Question

Plaskett's binary system consists of two stars that revolve in circular orbits around a fixed point half-way between them. This means that the masses of the two stars are equal (see the figure below). --- M If the orbital velocity of each star is v= 188 km/s and the orbital period of each is 11.7 days, calculate the mass Mof each star. (For comparison, the mass of our Sun is 1.99x1030 kg.)

Answer 1

Gravitational constant = G = 6.67 x 10^-11 N.m²/kg²

Mass of each star = M

Orbital speed of each star = V = 188 km/s = 188 x 10³ m/s = 1.88 x 10⁵ m/s

Orbital period of each star = T = 11.7 days = 11.7 x (24x60x60) sec = 1010880 sec

Radius of the orbit of each star = R

VT = 2$\pi$R

(1.88x10⁵)(1010880) = 2$\pi$R

R = 3.205 x 10¹⁰ m

Distance between the two stars = D

The distance between the two stars is equal to the diameter of the orbit.

D = 2R

D = (2)(3.205x10¹⁰)

D = 6.05 x 10¹⁰ m

The centripetal force for the circular motion of the stars is provided by the gravitational force between them.

$\frac{MV^{2}}{R} = \frac{GMM}{D^{2}}$

$\frac{V^{2}}{R} = \frac{GM}{D^{2}}$

$\frac{(1.88\ast 10^{5})^{2}}{3.025\ast 10^{10}} = \frac{(6.67\ast 10^{-11})M}{(6.05\ast 10^{10})^{2}}$

M = 6.411 x 10³¹ kg

Mass of our sun = M_s = 1.99 x 10³⁰ kg

$M =( \frac{6.411\ast 10^{31}}{1.99\ast 10^{30}})M_{s}$

M = 32.22 M_s

Mass of each star = 6.411 x 10³¹ kg = 32.22 solar masses