Astrophysicists in the 1960s tried to explain the existence and the structure of pulsars - extremely regular astronomical sources of radio pulses whose periods ranged from seconds to milliseconds. At one point, these radio sources were given the acronym LGM, standing for "Little Green Men," a reference to the idea that they might be signals of extraterrestrial civilizations. The explanation given today is no less interesting. Consider the following. Our Sun, which is a fairly typical star, has a mass of 1.99x10^30 kg and a radius of 6.69x10^8 m. Although it does not rotate uniformly, because it isn't a solid body, its average rate of rotation is about 1 rev / 25 days. Stars larger than the sun can end their lives in spectacular explosions called supernovae, leaving behind collapsed remnants of the stars called neutron stars. Neutron stars have masses comparable to the original masses of the stars, but radii of only a few kilometers. These stars emit beams of radio waves. Because of the rapid angular speed of the stars, the beams sweep past Earth at regular, very short, intervals. To produce the observed radio-wave pulses, the star has to rotate at rates from about 1 rev/s to 1000 rev/s.
a. Estimate the rotation rate of the Sun if it were to collapse into a neutron star of radius 10 km. The Sun is not a uniform sphere of gas, and its moment of inertia is given by I = 0.059MR2. Assume that the neutron star is spherical and has a uniform mass distribution.
b. Is the rotational kinetic energy of our Sun greater of smaller after the collapse? By what factor does it change, and where does the energy go to or come from?