(a) What is the average density of the sun? (b) What is the average density of a neutron star that has the same mass as the sun but a radius of only 20.0 km?

Radius of neutron star = 20km

Mass of sun and neutron star is same.

Depending on the terminology used, the mass density or fluid density of a fluid is defined as the ratio of its mass to its volume.

Here mass of both sun and neutron sun is same but differ in radius so average density will be differ in both the cases.

$$\begin{gathered} \mathrm{Mass}\mathrm{of}\mathrm{sun}\left({\mathrm{M}}_{\text{sun}}\right)=1.99\times {10}^{30}\mathrm{kg} \\ \mathrm{Radius}\mathrm{of}\mathrm{Sun}\left({\mathrm{R}}_{\mathrm{sun}}\right)=6.96\times {10}^8\mathrm{m} \end{gathered}$$

Substitute the above value in density formula,

${\rho }_{\text{sun}}=\frac{M_{\text{sun}}}{\frac{4}{3}\pi R_{\text{sun}}^3}=\frac{1.99\times {10}^{30}\mathrm{kg}}{1.412\times {10}^{27}{\mathrm{m}}^3}=1409\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$

$$\begin{gathered} \mathrm{Mass}\mathrm{of}\mathrm{neutron}\mathrm{star}\left({\mathrm{M}}_{\text{neutron}}\right)=1.99\times {10}^{30}\mathrm{kg} \\ \mathrm{Radius}\mathrm{of}\mathrm{neutron}\mathrm{star}\left({\mathrm{R}}_{\mathrm{neutron}}\right)=2\times {10}^4\mathrm{m} \end{gathered}$$

Substitute the above value in density formula,

${\rho }_{\text{neutron}}=\frac{M_{\text{neutron}}}{\frac{4}{3}\pi R_{\text{neutton}}^3}=\frac{1.99\times {10}^{30}\mathrm{kg}}{3.351032\times {10}^{13}{\mathrm{m}}^3}=5.94\times {10}^{16}\mathrm{kg}/{\mathrm{m}}^3$

Thus, average density of sun is$1409\times {10}^{3}kg/m^3$ .

Average density of neutron star is $5.94\times {10}^{16}kg/m^3$.