Lecture #5a-- Star Formation and the Jeans Mass



1) Fireworks and fossils
There are actually three quite separate ways to use observations to learn about the ISM. We can take look at the ISM at all different places at the current epoch of our own galaxy's life, and get a sense of all the different processes that can happen. We can look at the "fossilized" evidence of what the ISM has done in the past (like white dwarfs or planetary nebulae for the fate of low-mass stars, or x-ray binaries and supernova remnants for the fate of high mass stars) to contrast that with that it is doing now. Or we can actually look deep into the cosmological history of our universe and see galaxies much younger than our own, and try to see what their ISM was doing in the earlier epoch.

It's fortunate that we have these three different approaches to compare and contrast, for any one of them alone would not be able to provide a very complete view of what "ISM" is and how it has affected everything else. What we do see is a very dynamical ISM where the ages of most objects is similar or less than their sound-crossing times, i.e., where most objects are changing rather than in equilibrium. One of the most important of these changes is the formation and dissipation of a giant molecular cloud, and the star formation that happens in between.
2) The scale of gravitational instability
A self-gravitating ideal gas exhibits a constant dance between the pressure that would disrupt and explode it, and the gravity that would crush it. Whichever process has the "upper hand" depends on the scale at which you look-- small regions tend to be dominated by pressure and will expand to eliminate overpressures, while large regions tend to be dominated by gravity and will typically undergo gravitational collapse into stars unless something else disrupts them first (like supernovae of stars that were quicker to the punch).
3) The Jeans Mass
To estimate the minimum mass scale where gravity overcomes pressure, we equate something that characterizes the strength of gravity to a similar quantity that characterizes the strength of pressure. We can, for example, equate the free-fall time to the sound-speed crossing time, or equivalently, equate the escape speed to the thermal speed. When you do that, you get a result much like the virial theorem result for equilibrium, because equilibrium is the "tipping point" at the edge of gravitational collapse. When you do this, and you should, you will find that M/R is proportional to T, where M and R characterize the mass and size of the gravitationally unstable region. Since we normally know density rather than M/R, we can recast M/R as density to the 1/3 power times M to the 2/3 power, and equate all that to T and the constants to find the "Jeans Mass"-- the minimum mass that is unstable to gravitational mass. That is the mass that should characterize stars, and sure enough, it does, if you put in typical molecular cloud densities of 1000 particle per cc, and temperatures of about 10 K.
4) The Salpeter Initial Mass Function (IMF)
Not all stars have the Jeans mass-- any larger mass is also unstable, but will tend to fragment into a power law with the number of stars per mass interval scaling like M to the -2.3 power (the Salpeter IMF). These means most stars, and most of the mass in stars, is at the lower end of this power law, which is characterized by the Jeans mass and is roughly solar. However, there are also lower mass stars (and lots of them), which come from further fragmentation or other details of the gravitational collapse process (like radiative instability wherein cooling can be very rapid in small regions), so one can only get so far with the Jeans mass approach. Still, the Jeans mass does seem to characterize most stars at least roughly.
5) Problems in paradise
There is one very important outstanding challenge to this overall picture, which is that the angular momentum has been neglected. Just by chance, a collapsing "cloud core" (the mass destined for a single star system) should have a small amount of angular momentum, but as it contracts, this should require that a higher and higher fraction of the material be orbiting the same way. Eventually all the mass is orbiting the same way, at which point you don't have a star you have a disk, and you can't contract any more unless you figure out a way to get rid of the angular momentum. There do appear to be at least two important ways to get rid of angular momentum, both involving magnetic fields. One is called "disk locking", where a magnetic field connects the fast rotating star to regions far out in the slow orbiting Keplerian disk. That connection of a rigid field between fast and slow plasma transports angular momentum outward and allows the star to become a slow rotator. Another approach is jet formation, where field lines threading the disk do not connect to the star but rather to free space, and some disk material is "flung outward" along the field, like a bead on a wire, carrying away angular momentum in polar jets. These are ongoing research areas, and much remains to be learned.