Chapter 17, Problem 50Q

To determine

The factor by which the radius of the star and luminosity change, if in an outburst its surface temperature doubles, the average density decreases by a factor of 8, and its mass remains the same.

Answer to Problem 50Q

Solution:

The star’s radius becomes $2$ times its initial value and luminosity becomes $64$ times the initial value.

Explanation of Solution

Given data:

The temperature of the star’s surface doubles.

The average density of the star decreases to $\frac{1}{8}$ of the initial value.

The mass of the star will remain the same.

Formula used:

The expression for density is:

$\rho =\frac{M}{V}$

Here, $\rho$ is the density, $M$ is the mass, and $V$ is the volume.

The expression for the volume of a spherical body is:

$V=\frac{4}{3}\pi r^3$

Here, $r$ is the radius of the body.

The expression for the luminosity from the Stefan-Boltzmann’s law is:

$L=4\pi r^2\sigma T^4$

Here, $L$ is the luminosity, $\sigma$ is the Stefan-Boltzmann constant, and $T$ is the temperature of the body.

Explanation:

Consider the star’s surface temperature before outburst to be $T_1$ and after the outburst, it is $T_2$.

According to the problem, after the outburst, the surface temperature of the star doubles. Thus,

$T_2=2\left(T_1\right)$

Consider the average density before outburst to be ${\rho }_1$ and after the outburst, it is ${\rho }_2$.

According to the question, the average density of the star decreased by a factor of 8 after the outburst. Thus,

${\rho }_2=\frac{{\rho }_1}{8}$

Consider the star’s radius before the outburst to be $r_1$ and after the outburst, it is $r_2$; the luminosity of the star before the outburst to be $L_1$ and after the outburst, it is $L_2$; the mass of the star before the outburst is $M_1$ and after the outburst, it is $M_2$.

According to the question, the mass of the star remains the same after the outburst. Thus,

$M_1=M_2$

Recall the expression for density and write it for before the outburst

$\begin{array}{c}{\rho }_1=\frac{M_1}{V_1}\\ =\frac{M_1}{\frac{4}{3}\pi r_1{}^3}\end{array}$ ...... (1)

Here, $\frac{4}{3}\pi r_1{}^3$ is the volume of the star before the outburst.

Similarly, write the expression for the density of the star after the outburst.

$\begin{array}{c}{\rho }_2=\frac{M_1}{V_1}\\ =\frac{M_2}{\frac{4}{3}\pi r_2{}^3}\end{array}$ ...... (2)

Here, $\frac{4}{3}\pi r_2{}^3$ is the volume of the star after the outburst.

Divide equations (1) and (2).

$\frac{{\rho }_1}{{\rho }_2}=\frac{\frac{M_1}{\frac{4}{3}\pi r_1{}^3}}{\frac{M_2}{\frac{4}{3}\pi r_2{}^3}}$

Simplify the above expression.

$\frac{{\rho }_1}{{\rho }_2}=\frac{\frac{M_1}{r_1{}^3}}{\frac{M_2}{r_2{}^3}}$

Rearrange the above expression for $\frac{r_1}{r_2}$.

${\left(\frac{r_1}{r_2}\right)}^3=\frac{{\rho }_2{M}_1}{{\rho }_1{M}_2}$

Substitute $M_1$ for $M_2$ and $8{\rho }_2$ for ${\rho }_1$ in the above expression.

$\begin{array}{c}{\left(\frac{r_1}{r_2}\right)}^3=\frac{{\rho }_2}{8\left({\rho }_2\right)}\\ =\frac{1}{8}\end{array}$

Solving the above expression further,

$\begin{array}{c}{\left(\frac{r_1}{r_2}\right)}^3={\left(\frac{1}{2}\right)}^3\\ \frac{r_1}{r_2}=\frac{1}{2}\\ r_2=2r_1\end{array}$

So, the radius becomes 2 times the initial value.

Recall the expression for luminosity and write it for before the outburst as.

$L_1=4\pi r_1{}^2\sigma T_1{}^4$ ...... (3)

Similarly, write the expression for the luminosity of the star after the outburst.

$L_2=4\pi r_2{}^2\sigma T_2{}^4$ ...... (4)

Divide equation (3) and (4).

$\begin{array}{c}\frac{L_1}{L_2}=\frac{4\pi r_1{}^2\sigma T_1{}^4}{4\pi r_2{}^2\sigma T_2{}^4}\\ =\frac{r_1{}^2{T}_1{}^4}{r_2{}^2{T}_2{}^4}\end{array}$

Substitute $2r_1$ for $r_2$ and $2T_1$ for $T_2$ in the above expression.

$\begin{array}{c}\frac{L_1}{L_2}=\frac{r_1{}^2{T}_1{}^4}{{\left(2r_1\right)}^2{\left(2T_1\right)}^4}\\ =\frac{1}{4\left(16\right)}\\ =\frac{1}{64}\end{array}$

Solve the above expression further, for $L_2$,

$L_2=64\left(L_1\right)$

So, after the outburst, the luminosity becomes 64 times the initial value.

Conclusion:

So, the luminosity increases by 64 times and the radius increases by 2 times.