Chapter 16, Problem 44QP

(a)

To determine

The luminosity at ${10}^8\text{K}$.

Answer to Problem 44QP

The luminosity at ${10}^8\text{K}$ is $1.8\times {10}^{13}{L}_{\text{O}}$.

Explanation of Solution

In terms of solar units, the luminosity law is given by,

    $L\left(L_{\text{O}}\right)=R{\left(R_{\text{O}}\right)}^{2}T{\left(T_{\text{O}}\right)}^4$

Here, $L\left(L_{\text{O}}\right)$ is the luminosity in solar units, $R\left(R_{\text{O}}\right)$ is the radius in solar units and $T\left(T_{\text{O}}\right)$ is the temperature in solar units.

Radius in solar units is,

    $\begin{array}{c}R\left(R_{\text{O}}\right)=\left({10}^7\text{m}\right)\left(\frac{R_{\text{O}}}{7\times {10}^8\text{m}}\right)\\ =0.014R_{\text{O}}\end{array}$

Temperature in solar units is,

    $T\left(T_{\text{O}}\right)=\frac{T}{5780\text{K}}$

Using the expressions for $R\left(R_{\text{O}}\right)$ and $T\left(T_{\text{O}}\right)$ in the equation for $L\left(L_{\text{O}}\right)$,

    $\begin{array}{c}L\left(L_{\text{O}}\right)={\left(0.014\right)}^2{\left(\frac{T}{5780\text{K}}\right)}^4\\ =1.8\times {10}^{-19}{T}^4\end{array}$

Conclusion:

Substitute ${10}^8\text{K}$ for T to get $L$ in solar units.

    $\begin{array}{c}L\left(L_{\text{O}}\right)=\left(1.8\times {10}^{-19}\right){\left({10}^8\text{K}\right)}^4{L}_{\text{O}}\\ =1.8\times {10}^{13}{L}_{\text{O}}\end{array}$

Therefore, luminosity at ${10}^8\text{K}$ is $1.8\times {10}^{13}{L}_{\text{O}}$.

(b)

To determine

The luminosity at ${10}^6\text{K}$.

Answer to Problem 44QP

The luminosity at ${10}^6\text{K}$ is $1.8\times {10}^5{L}_{\text{O}}$.

Explanation of Solution

From (a), the equation for $L\left(L_{\text{O}}\right)$ is,

    $L\left(L_{\text{O}}\right)=1.8\times {10}^{-19}{T}^4$

Conclusion:

Substitute ${10}^6\text{K}$ for T to get $L$ in solar units.

    $\begin{array}{c}L\left(L_{\text{O}}\right)=\left(1.8\times {10}^{-19}\right){\left({10}^6\text{K}\right)}^4{L}_{\text{O}}\\ =1.8\times {10}^5{L}_{\text{O}}\end{array}$

Therefore, luminosity at ${10}^6\text{K}$ is $1.8\times {10}^5{L}_{\text{O}}$.

(c)

To determine

The luminosity at ${10}^4\text{K}$.

Answer to Problem 44QP

The luminosity at ${10}^4\text{K}$ is $1.8\times {10}^{-3}{L}_{\text{O}}$.

Explanation of Solution

From (a), the equation for $L\left(L_{\text{O}}\right)$ is,

    $L\left(L_{\text{O}}\right)=1.8\times {10}^{-19}{T}^4$

Conclusion:

Substitute ${10}^4\text{K}$ for T to get $L$ in solar units.

    $\begin{array}{c}L\left(L_{\text{O}}\right)=\left(1.8\times {10}^{-19}\right){\left({10}^4\text{K}\right)}^4{L}_{\text{O}}\\ =1.8\times {10}^{-3}{L}_{\text{O}}\end{array}$

Therefore, luminosity at ${10}^4\text{K}$ is $1.8\times {10}^{-3}{L}_{\text{O}}$.

(d)

To determine

The luminosity at ${10}^2\text{K}$.

Answer to Problem 44QP

The luminosity at ${10}^2\text{K}$ is $1.8\times {10}^{-11}{L}_{\text{O}}$.

Explanation of Solution

From (a), the equation for $L\left(L_{\text{O}}\right)$ is,

    $L\left(L_{\text{O}}\right)=1.8\times {10}^{-19}{T}^4$

Conclusion:

Substitute ${10}^2\text{K}$ for T to get $L$ in solar units.

    $\begin{array}{c}L\left(L_{\text{O}}\right)=\left(1.8\times {10}^{-19}\right){\left({10}^2\text{K}\right)}^4{L}_{\text{O}}\\ =1.8\times {10}^{-11}{L}_{\text{O}}\end{array}$

Therefore, luminosity at ${10}^2\text{K}$ is $1.8\times {10}^{-11}{L}_{\text{O}}$.