A certain triple star system, known as a trinary, consists of two stars, each of

Question

A certain triple star system, known as a trinary, consists of two stars, each of mass m, orbiting a third star of mass M (whose center is located at the center-of-mass of the three-star system) at a radius R. The two stars of mass m orbit directly across from one another, so that all three stars always form a straight line. Derive an expression for the period of revolution of the two stars. (Note: You must completely simplify your answer for full credit - this mainly means that any quantities, such as m, M, G, and R can appear at most once in your answer.)

Explanation / Answer

you need to figure out the forces since the two stars orbiting are directly opposite each other .. then they generate directly opposite pulls on the central star and that cancels any movment the central star might have done..

so basically the central star is motionless in this system

now you just need to figure out the forces on the outer stars

the one star that is in the center will generate a force of

Fc = G Mm/r^2

call this central force or

the other star will also generate some force whether its negligable or not will have to figure out later

Fm = G mm/(2r)^2

its 2r because the distance is the diameter or 2 times the radius from one star to the other


So basically the full force is

NET FORCE= (Gm/r^2 )* (M + m/4)

now you can use some rotational equation to figure out speed and period

v^2/r = a

since the force is known

Fnet/m = v^2/r
v^2 = r* Fnet/m
v = sqrt (r* Fnet/m)

we know distance is

circumferance = 2*pi*r
so
distance = 2*pi*r

time = distance / velocity = 2*pi*r/sqrt(r*Fnet/m)

=2*pi*r^1.5/sqrt(GM^2/m + GM)

thank you

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