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Question

A planet revolves is an elliptical orbit around the sun. The semi-major and semi-minor axes are $$a$$ and $$b$$, then the time period is given by.


A
T2b3
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B
T2[a+b2]3
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C
T2a3
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D
T2[ab2]3
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Solution

The correct option is C $$T^2 \propto a^3$$
 Using Kepler's $$2$$nd law,  $$A= \dfrac{L}{2m} T$$  where $$A= \pi ab$$ 
Also, $$b^2= a^2(1-e^2)$$
Now, at aphelion  $$\theta= 0$$  thus  $$a(1-e)= \dfrac{L^2}{GMm^2(1+e)}$$
$$\implies \dfrac{L^2}{m^2}= a(1-e^2)GM$$
Thus, from  $$\dfrac{L^2}{4m^2}= \dfrac{\pi^2 a^2b^2}{T^2}         \implies T^2  \propto a^3$$

Physics

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