A question that arose when Hubble began classifying elliptical galaxies by their eccentricity is whether the observed “flattening” is the result of the galaxy’s rotation. We can characterize this by comparing the energy of rotation to the energy of gravitation:
Experimentally, we have recently measured , so we know that elliptical galaxies are not rotationally supported. This calculation was just for stars. If we were to do this for gas, we would have to include the effects of pressure. However, most galaxies are not pressure supported, so this result should hold. Spiral galaxies are rotationally supported, with a ratio of about 10.\
Now we will talk about two kinds of systems: those with power law density distributions and disks.
Systems with Power Law Density Distributions
We’ll say that the density is given by:
We would like to derive expressions for the surface density , , and . is the distance from the center along the surface that we see. If is the vertical distance along the line of sight to the center of a galaxy, we can say:
Now , so . We then have:
For the special case that , we have:
This is called a singular isothermal sphere. In the more general case:
Note that for a singular isothermal sphere, is constant.
This requires . Thus, we have:
For disks, we need to write a Poisson Equation in cylindrical coordinates.
Using seperation of variables, we’ll say . Then
This last looks like a harmonic oscillator equation with constant :
We can then work out that:
We can read more about this at (Toomre, 1963, ApJ 138, 385), and about disks at (Freeman, 1970, ApJ 160, 811). From observations, we can say:
Where ’s are the modified Bessel Functions.
Thus we have:
To remind us of what these modified Bessel functions are:
For large x, and .\For small x, , and .\Thus, .\Plugging this into the equations we’ve derived, we find that as , a near-zero approximation gives us:
We can graph this, but it looks vaguely like a random distribution with a left bound. We have now developed the tools we need to be able to compute for the sphere and the disk, and since , we can compute the total of most galaxies. We can plot versus radius and observe that up to , is dominated by the bulge, between and we see a superposition of the two influences, and beyond , is dominated by the disk.
However, this predicts that as increases, . We know this is not the case: in fact, . This tells us there must be some matter there (which we are not seeing), that is behaving like an isothermal sphere–that is, it must have a density profile.