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SOLUTIONS TO EXERCISE 9
(a) The amount of energy released would be approximately equal to one neutron star falling onto the surface of the other neutron star from a large distance. The radius of a neutron star is about 10 km. If the other neutron star drops from a distance of 1000 km, then the energy released would be E = -GMm/r(start) – (-GMm/r(end)) = -(6.67 * 10^(-11)) * (4 * 10^30)^2 * (1/(10^6) – 1/(10^4)) = 1.1 * 10^47 J. This is a little more than ten times the energy produced by GRB031332, so if ten percent of the gravitational potential energy released were converted into radiation, this neutron star collision scenario would be a plausible way to create the gamma-ray burst. (b) The total amount of gravitational potential energy released would be the same if ordinary stars or diffuse gas fell onto a neutron star. But the timing would be very different; before that "ordinary" matter reached the neutron star's surface, it would be pulled into an orbiting disk of material, which would regulate a gradual flow of matter onto the neutron star. So instead of releasing the energy in a matter of seconds, it would take years or even millennia – not in the form of a burst. Only a very solid, tightly bound object like another neutron star could resist the tidal disruption as it fell in, thus allowing the gravitational potential energy to "burst" forth. |