Question

Two neutron stars are separated by a distance of 1.0 1010 m. They each have a mass of 1.0 1030 kg and a radius of 1.0 105 m. They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving at the following positions? (

a) Their separation has decreased to one-half its initial value. m/s

(b) They are about to collide. m/s

Answer 1

A. Initially, U = - Gm^2 /R, and since at rest, their moments are zero. Thus, by symmetry, they each must be moving with the same speed in opposite directions at any subsequent time.

At half that distance, U’ = 2U = - 2Gm^2 /R

dU = - Gm^2 /R

dKE = -dU

so for each star

K = Gm^2/2R = 0.5*mv^2

v = (Gm/R)^0.5 = (6.67*10^-11*10^30/10^10)

v = 8.2*10^4 m/sec

B. With a final separation distance of 2r = 2.0*10^5 m

dU = Gm^2 (1/R - 1/2r)

dKE = Gm^2 (1/2r - 1/R) = mv^2

v = [Gm(1/2r - 1/R)]^0.5 = [6.67*10^-11*10^30(1/(2*10^5) - 1/10^10)]^0.5

v = 1.8*10^7 m/sec