Finding the mass of a planet is a binary system

[Zynoakib]

Homework Statement

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. Assume the orbital speed
of each star is 220 km/s and the orbital period
of each is 14.4 days. Find the mass M of each star.

Homework Equations

The Attempt at a Solution

Kelper's 3rd Law

Mass of the planet = (4(pi)^2(r)^3)/GT^2

T = 1244160
r = 8.72 x 10^10
G = 6.672 x 10^-11

Mass = 2.54 x 10^32, but the answer is 1.26 x 10^32 which is exactly half of my answer

why should I further divide my answer by 2? Is it because there are two planets in the system? But if I use the same equation to calculate the Sun's mass, I just need to substitute all those unknown into the equation without the need to consider the mass of the Earth. So what's wrong?

Thanks in advance!

 

[haruspex]

Kepler's third law was originally in the context of planets orbiting the sun. In that context, the sun is so massive compared with the planets that the orbital radius is effectively the distance from the sun's centre. That won't be the case with two equal mass stars, so I'm not sure what the appropriate r is here. Also, how did you obtain r?

 

[Zynoakib]

Kepler's third law was originally in the context of planets orbiting the sun. In that context, the sun is so massive compared with the planets that the orbital radius is effectively the distance from the sun's centre. That won't be the case with two equal mass stars, so I'm not sure what the appropriate r is here. Also, how did you obtain r?

orbital speed of each star = 220 km/s
orbital period of each = 14.4 days.
Distance Traveled in one period = (220000)(14.4 x 24 x 60 x 60)m. Then, you can get the radius from 2(pi)r

I have solved the problem by using centripetal force= gravitational pull. So probably using Kepler's third law is wrong for binary system.

Thanks anyway