Three body, equal mass star system

Sarial

1. The problem statement, all variables and given/known data
There is no general analytical solution for the motion of a 3-body gravitational system. However, there do exist analytical solutions for very special initial conditions. The diagram below shows three stars, each of mass m, which move in the plane of the page along a circle of radius r.


Calculate the magnitude of the total gravitational force exerted on one of the stars due to the other two.
F total = ?

Period, T = ?


2. Relevant equations
F= Gmm/R^2
Trig


3. The attempt at a solution
My only attempt at a solution involved the assumption that the bodies are at 120 degrees, on the corners of an equilateral triangle, and trying to solve for the distance between the bodies in terms of r. Then using trig to solve for the y component of both forces and adding them.

Thanks in advance, guys.

Sarial

My most recent attempt, while I wait:

Equilateral triangle. 120 degree angles between each planet.

Law of sines:

r is distance between center and planet
R is distance between planets

r/sin(30) = R/sin(120)

R = r*sin(120)/sin(30)

G(m^2)/R^2

The force in the y direction on the top planet from the bottom left one should be

F*sin(60) = Fy

Both of the bottom planets forces should add, and the x forces cancel.

2*sin(60)*(((6.67x10^-11)(m^2))/(((r*sin(120))/sin(30))^2))

This, however, doesn't seem to be right. Any ideas?

Sarial

Bump? 20 mins left to answer, but I have to turn the work in on paper too. I really feel like my work I did in the reply should work, but it doesn't seem to.

Thanks~