A binary star is composed of two stars that orbit around their centre of mass under the influence of gravity. Consider such a system in which two stars have identical mass. In the centre of mass frame, each star moves in a circular orbit with a speed of 200 km/s. If the orbital period is 15 days, what is the approximate mass of the star?

a) 10^32 kg b) 10^30 kg c) 10^34 kg d) 10^28 kg e) 10^26 kg

Attempt:

Centripetal force = m*v^2/r

Gravitational force = Gm1m2/r^2 (m1=m2)

m*v^2/r = Gm1m2/r^2

v = rw; w = angular velocity

w = 2*pi/T (T, time period)

substituting

4*pi^2/T^2 = Gm/r^3

I am stuck here since r is not given.

Using the centre of mass equation, m*r=m*(R-r) [R is the total distance between the stars and r is the distance from COM to each star].

Don't know how to go beyond this. Please help with the solution!

## Best Answer

I seemed to have completely ignored that it was a circle. The solution is as follows:

T = 15 days = 15*24*3600 = 1296000 s; 2pi r = 200 * 10^3 * 1296000

r = 4.12*10^9 m; R = 2r = 8.24 * 10^9 m; mv^2/R = Gmm/R^2

m (approx) = 4.97 * 10^30 kg.