Today we’ll be talking about the Jean’s instability, ram-pressure stripping, and tidal forces (all topics pertaining to interstellar gas).
We start with steady-state fluid equations for mass and momentum conservation:
We will then perturb , and , keeping only first-order perturbations, and write down the differences between this perturbation and the steady-state solution:
We will then make the barotropic assumption that only. Then we can say , where is the (isothermal) sound speed of the gas. Using this in our previous equations, we have:
This equation, along with and the two equations above, we have 4 equations. Assuming that , , and , our equations become:
Taking of the first equation and combining it with of the second equation, we have:
For the wave equation portion, we’ll assume a solution , giving us a dispersion relation:
For large wavenumber , we have . As wavenumbers decrease, we arrive at an instability when . This is the Jeans Wavenumber. As a wavelength this reads:
This equation is simply a relation between the sound crossing time and the free-fall time . Let us remind ourselves that this relation is for collapsing gas. The Jeans criterion for instability in star systems is:
where is the velocity dispersion of stars.\
We may go on to ask what the Jeans mass at recombination might be:
At recombination, . The temperature over this time period goes from . Then using , and assuming , (and possibly remembering that gas at 1000K has a velocity dispersion of ) we have:
This gives us a Jeans mass of . This is about the mass of a dwarf galaxy of a globular cluster, suggesting that these objects were the first to form in our universe.\
We can apply this work on Jeans criteria to stars and star formation. Let’s try to figure out how hot the gas that forms a star can be, given observed densities of :
This gives us a temperature of . Note that we use the mass of because almost all of the hydrogen in these cool clouds is molecular. Observationally, we find temperatures around and densities around , giving us . Naively, we’d assume this should be the mass of most of the stars we see. We know, however, that most of the mass of stars is in stars. To explain this, we need for star formation to proceed inefficiently, with much of the mass being ejected during collapse.\
For rotating disks, we have a Jeans equation:
where is the epicyclic frequency. For gas in this system, we have the criterion for instability being , where Q is:
where is the surface density of gas. For stars we have:
For the Milky Way, we find , , and , giving us , which suggests that the galaxy is overall stable, but may become unstable in spiral arms. This work is recent by a guy named Toomre.
The idea here is that hot, static, x-ray emitting gas in the galaxy clusters imparts pressure on the galaxies moving around inside. The ram pressure that the galaxy imparts on the gas is , where is the velocity of the galaxy relative to the gas. This ram pressure can cause stripping of gas off of the galaxy if it exceeds the hydrostatic pressure of gas in the galaxy:
Using that , and we find that this effect is present in the Milky Way.