{M_1}{R_1^3}}\).
Step 3: Detailed Explanation:
For planet \(P_1\): Orbit radius = \(R\), Star mass = \(2M\).
\[ T_1 \propto \sqrt{\frac{R^3}{2M}} \]
For planet \(P_2\): Orbit radius = \(2R\), Star mass = \(4M\).
\[ T_2 \propto \sqrt{\frac{(2R)^3}{4M}} = \sqrt{\frac{8R^3}{4M}} = \sqrt{\frac{2R^3}{M}} \]
Calculate the ratio \(T_2 / T_1\):
\[ \frac{T_2}{T_1} = \frac{\sqrt{2 R^3 / M}}{\sqrt{R^3 / 2M}} = \frac{\sqrt{2}}{\sqrt{1/2}} = \sqrt{4} = 2 \].
Step 4: Final Answer:
The ratio of the time periods is 2.