Worksheet 9, Problem 2: Forming Stars

Forming Stars: Giant molecular clouds occasionally collapse under their own gravity (their own “weight”) to form stars. This collapse is temporarily held at bay by the internal gas pressure of the cloud, which can be approximated as an ideal gas such that P=nkT, where n is the number density (cm ́3) of gas particles within a cloud of mass M comprising particles of mass m ̄ (mostly hydrogen molecules, H2), and k is the Boltzmann constant, k = 1.4 x 1016erg K-1.
(a) What is the total thermal energy, K, of all of the gas particles in a molecular cloud of total mass M? (HINT: a particle moving in the ith direction has Ethermal =1/2 mv2 =1/2 kT. This fact is a consequence of a useful result called the Equipartition Theorem.)
(b) What is the total gravitational binding energy of the cloud of mass M?
(c) Relate the total thermal energy to the binding energy using the Virial Theorem, recalling that you used something similar to kinetic energy to get the thermal energy earlier.

(d)  If the cloud is stable, then the Viriral Theorem will hold. What happens when the gravitational binding energy is greater than the thermal (kinetic) energy of the cloud? Assume a cloud of constant density ρ.
(e)  What is the critical mass, MJ , beyond which the cloud collapses? This is known as the “Jeans Mass.”
(f)  What is the critical radius, RJ, that the cloud can have before it collapses? This is known as the “Jeans Length.” 

Let's take it from the top:

a) What is the total thermal energy, K, of all of the gas particles in a molecular cloud of total mass M? (HINT: a particle moving in the ith direction has Ethermal =1/2 mv2 =1/2 kT. This fact is a consequence of a useful result called the Equipartition Theorem.)

This looks pretty easy, but the equation needs a slight modification.  Since the terminal energy only accounts for particle movement in 1 direction, we need to multiply it by 3 for our three observable spacial dimensions.

\[K_{particle}=\frac{3}{2}kT\] 

Here: k is the Boltzmann Constant and T is the temperature in Kelvin.

We now must sum up all of the particles, so we will use the notation \(M\) to signify the total mass and \(\overline{m}\) to denote the average particle mass.

\[\boxed{K=\frac{3MkT}{2\overline{m}}}\] 

(b) What is the total gravitational binding energy of the cloud of mass M?

Okay, this is easy since we derived this on Worksheet 8, Problem 2.

\[\boxed{U=-\frac{GM^2}{R}}\]

(c) Relate the total thermal energy to the binding energy using the Virial Theorem, recalling that you used something similar to kinetic energy to get the thermal energy earlier.

From Worksheet 8 we know that viral theorem states that:

\[2K=-U\]

We have U and we have K, so let's plug and chug.

\[\frac{3MkT}{\overline{m}}=\frac{GM^2}{R}\]

This simplifies to:

\[\boxed{\frac{3kT}{\overline{m}}=\frac{GM}{R}}\]

(d)  If the cloud is stable, then the Viriral Theorem will hold. What happens when the gravitational binding energy is greater than the thermal (kinetic) energy of the cloud? Assume a cloud of constant density ρ.

Let's consider this.  The kinetic energy exerts a force outward on the cloud which is countered by the gravity potential exerting a gravitational force inwards. So logically, if the gravitational energy is greater than the thermal energy, the cloud will begin to collapse inward due to an unbalanced net inwards force.  This is a good thing, because over time, this will form a star!

(e)  What is the critical mass, MJ , beyond which the cloud collapses? This is known as the “Jeans Mass.”

When viral theorem holds, the cloud is in equilibrium.  Thus if we solve our viral theorem equation for M, we will know that this is the cutoff point for stability.

\[\frac{3kT}{\overline{m}}=\frac{GM}{R}\]

\[\boxed{M_J=\frac{3kTR}{G\overline{m}}}\]

(f)  What is the critical radius, RJ, that the cloud can have before it collapses? This is known as the “Jeans Length.” 

All that we have to do is rearrange our equation again.

\[\frac{3kT}{\overline{m}}=\frac{GM}{R}\]

\[\boxed{R_J=\frac{GM\overline{m}}{3kT}}\]

This problem done with collaboration with G. Grell, S, Morrison.