Question

The usual, simplified, derivation of the Kepler – Newton relation assumes that the object at the center of the orbit is stationary. This is equivalent to assuming it is much, much more massive than the object orbiting it. This is fine for bodies orbiting the Sun, Jupiter or Saturn, but there are other cases, such as binary stars where the masses are comparable. Such binary stars orbit around their Center of Mass in circular orbits, apart from some very strange cases with highly elliptical orbits which are not presently understood. For two stars of masses m and M separated by distance d (center to center), use Newton’s second law, the law of Universal Gravitation and the centripetal force to construct an expression for the mass of the stars (M+m) in terms of d, G, and the period T. (First explain how the periods of the two stellar orbits are related.)

Answer 1

for gravitational force on the object

Fg = Gm M/d^2 -------------------1

m1 and m2 are the masses and d is the center to center seperation between the masses.

accleration a = V^2/d

but V = 2pid/T

so

a = 4pi^2 d/T^2 -------------------------2

from 1 , ma = G m M/d^2

a = GM/d^2 --------------------------------3

equating 2 amd 3

4 pi^2 d/T^2 = GM/d^2

Solving for T,

T^2 = 4pi^2 d^3/GM

T = 2pi sqrt(d^3/(GM)

for a total mass of M+m, it is

same as T = 2p sqrt(d^3/GM

time perind is independent of the mass of sateliite/stars rather it depends on the mass of the Sun