Question

A neutron star has a density equal to that of the nuclear matter. Assuming the star to be spherical, find the radius of a neutron star whose mass is 4.0 × 10³⁰ kg (twice the mass of the sun).

Answer 1

Given:
Mass of the neutron star, M = 4.0 × 10³⁰ kg
Density of nucleus, d = 2.4$\times$10¹⁷${}^{}$

Density of nucleus, $d=\frac{M}{V}$

Here, V is the volume of the nucleus.

$$\begin{gathered} \therefore V=\frac{M}{d}=\frac{4\times {10}^{30}}{2.4\times {10}^{17}} \\ =\frac{1}{0.6}\times {10}^{13} \\ =\frac{1}{6}\times {10}^{14} \end{gathered}$$

If R is the radius, then the volume of the neutron star is given by
$$\begin{gathered} V=\frac{4}{3}\pi R^3 \\ \therefore \frac{1}{6}\times {10}^{14}=\frac{4}{3}\times \mathrm{\pi }\times R^3 \\ \Rightarrow R^3=\frac{1}{6}\times \frac{3}{4}\times \frac{1}{\mathrm{\pi }}\times {10}^{14} \\ \Rightarrow R^3=\frac{1}{8}\times \frac{100}{\mathrm{\pi }}\times {10}^{12} \\ \therefore R=\frac{1}{2}\times {10}^4\times 3.17 \\ =1.585\times {10}^4=15\mathrm{km} \end{gathered}$$