Chapter 26, Problem 38Q

(a)

To determine

The Jeans length if the temperature and density of a dark nebula is $30\text{ K}$ and ${10}^{-12}{\text{ kg/m}}^3$, respectively. Consider that the nebula only comprises hydrogen.

Answer to Problem 38Q

Solution:

The Jeans Length of the dark nebula comprising only hydrogen is equal to $1.0805\times {10}^{14}\text{ m}$.

Explanation of Solution

Given data:

The temperature of the dark nebula is 30 K.

The density of the dark nebula is ${10}^{-12}{\text{ kg/m}}^3$.

Formula used:

According to Jeans, an object will only grow if the fluctuation in density crosses the length that is called Jeans Length. Jeans length is calculated by the expression.

$L_{\text{J}}=\sqrt{\frac{\pi kT}{mG{\rho }_{\text{m}}}}$

Here, k is the Boltzmann constant, T is the temperature, m is the mass of a single particle of the gas, G is the universal gravitational constant and ${\rho }_{\text{m}}$ is the average density of matter in gas.

Explanation:

Consider the value of Boltzmann constant, universal gravitational constant and mass of hydrogen to be $1.38\times {10}^{-23}\text{ J/K}$, $6.67\times {10}^{-11}{\text{ Nm}}^2/{\text{kg}}^2$ and $1.67\times {10}^{-27}\text{ kg}$, respectively.

Recall the expression for calculating the Jeans length.

$L_{\text{J}}=\sqrt{\frac{\pi kT}{mG{\rho }_{\text{m}}}}$

Substitute $1.38\times {10}^{-23}\text{ J/K}$ for k, $6.67\times {10}^{-11}{\text{ Nm}}^2/{\text{kg}}^2$ for G, $1.67\times {10}^{-27}\text{ kg}$ for m, ${10}^{-12}{\text{ kg/m}}^3$ for ${\rho }_{\text{m}}$ and 30 K for T,

$\begin{array}{c}{L}_{\text{J}}=\sqrt{\frac{\pi \times \left(1.38\times {10}^{-23}\text{ J/K}\right)\times 30\text{ K}}{\left(1.67\times {10}^{-27}\text{ kg}\right)\times \left(6.67\times {10}^{-11}{\text{ Nm}}^2/{\text{kg}}^2\right)\times \left({10}^{-12}{\text{ kg/m}}^3\right)}}\\ =1.0805\times {10}^{14}\text{ m}\end{array}$

Conclusion:

Thus, the Jeans length of the dark nebula is found to be $1.0805\times {10}^{14}\text{ m}$.

(b)

To determine

Whether the density fluctuation, within a dark nebula only comprising hydrogen, will grow or not if it is several light years long. The temperature and density of the dark nebula is $30\text{ K}$ and ${10}^{-12}{\text{ kg/m}}^3$, respectively.

Answer to Problem 38Q

Solution:

The Jeans Length of the dark nebula is $0.01142\text{ ly}$ and the density fluctuation crosses this length, so the dark nebula will grow.

Explanation of Solution

Given data:

The temperature of the dark nebula is 30 K.

The density of the dark nebula is ${10}^{-12}{\text{ kg/m}}^3$.

Formula used:

According to Jeans, an object will only grow if the fluctuation in density crosses the length that is called Jean Length.

The relation between light year and meter is given by the expression,

$1\text{ ly}=9.46\times {10}^{15}\text{ m}$

Explanation:

Refer sub-part (a) for the value of the Jeans length, that is, $1.0805\times {10}^{14}\text{ m}$. Use the conversion formula to convert meter into light year.

$\begin{array}{c}1.0805\times {10}^{14}\text{ m}\times \left(\frac{1\text{ ly}}{9.46\times {10}^{15}\text{ m}}\right)\\ =0.01142\text{ ly}\end{array}$

The length of the nebula is several light years. According to Jeans, the density fluctuation is exceeding $0.01142\text{ ly}$, so it will grow.

Conclusion:

Since the density fluctuation of the dark nebula is greater than $0.01142\text{ ly}$, it will grow.

(c)

To determine

To show: The formation of protostars within a dark nebula is related to the growth of the nebula. The temperature and density of a dark nebula is $30\text{ K}$ and ${10}^{-12}{\text{ kg/m}}^3$, respectively.

Explanation of Solution

Given data:

The temperature of the dark nebula is 30 K.

The density of the dark nebula is ${10}^{-12}{\text{ kg/m}}^3$.

Formula used:

According to Jeans, an object will only grow if the fluctuation in density crosses the length that is called Jean Length.

Explanation:

The formation of protostars occurs when the density fluctuation crosses the Jeans length. From sub-part (b) it can be observed that the size of dark nebula crosses the Jeans length.

Hence, the formation of protostars will occur and it will grow till the point it becomes unstable.

Conclusion:

Since the Jeans length is crossed by the dark nebula, the formation of protostars will occur and after becoming unstable, due to increasing weight, it will collapse.