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. $T^2\propto b^3$
• B. $T^2\propto {\left[\frac{a+b}{2}\right]}^3$
• C. $T^2\propto a^3$
• D. $T^2\propto {\left[\frac{a-b}{2}\right]}^3$

Answer 1

The correct option is C $T^2\propto a^3$

Using Kepler's $2$nd law, $A=\frac{L}{2m}T$ where $A=\pi ab$

Also, $b^2=a^2(1-e^2)$

Now, at aphelion $\theta =0$ thus $a(1-e)=\frac{L^2}{GM{m}^2(1+e)}$

$⟹\frac{L^2}{m^2}=a(1-e^2)GM$

Thus, from $\frac{L^2}{4m^2}=\frac{{\pi }^2{a}^2{b}^2}{T^2}⟹T^2\propto a^3$