Question

Assume that the core of the Sun has one-eighth of the Sun’s mass and is compressed within a sphere whose radius is one-fourth of the solar radius. Assume further that the composition of the core is 35% hydrogen by mass and that essentially all the Sun’s energy is generated there. If the Sun continues to burn hydrogen at the current rate of $\mathbf{6}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{11}}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}\mathit{s}$, how long will it be before the hydrogen is entirely consumed? The Sun’s mass is $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{30}}\mathbf{}\mathit{k}\mathit{g}$.

Short answer

The required time is $5\times {10}^{9}years$.

Step-by-step solution

Describe the expression for the time needed for hydrogen to burn

The expression for the time needed for hydrogen to burn is given by,

$\mathit{t}\mathbf{=}\frac{\mathbf{m}}{{\displaystyle \frac{\mathbf{d}\mathbf{m}}{\mathbf{d}\mathbf{t}}}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Find the need for hydrogen to burn

Substitute all the known values in equation (1).

$$\begin{gathered} t=\frac{\left(0.35\right)\left({\displaystyle \frac{1}{8}}\right)\left(2\times {10}^{30}kg\right)}{\left(6.2\times {10}^{11}kg/s\right)\left(365\right)\left(24\times 60\times 60\right)} \\ =5\times {10}^{9}years \end{gathered}$$

Therefore, the required time is $5\times {10}^{9}years$.