Chapter 25, Problem 34Q

(a)

To determine

The mass density of radiation $\left({\rho }_{\text{rad}}\right)$ and its nature, whether it is matter-dominated or radiation-dominated at the photosphere of the Sun. It is given that the temperature at the photosphere of the Sun is $\text{T=5800}\text{K}$ and the average density of matter at the photosphere of the Sun is ${\rho }_{\text{m}}=3\times {10}^{-4}{\text{kg/m}}^3.$

Answer to Problem 34Q

Solution:

Mass density of radiation at the photosphere of the Sun is, ${\rho }_{\text{rad}}=9.5\times {10}^{-18}{\text{kg/m}}^3$, and this radiation is matter dominated.

Explanation of Solution

Given data:

Temperature at the photosphere of the Sun, $T=\text{5800 K}$.

Average density of matter at the photosphere of the Sun, ${\rho }_{\text{m}}=3\times {10}^{-4}{\text{ kg/m}}^3$.

Formula used:

Write the expression for mass density of radiation.

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

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation, $T$ is the temperature of radiation, $\sigma$ is Stefan-Boltzmann constant, which is equal to $5.67\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$ and c is the speed of light, which is equal to $3.0\times {10}^8\text{m/s}.$

Explanation:

Energy in the entire universe usually falls into two categories – matter or radiation. To calculate the weightage of matter and radiation in the energy, we find the mass density of the radiation, which is equivalent to the mass of the entire radiation divided by the volume in which that radiation is contained. If the value of mass density of the radiation comes out less than the average density of matter, then the radiation is matter-dominated, otherwise, it is radiation-dominated. Refer to the expression for mass density of radiation.

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

Substitute $5.67\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$ for $\sigma$ and $5800\text{ K}$ for $T$ in the expression for mass density of radiation.

$\begin{array}{c}{\rho }_{rad}=\frac{\text{4}\left(5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}\right){\left(5800\text{ K}\right)}^4}{{\left(3\times {10}^8{\text{ ms}}^{-1}\right)}^3}\\ =\frac{\left(22.68\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}\right)\left(1.13\times {10}^{15}{\text{K}}^4\right)}{\left(27\times {10}^{24}{\text{m}}^3{\text{s}}^{-3}\right)}\\ =\text{ 9}\text{.2}\times {\text{10}}^{-18}{\text{ kg/m}}^3\\ \end{array}$

Conclusion:

Hence, the radiation at the photosphere of the Sun is matter-dominated and value of its mass density is, ${\rho }_{\text{rad}}=9.2\times {10}^{-18}{\text{ kg/m}}^3$.

(b)

To determine

The mass density of radiation $\left({\rho }_{\text{rad}}\right)$ and its nature, whether it is matter-dominated or radiation-dominated at the center of the Sun. It is given that the temperature at the center of the Sun is, $\text{T=1}\text{.55}\times {\text{10}}^7\text{K}$ and the average density of matter at the center of the Sun is, ${\rho }_{\text{m}}=1.6\times {10}^5{\text{kg/m}}^3.$

Answer to Problem 34Q

Solution:

Mass density of radiation at the center of the Sun, ${\rho }_{\text{rad}}=4.8\times {10}^{-4}\text{}{\text{kg/m}}^3$ and this radiation is matter-dominated.

Explanation of Solution

Given data:

Temperature at the center of the Sun, $\text{T=1}\text{.55}\times {\text{10}}^7\text{K}$.

Average density of matter at the center of the Sun, ${\rho }_{\text{m}}=1.6\times {10}^5{\text{kg/m}}^3.$

Formula used:

Write the expression for mass density of radiation.

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

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation, $\text{T}$ is the temperature of radiation, $\sigma$ is Stefan-Boltzmann constant, which is equal to $5.67\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$ and c is the speed of light, which is equal to $3.0\times {10}^8\text{m/s}.$

Explanation:

As per Einstein’s law of mass-energy equivalence, that is, $\text{E}={\text{ mc}}^2$, to every mass, there is an energy associated, and vice-versa. Therefore, every object which has a mass also has a corresponding radiation. The nature of the radiation emitted from an object is determined by using mass-density of radiation. If the value of mass-density of radiation of an object is less than the average density of matter, then the radiation is matter-dominated, else it is radiation-dominated. Refer to the expression for mass-density of radiation.

