Galactic Density Profiles
We now ask the question: what is the form of for intermediate z (300 pc to 30-50 kpc)? We know that:
Then using Stokes’ Law:
Because the mass at intermediate z is dominated by stars (dark matter is still spherically symmetric around our system), we can treat all the mass as being concentrated in an infinite sheet at the midplane. Thus, this problem becomes identical to the electrostatics problem of the flux from an infinite charged sheet, and we can invoke Gauss’s Law and the symmetry of the problem to say that for a box drawn to include a section of the midplane, with faces parallel to the sheet, we have that:
where is the area of a face parallel to the sheet. Thus, we have
Where is the surface density of stars.
For large z,
Thus we have an overall picture of vs. z which increases linearly for small z, is flat for a long time, and then decreases as for large z. In the flat regime we can calculate:
Also, , so we can say , which has solution for . Thus, for a star close to the midplane (like our sun):
We can work out that the period of this oscillation is about 83 million years. It has been suggested that this period for midplane crossings corresponds to the period for mass extinctions on earth by comet/asteroid collisions.
For a midrange star, , so . This says that the period depends on where the star starts. We’ll say 1 kpc, which gives us the solution:
So we’ve worked out something of the motion of an individual star, but now we’d like to work out the distribution of stars in z for a population. We’ll assume that we have an infinite plane- sheet.
We’ll now change variables , and :
and differentiating, we have:
And this has the ultimate solution:
where, you recall, . From this messy equation we can infer the units of the following:
So we’ll define
as our scale height, giving us that
To find what z is when , we’ll define so that:
We find that this has solution , so:
From Binney & Merrifield, we have that in the solar vicinity , and , so we have for a thin disk:
For the thick disk, , so . From observations, we find that does not change with radius. Note that for small heights, is close to . Similarly, in the asymptotic limit of , we find that
which agrees well with observation ().