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Kepler's Law

A planet of m...

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 $r_{1}$ and $r_{2}$ , respectively. Find the relation between the time period of the planet in terms of $r_{1}$ and $r_{2}$ .

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Solution

Given the periheleon $= r_{2}$
distance of planet from sun $= r_{1}$
Planet's mass $= m$
mass of sun $= M$
Length of major axes of no elliptical orbit $= r_{1} + r_{2}$
Semi-major axis $= R = (\frac{r_{1} + r_{2}}{2})$
by Kepler's law
$T^{2} \propto R^{3} T \propto R^{\frac{3}{2}} T \propto {(r_{1} + r_{2})}^{\frac{3}{2}} T = 2 π \sqrt{\frac{{(r_{1} + r_{2})}^{2}}{8 G M}}$

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