Chapter 25, Problem 34Q

To determine

(a)

The mass density of radiation of the photosphere of the Sun and to explain whether the radiation is matter-dominated or radiation-dominated.

Answer to Problem 34Q

The mass density of radiation is $9.5\times {10}^{-18}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is matter-dominated.

Explanation of Solution

Given:

The temperature of the photosphere of the Sun is, $T=5800\text{K}$.

The average density of matter is ${\rho }_{\text{m}}=3\times {10}^{-4}\text{kg}/{\text{m}}^3$.

Formula Used:

The mass density of radiation is given by,

${\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}$

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation and $\sigma$ is the Stefan-Boltzmann constant.

Calculation:

The mass density of radiation is calculated as,

$\begin{array}{c}{\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}\\ =\frac{4\left(5.67\times {10}^{-8}\text{W}/{\text{m}}^2\cdot {\text{K}}^4\right){\left(5800\text{K}\right)}^4}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^3}\\ =9.5\times {10}^{-18}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

$\begin{array}{c}{\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}\\ =\frac{4\left(5.67\times {10}^{-8}\text{W}/{\text{m}}^2\cdot {\text{K}}^4\right){\left(5800\text{K}\right)}^4}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^3}\\ =9.5\times {10}^{-18}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

The mass density of radiation is less than the average density of matter. Therefore, the radiation is matter-dominated.

Conclusion:

The mass density of radiation is $9.5\times {10}^{-18}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is matter-dominated.

To determine

(b)

The mass density of radiation of the centre of the Sun and to explain whether the radiation is matter-dominated or radiation-dominated.

Answer to Problem 34Q

The mass density of radiation is $4.85\times {10}^{-4}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is matter-dominated.

Explanation of Solution

Given:

The temperature of the centre of the Sun is, $T=1.55\times {10}^7\text{K}$.

The average density of matter is ${\rho }_{\text{m}}=1.6\times {10}^5\text{kg}/{\text{m}}^3$.

Formula Used:

The mass density of radiation is given by,

${\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}$

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation and $\sigma$ is the Stefan-Boltzmann constant.

Calculation:

The mass density of radiation is calculated as,

$\begin{array}{c}{\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}\\ =\frac{4\left(5.67\times {10}^{-8}\text{W}/{\text{m}}^2\cdot {\text{K}}^4\right){\left(1.55\times {10}^7\text{K}\right)}^4}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^3}\\ =4.85\times {10}^{-4}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

The mass density of radiation is less than the average density of matter. Therefore, the radiation is matter-dominated.

Conclusion:

The mass density of radiation is $4.85\times {10}^{-4}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is matter-dominated.

To determine

(c)

The mass density of radiation of the solar corona and to explain whether the radiation is matter-dominated or radiation-dominated.

Answer to Problem 34Q

The mass density of radiation is $1.34\times {10}^{-7}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is radiation-dominated.

Explanation of Solution

Given:

The temperature of the solar corona is, $T=2\times {10}^6\text{K}$.

The average density of matter is ${\rho }_{\text{m}}=5\times {10}^{-13}\text{kg}/{\text{m}}^3$.

Formula Used:

The mass density of radiation is given by,

${\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}$

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation and $\sigma$ is the Stefan-Boltzmann constant.

Calculation:

The mass density of radiation is calculated as,

$\begin{array}{c}{\rho }_{\text{rad}}=\frac{4\sigma T^4}{c^3}\\ =\frac{4\left(5.67\times {10}^{-8}\text{W}/{\text{m}}^2\cdot {\text{K}}^4\right){\left(2\times {10}^6\text{K}\right)}^4}{{\left(3\times {10}^8\text{m}/\text{s}\right)}^3}\\ =1.34\times {10}^{-7}\text{kg}/{\text{m}}^{\text{3}}\end{array}$

The mass density of radiation is more than the average density of matter. Therefore, the radiation is radiation-dominated.

Conclusion:

The mass density of radiation is $1.34\times {10}^{-7}\text{kg}/{\text{m}}^{\text{3}}$ and the radiation is radiation-dominated.