Answer:
M = 4.61 10³¹ kg
Explanation:
For this exercise we will use Newton's Second Law where the force is the gravitational outside
F = ma
G M₁ M₂ / r² = m a
The acceleration is centripetal
a = v² / r’
Let's replace
The mass of the two stars in it, the distance between them is r and the distance around the center of mass is
r’= r / 2
G M² / r² = M v² / (r / 2)
G M / r = 2 v²
The linear velocity module is constant, so we can use the kinematic relationship
v = d / t
The distance of a circle of radius r ’is
d = 2π r ’= 2π (r / 2)
d = π r
We replace
v = π r / T
Let's write the two equations
v² = ½ G M / r
v = π r / T
r = v T /π
v² = ½ G M π/ vT
M = 2 v³ T / π G
Let's reduce the magnitudes to the SI system
v = 160 Km / s = 1.60 10⁵ m / s
T = 13.7 days (24 h / 1 day) (3600s / 1 h) = 1.18 10⁶ s
Let's calculate
M = 2 (1.60 10⁵)³ 1.18 10⁶ / (π 6.67 10⁻¹¹)
M = 4.61 10³¹ kg
