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 with arrow| = 160 km/s and the orbital period of each is 13.7 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 1030 kg.)

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