Question

Using Kepler's 3rd law and Newton's law of universal gravitation, find the period of revolution P of the planet as it moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun. Use G for the gravitational constant. Express the period in terms of G, Ms, R1, and R2.

Answer 1

Answer:

$P = \pi \sqrt{\frac{(R_1+R_2)^3}{2 \ GMS}}$

Explanation:

From the attached diagram below:

AC = a (1 + e) = R₂     -------- equation (1)

CD = a ( 1 - e) = R₁     ---------   equation (2)

⇒ 1 - e = $\frac{R_1}{a}$

$e= 1 - \frac{R_1}{a}$

Replacing the value for e into equation (1)

$a(1+1- \frac{R_1}{a})= R_2$

$= 2a - R_1 = R_2$

$a= \frac{R_1+R_2}{2}$

From Kepler's third law;

$P = 2 \pi \sqrt{\frac{a^3}{GMS}}$

$P = 2 \pi \sqrt{\frac{(R_1+R_2)^3}{8GMS}}$

$P = \pi \sqrt{\frac{(R_1+R_2)^3}{2 \ GMS}}$