Plaskett\'s binary system consists of two stars that revolve in a circular orbit

Question

Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal see figure below). Assume the orbital speed of each star is |V^vector| = 150 km/s and the orbital period of each is 11.3 days. Find the mass M of each star. (For comparison, the mass of our sun is 1.99 times 10^30 kg.) _______ If you know the orbital period and the speed of an object undergoing constant circular motion, how do you calculate the radius of the circle? solar masses

Explanation / Answer

v = 150 km/s = 1.5 x 10^5 m/s ; t = 11.3 days = 976320 s

We know that, centripital force is:

F = M v^2/r

gravitational force between two bodies

F = G M1 M2/r^2

for our case, F = G M^2/(2r)^2

the above two forces balances each other so

G M^2/4 r^2 = M v^2/r

M = 4 v^2 r/G

we do not have r so we have to eliminate it

vt = 2 pi r => r = vt/2 pi

M = 2 v^2 T/pi G

M = 2 (1.5 x 10^5)^2 9.76 x 10^5/3.14 x 6.67 x 10^-11 = 2.1 x 10^26 kg

Hence, M = 2.1 x 10^26 kg

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