Answer to Question #138088 in Mechanics | Relativity for Sneha

Question

A planet of mass M moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to r and the maximum distance to R. Making use of Kepler s laws, find its period of revolution around the Sun.

Answer 1

Explanations & Calculations

• According to Kepler's 3rd law, $\small T^2 \propto a^3$ : a = the semi major axis
• Accordingly the periodic time does not depend on the eccentricity as any time delay is caught up with the increased speeds on such location of the path.
• Therefore, an average value for the semi major axis could be written as $\small a = \frac{(r+R)}{2}$
• From the motion of a uniform circle by the application of Newton's second law towards center it could be written for the period, $\small T^2 = \large\frac{4\pi^2}{GM_0}r^3 \small \to T^2 \propto r^3$
• Therefore,

$\qquad\qquad \begin{aligned} \small T^2 &= \small \frac{4\pi^2}{GM_0}\times \frac{(r+R)^3}{8}\\ \small T &= \small \pi \sqrt{\frac{(r+R)^3}{2GM_0}} \end{aligned}$ : M₀ is the mass of the Sun