Question

two binary stars moving in a perfect circle orbit

The stars have the same mass, the distance between them is one billion km (1x10^9 km), and the time each takes for one orbit is 10.4 Earth years.

Determine the mass of each star (m1=m2).

(this problem DOESN'T uses the gravity formula with the G constant.

this problem must be solve using circular dynamics ecuations so please help!!!!)

Answer 1

Actually, the solution will involve the G constant. You will have to use it to compute the mass, unless you use Kepler's third law in a different form the involves the sum of the masses of the two objects, in terms of solar mass. "G" is already "built in" to that solution

Equal-mass stars revolve in orbits (circular in this case) about a point midway between the stars. The radius of each star's orbit is d/2, where d is the interstellar separation, 10^12 m.

Centripetal force = Gravity force
Let either mass be m.

G*m^2/d^2 = m*V^2/(d/2)= 2m*V^2/d
or G*m/d = 2*V^2

V*Period = 2*pi*d/2 = pi*d

Eliminate V from the first equation, using V from the second equation, and solve for the mass, m

V = pi*d/Period = 9573 m/s

m = 2*d*V^2/G