Question

A planet of mass m moves around the sun of mass M in an elliptical orbit. The maximum and minimum distance of the planet from the sun are r₁ and r₂ , respectively. The time period of the planet is proportional to

Moderate

Solution

Semi-major axis, $a = \frac{r_{1} + r_{2}}{2}$
Now, $T^{2} \propto a^{3}$
$\Rightarrow T \propto a^{3 / 2} \propto {(\frac{r_{1} + r_{2}}{2})}^{3 / 2} \Rightarrow T \propto {(r_{1} + r_{2})}^{3 / 2}$

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