••• Estimation of the radius of a neutron star. The following nonrelativistic calculation gives approximately correct values, but a rigorous calculation requires a relativistic treatment. (a) Estimate the gravitational pressure at the center of a star of mass M and radius R, assuming a uniform mass density. A fairly accurate estimate can be made from the following rough calculation: Consider the star to be made up of two hemispheres each of mass M/2. The centers of mass of the two hemispheres are roughly a distance R apart. The gravitational force between the hemispheres is then approximately F = G(M/2)2/R2. This force is spread out over the area of contact A between the hemispheres, and the pressure at the center is roughly p = F/A. (b) Starting with equations (13.16) and (13.25), derive an expression for the neutron degeneracy pressure in terms of R and M. [Hint: The mass density ρ he star, the number density n of neutrons, and the netron mass m are related by ρ = mn.] (c) The neutron star has a stable size because the inward gravitational pressure is equal to the outward degeneracy pressure. Derive an expression for the radius of a neutron star of mass M. If your expression is correct, it should show that the radius R and the mass M are related by R ∝ M−1/3. A larger mass produces a smaller radius! (d) Calculate the radius of a 2-solar- mass neutron star.
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