Chapter 14, Problem 4P

To determine

The density of Neutron stars of masses $1.4M_0$ and $3.0M_0$.

Answer to Problem 4P

The density of Neutron stars of masses $1.4M_0$ and $3.0M_0$ are $6.65\times {10}^{17}\text{kg}/{\text{m}}^{\text{3}}$ and $3.0\times {10}^{18}\text{kg}/{\text{m}}^{\text{3}}$ respectively.

Explanation of Solution

Necessary data is obtained from problem 2. It is found that from problem 2, radius of Neutron star having mass $3.0M_0$ is $7.8\text{km}$.

Write the relation between mass and density.

    $\rho =\frac{m}{V}$

Here, $\rho$ is the density, $m$ is the mass, and $V$ is the volume of H1 cloud.

Write the equation to find $V$.

    $V=\frac{4}{3}\pi R^3$

Here, $R$ is the radius.

Rewrite the equation for $\rho$ by substituting the above relation for $V$.

    $\begin{array}{c}\rho =\frac{m}{\frac{4}{3}\pi R^3}\\ =\frac{3m}{4\pi R^3}\end{array}$        (I)

Rewrite the above equation by substituting $1.4M_0$ for $m$.

    $\rho =\frac{3\left(1.4M_0\right)}{4\pi R^3}$        (II)

Here, $M_0$ is the solar mass.

Rewrite equation (I) by substituting $3.0M_0$ for $m$.

    $\rho =\frac{3\left(3M_0\right)}{4\pi R^3}$        (III)

Conclusion:

Case 1: Neutron star of mass $1.4M_0$

Substitute $1.99\times {10}^{30}\text{kg}$ for $M_0$ and $10\text{km}$ for $R$ in equation (I) to find $\rho$.

    $\begin{array}{c}\rho =\frac{3\left(1.4\left(1.99\times {10}^{30}\text{kg}\right)\right)}{4\pi {\left(10\text{km}\left(\frac{{\text{10}}^{\text{3}}\text{m}}{\text{1}\text{km}}\right)\right)}^3}\\ =\frac{8.36\times {10}^{30}\text{kg}}{12.56\times {10}^{12}{\text{m}}^{\text{3}}}\\ =6.65\times {10}^{17}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

Case 2: Neutron star of mass $3.0M_0$

Substitute $1.99\times {10}^{30}\text{kg}$ for $M_0$ and $7.8\text{km}$ for $R$ in equation (I) to find $\rho$.

    $\begin{array}{c}\rho =\frac{3\left(3.0\left(1.99\times {10}^{30}\text{kg}\right)\right)}{4\pi {\left(7.8\text{km}\left(\frac{{\text{10}}^{\text{3}}\text{m}}{\text{1}\text{km}}\right)\right)}^3}\\ =\frac{17.91\times {10}^{30}\text{kg}}{5.96\times {10}^{12}{\text{m}}^{\text{3}}}\\ =3.0\times {10}^{18}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

Therefore, the density of Neutron stars of masses $1.4M_0$ and $3.0M_0$ are $6.65\times {10}^{17}\text{kg}/{\text{m}}^{\text{3}}$ and $3.0\times {10}^{18}\text{kg}/{\text{m}}^{\text{3}}$ respectively.