Chapter 13, Problem 96P

(a)

To determine

To calculate: Compactness of Earth.

Compactness of Earth is $1.383\times {10}^{-9}$ .

Given:

Mass of Earth: $5.97\times {10}^{24}\text{ kg}$ .

Radius of Earth: $6.4\times {10}^6\text{ m}$ .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Earth}={\left(\frac{R_s}{R}\right)}_{Earth}$

Substituting the given values we get-

  $\begin{array}{l}{Z}_{Earth}=\frac{2G{M}_{earth}}{c^2{R}_{earth}}\\ Z_{Earth}=\frac{2\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times 5.97\times {10}^{24}\text{kg}}{{\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2\times 6.4\times {10}^6\text{m}}\\ Z_{Earth}=1.383\times {10}^{-9}\end{array}$

Conclusion:

Compactness of Planet earth is $1.383\times {10}^{-9}$ .

(b)

To calculate: Compactness of Sun.

Compactness of Sun is $4.238\times {10}^{-6}$ .

Given:

Mass of Sun: $1.99\times {10}^{30}\text{ kg}$ .

Radius of Sun: $6.96\times {10}^8\text{ m}$ .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Sun}={\left(\frac{R_s}{R}\right)}_{Sun}$

Substituting the given values we get-

  $\begin{array}{l}{Z}_{Sun}=\frac{2G{M}_{Sun}}{c^2{R}_{Sun}}\\ Z_{Sun}=\frac{2\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times 1.99\times {10}^{30}\text{kg}}{{\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2\times 6.96\times {10}^8\text{m}}\\ Z_{Sun}=4.238\times {10}^{-6}\end{array}$

Conclusion:

Compactness of Sun is $4.238\times {10}^{-6}$ .

(c)

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is $0.9934$ .

Given:

Density of Star: $4\times {10}^{17}\text{ kg}/{\text{m}}^3$

Radius of Star: $2\times {10}^4\text{ m}$

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Star}={\left(\frac{R_s}{R}\right)}_{Star}$

Substituting the given values we get-

  $Z_{Star}=\frac{2G{M}_{Star}}{c^2{R}_{Star}}$

Since $\rho =\frac{M}{V}$ then $M_{Star}=\frac{4}{3}\pi \rho R_{Star}^3$

  $\begin{array}{l}{Z}_{Star}=\frac{8G}{3c^2}\left(\pi \rho R_{Star}^2\right)\\ Z_{Star}=\frac{8\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times \pi \times 4\times {10}^{17}{\text{kg m}}^{-3}\times {\left(2\times {10}^4\text{m}\right)}^2}{3\times {\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2}\\ Z_{Star}=0.9934\end{array}$

Conclusion:

Compactness of Star is $0.9934$ .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Blackhole}={\left(\frac{R_s}{R}\right)}_{Blackhole}$

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore $R_s\approx R$

  $\begin{array}{l}{Z}_{Blackhole}=\left(\frac{R}{R}\right)\\ Z_{Blackhole}=1\end{array}$

Conclusion:

Compactness of Black hole is $1$ .

(b)

To determine

To calculate: Compactness of Sun.

Compactness of Sun is $4.238\times {10}^{-6}$ .

Given:

Mass of Sun: $1.99\times {10}^{30}\text{ kg}$ .

Radius of Sun: $6.96\times {10}^8\text{ m}$ .

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Sun}={\left(\frac{R_s}{R}\right)}_{Sun}$

Substituting the given values we get-

  $\begin{array}{l}{Z}_{Sun}=\frac{2G{M}_{Sun}}{c^2{R}_{Sun}}\\ Z_{Sun}=\frac{2\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times 1.99\times {10}^{30}\text{kg}}{{\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2\times 6.96\times {10}^8\text{m}}\\ Z_{Sun}=4.238\times {10}^{-6}\end{array}$

Conclusion:

Compactness of Sun is $4.238\times {10}^{-6}$ .

(c)

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is $0.9934$ .

Given:

Density of Star: $4\times {10}^{17}\text{ kg}/{\text{m}}^3$

Radius of Star: $2\times {10}^4\text{ m}$

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Star}={\left(\frac{R_s}{R}\right)}_{Star}$

Substituting the given values we get-

  $Z_{Star}=\frac{2G{M}_{Star}}{c^2{R}_{Star}}$

Since $\rho =\frac{M}{V}$ then $M_{Star}=\frac{4}{3}\pi \rho R_{Star}^3$

  $\begin{array}{l}{Z}_{Star}=\frac{8G}{3c^2}\left(\pi \rho R_{Star}^2\right)\\ Z_{Star}=\frac{8\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times \pi \times 4\times {10}^{17}{\text{kg m}}^{-3}\times {\left(2\times {10}^4\text{m}\right)}^2}{3\times {\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2}\\ Z_{Star}=0.9934\end{array}$

Conclusion:

Compactness of Star is $0.9934$ .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Blackhole}={\left(\frac{R_s}{R}\right)}_{Blackhole}$

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore $R_s\approx R$

  $\begin{array}{l}{Z}_{Blackhole}=\left(\frac{R}{R}\right)\\ Z_{Blackhole}=1\end{array}$

Conclusion:

Compactness of Black hole is $1$ .

(c)

To determine

To calculate: Compactness of Neutron Star.

Compactness of Neutron Star is $0.9934$ .

Given:

Density of Star: $4\times {10}^{17}\text{ kg}/{\text{m}}^3$

Radius of Star: $2\times {10}^4\text{ m}$

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Star}={\left(\frac{R_s}{R}\right)}_{Star}$

Substituting the given values we get-

  $Z_{Star}=\frac{2G{M}_{Star}}{c^2{R}_{Star}}$

Since $\rho =\frac{M}{V}$ then $M_{Star}=\frac{4}{3}\pi \rho R_{Star}^3$

  $\begin{array}{l}{Z}_{Star}=\frac{8G}{3c^2}\left(\pi \rho R_{Star}^2\right)\\ Z_{Star}=\frac{8\times 6.67\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\times \pi \times 4\times {10}^{17}{\text{kg m}}^{-3}\times {\left(2\times {10}^4\text{m}\right)}^2}{3\times {\left(3\times {10}^8{\text{ms}}^{-1}\right)}^2}\\ Z_{Star}=0.9934\end{array}$

Conclusion:

Compactness of Star is $0.9934$ .

(d)

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Blackhole}={\left(\frac{R_s}{R}\right)}_{Blackhole}$

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore $R_s\approx R$

  $\begin{array}{l}{Z}_{Blackhole}=\left(\frac{R}{R}\right)\\ Z_{Blackhole}=1\end{array}$

Conclusion:

Compactness of Black hole is $1$ .

(d)

To determine

To calculate: Compactness of Black hole.

Compactness of Black hole is1.

Given:

Compactness is the ratio of Schwarzschild radius to Actual radius.

Formula used:

Schwarzschild radius $R_s=\frac{2GM}{c^2}$ where $M$ is the mass of body.

Calculation:

Let compactness be denoted by $Z$ and for earth $Z_{Blackhole}={\left(\frac{R_s}{R}\right)}_{Blackhole}$

According to question the Schwarzschild radius of black hole is comparable to its actual radius therefore $R_s\approx R$

  $\begin{array}{l}{Z}_{Blackhole}=\left(\frac{R}{R}\right)\\ Z_{Blackhole}=1\end{array}$

Conclusion:

Compactness of Black hole is $1$ .