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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, mr=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!

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  • $\begingroup$ Yes but i am not sure how to make the substitution so as to get rid of the r. $\endgroup$ – rahul rj Mar 8 '17 at 9:19
  • $\begingroup$ That was trivial indeed, I was solely thinking in terms of centripetal force and gravitational force , and along the way completely ignored that its circle with C = 2pi r. Thank you! $\endgroup$ – rahul rj Mar 8 '17 at 11:00
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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.

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