Chapter 20, Problem 32Q

To determine

To discuss:

The effects on Earth if the Sun is suddenly replaced by a $1-M_{\odot }$ white dwarf.

The appearance of this white dwarf when viewed from the Earth.

Whetherit can be seen by naked eye or not.

Whether the surface temperature of the Earth would remain the same or not.

Explanation of Solution

Given data:

$\begin{array}{l}\text{Mass of the newly formed white dwarf}=1-M_{\odot }\\ \text{Surface temperature of the newly formed white dwarf}=100,00\text{0}\text{K}\\ \text{Surface temperature of the Sun}=\text{5,800}\text{K}\\ \text{Surface temperature of the Earth}=30\text{0}\text{K}\\ \text{Radius of the Sun}=696,000\text{km}\\ \text{Radius of the Earth}=6,378\text{km}\end{array}$

Formula used:

Luminosity can be found by using the formula

$L=4\pi \sigma T^4{R}^2\stackrel{}{\to }\left(1\right)$

Where,

$\begin{array}{c}L=\text{Luminosity}\\ \sigma =\text{Stefan}-\text{Boltzmann constant}\\ T=\text{surface temperature}\\ R=\text{Radius}\end{array}$

Stefan-Boltzmannlaw

$E=\sigma T^4$

Where,

$\begin{array}{c}E=\text{Energy},\\ \sigma =\text{Stefan}-\text{Boltzmann constant}\\ T=\text{surface temperature}\end{array}$

Calculation:

For the Sun

$L_{\odot }=4\pi \sigma T_{\odot }{}^4{R}_{\odot }{}^2\stackrel{}{\to }\left(2\right)$

Where

$\begin{array}{l}{L}_{\odot }=\text{Luminosity of the Sun}\\ \sigma =\text{Stefan}-\text{Boltzmann constant}\\ T_{\odot }=\text{surface temperature of the Sun}\\ R_{\odot }=\text{Radius of the Sun}\end{array}$

$\begin{array}{l}\frac{\left(1\right)}{\left(2\right)}\stackrel{}{\to }\frac{L}{L_{\odot }}=\frac{T^4}{T_{\odot }{}^4}\frac{R^2}{R_{\odot }{}^2}\\ \frac{L}{L_{\odot }}=\frac{{\left(100,000\right)}^4}{{\left(5,800\right)}^4}\frac{{\left(6,378\right)}^2}{{\left(696,000\right)}^2}\\ L=7.4L_{\odot }\end{array}$

The newly formed white dwarf would be $7.4\text{times}$ more luminous than the Sun.

Let $E_1$ be the energy radiated by the Earth’s surface today and $E_2$ be the energy radiated by the Earth’s surface if the Sun was suddenly replaced by $1-M_{\odot }$ white dwarf.

Using the formula,

$\begin{array}{l}\text{At present day}\to E_1=\sigma T_1^4\stackrel{}{\to }\left(1\right)\\ \text{When the sun was suddenly replaced by white dwarf}\to E_2=\sigma T_2^4\stackrel{}{\to }\left(2\right)\end{array}$

$\frac{\left(1\right)}{\left(2\right)}\stackrel{}{\to }\frac{E_1}{E_2}=\frac{T_1^4}{T_2^4}$

$\begin{array}{l}\frac{E_1}{7.4E_1}=\frac{{\left(300\right)}^4}{T_2^4}\\ T_2^4={300}^4\times 7.4\\ T_2^4=5.99\times {10}^{10}\\ T_2^{}=494.8\text{K}\\ T_2^{}=221{}^0\text{C}\end{array}$

Conclusion:

Even though white dwarfs said to be cooler, newly formed white dwarfs are much hotter and luminous than the Sun. Since the core temperature increases rapidly due to the Helium fusion, a typical newly formed white dwarf’s surface temperature is estimated to be about $100,000\text{K}$. Average size of a white dwarf is estimated to be the size of Earth. Since the mass of the newly found white dwarf is $1-M_{\odot }$, the gravitational attraction would be same as the Sun. Therefore, the Earth’s orbit would not be changed.

Even the present-day Sun cannot be viewed safely with unaided eye. Therefore, it is not safe to see a white dwarf by the naked eye. Besides, the surface temperature would be $221{}^0\text{C}$. No life will retain on Earth at this temperature.