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

Substitute $5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}$ for $\sigma$ and $\text{1}\text{.55}\times {\text{10}}^7\text{ K}$ for $T$ in the expression for mass density of radiation.

$\begin{array}{c}{\rho }_{rad}=\frac{\text{4}\left(5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}\right){\left(\text{1}\text{.55}\times {\text{10}}^7\text{ K}\right)}^4}{{\left(3\times {10}^8{\text{ ms}}^{-1}\right)}^3}\\ =\frac{\left(22.68\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}\right)\left(5.77\times {10}^{28}{\text{K}}^4\right)}{\left(27\times {10}^{24}{\text{m}}^3{\text{s}}^{-3}\right)}\\ =\text{ 4}\text{.8}\times {\text{10}}^{-4}{\text{ kg/m}}^3\\ \end{array}$

Conclusion:

Hence, the radiation at the center of the Sun is matter-dominated as the value of its mass density is, ${\rho }_{\text{rad}}=4.8\times {10}^{-4}{\text{kg/m}}^3$, which is less than the average density of matter for this radiation.

(c)

To determine

The mass density of radiation $\left({\rho }_{\text{rad}}\right)$ and its nature, whether it is matter-dominated or radiation-dominated at the solar corona. It is given that the temperature at the corona of the Sun is, $\text{T}\text{=}\text{2}\times {\text{10}}^6\text{ K}$ and average density of matter at the solar corona is, ${\rho }_{\text{m}}=5\times {10}^{-13}{\text{ kg/m}}^3.$

Answer to Problem 34Q

Solution:

Mass density of radiation at the solar corona, ${\rho }_{\text{rad}}=1.3\times {10}^{-7}{\text{kg/m}}^3$, and this radiation is radiation-dominated.

Explanation of Solution

Given data:

Temperature at the corona of the Sun, $\text{T}\text{=}\text{2}\times {\text{10}}^6\text{ K}$.

Average density of matter at the corona of the Sun, ${\rho }_{\text{m}}=5\times {10}^{-13}{\text{ kg/m}}^3.$

Formula used:

Write the expression for mass density of radiation.

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

Here, ${\rho }_{\text{rad}}$ is the mass density of radiation, $\text{T}$ is the temperature of radiation, $\sigma$ is Stefan-Boltzmann constant, which is equal to $5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}$ and c is the speed of light, which is equal to $3.0\times {10}^8\text{ m/s}$.

Explanation:

According to the mass-energy relation of Einstein, $\text{E}={\text{ mc}}^2$, there is an energy associated with every mass, and vice-versa. Therefore, to calculate the energy of the radiation, we find the mass density of the radiation. Mass density of any radiation is given by the mass of the radiation divided by the volume in which the radiation is contained. After evaluating the mass density of radiation, compare it with the average density of matter. If the former is less than the latter, then the corresponding radiation is matter-dominated, else it is radiation-dominated. Refer to the expression for mass density of radiation.

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

Substitute $5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}$ for $\sigma$ and $2\times {\text{10}}^6\text{ K}$ for $T$ in the expression for mass density of radiation.

$\begin{array}{c}{\rho }_{rad}=\frac{\text{4}\left(5.67\times {10}^{-8}{\text{ Wm}}^{-2}{\text{K}}^{-4}\right){\left(2\times {\text{10}}^6\text{ K}\right)}^4}{{\left(3\times {10}^8{\text{ ms}}^{-1}\right)}^3}\\ =\frac{\left(22.68\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}\right)\left(16\times {10}^{24}{\text{K}}^4\right)}{\left(27\times {10}^{24}{\text{m}}^3{\text{s}}^{-3}\right)}\\ =\text{ 1}\text{.3}\times {\text{10}}^{-7}{\text{ kg/m}}^3\\ \end{array}$

Conclusion:

Hence, the value of mass density of radiation at the solar corona is, ${\rho }_{\text{rad}}=1.3\times {10}^{-7}{\text{kg/m}}^3$, and the corresponding radiation is radiation-dominated.