Worksheet 9, Problem 2

Problem 2: 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=1.4×10−16ergK−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=12mv2i=12kBT. This fact is a consequence of a useful result called the Equipartition Theorem.)

 We are given the equation for the kinetic energy of one particle moving in i directions. In order to get the total energy we find the energy of one particle moving in 3 degrees of freedom and then multiply by the total mass divided by the average mass, which is the number of particles. 

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

This is simply the equation that we derived in the previous worksheet:

(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.

The last two steps gave us an expression for both kinetic and potential energy. Using the Virial Theorem, we can relate these two energies:

(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? Assumea cloud of constant density ρ.

If the gravitational binding energy were greater than the thermal energy, the cloud would not be in equilibrium and would begin to collapse on itself as the gravitational force pushes the particles closer and closer together.

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

We can solve for Jeans Mass using the relationship we came up with in the third part of the problem. We can isolate mass to find the threshold mass before the cloud collapses. As we found out in the previous step, if the kinetic energy is greater than the potential energy or vice versa, the cloud will collapse on itself. Thus, if M is larger than the other side of the equation, the cloud will collapse on itself.  

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

This is very similar to the last step and just involves a different rearrangement of the relationship from the third part of the problem.