Answer
(a) The gravitational force provides the centripetal force for each star to continue moving in a circle around the midpoint between the two stars.
(b) The mass of each star is $9.6 \times 10^{29} ~kg$
Work Step by Step
(a) The gravitational force provides the centripetal force for each star to continue moving in a circle around the midpoint between the two stars. Let $r$ be half the distance between the stars:
$v = \frac{2\pi r}{T}$
First, we need to find $T$:
$T = (12.6 ~yrs)(365 ~d/yr)(24 ~h/d)(3600 ~s/h)$
$T = 397353600 ~s$
We can use the expression for $v$ and the value for $T$ in the force equation. Let's consider the circular motion of one of the stars:
$\frac{Gm^2}{(2r)^2} = \frac{mv^2}{r}$
$m = \frac{4v^2r}{G} = \frac{16\pi^2 r^3}{GT^2}$
$m = \frac{(16\pi^2)(4.0 \times 10^{11} ~m)^3}{(6.67 \times 10^{-11} ~N\cdot m^2/kg^2)(397353600 ~s)^2}$
$m = 9.6 \times 10^{29} ~kg$
The mass of each star is $9.6 \times 10^{29} ~kg$
