# Vertical Structure of Thin Disks

## Astronomy 626: Spring 1995

As long as we are only concerned with the vertical structure on
scales `z` which are much smaller than the distance from the
center `R`, we can ignore radial gradients and pretend that our
disk has *infinite* extent. In this approximation, the vertical
structure of a stellar system becomes a 1-D problem.

The integrals in the following problems are fairly
straight-forward. You may use Mathematica or a similar system to
evaluate them, but I recommend that you explicitly check each
indefinite integral by differentiating to recover the integrand.

Each of these problems is worth `10` points.

Due Date: 3/21/95.

**16.** Using Gauss's theorem, calculate the gravitational
force produced by a sheet of matter with infinite extent and zero
thickness, assuming the surface density `Sigma` is the same
everywhere.

**17.** Now consider a sheet with infinite extent and finite
thickness. Let the mass density be

2 z
rho(z) = rho_0 sech (-----) ,
2 z_0

where `z` is the direction perpendicular to the sheet,
`rho_0` is the density at the midplane, and `z_0` is the
vertical scale-height. Calculate the vertical force `a(z)` and
potential `Phi(z)`, adopting the boundary condition
`Phi(0)=0`.
**18.** Suppose the distribution function has the form

-1/2
f(E_z) = rho_0 (2 pi sigma^2) exp(-E_z / sigma^2) ,

where `sigma` is the vertical velocity dispersion and
1 2
E_z = - v_z + Phi(z)
2

is the energy in vertical motion. Calculate the mass density
`rho(Phi)`. Note that the density is expressed as a function
of `Phi`.
**19.** Now compare the expression for `rho(Phi)` from
problem **18** with the implicit relation between `rho` and
`Phi` you got from problem **17**. These relations should
have the same functional form. Set them equal to derive a value for
`z_0` in terms of `rho_0`, `sigma`, and the
gravitational constant `G`. Congratulations! You have now
solved the CBE for a thin disk with a Gaussian velocity
distribution.

**Hint:** See BT87, Problem **4-25** for another way to
approach this solution.

Joshua E. Barnes
(barnes@zeno.ifa.hawaii.edu)
Last modified: March 7, 1995