Chapter 33, Problem 14P

To determine

The average density of nuclear star.

To compare: The obtained result with the density of white dwarf and with the density of the nuclear matter.

Answer to Problem 14P

The average density of the nuclear star is $5.35\times {10}^{17}{\text{kg/m}}^3$ .

The average density of the nuclear star is about $2.923\times {10}^8$ times larger than density of white dwarf.

The average density of the nuclear star is about $2.32$ times larger than density of nuclear matter.

Explanation of Solution

Given:

Assume the mass of nuclear star is equal to 1.5 times the mass of the Sun and is equal to $M_{\text{ns}}=1.5\times 1.99\times {10}^{30}\text{kg}$ .

Assume the radius of the nuclear star is $R_{\text{ns}}=11\text{km}$ .

Assume the mass of the white dwarf is equal to the mass of the Sun and is equal to $M_{\text{wd}}=1.99\times {10}^{30}\text{kg}$ .

Assume the radius of the white dwarf is equal to the radius of the Earth and is equal to $R_{\text{wd}}=6.38\times {10}^6\text{m}$ .

Formula used:

Assume the planet is spherical in shape.

The expression for the volume ( V ) of the Planet (spherical shape) is,

  $V=\frac{4}{3}\pi R^3$

Here, R is the radius of the sphere.

The expression for the density ( D ) of the planet is,

  $D=\frac{M}{V}$

Here, M is the mass of the planet.

Calculation:

The volume of the nuclear star is,

  $\begin{array}{l}{V}_{\text{ns}}=\frac{4}{3}\pi R_{ns}^3\\ V_{\text{ns}}=\frac{4}{3}\pi {\left(11\text{km}\times \frac{1000\text{m}}{1\text{km}}\right)}^3\\ V_{\text{ns}}=5.575\times {10}^{12}{\text{m}}^3\end{array}$

The average density of the nuclear star is,

  $\begin{array}{c}{D}_{\text{ns}}=\frac{M_{\text{ns}}}{V_{\text{ns}}}\\ =\left(\frac{1.5\times 1.99\times {10}^{30}\text{kg}}{5.575\times {10}^{12}{\text{m}}^3}\right)\\ =5.35\times {10}^{17}{\text{kg/m}}^3\end{array}$ .............. (1)

The volume of the white dwarf is

  $\begin{array}{l}{V}_{\text{wd}}=\frac{4}{3}\pi R_{wd}^3\\ V_{\text{wd}}=\frac{4}{3}\pi {\left(6.38\times {10}^6\right)}^3\\ V_{\text{wd}}=1.087\times {10}^{21}{\text{m}}^3\end{array}$

The density of the white dwarf is

  $\begin{array}{c}{D}_{\text{wd}}=\frac{M_{\text{wd}}}{V_{\text{wd}}}\\ =\left(\frac{1.99\times {10}^{30}\text{kg}}{1.087\times {10}^{21}{\text{m}}^3}\right)\\ =1.83\times {10}^9{\text{kg/m}}^3\end{array}$ .............. (2)

Compare the average density of the nuclear star and the density of the white dwarf.

Divide equation (1) by equation (2).

  $\begin{array}{l}\frac{D_{\text{ns}}}{D_{\text{wd}}}=\frac{5.35\times {10}^{17}{\text{kg/m}}^3}{1.83\times {10}^9{\text{kg/m}}^3}\\ \frac{D_{\text{ns}}}{D_{\text{wd}}}=2.923\times {10}^8\\ D_{\text{ns}}=2.923\times {10}^8{D}_{\text{wd}}\end{array}$

The density of nuclear matter is equal to density of nucleus of an atom and is equal to $D_{\text{nm}}=2.3\times {10}^{17}{\text{kg/m}}^3$ .

Compare the average density of the nuclear star and the density of the nuclear matter.

  $\begin{array}{l}\frac{D_{\text{ns}}}{D_{\text{nm}}}=\frac{5.35\times {10}^{17}{\text{kg/m}}^3}{2.3\times {10}^{17}{\text{kg/m}}^3}\\ \frac{D_{\text{ns}}}{D_{\text{nm}}}=2.32\\ D_{\text{ns}}=2.32D_{\text{nm}}\end{array}$

Conclusion:

Thus, the average density of the nuclear star is $5.35\times {10}^{17}{\text{kg/m}}^3$ .

Thus, the average density of the nuclear star is about $2.923\times {10}^8$ times larger than density of white dwarf.

Thus, the average density of the nuclear star is about $2.32$ times larger than density of nuclear matter.