Question

(a) Using elementary Newtonian mechanics find the period of a mass m 1in a circular orbit of radius r around a fixed mass m 2. (b) Using the separation into CM and relative motions, find the corresponding period for the case that m 2is not fixed and the masses circle each other a constant distance r apart. Discuss the limit of this result if m 2oo. (c) What would be the orbital period if the earth were replaced by a star of mass equal to the solar mass, in a circular orbit, with the distance between the sun and star equal to the present earth—sun distance? (The mass of the sun is more than 300,000 times that of the earth.)

Answer 1

SOLUTION:

By using newtonian elementry mechanics for a circular orbit of m₁ about a fixed m₂

Let w be the angular velocity of the planet

we have,

So,

but , T = 2 / w

thus, T = 2/ (Gm₂ / r³ ) ¹

For the motion of both the bodies

the distance of the body 1 from the centre of mass is = m₂ r / (m₁ + m₂ )

the equivalent mass is = (m₁m₂/ m₁ + m₂ )

hence,

so,

if m₂ is infinite then this period is 0 as the body revolves at infinite speed

if the earth was replaced by a star of mass same as sun.

then, T = T₀ /(2) 1/2

T=258.09 days.