The Time Scale of Star Formation and Stability

In this post I'll be doing #2 from this week's worksheet on star formation. The problem goes over deriving the time scale of star formation, and introduces the concept of the "Jeans Length." The question has multiple parts, which I'll do step by step.

The first part goes like this:

Consider a test particle in an e =  1 "orbit" around a point mass and use Kepler's Third Law to derive a general equation for the free-fall time. Frame this expression in terms of a single variable - the average density, p. What are the assumptions implicit in the last step? Are they valid?

 Let's begin by considering an e = 1 "orbit" (e = eccentricity) involving a the gas cloud (generalized as a point mass) and a test particle. In this case, we have an infinitely thin ellipse, essentially a straight line:

We can see that the total length of the "ellipse" is the radius of the cloud, R, which is twice the length of the ellipse's semimajor axis, a. Kepler's Law gives us the orbital period of a test particle traversing the ellipse, i.e. the time it takes for it to go back and forth on the line that is the ellipse:

where T is the orbital period, M is the mass of the cloud, and a is the semimajor axis.

We can see that the free-fall time, i.e. the time it takes for the test particle to move across the ellipse once, is half the orbital period. So, we have:

To put this expression in terms of density, we will use the following definition for a spherical gas cloud:

So, we have:

which gives us an expression for the time it takes for a self-gravitating cloud to collapse. We assumed a uniform density in the last step, which is valid, because the gas cloud is very diffuse, creating a near-uniform density throughout.

The second part of the question goes like this:

If the free-fall time of a cloud is significantly less than a "dynamical time," or the time it takes a pressure wave (sound wave with speed c_s) to traverse the cloud, the cloud will be unable to gravitationally collapse. Equate the free fall time to the sound crossing time and solve for the length variable R_j.

The time it takes for a sound wave to traverse the cloud (dynamical time) is given by the following:

If we equate the two times, we get the following expression for R_j:

which gives us an expression for the "Jeans Length," the subject matter of part three:

You have just performed an order of magnitude estimation of the "Jeans Length." A more careful derivation yields the following definition: 

 where c_s is the isothermal sound speed. Discuss the significance of this characteristic length.

The Jeans Length is intimately related to the collapse of a self-gravitating cloud of gas. When a cloud collapses, the edges of the cloud will create waves of pressure that traverse the cloud, opposing the gravitational self-collapsing force. If the pressure wave can reach the other edge of the cloud before the cloud collapses (i.e. in a time less than the free-fall time), then the cloud will stop collapsing. Thus, for clouds larger than the Jeans Length, the dynamical time (the time it takes for the pressure wave to traverse the cloud) will be larger than the free-fall time, and the cloud will collapse. On the other hand, for clouds smaller than the Jeans Length, the dynamical time will be smaller than the free-fall time, and the pressure waves will counteract gravity to prevent self-collapse.

Part four of the problem goes like this:

For simplicity, consider a spherical cloud collapsing isothermally with initial radius R_0 = R_j. Once the cloud radius reaches 0.5 R_0, by what fractional amount has R_j changed? What might this mean in terms of the number of stars forming within a collapsing molecular cloud?

From our above expression, we see that:

 So, if our initial radius halves in size, we have:

This tells us that the Jeans Length of the collapsing cloud decreases faster than the radius of the collapsing cloud.

What does this mean? If the Jeans Length is diminishing faster than the radius of the cloud, then the cloud will want to collapse faster and faster, since the difference between the Jeans Length and the radius is increasing. Because the density of the cloud is not perfectly uniform, smaller regions within the cloud that have higher density will therefore undergo their own gravitational collapse. This results in fragmentation, the process by which a large gas cloud fragments into smaller clouds that collapse into star clusters. (Note: the Jeans instability assumes an adiabatic contraction of the cloud, i.e. the temperature remains constant. This is not true in actuality).