Question

1. Binary Star (problem 66/chapter 8) [6] Consider two objects with equal mass M orbiting each other, such as a pair of binary stars. T-2nd'GM Show that the orbital period is given by distance between the objects [4] If the two stars are about 9AU apart, and each have a mass about 20% that of our sun, what is their angular velocity o? 12 a) where d is the b)

Answer 1

(a) Force between two objects, F = GM^2/d^2 (Gravitational force) Centripetal force, F_C = Mv^2/r, r = d/2 F_C = 2Mv^2/d v = squareroot F_C d/2M v = squareroot GM^2/d^2 (d/2M) = squareroot GM/2d Period, T = 2pi d/2/v = 2pi d/2 squareroot GM/2d T = pi d^3/2 squareroot 2/squareroot GM T^2 = 2pi^2 d^3/GM (b) w = 2pi/T = 2pi squareroot GM/squareroot 2 pi d^3/2 = squareroot 2 GM/d^3/2 w = squareroot 2 times 6.67 times 10^-11 times 0.2 times 2 times 10^30/(9 times 1.5 times 10^11)^3/2 w = squareroot 53.36 times 10^18/(13.5 times 10^11)^3/2 = 7.3 times 10^9/49.6 times 10^16 squareroot 10 w = 7.3 times 10^9/156.85 times 10^16 w = 4.654 times 10^-9 rad/s