Question

Binary Star (problem 66/chapter 8) [6] Consider two objects with equal mass M orbiting each other, such as a pair of binary stars. a) Show that the orbital period is given by T-2Tt'd'IGM where d is the 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? [2] b)

Answer 1

Force between two object = F = GM^2/d^2 This force acts as centripetal force on each object then GM^2/d^2 = MV^2_0/d/2 = 2MV^2_0/d V_0 = squareroot GM/2d so period = T = 2 pi times d/2/V_0 OR T = 2pi d/2 squareroot G/2d = pi d^3/2 squareroot 2/squareroot GM So T^2 = 2 pi^2 d^3/GM omega = 2 pi/T = 2 pi squareroot GM/squareroot 2 pi d^3/2 = squareroot 2 GM/d^3/2 OR omega = squareroot 2 times 6.67 times 10^-11 times 0.2 times 2.0 times 10^30/(9 times 1.5 times 10^11)^3/2 = squareroot 53.36 times 10^8/[13.45 times 10^11]^3/2 = 7.3 times 10^9/49.6 times 10^16 times squareroot 10 = 7.3 times 10^9/156.85 times 10^6 = 4.654 times 10^-9 rad/s omega = 4.654 times 10^-9 rad/s