Question

planet moves in an elliptical orbit around the sun. The mass of the sun is Ms. The minimum and maximum distances of the planet from the sun are R1 and R2, respectively.

Part A 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

Kepler's 3rd law is given by:

T^2 = 4*pi^2/(G*M)*a^3

a = semi major axis of orbit = (R1 + R2)/2

M = mass of sun = Ms

So, time period will be

T^2 = (4*pi^2*((R1 + R2)/2)^3)/(G*Ms)

T = sqrt [4*pi^2*(R1 + R2)^3)/(8*G*Ms)]

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(R1 + R2) 8GM T = 2π