Question

A binary star system consists of two equal mass stars that revolve in circular orbits about their center of mass. The period of the motion, T = 26.4 days, and the orbital speed v = 220 km/s of the stars can be measured from telescopic observations. What is the mass (kg) of each star?

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Answer 1

The force of gravity one star exerts on the other in terms of the diameter of the binary star system is:
F = G*m^2/d^2

And of course, d = 2*R. Thus:
F = G*m^2/(4*R^2)

This force causes centripetal acceleration:
a = v^2/R

Relation between v, R, and period T:
v = 2*Pi*R/T

Thus:
R = v*T/(2*Pi)

Substitute in each location:
F = G*m^2/(4*(v*T/(2*Pi))^2)
a = v^2/(v*T/(2*Pi))

Simplify:
F = G*(Pi*m/(v*T))^2
a = 2*Pi*v/T

Newton's second law for either one of the stars:
F = m*a

Thus:
G*(Pi*m/(v*T))^2 = 2*Pi*m*v/T

Solve for m:
m = 2*T*v^3/(Pi*G)

Data (translated to SI units):
T:= 86400*26.4 sec; v:=220000 m/s; G:=6.673e-11 N-m^2/kg^2;

Result:
m = 2.318*10^32 kg