Question 29.1: Sizing a Neutron Star Goal Apply the concepts of nuclear siz...
Sizing a Neutron Star
Goal Apply the concepts of nuclear size.
Problem One of the end stages of stellar life is a neutron star, where matter collapses and electrons combine with protons to form neutrons. Some liken neutron stars to a single gigantic nucleus. (a) Approximately how many nucleons are in a neutron star with a mass of 3.00 \times 10^{30} \mathrm{~kg} ? (This is the mass number of the star.) (b) Calculate the radius of the star, treating it as a giant nucleus. (c) Calculate the density of the star, assuming the mass is distributed uniformly.
Strategy The effective mass number of the neutron star can be found by dividing the star mass in \mathrm{kg} by the mass of a neutron. Equation 29.1
r=\,r_{0}A^{1/3} (29.1)
then gives an estimate of the radius of the star, which together with the mass determines the density.
(a) Find the approximate number of nucleons in the star.
Divide the star’s mass by the mass of a neutron to find A :
A=\left(\frac{3.00 \times 10^{30} \mathrm{~kg}}{1.675 \times 10^{-27} \mathrm{~kg}}\right)=1.79 \times 10^{57}
(b) Calculate the radius of the star, treating it as a giant atomic nucleus.
Substitute into Equation 29.1:
\begin{aligned} r & =r_{0} A^{1 / 3}=\left(1.2 \times 10^{-15} \mathrm{~m}\right)\left(1.79 \times 10^{57}\right)^{1 / 3} \\ & =1.46 \times 10^{4} \mathrm{~m} \end{aligned}
(c) Calculate the density of the star, assuming that its mass is distributed uniformly.
Substitute values into the equation for density and assume the star is a uniform sphere:
\begin{aligned} \rho & =\frac{m}{V}=\frac{m}{\frac{4}{3} \pi r^{3}}=\frac{3.00 \times 10^{30} \mathrm{~kg}}{\frac{4}{3} \pi\left(1.46 \times 10^{4} \mathrm{~m}\right)^{3}} \\ & =2.30 \times 10^{17} \mathrm{~kg} / \mathrm{m}^{3} \end{aligned}
Remarks This density is typical of atomic nuclei as well as of neutron stars. A ball of neutron star matter having a radius of only 1 meter would have a powerful gravity field: it could attract objects a kilometer away at an acceleration of over 50 \mathrm{~m} / \mathrm{s}^{2} !