Chapter 12, Problem 38QAP

(a)

To determine

The lifetime of a star; compare it with given values in Table 12.1.

Answer to Problem 38QAP

The lifetime of star is $56.6\text{billion years}$; it is nearly equal to the given value.

Explanation of Solution

Write the expression for lifetime of a main sequence star.

$\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{M_{\text{MS}}}\right)}^{2.5}$

Here, $M_{\text{Sun}}$ is mass of the Sun and $M_{\text{MS}}$ is mass of the main-sequence star.

The above expression shows that the lifetime of star is inversely proportional to the mass of main-sequence star. Greater the mass of star compared to the Sun is; less would be its lifetime.

Conclusion:

Substitute $0.50M_{\text{Sun}}$ for $M_{\text{MS}}$ in equation (I).

$\begin{array}{c}\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{0.50M_{\text{Sun}}}\right)}^{2.5}\\ =5.66\times {10}^{10}\text{years}\left(\frac{1\text{billion years}}{{10}^9\text{years}}\right)\\ =56.6\text{billion years}\end{array}$

Thus, the lifetime of star is $56.6\text{billion years}$; it is nearly equal to the given value.

(b)

To determine

The lifetime of a star; compare it with given values in Table 12.1.

Answer to Problem 38QAP

The lifetime of star is $0.11\text{billion years}$; it is nearly equal to the given value.

Explanation of Solution

Write the expression for lifetime of a main sequence star.

$\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{M_{\text{MS}}}\right)}^{2.5}$

Here, $M_{\text{Sun}}$ is mass of the Sun and $M_{\text{MS}}$ is mass of the main-sequence star.

The above expression shows that the lifetime of star is inversely proportional to the mass of main-sequence star. Greater the mass of star compared to the Sun is; less would be its lifetime.

Conclusion:

Substitute $6.0M_{\text{Sun}}$ for $M_{\text{MS}}$ in equation (I).

$\begin{array}{c}\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{6.0M_{\text{Sun}}}\right)}^{2.5}\\ =0.011\times {10}^{10}\text{years}\left(\frac{1\text{billion years}}{{10}^9\text{years}}\right)\\ =0.11\text{billion years}\end{array}$

Thus, the lifetime of star is $0.11\text{billion years}$; it is nearly equal to the given value.

(c)

To determine

The lifetime of a star; compare it with given values in Table 12.1.

Answer to Problem 38QAP

The lifetime of star is $56.6\text{billion years}$; it is nearly equal to the given value.

Explanation of Solution

Write the expression for lifetime of a main sequence star.

$\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{M_{\text{MS}}}\right)}^{2.5}$

Here, $M_{\text{Sun}}$ is mass of the Sun and $M_{\text{MS}}$ is mass of the main-sequence star.

The above expression shows that the lifetime of star is inversely proportional to the mass of main-sequence star. Greater the mass of star compared to the Sun is; less would be its lifetime.

Conclusion:

Substitute $0.50M_{\text{Sun}}$ for $M_{\text{MS}}$ in equation (I).

$\begin{array}{c}\text{Lifetime}=\left(1\times {10}^{10}\right){\left(\frac{M_{\text{Sun}}}{0.50M_{\text{Sun}}}\right)}^{2.5}\\ =3.6\times {10}^{-5}\times {10}^{10}\text{years}\left(\frac{1\text{million years}}{{10}^6\text{years}}\right)\\ =0.36\text{million years}\end{array}$

Thus, the lifetime of star is $0.36\text{million years}$; it is nearly equal to the given value.